The following question which appeared on an Edexcel GCSE maths paper (for readers outside of the UK, this is a test that would be taken at the end of high school, at the age of 15 or 16) has gone viral, with many students complaining about its difficulty.

There are

nsweets 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.

Show that

n^{2}−n− 90 = 0.

I sympathise with the students complaining about this question’s difficulty. I don’t think it is an easy question for GCSE students. I found it easy to answer, but I’m studying for a Maths degree.

However, there are different kinds of difficulty. Sometimes a question is difficult because it involves the application of knowledge which is complex and/or less prominently featured in the syllabus, which makes acquiring this knowledge and keeping it in your head more difficult, and also makes it harder to realise when this knowledge needs to be applied. This was not one of those questions. The knowledge required to complete this question was very basic stuff which I expect most of the students complaining about it already knew. As far as I can tell, the following mathematical knowledge is required to answer this question:

- If
*m*of some*n*objects have a certain property then the probability that one of these*n*objects, picked at random, will have that certain property is*m*/*n*. - The combined probability of two events with probabilities
*x*and*y*is*x**y*(as long as the events are independent of each other, although you could get away with not understanding that part for this question). - Basic algebraic manipulation techniques, so that you can see that the equation (6/
*n*)(5/(*n*− 1)) = 1/3

can be rearranged into *n*^{2} − *n* − 90 = 0. All this takes is knowing how to multiply the two fractions together on the left-hand side (new numerator is the product of the old numerators, new denominator is the product of the old denominators), then taking the reciprocals of both sides and bringing the 30 on the left-hand side over to the right-hand side (there are different ways you could describe this process).

Certainly for some of the students, the knowledge was what was at issue here, but I think the main challenge in this question was the successful *application* of this knowledge. Most of the students knew the three things listed above, but they failed to see that this knowledge could be used to answer the question.

Unfortunately for the students, failure to apply knowledge successfully is in many ways a much more serious failure than failure to possess the appropriate knowledge. If you fail due to lack of knowledge, there is an obvious step you can take to prevent further failure in the same situation: acquire the knowledge that you lack, by having somebody or something teach you it. Crucially, when you successfully learn something you not only end up knowing it, but you also know that you know it.

If, on the other hand, your problem is that you failed to apply your knowledge successfully, then it is much less clear what the next step you should take is. And, also, you never know whether you are capable of applying your knowledge in every situation where it might be useful, because there are usually a whole lot of different situations where it might be useful and it is impossible for you to be familiar with them all. This is why cramming for tests is not a good idea. Cramming may be the most efficient way to obtain the knowledge you need for a test (I don’t know whether this is actually true, but I don’t think it’s impossible) but it certainly won’t help you with *applying* your knowledge.

There is one relatively straightforward way to become better at applying your knowledge. If you remember how you applied a certain piece of knowledge in a previous situation, then, if the exact same situation occurs again, you won’t have any trouble, because you will just apply the knowledge in the same way again. In a similar way, the more similar the new situation is to the old one which you remember how to deal with, the more likely you are to be able to successfully apply your knowledge.

But, in a way, this is a *trick*. I am about to get a bit esoteric here, but bear with me. Let’s say your mind has two “departments” which work together to solve a problem. One of them is the Department of Knowledge-Recall, or the DKR for short, and the other is the Department of Knowledge-Application, or the DKA for short. I think that when you carry out the strategy above, what is happening in your mind is that the DKA is pre-emptively passing on the tough part of the work to the DKR for them to deal with. If you remember how you used a certain piece of knowledge (let’s call it *X*) to deal with a previous situation, then that memory, in itself, has become knowledge (let’s call it *Y*). The DKR has worked so that *Y* is easily recalled. When the situation comes up again, the DKA’s task is really easy. It’s looking at the most prominent pieces of knowledge that the DKR has made you recall, and it notices that *Y* has a kind of direct link to this situation. If the DKR hadn’t worked to make *Y* a piece of easily-recalled knowledge for you, the DKA would have to do more work, sifting through the knowledge that the DKR has made you recall, possibly asking the DKR for more, inspecting them more closely for any connection to the situation at hand.

