Reasoning in Maths Reasoning is fundamental to knowing

Reasoning in Maths Reasoning is fundamental to knowing and doing mathematics. How would you define reasoning?

Reasoning in Maths • Reasoning could be thought of as the 'glue' which helps mathematics makes sense. • ‘all pupils will: reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language. ’ National Curriculum 2014

Reasoning in Maths In order to explore this aim, three questions need to be answered: • When is reasoning necessary? • What do we do when we reason? • How do we support children to develop their reasoning skills?

When Reasoning is necessary When faced with a mathematical challenge • Make use of prior knowledge • What methods would be useful? • What problems like this have I done before? • Reasoning is complex and unique – individual past mathematical experience.

When Reasoning is necessary When logical thinking is required Hundred Square Challenge

When Reasoning is necessary When a range of starting points is possible That number square!

When Reasoning is necessary When there are different strategies to solve a problem Coded Hundred Square

Coded Hundred Square One child’s reasoning: “I knew that the square went from 1 to 100. That meant there was a number that had 3 digits (100). I worked out that all of the numbers on the first row only had one digit so I started from the first row and worked my way on. ” A different method: “The way I worked it out was easy I just forgot about the numbers and the code and I just fitted the shapes in the way that they would go, and at the end I worked out the code and the numbers were right. ”

When Reasoning is necessary 5. When there is missing information Amy thinks that she is missing some dominoes from her set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing?

When Reasoning is necessary 6. When selecting a problem-solving skill In order to solve a problem, we need to draw on one or more problem-solving skills, such as: • • Working systematically Trial and improvement Logical reasoning Spotting patterns Visualising Working backwards Conjecturing

When Reasoning is necessary • There are three baskets, a brown one, a red one and a pink one, holding a total of ten eggs. • The Brown basket has one more egg in it than the Red basket. • The Red basket has three fewer eggs than the Pink basket. • How many eggs are in each basket?

When Reasoning is necessary 7. When evaluating a solution in context Having come to a solution, it is valuable to reflect on whether the answer is 'sensible' in the context of the problem. Have a go at the two questions below: 56 children are going to a football match. A minibus can take 12 children. How many minibuses do I need? Eggs are sold in boxes of 6. I have 37 eggs. How many boxes can I sell?

What do we do when we reason? • • • Evaluate situations Select problem-solving strategies Draw logical conclusions Develop solutions Describe solutions Reflect on solutions

Progression in reasoning Step one: Describing Step two: Explaining Step three: Convincing Step four: Justifying Step five: Proving

Sealed Solution: A set of ten cards, each showing one of the digits from 0 to 9, is divided up between five envelopes so that there are two cards in each envelope. The sum of the two numbers inside it is written on each envelope: 7 8 13 14 What numbers could be inside the "8" envelope? 3

One solution • At first I was randomly picking numbers, and on my first attempt doing it I found a solution: 0 + 7=7 5 + 3=8 9 + 4 =13 6 + 8=14 2 + 1=3. And then I tried using a system from then on, of adding a number to the smaller number then subtracting one from the smaller number, but it did not go very well because when I converted 5 + 3 to 4 + 4 I realised that you cannot do that. Then I found out something quite clever that from one onward: each 2 numbers have the same amount of possibilities (I think by this Ieuan means that pairs of consecutive numbers have the same number of possibilities). For example 2 and 3 have two possibilities, 4 and 5 have three, 6 and 7 have four, 8 and 9 have five and it goes on forever! So I wrote down all the possibilities for 7, 8, 13, 14, 3. Then I shortened it, so if I use 13 as an example: 13+0, 12+1, 10+3, 9+4, 8+5, 7+6. Then I would take off 13+0, 12+1 and 10+3 and do that for all the rest! So when I had all the possibilities I did two attempts without succeding then I got one and I started explaining it on here, but I realised I had found the same as my first attempt. Then I did one attempt and I found another: 3+0, 8+6, 9+4, 7+1 and 2+5.

Another solution • We started off by thinking of all the possible ways of making the totals. This took a long time. We thought that it would be best to make the biggest totals first, using the bigger numbers to make them: 14 = 9 + 5, 13 = 6 + 7, 1 + 2 = 3, 4 + 3 = 7 and 8 + 0 = 8. Some of us did it the other way round, making the smallest totals first, with the smallest numbers: 1 + 2 = 3, 4 + 3 = 7, 8 + 0 = 8, 7 + 6 = 13 and 9 + 5 = 14. We could also come up with pairs randomly but it's quicker to use a strategy. 7 + 0 = 7, 5 + 3 = 8, 9 + 4 = 13, 6 + 8 = 14 and 1 + 2 = 3.

A more refined solution

Communicating reasoning • • • I think this because. . . If this is true then. . . I know that the next one is. . . because. . . This can’t work because. . . When I tried xxxx I noticed that. . . The pattern looks like. . . All the numbers begin with. . . Because xxxx then I think xxxx This won’t work because. . .

Conclusions • Developing excellence in reasoning with young learners is a complex matter; • Need to think about the reasoning itself; • And the ability to communicate that efficiently and elegantly; • Help children to develop complete chains of reasoning; • This aspect of mathematics will help us to deepen and extend all learners.

Useful links • https: //www. stem. org. uk – for maths and science resources. • https: //nrich. maths. org - a large amount of open ended problems which provide opportunity for reasoning.