A few months ago, @reflectivemaths (Dave Gale in real life) tweeted the following:

An excellent question, much as it pains me to give Dave credit for anything.

My personal view: the student has given a perfectly reasonable response to the question, one that is – if anything – more mathematical than the answer the mark scheme presumably suggested, in the great tradition of “there exists a sheep in Scotland that’s at least half white”.

But this post isn’t about the answer to the question: it’s about the purpose of GCSE (and maths exams in general).

In the ensuing Twitter conversation, @7puzzle made the point that many students dislike maths lessons as they feel they may be tricked, which – I think – hits the nail on the head: the GCSE tries to wear two hats at once.

Hat #1 is a test of the numeracy necessary for living as a competent adult. This is the hat that requires common-sense thinking, using rules of thumb, in preparation for everyday life.

Hat #2 is a test of formal mathematical thinking, using incontrovertible rules to make sure everything is watertight, in preparation for further study in maths or maths-based subjects.

And herein lies the problem: this question – like many others – tries to wear both hats at once. The conversation was – from memory – more or less evenly divided between people saying “it’s obvious you have to use the table to answer the question” and people saying “that’s a perfectly reasonable objection”.

@aPaulTaylor took it a step further: what if the student had said “can’t tell – what if the table was mis-transcribed?” This, to a mathematician, remains a perfectly valid objection: we don’t know absolutely for sure that the table is right, and so mathematically “can’t tell” is appropriate.

There’s a difference between “true (mathematically proven)” and “true (common-sensibly uncontroversial)”, and there’s no indication in the question which kind of truth is expected. As long as maths exams try to wear both hats at once, there will be confusion about ‘real world’ questions and what, precisely, is expected.