Astute readers will notice that it’s been a very long time (five months!) since my last post. This is because I’ve been on sabbatical for this time, and not teaching. While this has been a wonderful opportunity to get to grips with some research about the use of technology in mathematics education, it has meant that I haven’t been teaching. And it’s teaching, and the thoughts and ideas generated by teaching, which drives this blog.
Anyway, two days ago I had the pleasure of going to a seminar by Conrad Wolfram, who’s the brother of Stephen Wolfram, the creator of Mathematica. The seminar was titled “Mathematica, computer-based math, and the new era of STEM”. In my home state of Victoria, Australia, the Education Department (which overseas K-12 education; higher education being the province of the Federal government), is making Mathematica available to all government schools. Wolfram, who is a fine polemicist, made several points:
- Mathematics as taught in schools, and mathematics as is practised in the “real-world” are two different subjects: schools are primarily computation and calculation; real-world is more about modelling.
- Mathematics is useful primarily for three things: (1) technical jobs, (2) everyday living, (3) logical mind-training. (Although it occurred to me that a good solid dose of philosophy would probably do at least as well a job for (3). )
- Solving a problem mathematically requires four steps: (1) define the question, (2) formulate the problem in mathematical terms, (3) perform the mathematical computations required, (4) interpret the result in the original problem context. Schools are mainly about step (3), which is the one best done by computers, whereas steps (1), (2) and (4) which require the greatest human ingenuity, are barely touched on.
- Mathematics should be taught in a manner which is computer-based, and not just computer assisted.
- We do need to teach some basics. Counting, arithmetic (including multiplication tables), and maybe a few other things, should be part of everybody’s mental box of tools.
The common complaints against computers in mathematics teaching:
“Computers dumb mathematics down”
“It’s hand-calculation which teaches understanding”
are fallacious and easily refuted. (And in fact the literature bears him out.)
- Coding (computer programming) should be a core component of any mathematics course.
You can watch Wolfram’s presentation online here.
Wolfram made the comparison between mathematics as currently taught, and Ancient Greek. One hundred years or so ago, at least in the UK, Ancient Greek was a necessary part of a “proper” education. You couldn’t consider yourself educated unless you had a good solid grounding in the classics (Ancient Greek, and of course Latin). No doubt the policy-makers and curriculum designers of the time feared some great cataclysm would occur if the populace weren’t grounded in the classics. And who knows, maybe they were right? Maybe studying the classics really is superb training for the mind. However, you’ll notice that classical languages do not any more form any part of a core curriculum. And yet the world goes on. Wolfram considers modern school mathematics (which he calls “proxy math”), similar to Ancient Greek. It’s probably quite good mind-training, but it’s mostly dull, engenders enormous resentment in most students (and not a few teachers), and is poor grounding for the modern world.
I think I agree with most of what he says. Policies tend to me made by old codgers (like me) who form part of the few who succeeded in school mathematics. We enjoyed our algebra, our trigonometry, our calculus, and simply playing with symbols. And because it worked for us, we extrapolate to assume it should work for everyone.
I wondered also – where does this leave research mathematicians – people whose job it is to apply mathematics in new ways, and also to create new mathematics, and to build new bridges between existing mathematical theories? Suppose we were all trained Wolfram-style. Would we have the ability, the knowledge, the expertise and the experience to launch into a research career? In fact I think we would. Suppose, for example, you got interested in differential equations: and decided to research the existence and solution properties of a particular family of DE’s. Suppose also that you knew what a DE was, and you could apply them (using your CAS “DSolve” function), but you’d never actually wrestled with one by hand. I reckon you could pick up the necessary skills as you went along. Using a case study of one (me!) – both my research degrees: Masters, PhD, were in areas about which I knew nothing when I started, and I picked up what I needed as I went along. I have never formally studied mathematical logic, but I learned what I needed for my Masters degree in an area which required some recursion theory. Likewise I picked up signal and image processing for my PhD, as well as a lot of topology.
I think Wolfram’s ideas have great merit. I think that school mathematics is a failure. A few people enjoy it and succeed, many don’t enjoy it but struggle through it enough to move to the next level; many dislike – and even despise – the subject. We need to reconsider the curriculum from the ground up – to strip everything away to a blank slate, and start from scratch. What do we want our school leavers to know, and why? Is it necessary for the entire population to be schooled in trigonometry? Could we replace a lot of current mathematics with a new computer-based curriculum? Is the current syllabus very much “Ancient Greek” – it’s good for you?
I don’t know who’s asking these questions, but I think they need a good airing.