Stephen Wolfram has some serious credibility in the Mathematics world. He wrote Mathematica. He created Wolfram Language. He created Wolfram Alpha and many other things. If you don’t know what any of these are then look them up. You can also look up his TED talk. It is fair to say that his language and his maths software have revolutionised the actual use of real maths in the real world. No engineer, scientist, statistician or indeed anyone needs to crunch the number, the algebra or the calculus anymore. There are tools to do it.
He came to our school, Markham College in Lima – Peru, while visiting his Wolfram research team for South America that has its home in Lima. He chatted with a group of us before giving an inspirational talk to over 300 students. In the conversation he described how computational thinking was taking over from mathematical thinking. He despaired that Maths education in schools and universities was still mired in the formal processes and that not only was it increasingly irrelevant but was designed to put kids off Maths. These are my words for his ideas, but I am not too far off I think. In any case it set me thinking about what he was saying. Is it really so radical? Does it question what we do in schools? Is it about time we changed? The answers to these questions are No, Yes and Yes. I will explain.
Wolfram began as a physicist in the white heat of the 70s and 80s when those looking at particles discovered or invented (take your pick depending on your philosophy) Quarks and their friends. He told us that some of his work still used at CERN in testing the bits that fly off when they collide particles at near light speed. He maintains that particle Physics has not really moved on from there in the last 30 years, which is an opinion I am not qualified to judge on.
What has changed has been computational thinking, partly due to the technological advances that have allowed Mathematica, for example, to become so sophisticated and powerful. It is also due to a change in the way people have thought about how to solve problems. It has taken a long time. I remember reading an article by David Tall in about 1982 (I guess) saying that the way we teach calculus was redundant already. He foresaw correctly that no practical mathematician would solve a differential equation analytically. They would use numerical methods, and the technology would give them a much more useful answer. The arguments were largely threefold. Numerical methods are easier to use. They do not require a set of clever tricks and they are universal. This means that computational methods can integrate Sin(Sin(x)) just as easily as they can integrate Sin(x). Note that there is no analytical solution for the former. No function that we can write in algebraic syntax differentiates to Sin(Sin(x)). The classical mathematician has to give up. The computational mathematician doesn’t see the problem.
Wolfram also argues that the mathematical way of thinking puts a lot of children off the subject. In his book on coding the concept of a variable does not arise until some 30 chapters in! This is shocking to a mathematician who would hinge understanding of the processes to an understanding of variables.
What can we draw from this? I certainly do not believe Maths is dead, but have long thought that the formal, syntactical way we teach it puts people off. As many mathematicians will have experienced there is an often repeated response when “normal” people hear what my subject is. “! never understood Maths” or even “I hated Maths” often followed by “I know it’s important, I just can’t do it”. The sad times are when this comes from students who steadfastly refuse to see the beauty that we see in the subject. I agree with Wolfram that the way we teach it contributes greatly to this.
I was talking to a friend and colleague on the weekend about this. He is a literature teacher and struggled to understand what I was saying about Maths teaching until I gave the following analogy. Imagine, I suggested, that in teaching literature you spent years on vocabulary, grammar, syntax, the techniques of narrative, of analogy, of plot and character construction but did not allow the students to read any book or poem until these were mastered. This is how we, all too often, teach Maths. We constrain our students to master numerical, algebraic and calculus techniques long before we let them loose on any voyage of mathematical discovery. Of course my literature minded friend found the idea that years would be spent on techniques before discovering the books was ridiculous. So why do we do this?
Perhaps the answer is that we Maths teachers are part of the small group of people who have succeeded through our ability to understand, appreciate and dare I say love this formal manipulation and expression. It is, then, no surprise that we cling to this as being the heart of Mathematics; its soul and its very nature. Wolfram might then respond that it was time it was mercifully killed off.
But, if so, is that the death of Maths? Well I guess that depends on how you define Maths. I sympathise with Wolfram’s story that he never knew his times tables until middle age, as I had a very similar experience, so perhaps my view of Maths is not the normal one. I got very excited in the early 80’s when I discovered the fascination and beauty of exploring iterative processes, as Wolfram did with cellular automata. Mandelbrot and Feigenbaum were my heroes. I created pictures and made music, some of which can still be heard on these pages. Mathematics was about the exploration of the consequences of the rules we set ourselves. It was a journey into the unknown that drew me along for hours on end. It still does!
One key question for educators is whether we need the great technological tools to be able to discover Mathematics. I would say the answer is a resounding No, although there are some wonderful things that the technology helps us to do. Investigative, exploratory Maths can come from a pencil and paper. The skills acquired in these explorations are the same as when looking at iterative processes on a computer, just the tools are different.
So, Maths is not dead. It just needs to wake up and smell the changed planet in which it resides.