7 reasons why kids right to have gone apeshit over this GCSE maths question

A maths question posed to 16 year olds in England caused a Twitterstorm, as teenage wrath came down upon the examiners for being deliberately obscure, oblique and irrelevant. A petition has reached 15,000 signatures (impressive number), asking for the grade boundaries to be changed. Something freaked these students and it's worth reflecting on what could have gone wrong. After all, in previous exams, impossible to answer questions have been set.

The maths Taliban would love every child to be able, like them, to solve this conundrum and their reaction has been to blame the students and claim that here's nothing wrong here. But i think there is fire behind this smokescreen - it is a poor test item. It is also a touchstone for educational debate.

1. Bait and switch

Although it sets you up with a probability question, they bait you with probability, then switch to algebra and throw in a quadratic. To understand why this is a problem you have to understand that the maths GCSE curriculum is taught in silos – number, algebra, ratios, geometry, probability and statistics. Maths is not (or rarely) taught in terms of the broader mathematical thinking skills required to answer this type of question. Kids are taught to identify the ‘type’ of question in relation to the curriculum categories they’ve been taught and apply the rules they’ve been taught in that one domain. I understand that this is not right but it's the reality. That, in iteself, I suspect, explains the uproar.

2. Final probabilty is given to problem

Rather puzzlingly, they use the all too predictably infantile, Beano-like, how many sweets are in the bag (really - sweets?) story. This is a bit pathetic and an artefact that seems to only exist in the minds of exam setters. They are then confused by being given a known probability (1/3). This sets up dissonance in the mind of the learner. They think - hold on – I though we were being asked to find out how many sweets are in the bag (solve a probability problem) now I’m being given a final probability – that doesn’t make sense. By giving the final probability in the question, you confuse the student. They’re asked to fill out the middle of a sandwich, which is rarely real, and not what they’ve been doing for the last four years. In fact it’s rare in mathematics. I suspect that a lot of students would have been panicked by this on first reading.

3. Trap

As an assessment item it is poor, as it lays a trap for students. The appearance of a quadratic equation in the question n² – n – 90 = 0 suggests that this needs to be solved - that's almost what asked for in the final statement. Most of the young people sitting this exam would have been drilled in solving equations like this. Again, I'm not saying this is right, but it's how it's done. This was part b) of the question but many would have been thrown trying to answer part a) and not got that far.

4. Show what?

Show what? Another problem lies with the word ‘show’. What does that mean in maths? Draw a diagram? Solve the equation? Come to a conclusion? Provide a proof? It’s a vague term that should have been made much clearer. I know this is common in GCSE papers but more exact instructions would help. For a good discussion of the 'show' issue in maths see this TES piece.
5. Position
The question was not at at the end of the paper but earlier than expected for such a difficult test item (19/25). Students are taught that the questions ramp up in difficulty, with questions like these right at the end. By appearing relatively early in the paper it would have stumped many and lost them time and confidence for the completion of the last six questions. This, for me, is the one reason the grade boundaries should be reconsidered.

6. Irrelevant

I’d set a high probability on the fact that the majority of politicians, supporting the maths obsession, and people working in the Department of Education, wouldn’t be able to and don’t have to be able to complete this problem. I’d say with huge certainty that the vast majority of adults, functioning well in their lives and jobs, would not be able and don’t have to be able to complete the problem. The vast majority of adults will never have to do algebra anywhere near this level of sophistication in their lives. I’d go even further and say with a very high probability that no human being has ever thought of this particular exercise as a problem, never mind used it in any practical sense. Let’s suppose we put relevance issues to one side, it is still a poorly designed test item.

7. Non-functional

Even those who do go on to higher education do not need to be able to solve this problem. In fact, the question may have skewed the Edexel exam away from the other boards Maths exams, unfairly hitting those who just happened to find that their school went with this exam board. This is a serious weakness in the English education system. We have been driven towards abandoning vocational skills, which largely need ‘functional’ maths, not the obscure algebra embedded in the GCSE curriculum. We are now locking students into a ‘resit until you pass’ model that does nothing more than make them feel like failures and turns them off progression in their education. The sooner we have a functional (not foundation) maths qualification, alongside the GCSE the better.

Maths in schools

The issue should be made clearer when the stats emerge for student performance on this question and the paper as a whole. I’ve taught maths like this to teenagers who don’t see the point of it all, and they have a point. First, as I have argued, this question raises the question of the quality of our exam boards and examiners. We have had impossible to answer questions and although this is just a badly constructed test item, quality control is clearly an issue. More importantly it is a symptom of something far worse – our modern obsession with maths in schools.
Quality

We really do have to ask whether we need multiple examination bodies in England. It increases the probability of mistakes being made and make no mistake, they are being made. This is not a one-off incident. AQA this year posed a chemistry question, then gave the full answer just a few questions later. We have had OCR ask and impossible to ask Maths question in an AS paper. The question as printed asked candidates to verify the shortest route, for two given conditions, giving values of 32.4 + 2x km and 34.2 + x km. These values should have been 34.3 + 2x km and 36.1 + x km respectively. The error was not to have included twice the journey between A and B (0.9 km) and the journey between F and G (1.0 km) in the values given. More errors were found found in A-level physics and a GCSE Latin paper from the OCR exam board and an AQA maths GCSE foundation paper. AQA admitted they had another rogue question in this year's A-level paper. Ofqual have proven to be toothless, all too ready to defend the hapless examiners and examination boards, when students were clearly affected by these errors. The quality control is clearly insufficient, realt tests by real students is nor rigorous enough and poof-checking lax.

Solution

1. Express two probabilities:

Hannah takes the first sweet, so there is a 6/n chance it will be orange.

(6 oranges and n sweets)

Hannah takes a second sweet, so there is a 5/(n-1) chance it will be orange.

As there are only 5 orange sweets left out of a total of n - 1 sweets.

2. Calculate two probabilities as one:

The chance of getting two orange sweets in a row is the first probability multiplied by the second.

Which is 6/n x 5/n–1

3. Express as a single equation:

We’re told that the chance of Hannah getting two orange sweets is 1/3.

So: 6/n x 5/n–1 = 1/3

4. Rearrange equation:

 (6x5)/n(n-1) = 30/(n2 – n) = 1/3

Or 90/(n2 – n) = 1

So (n2 – n) = 90

Hence: n2 – n – 90 = 0