Are maths questions written poorly?

Is it as much about translation as maths ability...
23 May 2023

Question

The cost of a mobile phone is x pounds and a television y pounds. When both prices are increased by 40 quid, the ratio for the cost of the mobile phone to the cost of the television is 15 to 22. When both prices drop by a hundred pounds, the ratio for the cost of the mobile phone to the cost of the television is 8 to 15, find the values of X, the mobile phone, and Y, the television.

Chris Smith asked mathematician Ems Lord, astrobiologist Lewis Dartnell, and space scientist Xander Byrne to take a look at this GCSE maths question...

Ems - A good maths question is one you want to solve. And looking at this question, I think my first reaction was, and I don't think I'm the only one in this room who thought the same way, "how fast can I get out of here." It's not something that you wake up and think that's the sort of question I want to solve when I do my maths. And I think it's really important that we set engaging questions and ones that link to real life.

Chris - I was laughing because we've got a brainpan in here the size of a planet. The IQ of you lot combined is into hundreds and hundreds. All of you were scribbling. Xander, what do you think of this question?

Xander - Halfway through reading this one, I just got bored. It's so long and I think it said it was four marks for this question?

Ems - I think with something like this, you can't lose interest in it. You've got to be really engaged. So I think that's the first thing, have a straightforward question with some really great maths in. We also have to think about the reading level because not everybody who is great at maths is necessarily great at the reading. And this takes a lot of comprehension to get your head around. And let's face it, reading comprehension has been a lot in the news with younger children in the last week and the publication of the SATS paper today. So, you know, on the comprehension side it's challenging. I can see they've tried to do real life, 'mobile phones, televisions', but really? It's probably an example of a question written by a mathematician trying to think about real life, but perhaps not quite with the children. And I think there was a good question a few years ago about sweets and when somebody was sharing some sweets and that also hit the headlines with everybody trying to solve that one when they came out of the exam room.

Lewis - I've got an answer and I'd be curious if you've got the same answer, but what I like about this kind of question is it's challenging how well you can basically do translation between different languages. How can you take this slightly awkwardly phrased English sentence and then write it as a maths sentence. And you get two equations, simultaneous equations, and you merge them together and out the end pops two answers, two numbers, and it's kind of satisfying. You've solved a puzzle. Although I really couldn't care less about the answer.

Chris - What is really interesting is that both of you have sort of said it's about translation, these questions. It's about taking what someone has set as a challenge in English and you've got to translate it into maths speak, but your maths speak. Ems, can you put us out of our misery now and tell us how to do it?

Ems - Now you've got somebody here who's the director of NRICH, and what we try to do is give hints and tips. So when I've been scribbling, I've been trying to think of the different ways that students might do this to give hints and tips. And what I would say is one of my colleagues, who was also called Chris, had a really lovely way of solving these problems when the numbers get a little bit challenging. And he said, think of something that's a little bit simpler. So forget about adding on these numbers, maybe you've got a 20 pound note and you're going to be sharing it in a simple ratio, maybe two to three. How are you going to do that? So make it simple to begin with. And once you've had that chance to play with it on a much more simple level and you can put it into your own words a little bit more easily, then you can do what we've all been doing here, scribbling away with a simultaneous equation. So that would be my answer.

Chris - You have to put us out of our misery and tell us how to do it and what the answer is.

Lewis - So I wrote two sentences in maths. My first sentence was X plus 40 over Y plus 40 equals 15 over 22. And then similarly with the second sentence in maths language: X minus a hundred over Y minus a hundred equals eight over 15. And then you multiply it all out and you get X equals some gubbins and then you substitute that gubbins into the other equation and then you churn through the gears of maths and it's like four or five lines of scribbles until the answer just plops out at the end.

Xander - More or less. Sometimes you can kind of see tricks with this. I've been doing an obscene amount of maths for my degree, so occasionally I spot shortcuts. One equation subtracted from the other gives seven x minus seven Y equals something and then you can divide by seven and get it. I have a simpler thing in terms of just x and y rather than any factors out the front. So there are shortcuts you can get with enough experience.

Chris - I'm relieved that you solved it. It was the English that killed it rather than the maths. It's not hard maths, it's just hard to turn into maths speak, I think.

Ems - And I think that shows the skill in writing questions and that sometimes perhaps even though folk are very well meaning they don't always get it quite right.