Stop screaming. You aren’t screaming out loud, but your eyes are screaming. The screams reverberating around the inside of your skull are making your irises quiver. I see it a lot, but it doesn’t get any less disconcerting. And by the way, this isn’t my fault; the exam is completely fair. All the questions – right up until me, the final question – have been given in class or on assignments, with minor modifications. (They literally just changed some of the numbers.)

You are angry with me because I’m not like the other questions you’ve seen. I’m an unfamiliar stranger who has arrived with the apparent intention of fucking your life up. Well, I am a little different, but you shouldn’t hold that against me. You’ll just have to think a little. And yes, I know, a calculus final is not really the time or place to be doing any serious thinking.

I wouldn’t be angry with the professor. It isn’t exactly making her day to see what the past semester of work actually amounts to. But, for the test to mean something, she needs to push you guys a little further and see how far your understanding goes. OK, OK, you are screaming with your eyes again. Look, why don’t you just start by drawing a diagram? Didn’t the professor beg you all a thousand times to start answering a question by drawing a picture? There you go. The curves are a little rough, but I can already see that you are a little more in control now. Ah – you’ve even started computing the points of intersection. We are dealing with a pro.

Getting back to what I was saying, I don’t want to get stuck into some elitist-slash-meritocratic claptrap, but if we present mathematics without the challenges, then no one is going to be walking away feeling satisfied. Just think how dope it must have been for Isaac Newton, a man with possibly heretical religious ideas and a creeping sense that the gradient might somehow relate to area, to stumble upon the fundamental theorem of calculus. Or for Pythagoras to wake up one day as a man haunted by triangles with perplexing little squares in one corner, but go to bed with a snazzy looking equation. Or for Euler to finally settle a popular pub argument in Koenigsberg by inventing graph theory and making the first foray into topology.

OBVIOUSLY, we aren’t looking for you to make similar kinds of breakthroughs. Almost no one makes those kinds of breakthroughs. Take a closer look at those breakthroughs, and you might find the reality to be a little more complicated. But you should still be making little personal breakthroughs. This is a big ask right now, but I swear that to solve me, the final question on your exam, you just need to apply what you were taught in lectures. On that topic, I’m pretty certain there was a formula that might come in handy right now. You’ll get a few points thrown at you just for writing it down. That is how needy the mark schemes are.

That is how you should see your prof: as someone who has been laying out in front of you a long trail of little breakthroughs.

That is a formula, but maybe you should consider another one.

Yes, we both understand that the formula was not the correct one. But it isn’t necessary to obliterate it from the page so thoroughly. A simple crossing out would suffice. Christ. The grader is going to imagine that you wrote something really obscene under all of that.

Great – that is the formula we were looking for. Now you’ll want to plug some numbers in. I don’t want to be glib about it, but there are only so many ways you can insert the numbers, and one of them will work. Well, yes, maybe. OK, that’s promising. And now we apply the fundamental theorem of calculus. Oh God, that isn’t an indefinite integral, so you don’t need that constant at the end. Fine, obliterate away.

I can see that you are one of those undergraduates who do arithmetic like you are reading a tarot.

You’ve got a final answer, which is good. Yes, put a box around it if you like. Or a big heart. Fine. I think the grader isn’t going to have any problems locating it. Perhaps you should go back and check all your arithmetic. You still have a good ten minutes of the exam left, and I know for a fact that you guys can usually pick up a good five points just by double-checking.

Or you could start drawing your favorite animé character on the back page. Up to you.