1. Session 5: Mathematical Thinking
Mastery Unlocked
Session 5: Mathematical Thinking
The key message is that reasoning and mathematical thinking is not the “icing on the cake” which only our rapid graspers can do. Reasoning about concepts and working with them rather than being passive receivers of them is how children come to develop conceptual understanding.

2. The 5 big ideas
Much of this presentation has been based on the work of the nrich and their series of articles around developing reasoning in KS1 and KS2. Make sure you acknowledge this.

3. What do we mean by mathematical thinking?
Mathematical thinking involves:
Looking for pattern in order to discern structure
Looking for relationships and connecting ideas
Reasoning logically, explaining, conjecturing and proving
Mathematical thinking is central to deep and sustainable learning of maths
Concepts need to be thought about, reasoned with and discussed in order to become secure. They can’t be passively received
How is mathematical thinking reflected in the aims of the National Curriculum?

4. Mastery is for all…..
Mathematical thinking is not just for your, “most able” children. (fast graspers)
All children need to be exposed to questions which encourage them to develop good mathematical thinking skills.
Pre-tech sessions can be vital for struggling learners in developing their reasoning

5. National Curriculum Aims
become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions

6. Reasoning When is reasoning necessary?
How do children progress in reasoning?
How can we develop reasoning skills in children?
“Reasoning is the “glue” that helps mathematics make sense.” nrich Reasoning: Identifying Opportunities

7. When is reasoning necessary?
When first encountering a new challenge
When logical thinking is required
When a range of starting points is possible
When there are a range of strategies to solve a problem
When there is missing information
When selecting a problem solving skill
When evaluating a solution in context
When there is more than one solution

8. Progression in reasoning
Describing: I can describe what I did.
Explaining: I can offer some reasons for what I did.
Convincing: I am confident that my reasoning is correct (even if it’s not!) and can try and convince you that I’m right.
Justifying: I can use words like, “therefore,” “that leads to” to justify a correct logical argument with a complete chain of reasoning.
Proving: I can make a watertight argument that is mathematically sound.
Inductive reasoning is reasoning in which the premises are viewed as supplying strong evidence for the truth of the conclusion. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.
Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements to reach a logically certain conclusion. It differs from inductive reasoning and abductive reasoning.

9. Developing reasoning
Encourage participants to try the problem (it’s aimed at KS1) and then discuss the thinking involved.
This is an example of reasoning when selecting a problem solving skill.
Low threshold – small numbers
Lots of different approaches to solve.
What does a child get out of this type of problem?
It is important that children are not cognitively overloaded with trying to understand an elaborative problem as they practise and refine their skills in reasoning. Questions must be clearly worded and unambiguous.
What happens to the area of a square when we double the length of a side?
Fermi questions
Create opportunities to compare and discuss different solutions.
Create an environment where children feel safe to take a risk and explain their reasoning

10. Developing reasoning Maze 100
In this maze there are numbers in each of the cells. You go through adding all the numbers that you pass. You may not go through any cell more than once.
Can you find a way through in which the numbers add to exactly 100?
What is the lowest number you can make?
What is the highest number you can make?
Maze 100
This is an example of reasoning when there is more than 1 solution.
Having found one route challenge the children to prove they’ve found the right solution. Is there more than one way to answer the question?
Do children use reasoning to tweak the solution they have already found or do they start from scratch?
If the challenge is to find all the possible solutions, the reasoning might involve having a system which ensures none are left out.

11. Progression in reasoning
Present the attendees with a problem to solve then look at children’s reasoning. Ask how far along the scale they are. What could we do to move them on?
This is a good problem because it’s impossible! Children need to look at the numbers to realise that all of the numbers in the bags as well as the total 37 are odd. They need to prove that an even number of odd numbers can’t make an odd number.
The top of the reasoning chain is proving why its impossible.

12. Where on the progression in reasoning scale is this solution?
Imogen, Molly, Hannah, Charlotte, Elodie and Jess from Sandbach High School wrote: “We quickly got 36 and 38. Only when we looked at the other possible totals from 10 to 70 did we spot the answers were always even numbers! This puzzle is impossible to prove!”
Some convincing

13. Where on the progression in reasoning scale is this solution?
From Josephine at St Wilfrid's Ripon we had: “This task is impossible because if you add an odd number to an odd number you will always get an even number. We would need an odd number and an even number to make an odd number. eg: 7+7=14, even     4+5=9, odd        9+9=18, even     3+2=5, odd        7+3=10, even     6+7=13, odd”
Here the child is moving into proof by showing that you cannot make an odd number with an even number of odd numbers. They are using a mathematical generalisation.
This is much higher along the chain of reasoning, but to develop into a proof it needs to show the generalisation holds true for all examples. The child would need to show that two odd numbers always make an even number and that ten odd numbers is make up of five pairs of odd numbers, therefore there will be a sum of 5 even numbers, which must be even.

14. How do we help children to communicate their reasoning?
Model the words and give sentence starters:
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….I noticed that
The pattern looks like
All the numbers begin with

15. Key questions to ask children
What is the same and what is different?
Do you agree? Is that correct?
Is this always, sometimes or never true?
Have you found all of the possible solutions?
Is there a way to prove you have all the solutions?
Can you prove your solution is correct?
Will this always work?

16. The 5 big ideas
Much of this presentation has been based on the work of the nrich and their series of articles around developing reasoning in KS1 and KS2. Make sure you acknowledge this.