**With many thanks to Nick Hart, deputy head at Penn Wood Primary School in Slough, for sharing how teachers in his school deepen children’s understanding of mathematical concepts.**

It is important to say from the outset that depth of understanding of maths should be a goal for all and not just the most able children. Differentiating for the most able should not necessarily be fundamentally different tasks. Certainly, those children should have opportunities to deepen their conceptual understanding of an area of maths but why not have that expectation of all children?

In my school, we don’t set or group children by ability but children who grasp basic concepts quickly are challenged to work at a deeper level. Great teaching in the first place and, importantly, over sufficient time, lays the foundation for most to work at depth with a concept. If a child is working on multiplying numbers by 10, 100 and 1000, and they only ever do basic procedural practice, they may technically have met the objective but will not have mastered it. This is perhaps the reason why some children’s long term retention does not match their short term performance – they are not thinking about the concept deeply enough to internalise it reliably.

__The underlying structure of a problem__

Aiming for fluency in the procedural basics is the necessary first step. While those children that need it work on fluency of recall or perfecting a procedure, many children can linger with a concept. I would suggest that a prerequisite for deeper understanding is to teach the language needed to talk about the underlying structure of a problem. Novices and experts think about problems in different ways. Take these three questions from a unit of work on times tables:

*Eggs are sold in boxes of 6. Tim bought 4 boxes. How many eggs did he buy?*

*Eggs are sold in boxes of 6. Ajay bought 7 boxes. How many eggs did he buy?*

*Eggs are sold in boxes of 6. Mina bought 18 eggs. How many boxes did she buy?*

A novice might look at those questions and say that they are all about boxes of eggs. This is an example of noticing the superficial structure of problems. An expert might look at the same questions and be able to explain that the first two questions are such that the *whole *is unknown but the* number of equal parts* and the* size of each part* are known. They might also notice that in the last question, the number of equal parts is unknown while the whole and the size of each part are known. This is an example of them seeing the deeper structure of problems. If provided with the right modelled language, and good objects to manipulate, all children can think with this precision. One possible task to work on this kind of understanding is not simply to solve problems like this but to sort them:

__Changing the unknown__

Setting the standard for how to talk about what is unknown can lead very easily into deeper understanding. A staple challenge for children in maths should be to change what is unknown in a problem. Now although this is work of a greater depth than the basic procedure, it will still need to be modelled thoughtfully. The language will help, but representations like bar modelling can be used to show children the difference between the basic concept and the changed unknown. Here’s an example for multiplying by 10; if a child can differentiate between the three underlying patterns and use these patterns to solve varied problems then they’ll have a much more secure understanding of a concept:

**Variation**

Back to the times tables questions:

*Eggs are sold in boxes of 6. Tim bought 4 boxes. How many eggs did he buy?*

*Eggs are sold in boxes of 6. Ajay bought 7 boxes. How many eggs did he buy?*

*Eggs are sold in boxes of 6. Mina bought 18 eggs. How many boxes did she buy?*

*Insaf fills a sticker book with 48 stickers. On each page, he arranges the same number of stickers. There are 8 pages in the book. How many stickers on each page?*

The first two problems give children a chance to build procedural fluency through minimally different details. The repetition of the context may also help the novice to develop an understanding of the concept in one context without too much cognitive load. However, presenting the last two questions together provides variation and therefore requires more flexible thinking. The first two questions demand little cognitive load; the last two demand more. The more proficient children become, the more varied their problems should be so as to develop conceptual fluency, that is the picking out of the deep structure from varied problems. These problems will vary not just in what is unknown but also the context in which the problems are set and the wording used.

**Representing a problem in different ways**

When children first learn a concept, they’ll probably use one representation. One way of encouraging deeper thinking is to get them to represent the same problem in a different way. They might draw a bar model or variations of the abstract representation:

More important than the multiple representations is the talk around them. Children could be encouraged to make links between the representations. In the picture below, for the example of 3 x 4, children could be asked to create and then talk about what is the same and what is different between the various representations. This could progress to comparing similarities and differences between the representations of 3 x 4 and 4 x 3:

For those children that grasp concepts quickly (and it may well be different children in different topics during different times of the year), challenging them through depth of thinking should be the first step. If our subject knowledge and task design is strong enough, we can go a long way to giving all children the opportunity to work in this way and in doing so helping them to build truly secure mental models of mathematics that will stick.