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Reasoning and problem solving
The National Curriculum in England has three aims for maths: pupils should become fluent, reason mathematically and solve problems. Fluency is knowing facts and methods. Reasoning is explaining, justifying and spotting patterns: "how do you know?". Problem solving is using what you know in a situation you haven't met before.
Reasoning doesn't need a special lesson. It grows from the questions you ask and the talk you expect, every day. The ideas below fit into any topic and any year group.
Question stems that get pupils reasoning
Keep a few of these on a lanyard card or the wall and use them until they become habit:
- What do you notice? What do you wonder?
- What's the same and what's different?
- How do you know? Can you prove it?
- Is there another way? Which way is quicker?
- What would happen if…?
- Give me an example. Now one that nobody else will think of.
- Is this always true, sometimes true or never true?
- Where has this pupil gone wrong?
Convince me
Give pupils a fact they already believe, and ask them to convince you (or a sceptical partner) that it's true. They have to go beyond "because I know it".
- Year 2: Convince me that 7 + 8 = 15. Double 7 is 14, and 8 is one more than 7, so the answer is one more than 14.
- Year 4: Convince me that 6 × 7 = 42. 5 × 7 = 35, and one more 7 makes 42.
- Year 6: Convince me that ⅗ is bigger than ½. Half of 5 fifths is 2½ fifths, and 3 fifths is more than that.
Pictures, cubes and number lines all count as proof. Some of the best explanations are drawings.
True or false?
A statement on the board, and pupils decide and explain. Mix true and false ones so pupils can't guess the pattern.
| Year | Statement | Answer |
|---|---|---|
| Reception | If I spread my 5 cubes out, I have more cubes. | False. There are still 5; they take up more space. |
| Year 1 | 7 + 3 = 3 + 7 | True. Both make 10; you can add in any order. |
| Year 2 | 15 − 6 = 6 − 15 | False. Subtraction can't be done in any order. |
| Year 3 | When you add 10 to a number, the ones digit stays the same. | True. 46 + 10 = 56: only the tens digit changes. |
| Year 4 | Every multiple of 6 is also a multiple of 3. | True. 6 is 2 lots of 3, so 6, 12, 18… are all in the 3 times table. |
| Year 5 | 0.65 is bigger than 0.7 because it has more digits. | False. 0.7 is 7 tenths; 0.65 is only 6 tenths and 5 hundredths. |
| Year 6 | Multiplying always makes a number bigger. | False. 5 × 1 = 5, 5 × 0 = 0 and 5 × ½ = 2½. |
Spot the mistake
Show a worked answer with an error in it, and ask pupils to find it, explain it and put it right. It's less threatening than correcting their own work, and it targets the misconceptions you know are coming.
- Years 2 and 3: 52 − 27 = 35. In the ones, the pupil took 2 away from 7 instead of 7 away from 2. Exchanging a ten gives 12 − 7 = 5 ones and 4 − 2 = 2 tens: 25.
- Year 3: ¼ + ¼ = ²⁄₈. One quarter and one quarter is two quarters, ²⁄₄. The denominator doesn't get added.
- Year 4: 450 rounded to the nearest hundred is 400. Halfway rounds up, so it's 500.
- Year 6: 0.4 × 3 = 0.12. 4 tenths × 3 = 12 tenths, which is 1.2.
Make the mistake belong to someone else: "a pupil in another class wrote this". Pupils will happily pick holes in a stranger's work, and the ones who would have made the same mistake learn from it without feeling caught out.
Open-ended problems
Problems with more than one answer, or a clear answer but many routes, let every pupil start and give confident pupils somewhere to go.
- Any year: The answer is 12. What could the question be? (Reception might say 10 + 2; Year 6 might say 3 × 4 or 25% of 48.)
- Year 1: How many ways can you make 6 when you roll two dice? 1 + 5, 2 + 4 and 3 + 3, or 5 ways if 5 + 1 and 4 + 2 count as different. Arguing about which is right is the reasoning.
- Year 3: I paid exactly 21p using only 2p and 5p coins. Which coins could I have used? One 5p and eight 2p, or three 5p and three 2p. You need an odd number of 5p coins, because 21 is odd and 2p coins always make an even amount.
- Years 5 and 6: A rectangle has a perimeter of 20 cm and whole-centimetre sides. What could its area be? The length and width add to 10 cm, so the areas are 9, 16, 21, 24 or 25 cm². The square (5 cm by 5 cm) gives the biggest.
Bar models for word problems
A bar model is a drawing of the problem using rectangles. It helps pupils see what they know and what they need to find, before they choose an operation. Younger pupils start with a part-whole model: "There are 23 children in the class and 14 are girls. How many are boys?" The whole is 23, one part is 14, so the other part is 23 − 14 = 9.
Bar models also work for fractions. In Year 6: "Tom spent ⅖ of his money and had £18 left. How much did he start with?" Draw a bar in 5 equal parts and shade 2 as spent. The 3 parts left are £18, so one part is £6 and the whole bar is 5 × £6 = £30. Check: ⅖ of £30 is £12, and £30 − £12 = £18.
Sentence stems for explanations
Many pupils know the answer but not how to say why. Sentence stems give them the words. Model them, display them, and expect them in full-sentence answers:
- I know that… so…
- First I…, then I…, so the answer is…
- It can't be… because…
- I agree (or disagree) with… because…
- This is always / sometimes / never true because…
- My answer is sensible because it's close to my estimate of…
Pair them with talk partners: pupils say the sentence to a partner before anyone answers in front of the class.
Quick answers
Reasoning is explaining and justifying: why something is true, or why a method works. Problem solving is applying maths to a new situation. They overlap: solving a problem well usually needs reasoning along the way.
No. Every pupil can reason about maths they are secure with. Use smaller numbers or concrete objects so the calculation doesn't get in the way of the thinking.
The Year 6 SATs in May have one arithmetic paper and two reasoning papers. The reasoning papers mix word problems, explanations and multi-step questions across the whole curriculum.
Give thinking time, then partner talk before whole-class answers, and offer sentence stems. Explaining to one person first is far less daunting than explaining to thirty.