1953 - ON GENIUSI flunked mathematics in High School and at Glasgow University and was one of those strange scholars who was not embarrassed (indeed almost boasted) about my inability to understand mathematics.
Reveal Technique to Destroy Mystique
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.
That "extra" step embodies the universal principle which applies to all equations. Once you understand this simple principle, algebra is easy.
X + 4 = 7 X/4 = 7
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.
Let's destroy the mystique by revealing the technique. Mathematics is just a section of the missing operating manual. It is an important section because it provides the precision in communication which is required by science. The failure to teach logic drives a wedge between the two cultures - the artists who speak everyday language and the scientists who speak mathematics. We are all artists, we are all scientists, and we should not be browbeaten into believing that we are incapable of acquiring all the tools we require.
1 I had the good fortune of having Piaget to myself at a party after he had given a talk, because I could speak some French and he refuses to speak English. He asked if I remembered the first question after his talk. Yes, indeed, after he had described the stages of cognitive growth, the question was "How do you speed it up?" Piaget said: "That's the American question. Every time I give a talk here, that's the first question I'm asked". His answer, of course, was leave the kids alone. Let nature take its time.