There’s a good chance the above paragraph made no sense. But basically, I’m trying to say that familiarising yourself with the situations where you need to apply your knowledge works because it is a process of acquiring specialised knowledge about where you need to apply your knowledge—it is not *truly* applying your knowledge. But probably the more important point to make is that this process is inefficient. It would be much simpler if you could simply apply your knowledge to unfamiliar situations straight away. And it’s often impossible to familiarise yourself with every conceivable situation, because the range of conceivable situations is so vast.

Students taking tests try to carry out the familiarisation process by doing past papers. This is often effective because the range of questions that can be on a paper is often quite limited, so it really is possible to familiarise yourself with every situation where you might need to apply your knowledge. But this isn’t a good thing!

As I argued (somewhat incoherently) above, the skill of being able to apply your knowledge is only separate from the skill of being able to recall your knowledge if it includes the sub-skill of being able to apply your knowledge to *unfamiliar* situations. That particular sub-skill is one which is hard to improve. Arguably, this sub-skill is what we mean by “intelligence” (the word “intelligence” is used to refer to a lot of different things, but this might be the most prototypical thing referred to as “intelligence”). It’s certainly possible to get better at this sub-skill, but I don’t know if it is possible to get better through conscious effort.

But intelligence (by which I mean, the ability to apply your knowledge to unfamiliar situations) is often the main thing the tests that students take are trying to assess. After all, there are few vocations, if any, that a person cannot do better at by being more intelligent. You don’t always need intelligence to get to an acceptable level, sure, but intelligence always helps. The purpose of the GCSEs is not *just* to find out which students are more or less intelligent—they are also supposed to increase the amount of people who have useful knowledge—but it is one their main purposes.

That’s why I don’t think this question was unfair, as some students have been saying. Yes, it was quite different from anything that was on the past papers. But that was the whole point, to test people’s ability to apply their knowledge in unfamiliar situations. It is natural to be disappointed if you couldn’t answer the question and to complain about your disappointment, but saying that it was unfair is a step too far. It did what it was meant to do.

I think there is possibly a genuine problem here, though. If questions like this which strongly tested intelligence (as defined above) were usual on the GCSE papers, you wouldn’t expect this one to become such a topic of interest. Perhaps the GCSEs have suffered in the past from a lack of questions like this, which has affected students’ expectations of what these tests should be like. I should be clear that I have no idea whether this is true. I took my GCSEs in 2011, but I don’t remember what the questions were like in this respect.

PS: I’ve seen some people saying “this is easy, the answer is 10”. These people are making fools of themselves, because the answer is not 10, in fact that doesn’t even make any sense. The question is asking for a proof. (“Show that…”) It seems these people have just seen the quadratic equation at the end and assumed that the question was “Solve this equation” without actually reading it. Perhaps this is evidence that the question really isn’t easy. Or maybe these people just aren’t thinking about it very much.

It is a really easy question – I’ve almost completely forgotten maths, but even I could do that one. [Well, I couldn’t be bothered to actually work out the quadratic and I’m probably not 100% sure on the quadratic formula, but that’s just a memory thing. Anyway, when I did maths we had formula sheets provided in at least some exams…]

Personally, I think the key to making it really easy is that you’re given both the start AND the end-points. Given a blank ‘phrase this as a quadratic’ I’d probably have had to think about it, but when it’s there on the page it’s obvious. n^2-n-90=0 is obviously n(n-1)=90, and n and n-1 are obviously the denominators of the probabilities, so it’s clear what you have to do. I suspect a lot of why some people struggled is that they couldn’t approach the problem in a non-linear way – a lot of people, as it were, only ever start from the beginning and never think of starting from the end and working backwards.

The more general question you raise is an interesting one, though. I think there are actually three aspects of intelligence involved in your ‘knowledge application’: judgement, analogy and innovation. ‘Judgement’ is the bit that looks at the situation and sees which method has to be applied. Analogy is the bit that, when you don’t have a method to apply directly, works out how to alter an existing method to fit this situation. Innovation is the bit that invents a new method out of whole cloth. I think these three are quite different things. Personally, I think (no boasting intended) that I’m very good at analogical thinking, but less good at judging, or perhaps discerning is a better word, which methods to apply, and no good at all at innovative thinking – I’m no good at those mensa-type lateral thinking puzzles, for instance, where I don’t have a model to go on. I don’t know where to start with them. Fortunately, innovative thinking is very rarely needed in real life, if you’re good enough at reasoning from analogy and at applying methods of analysis…

