Ensuring that all pupils can “reason mathematically” is one of the key aims of the primary maths curriculum. This tends to be interpreted as a focus on numerical reasoning: arithmetic plus number sense.

There is no doubt that this is important; number sense allows children to think about relations between operations and about number meanings, allowing them to understand, relate and connect numbers. These skills are fundamental for mathematical success.

However, the national curriculum fails to emphasise the importance of quantitative reasoning: the ability to reason about quantities and relationships between quantities.

According to the Organisation for Economic Co-operation and Development, within the broad scope of mathematical reasoning, quantitative reasoning is especially fundamental. It not only underpins other forms of mathematical reasoning, but is critical for success in later maths learning and in many real-world contexts.

In fact, the Royal Society has pointed out that today’s industrialised and data-driven society demands more quantitative reasoning than ever before. To meet this challenge, it recommends “a strong focus on general quantitative literacy for all learners, at every stage and level”.

Yet quantitative reasoning has received limited explicit attention in curriculum policy and classroom practice in England, partly because it is hard to assess reliably with young children.

So, are we in danger of teaching what is easier to assess, rather than what’s most important?

Assessing quantitative reasoning

In prior research, commissioned by the Department for Education, we found that quantitative reasoning tests, which demand substantial reasoning but only very simple arithmetic and are given at ages 8-9, predict pupils’ performance in key stage 2 and KS3 tests better than tests of arithmetic skill.

Further research showed that our quantitative reasoning test given in the spring term of Year 1 predicted KS1 outcomes.

This evidence demonstrates quantitative reasoning’s role in later mathematical success. At the same time, it underlines that it is possible to measure quantitative reasoning reliably, including with younger children who are not yet proficient in reading or writing.

Assessing quantitative reasoning requires carefully designed tasks that capture pupils’ ability to reason without overloading them with complex language, calculations or memory demands.

By embedding quantitative reasoning more directly into curriculum documents and teaching resources and providing teachers with reliable tests to assess their pupils’ progress, we can help to foster these skills from an earlier age, giving children a stronger foundation for future learning.

Until that happens, what do schools need to know about quantitative reasoning, and how to teach it?

How do you teach quantitative reasoning?

Quantitative reasoning is the ability to think about quantities and relations between quantities in situations and make logical inferences on the basis of relations between quantities.

In primary school, children need to learn about two types of relations between quantities: additive reasoning (part-whole relations between quantities) and multiplicative reasoning (fixed ratios between quantities). The relation we use to solve a problem depends on the question we ask.

Let us think about relations between quantities in the following example:

What can teachers do to help children develop quantitative reasoning? The following steps could help:

1. Provide opportunities for children to learn about relations between quantities through activity and thinking.
2. Present the problems as a cartoon and encourage the children to act out the problems using manipulatives and diagrams.
3. Help the children to get started and to be systematic in their approaches to solving problems.
4. Ask children to explain their reasoning in order to think about the actions and the relations between quantities.

In additive reasoning problems, help the children to go back and forth and to reason about doing and undoing actions using different types of problems. For example:

• There were five cakes on the plate. The child ate three. How many were left?
• There were five cakes on the plate. The child ate some, and there are two left. How many did the child eat?
• There were some cakes on the plate. The child ate three. There are two left. How many were on the plate at the start?

Problems that require children to think of undoing actions are called inverse problems. Cartoons help children to go back and forth and to reason about doing and undoing actions. For example, in the problem below, the question is about “undoing” the action to get back to the starting point.

In multiplicative reasoning problems, there are two different quantities. Teachers can help children by giving them two different representations, one for each quantity, so that they can establish a one-to-many correspondence between children and cars.

The Reasoning First Team at the University of Oxford is Professor Gabriel Stylianides, Professor Terezinha Nunes, Dr Rossana Barros Baertl and Louise Matthews.

For further information about quantitative reasoning, visit the website

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