In the final panel discussion at the Psychology of Mathematics Education meeting in Montreal last July, Hermine Sinclair (in remarks delivered in her absence by Carolyn Kieran) posed a paradox that she saw underlying much of the wrangling about constructivism that had occurred at the meeting. Everyone accepts the view that there can be no end to human knowledge, that there will always be something new to be added to what we know. The more difficult view is that our present-day knowledge will change. Learning entails both the conservation and the transformation of our knowledge. The paradox is that although change occurs when we push our ideas until we encounter a conceptual gap or obstacle, which signals that our ideas are in some way false, we can change those ideas only by remaining completely convinced of their truth. Only on looking back can we see how our knowledge changed and how it remained the same.