But I’m not convinced that these things aren’t learnable. I think an example here is language. When you start to learn a language, you think like a GCSE student – you see the situation, and you pick which of your existing stock of phrases best fits the situation. Later, you start taking your phrases and modifying them to fit the situations better, but you still begin with that stock of phrases and reason by analogy. But eventually, everybody becomes fluent in the language (everyone, with the exception of a few highly traumatised or brain-damaged people, learns at least one language to a reasonable degree of fluency…), and they are then extremely adept at (subconscious) analogical applications of their skills. After all, nobody has ever said this paragraph before, so I didn’t learn it out of a book: I had to take all the words and constructions from other sentences and fit them – in an instant, without even thinking about it in the slightest – to the current situation.

So I think that we CAN learn to analogise correctly in particular spheres. But often GCSE students don’t. I think that this is less a matter of intelligence and more a matter of knowledge. Which is to say that what people do with exams is treat them as a matter of ‘fact recall’, and try to memorise the facts, including facts like ‘this is how you answer this sort of question’. This can be an efficient way to pass exams, so long as you, in essence, know the questions in advance. What the education system is meant to provide, however, and what unexpected questions test, is the students’ FACILITY with the subject at hand, which I think is a matter not of wrote knowledge but of understood knowledge. I think our exams are actually quite bad at testing understanding, and I suspect many of our teachers are very bad at encouraging understanding, in part because our exams do not encourage it.

I suspect a big part of that, in turn, is a distinction between knowledge of methods and knowledge of identities. Knowledge of methods is extremely important, but knowing which methods to apply and how to modify methods relies at some level on knowing how things actually are. To take a trivial example: I found that question easy because I know instinctively that n^2-n is the same as n(n-1) – I can exchange the two forms at will; and I think this is different from the methodical knowledge about how to factorialise a quadratic. Because if you know ‘this is how to factorialise a quadratic’, you do not know what to do with n^2-n-90=0 until someone tells you to factorialise it! Whereas from understanding the identity you can move from one expression to another: THIS is the same as THAT, now can I do anything useful with THIS? This is how a lot of analysis of things progresses in reality: you don’t go “oh, I know how to solve problems like this!” right away, you go “ok, well that’s equivalent to this, and this is equivalent to such-and-such, and then… oh, it’s obvious now”. [Then difficulties arise when you try to apply two different analogies that yield totally different implications…] Trying to approach a maths problem armed only with a stack of methods to apply and no intuitive awareness of identity is like those English pupils who are asked whether a poem is in iambic metre or not but who, when asked to read the poem, read it out in a steady monotone, not understanding that the ‘poem’ is only something someone is saying, ordinary english words and phrases, and that ‘iambic metre’ is just the normal English pattern of stress…

Anyway, not sure where I’m going with this anymore, which I guess is rather apropos, as that’s kind of the pitfall of reasoning analytically off the top of your head… thanks for the opportunity to think about the question, though!

Ha, I wasn’t quite sure where I was going with my thoughts about knowledge and knowledge application either. It’s a complex subject, and you’re probably right that my ‘knowledge application’ may be best viewed as a composition of different skills. Your point about identities is an interesting one. Being able to recognise that two things are the same and freely interchange between them is definitely important in mathematics. You could say that’s what a lot of higher-level mathematics concerns itself with: abstract algebra, for example, is about constructing a kind of mental apparatus that allows you to view two apparently quite different structures (that is, sets of entities together with operations that can be carried out on those entities), such as the Euclidean plane and the set of all real-valued functions on

R, as the same or similar. I think one of the reasons people sometimes fail to think in terms of identities might be that they don’t have a clear idea of what the mathematical expressions mean in the first place, and without that the distinction between identity and non-identity is no longer meaningful. That might be due to bad teaching, or due to the tendency to resort to guessing the teacher’s password.Thanks for your interesting comment!

Should part of the problem read,”The probability that Hannah eats two ORANGE sweets is 1/3.”? The probability that she eats two sweets is 1.

Ah, that’s right! It’s fixed now. Thanks.