## Sunday, 12 March 2017

I was marking mock exam papers this weekend (in fact that's pretty much the only thing I did this weekend - whose idea was it to introduce a third paper? And whose idea was it to put 31 students in my class?) when during a moment of extreme boredom I read the front of the exam paper.

I was going to blur out the student's name to protect his privacy, and then I realised he had (very considerately) already made his name unreadable. Unfortunately for me, he also made all of his working out illegible too.

In the section that says "instructions to candidates", it says:

"Make sure you know what you have to do before starting your answer."
Does anyone else disagree completely with this advice?

When trying to solve a maths problem, you will very rarely know what you have to do before you start answering it. For example, take the last question of this paper:

When I look at this question, I don't know exactly what the steps are and what the proof looks like. I know it will involve some angle rules, but which ones and in which order, I don't know. It would be completely stupid to sit there looking at the question for fifteen minutes trying to work out exactly what I need to do before starting my answer. The sensible thing to do would be to write down something you do know. For example, you know that OQ and OR are both radii and hence equal in length. After that, you might be a bit stuck, so you start trying things. Eventually, you might realise that drawing in the line PO would be helpful. At this point, you still don't know how you're going to complete the proof. You give some angles some letters, so that you can write some relationships more easily. You could let PQO=x, and therefore QPO=x too. Let ORP = y, so OPR =y. Then you might decide to work out QOP and ROP, not because you know that it's going to help, but because you know how to do it so you may as well. QOP= 180-2x, ROP = 180-2y. Then you work out QOR, because, well, it's there. QOR = 360-(180-2x)-(180-2y) = 2x+2y. Then you realise that QPR= x+y which means that you have proved it. Then you do a happy dance and probably write your maths teacher a note like "circle theorems is bae" (yes one of my students actually wrote that).

At no point during this proof had you worked out what you needed to do to answer the question. You tried some stuff, and some stuff led to other stuff, and eventually the problem was solved. This is how maths problems should be solved. If you know exactly how to answer the question before you start answering it, it's probably not a very interesting question.

I'm wondering whether the creator of this exam question intended for the students to know what they have to do before answering it. That would mean they were expecting students to have memorised by heart proofs to all of the circle theorems. What a depressing thought that is. How is regurgitating a bunch of proofs a sign of good mathematical thinking?

My head of department has a speech he likes to give students on this topic. (Actually, he has lots of speeches he likes to give, as his (and my) students know all too well). He says that it's like untangling your earphones. Earphones, as we know, automatically tangle themselves as soon as you stop using them. And when you want to use them, you have to spend a good five minutes untangling them. When you do this, do you look at the ball of entwined wires in your hand and think, "right, first I'll move that bit under that bit, then I'll pull this end through there, then I'll twist that bit round..."? Of course you don't, that would be silly. Instead, you get stuck in trying to untangle them. You pull bits and you twist bits and you play around with it. And eventually, no matter how bad the tangling was, you always succeed in getting them straightened out. This is exactly how you should tackle a maths problem.

So my advice would be to ignore the advice on the front of the exam paper. Well, except the bit that says "read each question carefully". Oh and also the bit that says "You are reminded of the need for clear presentation in your answers" (which several of my students definitely ignored).

Emma x x x

1. I completely agree (and I'm going to ask for a copyright fee each time you quote me ��).
The only defense for this statement would be if the instruction is intended to mean, "Make sure you understand what you are trying to find out."
I don't remember seeing this on previous papers, but perhaps I just haven't looked carefully enough (obviously not as easily bored as you, though how you can be bored when you've got 93 papers to mark is beyond me!).

2. And, by the way, I think you'll find it was your suggestion to have 31 in your class?

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