Last Thursday in the UK around half a million 15 and 16-year olds took a GCSE maths exam, specifically the second paper in the non-calculator exam. By Friday the exam was trending on twitter (#EdexcelMaths), with one particular question attracting attention:
There are n sweets in a bag.
6 of the sweets are orange.
The rest of the sweets are yellow.
Hannah takes at random a sweet from the bag.
She eats the sweet.
Hannah then takes at random another sweet from the bag.
She eats the sweet.
The probability that Hannah eats two orange sweets is 1/3.
(a) Show that n2 – n – 90 = 0
I had a quick attempt and after one unproductive sidetrack I got the answer. So why am I writing about this? Because it fits in with the other posts on assessment I’m doing, and to explore some of the issues around it.
First, the actual mathematical content is pretty straightforward – you only need to know how to do three things: calculate a probability without replacement, multiply fractions and rearrange an equation. This is hardly Sheldon Cooper territory.
The exam board has two tiers for the qualification (foundation and higher) and probability without replacement is only explicitly mentioned for the higher tier. The exam has been quoted as saying the question was aimed at those students who would achieve the highest grades (A and A*), and I think grade discrimination is a fair approach. I did ask my daughter (who’s currently revising and taking A-level maths) and she said she wasn’t sure she would have been able to answer it at 16. For non-UK readers, GCSE exams are taken at the end of compulsory schooling and A-levels are taken at 18, typically as a route to studying at university.
So why my unease with students finding it difficult? There’s always the charge of dumbing down levelled at exams but I don’t think that’s it. True, when I did my maths exam at that age the syllabus included calculus of polynomials and their applications, which now is only introduced at A-level, but they were different qualifications – GCSEs were only introduced after my school career had ended. I think my unease comes from the fact that I think this shouldn’t have been seen as a difficult question. Donald Clark has blogged seven reasons why he agrees with the children and thinks it wasn’t a fair question, some of which I agree with and most that I don’t.
There’s a couple of factors involved here. I recall reading a study where they looked at who could answer questions with the same maths content but that were written in different ways. That study found that questions written as word questions rather than equations were consistently harder to answer, even though there was no difference in the actual mathematical content. Secondly, I think it’s the way that maths is taught as rules and recipes to follow rather than a creative problem solving activity. This is not a criticism of the teachers because I think that it’s taught that way precisely because of the pressures that have (politically) been placed on education. As I’ve mentioned before I’m a big fan of Jo Boaler’s approach and it’s emphasis on flexibility and application of technique rather than stamp-collecting formulae. Donald Clark makes the distinction between functional maths (maths for a practical purpose such as employment) and the type of maths typically found in exams, but I think that’s a false dichotomy in this case. As Stephen Downes said “… what this question tells me is the difference between learning some mathematics and thinking mathematically.”. The difference between functional and theoretical maths (at this level) starts to disappear when we think mathematically – maths becomes a toolbox of skills to be applied to the problem at hand, rather than a particular formula in a particular topic to be remembered.
And if you’re wondering what the answer was: