Developing Maths Reasoning in KS2: The Mathematical Skills Required And How To Teach Them

Developing maths reasoning skills at and before KS2 is crucial to succeeding in the new curriculum and its maths mastery approach to learning.

Moreover, as the results of the KS2 SATs, with their focus on maths reasoning questions have shown us, pupils need strong applied reasoning with their maths facts to be able to succeed.

My approach to primary school mathematics teaching and learning is that it should be about exploring, reasoning and challenging thinking rather than learning rote/abstract rules for calculations and facts.

Though I recognise that fluency in maths and memorising key number facts is essential in Key Stage 1 and Key Stage 2 mathematics, to acquire the basics, these skills ought to be used and applied in real life contexts.

The questions from the KS2 SATs Papers 2019 seem to align with my belief. To succeed in the national curriculum tests, it is clear that children require deep knowledge of facts and mathematical concepts. Moreover, they need to be able to use and apply these facts to a range of contexts, and different types of word problems, including the more complex multi-step and two-step word problems

What is reasoning in maths?

Let’s start with the definition of maths reasoning. Reasoning in maths is the process of applying logical and critical thinking to a mathematical problem in order to work out the correct strategy to use (and as importantly, not to use) in reaching a solution.

Reasoning is sometimes seen as the glue that bonds pupils’ mathematical skills together; it’s also seen as bridging the gap between fluency and problem solving, allowing pupils to use their fluency to accurately carry out problem solving.

In my opinion it is only when we teach children to reason and give them the freedom to look for different strategies when faced with an unfamiliar context that we are really teaching mathematics in primary school.

Why focus teaching and learning on reasoning?

Logical reasoning requires metacognition (thinking about thinking). It influences behaviour and attitudes through greater engagement, requesting appropriate help (self-regulation) and seeking conceptual understanding.

Reasoning promotes these traits because it requires children to use their mathematical vocabulary. In short, reasoning requires a lot of active talk.

It is worth mentioning that with reasoning, active listening is equally important and if done right can also ensure increased learning autonomy for pupils.

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The theory behind mathematical reasoning at KS1 & KS2

The infographic (below) from Helen Drury cleverly details what should underpin mathematics teaching and learning. It’s a good starting point when you’re thinking about your mathematics curriculum in the context of fluency reasoning and problem solving.

John bee reasoning and problem solving 1

I’ve also been very influenced by the Five Principles of Extraordinary Math Teaching by Dan Finkel

These are as follows, and are a great starting point to developing maths reasoning at KS2

1. Start mathematics lessons with a question
2. Students need to wonder and struggle
3. You are not the answer key
4. Say yes to your students original ideas (but not yes to methodical answers)
5. Play!

See also this free guide to KS2 maths problem solving and reasoning techniques.

How to make reasoning central to Maths lessons in KS2

Pose lesson objectives as a question to KS1 & KS2 children

A  ‘light bulb’ idea from my own teaching and learning was to redesign learning objectives, fashioning them into a question for learning. Instead of ‘to identify multiples of a number’, for example, I’ll use ‘why is a square number a square number?’.

Phrasing LOs as a question instantly engages and enthuses children, they wonder what the answer is. It also ensures that they show their reasoning in a model or image when they answer.

In this instance – interestingly – children knew the process to calculate square numbers but could not articulate or mathematically reason why it worked until after the session.

It seems denying children answers allows them time to think, struggle and learn.

Ban the word ‘yes’ in Maths lessons

One of the simplest strategies I have found to make reasoning inseparable from mathematical learning is to ban the word ‘yes’ from the classroom.

Instead, asking children to reason their thoughts and explain why they think they are right can allow for greater learning gains and depth of understanding. Admittedly, this is still a work in progress and easier said than done.

To facilitate this, I always tell my children that I am not the answer key.

Using my example of square numbers, I allowed children time to struggle and wrestle with my question without providing an answer or giving hints. Instead, I questioned to unpick understanding at the beginning of the lesson and brought together mathematical ideas during a whole class discussion.

After a short discussion on how children might show or visualize a square number we began to show a model using arrays, like below:

John bee reasoning and problem solving 2 1

The children working at greater depth were encouraged to consider cubed numbers and show how they might be represented using multi-link cubes without any input from me. This made sure links were made between mathematical knowledge, mathematical vocabulary, and learning.

Use ‘sometimes, always, never’ classroom activities

A ‘sometimes, always, never’ activity is another great way to foster reasoning and problem-solving skills. Take the image below:

John bee reasoning and problem solving 3 1

Here, children are first required to sort the statement into always, sometimes or never being true. The next day, they are moved on to the lesson with the title phrased as a question. So Not ‘to identify patterns’, but ‘how does this pattern work?’ with a pattern already presented on the board.

The children, instantly engaged, begin conjecturing, making predictions and thinking about the next patterns in the sequence (this lesson was actually inspired by an Nrich activity).

5 tips for developing mathematical reasoning in the KS2 classroom

While small changes will not provide the framework you need to properly embed reasoning in the classroom when implemented alongside ideas such as those mentioned above. These tips can help instil greater depth in maths in your class for all ability levels.

1. Start lessons with a question.

2. Start lessons with a provocative mathematical statement and challenge your class to provide the mathematical proof: “ N will always = N” or “Multiples of 9 always have the digital sum of 9”.

3. Present answers to SATs question as a puzzle to generate discussion. When framed like this, children like to ‘come up’ with what the question could be:

John bee reasoning and problem solving 4 1
John bee reasoning and problem solving 5 1


4. Grouping children in threes is the magic number when working through problems. Child one talks through the problem. Child two writes down everybody’s reasoning. Child three actively listens and watches.

5. Include reasoning prompting posters around the classroom. The image below, for example, can be useful to children who are starting to formulate thoughts, predictions and assertions.

John Bee reasoning and problem solving 6

As the most recent KS2 Maths SATs tests proved, your pupils will need an in-depth understanding of facts and concepts to truly succeed. Plus, they will need to be able to use and apply that knowledge to a range of contexts. As such, it’s clear that we need to provide them with a strong foundation of reasoning skills to give them their very best shot at the assessments they must face.

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