ABSTRACT
Although the decisions concerning the choice of goals prior to mathematics teaching cannot be made on purely rational grounds, the decision-maker does have choices concerning the course of action that he or she follows in developing rational grounds for those decisions. The mathematical choices can be made based on a mathematicians’ view of mathematics. After making such a choice, two distinctly different paths can be followed. First, a curricular analysis of the mathematics under consideration can be undertaken and a logical sequencing of the mathematical topics for classroom instruction can be produced. This way of operating is traditional and leads to the notion of hard constraints and the paradox of happy agreement as explained by Lewin (Ch. 4). Second, an analysis of the mathematics under consideration can be undertaken for the purpose of orienting the curriculum developer. The second of these two paths is illustrated by Brink (this volume, Ch. 9). Rather than begin with the assumption that plane Euclidean geometry should be the mathematics of choice, Brink instead assumed the attitude that ‘not only one geometry was taken as the holy truth. This opens up possibilities for multiple conceptions and it helps keep the student’s ideas viable’ (p. 123 above).
