Could ‘mathematical reasoning’ improve primary maths teaching?

The ability to reason mathematically is one of the core aims of the primary curriculum.

However, it has previously been highlighted that maths lessons often place greater emphasis on numerical calculations than on the reasoning about the relationships between numbers and quantities that underlie them.

A recent Education Endowment Foundation (EEF) trial of a University of Oxford programme found explicitly teaching children as young as six how to reason about maths can lead to an extra two months of learning for children from the poorest backgrounds.

Replacing one maths lesson a week with an explicit mathematical reasoning session in a trial that involved 238 schools and 6,291 Year 2 pupils also found an extra one month’s progress for children who were not eligible for free school meals.

But what does teaching mathematical reasoning look like in practice?

What is mathematical reasoning?

Mathematical reasoning is about encouraging children to think about relationships between numbers and quantities and use this understanding to make sense of a maths problem, rather than just learning calculations.

Professor Terezinha Nunes, emeritus professor of educational studies at the University of Oxford and one of the programme leads, explains that reasoning involves making inferences.

In order to make an inference based on numbers, you have to already know something about the relationship between those numbers. So pupils must already understand the concepts of “more” and “less” and be able to count.

“For example, if you know that two comes before three, and three comes before four, you can infer that two comes before four because you already know those two relations,” she says.

You can reason about relationships between both quantities and numbers, adds Gabriel Stylianides, professor of mathematics education at the University of Oxford, also a programme lead.

For example, when learning about odd and even numbers, pupils will see over time that when they add two odd numbers, they get an even number.

“It’s natural for the pupils when they identify this pattern to wonder, will this always be true?” Stylianides says. “There is an infinite number of odd numbers - they cannot check them all. The challenge here is for the teacher to help them reason through these relationships and show it will always be true, given what children already know.”

Mathematicians would prove this using algebra, he adds, but younger children won’t yet know about algebra.

How do you teach mathematical reasoning?

So what could teachers do instead? In the example about even and odd numbers, a teacher could help pupils represent the different number types using tiles: odd numbers as a group of two with one left over, and even numbers as groups of two, with none left over.

When two odd numbers are added together, the two leftover tiles can be grouped together, making an even number.

Another example might be to use mathematical reasoning to introduce division through the activity of sharing.

Start by giving the children a number of objects and ask them to share them in equal amounts. They might start by giving one object to each child, and then a second.

“As they do this, and realise they can give two objects each because there is enough, they start generalising the way in which they can do that,” Nunes explains.

She adds: “But the crucial thing is that, at the end, the teacher asks them the question that is going to get them to go between the actions. So we ask them, how many sweets per child? That introduces the idea of ratio.

“This going from doing something to thinking about the relation between quantities is the crucial step to get the children to understand mathematical reasoning.”

The lesson can then explore things in the other direction - by saying we know how many children there are and we are going to give each child three sweets, and working with the class on how this allows us to work out how many sweets there are altogether.”

“They’re not calculating. They’re thinking about relations around ratio,” Nunes says.

Aren’t teachers doing this already?

Primary teachers are likely doing similar maths activities in their classrooms already, so may feel they’re ticking the mathematical reasoning box.

The important step that might be missing is the exploratory mathematical discussion that accompanies the activity.

Take the sweet sharing example above, where children have explored how many sweets per child. If the teacher wants to focus the lesson on quantitative reasoning, rather than going straight on to focus on the arithmetic of division and multiplication, there needs to be discussion of how this would change if more children were sharing the sweets, or how many more sweets would we need to give each child one more?

In this way, pupils will go beyond what they already know to reason based on their understanding of quantities.

In Nunes and Stylianides’s view, mathematical reasoning, particularly quantitative reasoning, does not always receive sufficient attention in primary maths teaching, as maths lessons often focus on “here’s a problem, what arithmetic do we do to solve the problem?”, rather than understanding the mathematical relationships involved.

“These concepts are well established in the literature as hard to teach and hard to learn - not because pupils don’t have the capabilities, but because it’s very hard to find good ways to support teachers to engage their pupils in these ideas,” Stylianides says.

Nunes adds: “When we work with teachers, they are quite surprised at how many ratio situations can be explored with young children who haven’t been taught about multiplication or division.”

How can teachers get better at teaching mathematical reasoning?

It’s difficult to expect teachers to come up with a scheme of work on their own that incorporates the principles behind mathematical reasoning and develops them in a systematic way, the pair say.

That’s why the intervention provided teachers with resources, including lesson plans, workbooks and slideshows. There were also handbooks for teachers to help them understand the rationale behind the activities, which included suggested ways to implement them, ways to scaffold their pupils’ learning, questions to ask and directions to online games pupils could go to for further practice.

“You need a package like this, and also training for teachers,” Stylianides says.

Just over half of the teachers who took part in the project could not think of a way to support a child’s reasoning on a problem before they took part. After taking part in the training, just over half said they could think of good ways to support children’s numerical reasoning.

The EEF trial, which was delivered by the National Foundation for Educational Research, was designed for Year 2 children.

However, the Oxford team are working on activities for four- and five-year-olds that apply the same principles.

Nunes hopes that successfully implementing these would help to reduce the attainment gap in maths right as children are arriving at school.

“We want to close that attainment gap in maths at an early age, when the impact is multiplied. If you close that gap in Year 2, it helps it not widen and widen and widen as children go through school. Because this programme was so effective, it has the potential to do that,” she says.