That’s the title the editor gave to a letter I had published in the Education Age (21 May 07), commenting on an opinion piece by my University of Melbourne colleague Marty Ross. Since they don’t make the letters to Education Age available online, I’m putting it up here.
Marty’s piece generally was very good. He and I have no deep disagreement; but this letter puts into relief one point where we’d differ at least in emphasis.
According to Marty Ross, the purpose of teaching mathematics is “training in logical thinking, learning to reason about anything.” But mathematics is a poor way to achieve this goal, and there are better reasons to study the “queen of the sciences”.
Ross is recycling an idea as old as Euclid himself, for we can find in Plato the view that reasoning is like a muscle, which can be strengthened through training in some formal discipline such as mathematics or grammar.
The view is plausible, even appealing, but it is misguided. It violates a key insight of research into cognitive skills, known as the “problem of transfer”: skills learned in one situation “transfer” to other situations far less than we would expect. Studying mathematics may help students learn to prove theorems, but quite different kinds of reasoning are usually needed in everyday life and the workplace.
There is a better way. General “informal” reasoning skills can be taught directly, i.e., as skills in their own right. In recent years, some remarkably effective methods for doing this have emerged. Crudely, to Ross I would say: if you want to improve general reasoning, why not teach general reasoning? Why teach something else, in the forlorn hope that some general reasoning skills will result?
Unfortunately, effective direct approaches are not widely used. In addition to its mauling of mathematics, a major problem in the Victorian education system is its failure to systematically teach the fourth R, reasoning.
Lets not try to make mathematics carry a burden it cannot bear. Lets instead teach mathematics to give all students at least a chance to appreciate the profundity, and beauty, of the most magnificent achievements of the human intellect. Meanwhile, lets teach general informal reasoning skills using more direct and effective approaches.
A quick response to this reading:
This is quite a strange moment. Only last night I was discussing the current state of education on a balcony 20 stories up overlooking a tropical forest (I work in education in Singapore) with an advertising executive. Today, while setting up a PD meeting about using Rationale, I come across this posting. As a teacher of the IBO’s Philosophy, TOK, History programmes and a subject that seeks to introduce critical, evaluative and creative thinking (CEC) to younger students I have thought long and hard about the needs of my students at this present time and our response to these needs.
Anyway, to pass some time at a BBQ, the ad exec asked me to give him a summary of the state of education, partly to please his wife, a teacher, but also because of his expressed frustrations about young employees. I pointed out quite a few things – including one on the absurd emphasis on the WW1 and WW2 in History subject schemes. One of them in particular was the time devoted to Maths in every school, and every school’s obsession with it as a subject. My belief has always been as expressed in this letter:
1) The role of Maths in our thinking is a legacy of Plato at al.
2) The emphasis on Maths is partly due to 1) but also due to the legacy of the establishment of the framework of modern education during the industrial revolution.
Maths as part of an education was conceived to teach critical thinking, as the type deemed necessary at that time. These days the need is still here but reasoning has changed significantly and yet we are adhering to these legacies. In an international school where we have immense freedom define the needs of an emerging generation, and adjust/change our education system to cater for it, the traditionalist still hold sway. I am constantly battling with the lack of appreciation by very bright students of the art of thinking, and especially thinking in slow motion. I am an international examiner, examining students from all four continents, and one thing that they struggle with the reasoning. As in changing environmental thinking (ironically a strong part of my teaching) we need to change thinking about education, for the same reason – the future.
My one point is that each subject should seek to dissolve its boundaries, even at the ‘pre-university’ qualification level and integrate or weave in reasoning as one of the primary goals of their learning; before specialising in content for the final judgment day of exams. It is a skill or skills that all subjects can, and should, be conspicuously delivering, extending and assessing. And it is universities who should be supporting this change by simply changing their expectations at entrance – we can do the rest.
Does anyone have the vision and steel to change something so ingrained in Eastern and Western society’s thinking? It’s a tough one. This of debate is essential to the broadening of people’s understanding and a good read at the end of (my) academic year.
Chris, I am not sure but I think you are saying maths is a subject that shouldn’t be taught at the level it has been, because its purpose long ago was to teach reasoning, not maths.
If this is what you mean, its my humble opinion that you may be missing something. But probably I misunderstood.
I want to introduce a new term. “Systems thinking.” It requires both mathematics and logic. Consider the question: “what do we know how do do very well indeed but no one person knows how to do?” Some examples: we can make airliners and run airlines with astonishing reliability given what we ask them to do. We fabricate reliable microchips and computers by the billions and so on. Good mas manufacturing is an extrordinary application of systems thinking. It is highly detailed and tightly logical. It contains a huge mix of expertise. Certainly we need mathematicians and logical thinkers. The logical thinkers are turned out in abundance by computer science and its allied disciplines. Logic in a philosophy department trivial in comparison to what the good analyst programmer deals with on a daily basis, including the problems of the way people talk.
What is crucial to systems thinking and what should be a subject of much more debate, is the ability to think through chains of interactions – cause and effect – on many scales. Mathematics is crucial here. There is no better challenge to intuiton than to try and control a dynamic system with many variables. Or look at what a model of a compost heap or ecosystem involves. However most people do not get to the level of mathematics where they teach such profoundly useful ideas. Nevertheless a very good example is congested traffic. This is not some chaotic beast but a highly predicable social phenomenon. You can teach that and many people will be interested. It is a lesson in systems thinking.