 # Why we need to focus on conceptual maths

Does any school subject divide public opinion as much as maths?

Earlier this year, prime minister Rishi Sunak reignited the debate on the purpose of maths when he said he wanted every student in England to study it until they were 18.

While some were delighted by the idea, others saw it as a waste of time, believing that the maths we learn in school, particularly at GCSE and beyond, has little bearing on most people’s day-to-day lives.

But that belief stems, I think, from a misunderstanding about what maths is, and what it is not – or, at least, what it shouldn’t be: a collection of facts and procedures. Presenting maths in this way is what leads to the belief that studying maths is ultimately a recall exercise, with pupils expected to memorise disparate facts and processes that have little real-world relevance.

The reality, though, is that these facts and processes arise from the concepts that underpin the subject. So, if we really want to engage children in maths and equip them with skills that last, conceptual maths is what we should focus on.

More teaching and learning:

What does this look like in practice?

Consider percentages: many pupils are successfully taught to calculate simple percentages of a given value. However, percentages are just one example of what is called a “multiplicative relationship”. Exactly the same relationship exists within (most) unit conversions, within pie charts, within speed calculations, right the way up to algebraic proportion relationships.

What pupils aren’t often taught is the deep multiplicative structure that ties all of these different “topics” together. One way to do this is using a consistent representation, such as a ratio table, another is dual number line.

The use of a consistent representation allows pupils to see how the same structure is being applied to all of these “different” questions. In this case, the existence and use of the functional multiplier (the value is always 0.4 times the percentage) or the scalar multiplier (the second row of the table is always one one-hundredth the size of the first row) are the crucial aspects of the multiplicative relationship we would want pupils to learn about, and they are clearly identifiable by pupils and teachers in both representations.  The idea is that if students are able to see how the facts and processes arise and function, they will be better able to assimilate new maths knowledge and link that knowledge to what they have previously learnt. You will know they are beginning to consistently recognise the structure of an underlying concept when they start to say things like “Oh, that’s just like when we did…” when you introduce them to a new procedure.

Being able to teach the underlying concepts of maths relies on teachers developing students’ knowledge in three ways.

The first is to make sure the teachers themselves have a secure understanding of how concepts develop and link together. They need to be able to make explicit, for example, the links between percentages and unit conversions, and appreciate how to represent concepts and make sense of them, such as through the ratio tables and dual number lines above.

They also need to change how they think about questioning; questions should be used to draw attention to and reinforce the concepts that underpin tasks.

For example, if we want pupils to see percentage as a proportional idea, we might devise questions like:

• One quantity is four times greater than a second quantity. Work out the percentage of the second quantity the first represents.
• Two quantities are in the ratio 3:5. What percentage of the second quantity is the first?

It’s also worth including questions that would be better solved using the functional multiplier between the percentage and the value, such as: “30 per cent of a quantity is 120, work out 73 per cent of the quantity”.

Last, teachers need to know how the procedures that we teach pupils to carry out arise from the concepts that they are associated with. In the case below, the standard procedure is multiplication by the equivalent decimal. However, these can be explained as scalar multipliers which turn 100 per cent into each of the other percentages, for example: The idea here is to help pupils recognise that no matter what the value associated with 100 per cent, if we want to find 40 per cent, we will always multiply by 0.4; to find 44 per cent we will always multiply by 0.44 and so on.

Every concept in school-level maths has these links, representations and the like that can be explored with and taught to pupils so that their journey through school level maths is coherent and makes sense.

Of course, to take full advantage of this approach to maths teaching we would need a curriculum properly resourced and structured over a pupil’s entire time at school. However, schools and teachers can take steps on their own to ensure that they are teaching pupils to make sense of mathematics and mathematical ideas rather than just carry out calculations and follow processes.

If we do that, students will understand what it means to learn about mathematics rather than just do mathematics – and it’s this that will lead to more of them wanting to study the subject until the age of 18.

Peter Mattock is an assistant principal at Brockington College in Leicestershire and author of Conceptual Maths and Visible Maths 2 weeks Create your