This is something that struck me the other day when someone asked me about the difference between university maths and sixth-form maths: every time a student moves between educational levels, “what maths is” undergoes a dramatic change.

This is based on my memories of school and is likely to be wrong in the details. Is it wrong in a more structural sense? Let me know in the comments.

There’s an awful lot of number work, memorising times tables, dealing with decimals, weighing and measuring - the numeracy stuff that Education Secretary after Education Secretary has insisted be drilled into kids. It’s understandable that students arrive at the big school thinking maths is all about numbers and repetitive exercises.

All that numeracy stuff? Well, it’s useful, but it’s not what maths is all about. Now you’ve got a big step up. Suddenly there’s algebra and graphs and stuff. Instead of it all being about numbers and repetitive exercises, it’s letters and squiggles and repetitive exercises (and a dozen mnemonics that take more effort to remember than it would to learn the underlying principles).

All that algebra stuff? Well, it’s useful, but it’s not what maths is all about. Now you’ve got a big step up. Suddenly you’re differentiating and integrating, gradients that sometimes work the same as they did at GCSE, and a dozen new mnemonics that take more effort to remember than learning the underlying principles.

All that calculus stuff? Well, it’s useful, but it’s not what maths is all about. Now you’ve got a big step up. Suddenly, it goes from “show the train is going at 50 metres per second” to “show that there exists a train.” The proofs you’ve learned at A-level are only limited help, and things become yet more abstract.

Here’s what happens: if you struggle with primary school maths, you get the belief you’re bad at maths and you persevere because you’re forced to. If you’re good at it, you get to the next level and find out that what you’re good at isn’t really maths, and it gives you another area in which to struggle - and the pattern repeats over and over through the system. (In fact, when I got into my PhD programme, I wasn’t particularly surprised to learn that much of what I’d done in my degree wasn’t what maths was all about. Now I had a big step up.)

I understand that GCSE (an exam for everyone) and A-level (an exam for specialists) have different purposes - but I can’t help but feel that having a clearer and smoother transition between the levels would make for a much higher retention rate of mathematicians.

I’d love to hear your opinions on this - both in terms of whether my assessment is nonsense, and what we can do about it if it isn’t.