## Mathematical Style

Yesterday I read this post from Tom Bennison (@DrBennison). The post was written to start a conversation for a twitter chat that I unfortunately couldn’t make. It did, however, make me think.

He was questioning Wetherby mathematical elegance and style should be assessed at A level. Suggesting that solutions with more elegance should be awarded more marks.

Bizarrely, the example he used was almost exactly the same as a discussion if had with a year 13 class not long before I read his post. His example was finding the midpoint of a quadratic. He looked at two methods – completing the square and differentiation – and suggested that as CTS is more elegant that should be worth more.

I agree immensely that CTS is a preferable method with far more elegance, but I don’t think the marks should be different depending on the method you choose. I feel that we should be encouraging mathematical thought, trying to create young mathematicians who can apply themselves to a problem and find their own way through. I feel if we start assigning marks for elegance and style them we would be moving towards the “guess what’s in my head” style of assessment that I feel we need to be moving away from. The way to do well would be to spot from a question what the examiner wants, rather than to apply the mathematical tools at ones disposal and find a solution.

Back to that Y13 lesson I mentioned, we were looking back over some C3 functions work and one of the questions involved finding the range of a quadratic function – so obviously it was necessary to.find the minimum. A discussion ensued as to how to do this with students coming up with 3 valid methods. The two mentioned above, both of which I find quite elegant, although I do much prefer CTS. The third method was suggested by one student who said “it’s -b/a – you just do -b/a” I knew what he meant – he was saying that this was the x value where the minimum occurred and that you put that in to find y, but he didn’t really understand what it was or why. He’d come across the method online and has learned it as a trick. When I showed him it came from completing the square and looking at it as a graph transformation, I saw the light bulb come on.

It is an interesting discussion. Some methods are far more elegan, and some are just algorithmic tricks. I think that the lack of understanding with these tricks will lead to marks being lost. So perhaps this will self regulate.

*I’d love to hear your views in this, which way would you tackle finding the minimum of a quadratic? And do you think we should assign marks to elegance and style?*

I agree CTS is quicker for quadratics but it only works for quadratics. I’d argue that what you are teaching with differentiation and stationary points is a general principles that you can apply to any function you can differentiate. So in that sense you could say CTS is an algorithmic trick and differentiation is better, more general method.

I’m just playing devil’s advocate really to show that “elegance” can be a bit subjective and dependant on the context.

It could indeed be seen that way. Although I’d argue it’s still not a trick, in the sense that it gives you more information about the function. Differentiation is definitely more applicable to other higher order functions.

We have talked about “tricks” before and I think we are all in agreement that they should be avoided on the principle that they don’t teach understanding. I still have students who insist on synthetic division.

Awarding more marks for a perceived better method is not going to help anybody. Congratulating pupils for seeing elegant solutions is however a valid contribution to help their progress in mathematics.

The chain rule is something you end up memorizing. Do you consider that a “trick” if the person doesn’t define f(u) and f'(u), but rather does it automatically? Define what you consider a “trick”. If only a shortcut is taught with no context to how one arrived at it, yes, that would lack some understanding. But if context is given, what’s wrong with a shortcut? Are you against students doing “cancelling” when multiplying fractions, because it’s a “trick”? If a student can do a variety of problems correctly, how do you conclude the student lacks understanding and is relying on “following a recipe” and being parrotlike?

I think generally your definition of trick there is mine. A trick is a shortcut without context.

I’ve further fleshed out some thoughts here: https://cavmaths.wordpress.com/2016/05/24/dialogic-teaching-and-problem-solving/

Thanks. You might also want to read my thoughts on it here: http://www.ams.org/notices/201310/rnoti-p1340.pdf

Thanks!

I’d almost argue that A levels aren’t hard enough to do this. Since a lot of the questions are fairly similar conceptually across exams, this surely would only lead to enforcing the idea of there being one “good” method (the one which the exam board has deemed most elegant), which is not what we’re aiming for at all. This kind of marking may have a place in exams which involve more problem solving (for example, STEP exams, Olympiad papers), but even then, the subjective nature of elegance makes this problematic. I believe the BMO2 papers actually do have an award for elegant solutions: https://bmos.ukmt.org.uk/news/20131020-00-bradleyprize.shtml which I quite like.

Elegance has its own reward: as well as leading to more satisfying answers, it’s also likely to cut down on the time taken to do a question, I don’t think it needs marking. At a higher level, not all solutions can be elegant. Some are messy and crude. Essentially, “brute force” methods are used at a University far more than I would’ve thought. We sell maths on being elegant, it’s often what makes it appealing and “beautiful”, but getting hung up on it early on is not necessary. With the increasing involvement of computers within maths, a lot of the ways of working are becoming inherently less pretty: just think of Four-Colour Theorem.

I wouldn’t be against marks for mathematical presentation though. Not quite the same as elegance, but essentially ‘quality of written communication’ marks at A level would be good.

Thanks for the comments, you have eloquently expressed the way my thoughts are currently leaning and I certainly agree with you regarding presentation!

I think we should spend time talking about elegant maths and what constitutes a ‘better’ route to a solution than others – this is something a more experienced mathematician can see and appreciate. I discussed it a little with my year 11 students the other day when I said I wanted to factorise the expression in the numerator of an algebraic fraction (even though it couldn’t be simplified further). I explained how it’s no “more correct” but it is prettier – that prompted a good chat about what constituted “pretty” maths.

Having said that, I can’t see a fair way to examine elegance so I can’t see how it’s something that could feature on a test. It comes down to the idea that what happens in the classroom has to go over and above what is (or can be) examinable.

I would certainly agree there. I definitely think the idea of elegance should be discussed and debated in the classroom- particularly with those more able mathematicians. But I too can’t envisage a way to assign credit fairly.