My first teaching assignment was all the High School mathematics, physics, and chemistry in Seven Islands, a remote community in Northern Quebec. In my panic, I read the preface to Euclid's Geometry. Euclid invites us to a game of let's pretend the following things are true (he gives us the axioms - a straight line is the shortest distance between two points, etc.) and then see what follows (he gives us the various theorems - the square on the hypotenuse is equal to the sum of the squares on the other two sides, etc.). I learned in an evening what I could not learn over four years in High School. My teachers had not revealed the basic principle to me. The Pythagorean Society, which considered those theorems as mystical secrets and killed anyone who revealed them, is alive and well in High School common-rooms.

      My second teaching assignment was Grade 8 Maths in a High School in Montreal. I had a special gifted class (I was the only one in the class who was not gifted). Elementary school teachers had enriched the curriculum for my gifted students by teaching them some algebra even although it was not on the curriculum. So they had learned to solve equations by "putting it over the other side and changing the sign". This little mantra works when the equation is X + 4 = 7 but not when X/4 = 7. When I introduced the principle that an equation is an balance and, therefore, one must do the same to both sides to retain the balance, they complained about the "extra" step.

      My doctoral thesis was based on the work of Jean Piaget. He argues that cognitive development passes through stages - sensorimotor (0-2), concrete operations (2-11) and abstract operations (11 on). Much of our problem with mathematics is due to presenting abstract mathematics before we are ready for it.^₁ We may manage to pass courses by memorising mantras but never fully understand the underlying principles. Maths, which is simply everyday language made more precise, remains a mystery to us for the rest of our lives. My thesis was on the development of understanding of the meaning of logical operators in propositional reasoning. Curious about the principles of propositional reasoning, I took a course in logic. This I found was the missing link between everyday language (which I understood) and mathematics (which I didn't understand). Once again, the Pythagorean Society was hiding its secrets from me!

      At UC Santa Cruz, I consulted briefly in a course called Maths Without Anxiety for women who were having re-entry problems in university,. Feminists seeking explanations for the under-representation of women in the professions had found that the bottleneck was High School Mathematics which was a prerequisite for professional programmes. Girls had learned in High School that it was not cool to be good at mathematics. They turned suddenly stupid. Failure in mathematics had nothing to do with any innate inability for women to learn mathematics. As this course demonstrated, once the anxiety was removed, the maths was learned.