Traditionally, there has been a certain amount of detachment between teachers of mathematics and cognitively oriented educational scientists who endeavored to develop theories about the learning of mathematics. At present, however, there are signs of a rapprochement, at least on the part of some of the scientists, who have come to realize that their theories must ultimately be evaluated according to how much they can contribute to the improvement of educational practice. Healthy though this realization is, it at once raises problems of its own. At the outset there is the research scientists” inherent fear of getting bogged down in so many practical considerations that it will no longer be possible to come up with a theory that may satisfy their minimum requirements of generality and elegance. Then, when scientists do come up with a tentative theory, there is the difficulty of applying it in such a way that its practical usefulness is demonstrated. This would require either scientists” direct involvement in teaching or the professional teachers” willingness and freedom to become familiar with the theory and to incorporate it into actual teaching practice for a certain length of time. In both cases, it will help if scientists and teachers can establish a consensual domain. In other words, they must come to share some basic ideas on the process of education and the teaching of mathematics in particular.

Key words: education, cognition, development

The notions of prediction and retrodiction, and the role they play in the natural sciences, are discussed. These notions derive from our perception of the fundamental category of time, as an ordering scheme for our experiential world. The issue of philosophical determinism vs. human free will is examined from a perspective of radical constructivism, and contrasted with the issue of solipsism vs. shared experience; and it is argued that both philosophical determinism and solipsism may be rejected, on the same (existential, not logical) grounds. Both prediction and retrodiction are discussed, in the context of some sciences (notably, classical and quantum physics), and are shown to be realisable only to a very limited extent. Some consequences of this, for the ability of science to forecast what will happen in the future, or to infer what has happened in the past, are reviewed. It is concluded that scientific knowledge, of both the past and the future, is (and must be) constructed in the present, based on presently observed data and theoretical arguments. Hence there can be no conception of true knowledge, either of the future or of the past.