Sunday, January 20, 2008

Mathematics as the science of reasoning

Last year, a music teacher at my school photocopied an article and placed it in my pigeonhole. I picked up the article and saw the title "Maths minus reason = failure". I assumed it was an apology for everything being taught with its real life applications. Gladly, I was wrong. The article was written by Marty Ross who does much to popularise mathematics in Victoria. His partner in this is Burkyard Polster. You might think of the pair as the Myth Busters of Mathematics - geeky, intelligent and entertaining.

Back to the article. Ross states: "Clearly the purpose for teaching mathematics cannot be primarily to convey facts and formulas that are rarely used and almost immediately forgotten. The true purpose is to teach the reasoning by which these facts can be established."

Ross goes on to lament the way in which the Pathegorean theorem is taught in Victoria. It is a formula to be accepted and applied over and over. He states that "the fundamental reason to teach the Pythagorean Theorem is because of its central role in Euclidean geometry, that monumental body of work immortalised in Euclid's Elements." And here is what he says about the Elements:
The Elements is the most successful textbook of all time ... But its popularity was not because of some universal love of geometry ... . The real lesson was the process by which these geometric truths were obtained. The Elements is a brilliant, extended display of reasoning, beginning with a small number of accepted truths and proving all that follows.
Ross then presents a pictorial proof of the theorem and expresses his sadness at the fact that no emphasis is based on the beauty and simplicity of such mathematics. Instead, the curriculum documents present maths as a list of facts to be remembered and prescribe an ever increasing use of technology.

Ross then puts on his university lecturer's hat and says that, despite the fact that he teaches those students to need or choose to do maths at a university level, he finds that they "enter university viewing mathematics as no more than a collection of facts, upon which they have the most tenuous grasp. Albert Einstein said: 'Any fool can know. The point is to understand.'"

Reading Ross' description of Euclid's Elements is like reading a synopsis of my year 11 Geometry textbook. I was educated under an old-fashioned adaptation of the French system. We learned many theorems, always with their proofs. The test was always guaranteed to throw a new fact or a theorem at us with the simple question "demontrer que ..." or "show that ...".

I have often heard people say "how could maths be made different?" and each time I have thought "Oh, believe me, we could be teaching very different stuff."