The idea of ‘understanding’ is notoriously slippery, and much-discussed.

If you can answer a standard question on a topic, does that mean you understand it?

I think most maths teachers would say ‘no’, distinguishing between following a method and understanding a concept and its connections.

In my own mathematical experience, an obvious case for me is with trigonometry. As a student it truly seemed as though the trig tables were mystic runes which could be used to give correct answers if used in the right way (and as an adult, the tables evolved to magic buttons on a calculator). I had no idea why they worked, but could always make them do what I needed.

I remember being in Sheffield, in my third year of teaching, when I first read about the unit circle (almost certainly in a Mike Ollerton book). It was overwhelming to finally realise why people talked about ‘ratios’ as part of trigonometry. Even now, I know that I am quite far behind my more mathematically proficient friends who find that trig graphs illuminate and simplify problems. I still concentrate quite hard to remember why they make sense and why they are useful, as I have a shameful preference for static equations over dynamic graphs….but don’t tell that to anyone with a maths degree! I am pretty sure I still don’t ‘understand’ trigonometry to the degree my colleagues do, even if I enjoy and can complete most problems which use it.

I’m sure you can think of many pupils who are similar – they can solve simultaneous linear equations but couldn’t explain how or why it works, or what its special features are, or would use the same sledgehammer approach no matter the question. By the latter, I am thinking of how likely most pupils are to use the standard approach to tackle this question:

3x + 6y = 33

3x + 7y = 38

and get bogged down in subtracting, rearranging, substituting and solving, and not spot that it must be the case that y = 5. For me, this suggests a gap between procedural proficiency, and the flexibility that signals ‘understanding.’

Considering all this, I think it is fascinating that almost everything we call a “check for understanding” is really a “check for procedural mimicry.”

Consider how most assessment for learning takes place in maths. It is usually a version of:

- Tell me the answer
- Tell me which answer is correct (e.g. if it is multiple choice)
- Tell me how you got the answer (i.e. describe the procedure you followed)
- Show me how you got the answer (e.g. on a whiteboard, or looking in books)
- Tell me what is wrong with this person’s procedure (e.g. ‘spot the mistake’)
- Tell your partner what is wrong with their procedure (e.g. peer marking)

This is not to say procedural mimicry is not important. It can be very useful as part of learning, especially for multi-step processes where we need to build up muscle memory and make many little decisions.

I think that often we don’t ask questions which check for understanding for two reasons:

- We’re not clear, as teachers, what it would mean to understand something (we haven’t specified the small pieces of knowledge, connection and structure which form understanding), which means that…
- We don’t teach those things, either explicitly or by drawing them out. As a result, asking about them is almost always pointless, or it is blind luck if a pupil has got there by themselves and can answer a question which explores understanding.

A step that helped me was to plan backwards from questions I wanted my pupils to do by asking myself:

- How did I know to do that?
- Why do these steps make sense?
- What is happening when I do these steps?
- Why does this answer make sense?
- What other bits of maths are similar to this?
- What happens if we change this thing? Can we predict the effect without calculating?
- What would we have to change in the question to get a specific kind of answer? (this is the reverse of the previous question 😊 )

…and then plenty of topic-specific questions.

Although I am a bit sad that ages 15-24 I was using trigonometry without understanding it, it is exciting to still have a journey of understanding with superficially simple maths. I hope it is the same for you, and your pupils can have these moments before their mid-20s!