The thirty-second yearbook of the NCTM (Jones 1970) deals with the history of mathematics education in the United States and Canada. Scattered throughout the book are references to psychological and educational theories and their impact on mathematics education. The name of Piaget occurs on seven pages, and some of his books are named; but the volume does not spell out the precise ways in which Piaget's work is of value to the mathematics educator. Personally, I have been greatly influenced by his work, details of which have been published in a vast array of books and papers over almost fifty years, although it would be true to say that it is only in the last ten years that there has developed a widespread interest in his work in the United States. I believe that his position regarding the acquisition of certain kinds necesof new knowledge is of more value to the mathematics teacher than any other position at the moment, although I affirm with equal conviction that his theory does not cover a11 the facts and that one day it will be replaced or subsumed by a more allembracing one. Some of the strengths and inadequacies of his theory in relation to mathematics learning have been given by Lovell ([a] 1966) and Beilin (1971). Although it would be wrong for me to outline Piaget's cognitive-development system here, I must make six points that are the assumptions, so to speak, on which the remainder of my paper rests.