Developing reasoning: How to improve Maths reasoning skills in primary school pupils

By MrBeeTeach 

Developing primary Maths reasoning skills for Key Stage 1 and Key Stage 2 pupils: a classroom how to. John Bee discusses approaches to metacognition in Maths lessons and how to embed reasoning into your teaching and learning, with tips from his own practice.

It is my firm belief that primary school (and indeed all) mathematics should be about exploring, reasoning and challenging thinking. Rather than learning rote/abstract rules for calculations and facts.

Though I recognise that memorising key 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 KS2 SATs Maths test this year seems to align with my belief. To succeed in the national assessment, it is clear that children require deep knowledge of facts and concepts. Moreover,  they need to be able to use and apply these facts to a range of contexts.

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.

The theory behind 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 if you’re looking to redesign your mathematical curriculum with a focus on reasoning and problem solving.

John bee reasoning and problem solving 1

With her theory in mind I’ve created five simple principles for embedding reasoning in your primary Maths teaching and learning:

1. Start 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!

These points are discussed in greater depth below, but you can read more on how to get to grips with KS2 Maths problem solving and reasoning here.

How to make reasoning a key part of Maths lessons

 
Pose lesson objectives as a question to KS1 & KS2 children

A  ‘light bulb’ idea from my own T&L 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 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 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 giving providing an answer or giving hints. Instead, I questioned to unpick understanding at the beginning of the lesson and brought together 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 concepts, 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. 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. Not ‘to identify patterns’, but ‘how does this pattern work?’ with a pattern already presented on the board.

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

Reasoning in the classroom - 5 final tips

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 learning in your class.

1. Start lessons with a question. See my 5 principles of extraordinary mathematics mastery to get the discussion flowing.

2. Start lessons with a provocative statement to prove or disprove: “ 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  or John bee reasoning and problem solving 5  1

4. Grouping children in threes is a magic number when working through. problems. Child one talks through the problem. Child two writes down everybody’s reasoning. Child 3 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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If you're looking for Maths specialist support in your primary school, we're offering a free trial of our 1-to-1 Maths tuition for your pupils. They'll work online with their own Maths specialist tutor on a lesson selected by the class teacher. We think your pupils will love it, but we know you'll want to see it in practice before making any decisions.

Further reading:

Get to grips with KS2 Maths problem solving: The Ultimate Guide to Problem Solving Techniques

5 ways to use topical and contextual problem solving to boost engagement in Maths

Using bar modelling techniques to solve multi-step KS2 SATs problems

How to teach the bar model method to ace arithmetic and word problems in KS1 and KS2 Maths

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