There’s been a lot of talk on social (and other) media about “that maths question”. A problem in last week’s GCSE Maths paper that stumped a whole cohort of candidates.

I suppose the resultant Twitter storm is the modern day equivalent of sitting outside the school gates discussing the exam that we had just taken.

I always tried to avoid these discussions, and am still not an advocate of them.

Exam post mortems are rarely useful, and are never going to change how you have answered the paper. They either make you feel bad because you didn’t spot the answer that your friends did, or make your friends feel bad because they didn’t spot the answer that you did.

The difficult question started like this:

A bag contains an unknown number of sweets. Six are orange; the rest are yellow. Hannah took two sweets from the bag and they were both orange.

So far, so good. It’s looking like a standard probability question, and the next bit is going to ask “what is the probability of this happening”.

Except … it wasn’t. The next bit was:

The chances of this happening are 1/3Prove that n

^{2}– n – 90 = 0

**Do what**? I can see why this caused such confusion. It looks like a non-sequitur. OK, once you have worked out the equation for the probability, it cancels down to this quadratic equation, but maybe there should have been an intermediate step to tell the students to do this.

I spent a few minutes working this out yesterday. My first attempt went off at a tangent, and I filled a page of A4 with equations that proved that 1^{2} = 1 – not a whole lot of use, other than to provide amusement to my tame maths teacher.

But then I got there:

Initially there are n sweets in the bag of which six are orange. So the probability of picking an orange sweet is 6/n.

There is then one fewer sweet in the bag, and only five orange ones. So the probably of picking second orange sweet is 5/n-1.

Multiply these together, and you get the probability of picking two orange sweets:

We’ve been told that this is 1/3, so:

Multiply both sides by 3:

Multiply both sides by (n

^{2}-n)And finally subtract 90 from both sides

W^{5}(WWWWW – Which Was What We Wanted). Or QED (Quite Easily Done), if you prefer.

Apparently the next part of the question was to solve the equation – having been given the quadratic equation.

So not rocket science. Not even algebra, in my book – it’s simple substitution.

“In my day we were expected to know the quadratic equation.” Though we probably weren’t as it would have been in the Tables Book (which don’t exist anymore as calculators do it all) along with the rest of the formulae that we might need.

“Come on, Page 8, the refusal formulae” was the familiar cry from Ratty Saunders when we were faced with a trigonometrical problem.

“Why do you call them the refusal formulae?” we once dared to ask. “Because generations of schoolboys have refused to learn them” was the answer.

#### Was the old O-Level paper easier?

I dug out my O-Level paper to see what sort of probability or algebra questions we were expected to solve.

A number is chosen at random from the numbers 2, 3, 4, 5, 6, 7, 8, 9. What is the probability that it is divisible by 3? What is the probability that it is either a prime number or divisible by 3, or both?w = (x – y) / 2a. Express x in terms of a, w and y

After modification, a car will do 25% more miles to the gallon. Calculate the resulting percentage saving in fuel for a given journey.

Solve the inequality 5 – 3x > (x / 2)

These don’t look particularly more difficult than today’s GCSE questions – in fact, it could be argued that these examples are easier, as it specifically tells you what you need to do.

We can only realistically compare exam papers if they are assessing the same syllabus. Maths is a huge subject, and topics drop in and out of the syllabus more frequently than … fill in your own metaphor here.

#### Still carrying your “Little Oxford Dictionary” ?

I took O-Levels in Maths (in year 10) and Advanced Maths (in year 11), followed by A-Level Maths and (for a while, until I saw the light (or, rather, I couldn’t)) Further Maths. It’s all a bit of a blur as to which topic was covered at which level.

I know we did matrices at O-Level – they are not in the current syllabus (but even back then, the syllabus differed between exam boards. A friend didn’t see matrices until he got to university – where he studied maths). But we didn’t do the Circle Theorems, which are.

Also, technology has moved on in ways that we couldn’t even imagine back in my schooldays. My mobile phone (still a dumb-phone by the way) has a more comprehensive spell-checker than the “Little Oxford Dictionary” that I used to carry round. It has a better calculator than my TI-33 too. Pretty much everything is available on the internet at the click of a few buttons.

#### Schooling today

Schooling today isn’t so much about remembering facts; it’s more about knowing where to find the facts and assessing their usefulness. It’s not so much about knowing stuff; it’s more about knowing what you need to know and where to find it.

I was once told that children don’t go to school to learn; they go to school to learn how to learn. That’s still the case – but how we learn things has changed. For all of us, not just schoolchildren.

Note: Back to Hannah’s sweets, here is the solution by Rob Eastaway, author of *Maths for Mums and Dads*.

#BBCtrending: Here’s the solution to that difficult maths problem – BBC News

Mike Sedgwick says

I had a go at it too. I did the Bayesian part and then decided that nobody but a theoretical mathematician would care about whether 0 = n^2 – n – 90. So I lost interest (and I couldn’t do it anyway.)

Bayes is important and we saw this when a mother was jailed because both of her infants died a sudden death. The chances of one dying is very small, or two dying is very small x very small = infinitessimal. Therefore the mother must had done them in.

BUT that is only true if the 2 infants are exposed to the same small probability and are independent from one another.

These siblings had the same genes, same environment, same diet, same climate etc so they were not independent. The judge, the lawyers, and the expert witnesses did not know about Bayes.

The mother was freed on appeal when the law had had a lesson in statistics.

It can never be proved but both infants may have had the same gene leading to a cardiac rhythm abnormality and to cardiac arrest.

Janet Williams says

Did you tell the judge all these? Did the jury understand you?

Ruby says

Although it works out quite nicely when solved in a logical and step-by-step process, it is not immediately obvious how the end relates to the beginning. I solved it fairly easily, but I have 20+ years experience of solving problems – I’m not sure I would have found it so easy if I were a 16-year old sitting the paper.

It also combines algebra with probability – and not just simple probability, but conditional probability. It relies on knowing that the chances of taking two orange sweets are not 2 out of n, as might be thought.

I suspect that the students are more used to seeing the problem the other way round – given the number of sweets, what is the probability of picking two orange ones?

Mike Sedgwick says

I was not involved in the case but when it was reported I felt someone should say something. I did not have the confidence to make a strong argument and follow it through. I do not know enough stats.

However someone did and the outcome was changed.

Science and the law are uneasy together. We ask the law to decide scientific truths sometimes and they cannot do it e.g. Vatican vs Gallileo. The case was only finally settled about 20 years ago but the scientific truth was there for all to see who could see. An extreme case, I know.

Belinda Wong says

About Hannah’s sweets. The chance of any particular sweet being orange is 6/n. That applies to each and every one of them whether they’re in the bag or removed from the bag. The probability of any two of them both being orange is 36/(n x n). That probability can never be 1/3, it’s impossible. The examiners got it wrong and it’s no wonder the students were confused.