It has increasingly been the case that when I think about how to teach - or, more specifically, about how to teach maths - I think about that Swedish palace of flatpack furniture: Ikea. Now, this is not a comment on production-line education, rather about how useful the purveyor of affordable meatballs can be in influencing our pedagogy.

You see, the process of building any item in its vast bank of products would be almost impossible if they simply gave you an Allen key, a bag of unfamiliar fixings and a list of written instructions. They’ve made the task almost manageable - be it still plagued by orientation and logistical issues (but perhaps that’s just me) - by including simple diagrams.

In maths teaching, diagrams make for a much better mathematical solution. So why don’t we take a leaf out of Ikea’s book and push them more?

I have been using the phrase “when it gets tricky, draw a piccy” for many years. It started out as just something funny to say. But I found myself using it more and more when teaching difficult topics or topics that require a greater depth of understanding. A picture often helps me, but I often find that students are reticent to draw a picture themselves.

Could it be that they don’t know what to draw, or is it simply that they want to follow a set of rules to get to an answer, rather than figuring it out themselves?

I have been using the phrase ‘when it gets tricky, draw a piccy’ for many years

Following a set of rules given by the teacher is something that students seem to be drawn to like a couple with a new flat to an Ikea Billy bookcase. I often hear, “Don’t ask me a question. Just tell me what to do.”

Of course, they would say that. It’s easier for them to follow a set of rules, but can they remember those same rules later? Often, as we know all too well, the answer is no.

I think the reason for this is not due to “retention” as is often stated, but a lack of deep understanding. It’s the old procedural vs conceptual understanding debate again.

Procedural understanding, or rules without reasons, and conceptual understanding, or knowing what to do and why to do it, has been discussed time and time again. However, I think one element of conceptual understanding is often overlooked: how a visual representation, a “piccy”, of the maths helps to promote deeper understanding and enables students to see what to do next in a way that words alone cannot.

Take this question from a 1968 O-level paper:

A quadrilateral ABCD is inscribed in a circle, the tangent at A to the circle making an angle of 42° with AB. The angles BDC and CAD are 15° and 67° respectively. Calculate the angle ABD.

Now, I managed to read about half of the question while trying to visualise what was going on. But, I’ll admit, by the time I got to an angle of 42° with AB, I struggled.

However if a diagram was there, it would have been a much more manageable task.

Drawing the right conclusion

Drawing a picture might seem an obvious thing to do, but without the knowledge that the drawing doesn’t have to be accurate, or to scale, or even the “right way round”, students will, more often than not, be just as likely to give up as I was.

So, what if we spend time in our curricula teaching students how to be tooled up with a set of conceptual pictures which can help in a variety of situations? Maybe then, we can promote deeper understanding and the resilience required to tackle the sorts of questions seen in the new-style GCSE, which are erring back to the old-style O-level questions more and more.

One way of doing this is to spend the time in Years 7 to 9 developing the concrete - pictorial - abstract techniques like the bar model, to aid learners to follow my mantra of “when it gets tricky, draw a piccy”.

Learners can be offered a wide variety of pictorial skills to model different situations, whether it be proportional reasoning using a bar model, or directed numbers using a number line, or drawing a picture that enables learners to identify that the question was actually about Pythagoras’ theorem rather than an actual tent pole, or - in the case of the O-level question - recognising that the alternate segment theory would help to solve the question.

This is exactly what I have tried to achieve in my school this academic year. I have ensured that our programmes of study include opportunities for all learners of all ages to engage with a visual representation of the mathematics being taught whenever possible. That is from the lowest prior attaining groups all the way through the spectrum to the highest - across all key stages, including post-16 learners.

It leads to questions like: does this picture help? How does it help? Could you draw something similar next time you encounter a similar problem?

Long term, I am convinced that this approach will enable students to become more adept at problem-solving and have a greater conceptual understanding of maths. Maybe they would have a more positive experience of maths as a result. And maybe we’d also be assisting them in the future, should they ever have to construct a flatpack wardrobe from Ikea.

Graham Walton is head of mathematics at Tupton Hall School in Derbyshire
@mr_g_walton