This year I am tutoring some maths again. I’ve missed it and it was nice to find my brain in decent shape – after 5 years away from teaching this material.
But as I was preparing for the tutorial, there was an integral substitution that didn’t come immediately. I thought: ‘OK, shall I thrash around for a bit, use some rough paper, maybe get out a few old text books?’ But all decent mathematicians like a short cut, so I just thought, ‘Nah, type it into google and see what happens.’ So without any attempt at intelligence I typed in ‘How do you integrate Cos2x divided by 1+Sin2x? ‘ . Up popped YouTube vids and a written description that I could mash together to give me the solution in about 2 minutes. Bing. Wow.
Even just three years ago if I had been well and truly stuck on that problem it might have taken me days to sort out. First trials with the rough paper, then, if that was no good, out with the textbooks, then calls to friends and finally (if I were a student) a trip to the tutor in their office hour to try to get the solution. Not anymore.
What does this mean for teaching and learning maths? And what, if any, wider issues does it throw up? This experience is fresh in my mind so what follows is still very much being processed, but I’d like a first stab at some thoughts.
The first thing I’d say is that learning a great deal of stuff is getting easier and, yes, more fun. There is delight in being able to interact with the web and get the solutions you are looking for – plus extra stuff you may not have known about but which interests you.
However, there is more than just the fun part to think about.
Here are some issues:
1. Do we need to teach standard year 1 linear algebra and calculus if asking the computer the right questions gives immediate answers?
2. If we don’t teach such material, what are the implications for those who really want to go on to be mathematicians, and what are the implications for those of us who want to use mathematics but don’t want to be mathematicians? Should we separate these two things?
3. (A wider point) What, if anything, are we losing by not having to sit with pencil and paper to try to work out such maths problems?
Let’s say we took a radical step and decided we would not teach any of the traditional A- level and year 1 undergraduate maths. What would happen? Well, first we’d need to think what to put in its place. Maths is important. I’m not suggesting we chuck out maths, I’m asking if we need to continue to ask students to work in a certain way, calculate in a certain way, maybe even think in a certain way if so much stuff up to the level of undergraduate work can be done by asking the computer. I have no doubt that any new work we devise could be rigorous and deeply engaging – I mean what we could ask the students to do could be rigorous and academically challenging. The technology pushes us to think more deeply about what we teach and to ask the right questions of ourselves and the subject. Saying that the technology forces us to dumb down is a cop out for those who do not think through what the technology is offering and how it affects us.
I guess the closest comparison I have in my personal experience for losing something in maths education is the demise of compass and ruler constructions for geometric proofs. I just about caught the tail end of these sorts of questions in school but I’m pretty sure they are completely gone now. Has the field of mathematics suffered? Have brains deteriorated as a result?
There are many, of course, who say that brains have deteriorated and many who say that in losing Euclid we have lost a great deal. I agree that Euclid can change your life for the better, but I am dubious as to whether dropping him and the compass and ruler constructions from the maths curriculum is a major cause of decline in mathematical excellence at the top end of the subject or more general intellectual decline among those who are not mathematicians.
What about splitting ‘pure’ maths from maths methods in our teaching? There seems to be quite a bit of intellectual (ontological?) value in this, but it would be hard to implement.
Basically, if you want people to know what maths is about – that is maths as understood by working mathematicians – you need to teach them about proof, axioms, estimations and the like. Maybe one could keep such an education strictly to pencil and paper: logic, proof, numbers – even a return to Euclid for an insight into geometry?
But other people want to use maths, not worry too much about the proofs, or even the calculations; and the technology is now arriving at the point where, so long as you understand what you are asking and can frame the question correctly, the computer can get you the result of the calculation, without your having to do the calculational work, let alone understand any of the reasons why the formulae work as they do, how they have been proved etc.
In some ways this is the million dollar question: ‘Can you learn to frame a mathematical question correctly without going through years of algebra and calculus?’. Conrad Wolfram has been touching on these issues for a while. He is way ahead of most of us, of course. I was sceptical, but my first brush with ‘calculus by computer’ last night has me thinking again.
I’d say we don’t know the answer to this question yet, but I would venture that some people have a good grasp of mathematical concepts (e.g. rates of change, symmetry, limits) without being able to work out mathematical problems. Do David Harvey and Niall Ferguson know maths? And yet they seem to grasp many mathematical concepts related to economics very well. What about Brian Appleyard or Bill Bryson on science?
What seems different about this proposal to consider dropping calculus is the size of the change. It is one thing to drop pencil and compass proofs and Euclidean geometry, it is quite another to abandon calculus and linear algebra – I mean abandoning a syllabus in which one does not have to work things out with pencil and paper in this way. But maybe this is just being romantic. Maths moves on, calculational tools move on, technology moves on. Big Time.
I repeat, this is not about dumbing the brain down per se. To keep the standards up one would simply have to change the questions. If you get this right, rather than a 17-year-old grinding through hours of differentiation and integration on dry, unrealistic equations which model very little in the real world, you could ask them to solve a real-world problem of a much more sophisticated sort by asking the correct questions and working with the computer in the appropriate way. Not easier brain work, just different.
True, if the student were asked ‘Why does this calculation that you have done work?’, they might not be able to answer in a way which we expect. But I think it would be wrong to say that their answers lacked depth. How many students who have studied maths for many years are currently able to answer the ‘Why?’ questions of maths? If the student on the new curriculum is able to offer similar examples, analogies etc which demonstrate that he has used the right modelling tools, that is surely as much evidence of understanding as saying ‘ it works because that’s the way I learned to do calculus’.
One thing I instinctively feel sure about. If I were 17-18 and every question on my maths homework could be answered by typing it into google I would be demotivated to do the maths without a very good explanation as to why I needed to do it. If that explanation is: ‘You need to do this so that you can calculate results relating to economics, engineering, physics etc’ I would listen, but I would question more. ‘Can’t I be smart and learn how to ask the computer to model fluid flows, tension in structures, financial derivatives, population shifts, astronomical movements etc without doing all these calculations myself? Can’t I grasp the principles, the pictures, the graphs, without the algebra?’
On the other hand, if I loved proof and the idea of ‘being a mathematician’ I would want a teacher to explain to me really what this entailed – and this would also lead me to a different kind of study from that currently offered at A-levels. Is there scope for two, separate, rigorous maths syllabi here?
Photo by Robert Scarth under CC license