Constructivism has made it relatively easy to recognise good learning: an active process on the part of the learner, in which meaning is constructed through both individual and social negotiation. It may, however, have had the opposite effect concerning good teaching. In some ways, a constructivist perspective makes us afraid to teach, since this so often equates with transmission of content and domination of the learning process. What is the role of the teacher within the constructivist classroom? Is it merely to stand back and to allow student interest and curiosity to motivate and direct enquiry. If so, then little learning might be expected to follow. Good teaching remains as important as ever, and the demands on the teacher as great. What has changed in recent years has been the nature and perception of the teacher's role. No longer the source of all knowledge and direction, the teacher assumes a more dificult task, as the architect and facilitator of an effective mathematics learning environment.

From a constructivist perspective, learning is not achieved through the transmission of knowledge from teacher to students; rather, each student constructs his or her own meaning from the learning experiences encountered, meaning which undergoes a process of personal and social negotiation before it is internalised. Traditional exposition models of instruction assumed that the same message was received by the thirty or so different students to whom it was transmitted; constructivism denies the likelihood of such uniformity. A single instructional message may well be interpreted in many different ways. The teacher's role must still involve providing meaningful and carefully planned learning experiences; of even greater importance, however, becomes the responsibility for providing the means by which students may question what they have experienced, may compare and contrast their perceptions with others (especially their peers) and may negotiate meaning which is consistent with their existing understandings and with those intended by the teacher.

When viewed from the Piagetian perspective of constructivism, the process of learning becomes a spiral of equilibrium -> disequilibrium -> re-equilibrium. The role of the teacher, in such a view, is to attempt to induce "perturbations," creating just enough disequilibrium in the learners to encourage re-equilibrium (Doll, 1986):

The teacher must intentionally cause enough chaos to motivate the student to reorganise. Obviously this is a tricky task. Too much chaos will lead to disruption (Bruner, 1973, Chapter 4), while too little chaos will produce no reorganisation. (p.15)

The traditional "sequence" of instruction assumes learning to be linear and common to all learners within the group. A constructivist view, however, presents human learning as complex and branching, not simple and linear. Individuals learn in different ways: not all at the same times, nor in the same straight lines. Students working within a learning environment developed in this way might be expected to do so in different ways, at different rates, and to make different decisions along the way, concerning their style of investigation, and their path through such a program.

This paper examines the nature of this complex construct. It proposes a simple model, identifying effective mathematics teaching with the creation of a learning environment which strikes a balance between instruction and enquiry as one dimension, between process and product as another, and between challenge and support as a third critical dimension. Figure 1 offers an illustration of such a model. Note that this three dimensional view is not intended to be exclusive. It does not suggest that there are not other dimensions associated with effective mathematics learning. It merely draws attention to three aspects of mathematics teaching and learning which are considered significant. In particular, it places the third dimension, that of challenge and support, within a context defined by the other two more familiar distinctions.

Figure 1: A three-dimensional model of mathematics teaching and learning

This paper begins with some theoretical considerations, briefly examining three theoretical models, and then an attempt is made to draw together themes common to these which inform our consideration of an effective learning environment. These form the basis for recognising those features most important in creating an efective mathematics learning environment.

Challenge & Support Index. The contribution of Vygotsky.