A Foray Into Reasoning And Proof 

Kathryn Jones It was a pleasure to listen to Dr Audrey Curnock speaking on the importance of reasoning and proof when teaching mathematics in schools. 

The workshop began with an icebreaker where we were asked to imagine that we were in a swimming pool and we had to decide what depth we thought we would be in in relation to how confident we felt about our mathematics teaching so far. It was interesting and fun to interact with other mathematicians and hear their reasons of why they were either in the shallow end or deep end of the pool. 

After this, we discussed the importance of reasoning and proof in mathematics making reference to the current overarching aim of the national curriculum and how it is important to encourage this critical thinking in mathematics.

The first part of the session was related to reasoning in mathematics. We were given two mathematical problems to work out and discuss the skills we needed to solve them and what misconceptions students could have. After discussing these problems and feeding back, we discussed how many people worked out the answers differently, and this reinforced the fact that there is no one way of doing things and it is important that answers are open to students as this will encourage a more investigative approach to solving problems. Reasoning requires thinking about something logically and is a crucial pre-curser to developing the skills needed to understand proof. 

Practical suggestions for maths teachers to promote reasoning:

- Ask students to explain and discuss answers with peers.
- Ask students to show their answers in different ways.
- Encourage students to not be afraid of mistakes, as they will learn from them.
- Get the students to complete more questions on reasoning after each topic. 

The second part of the session was about proof. We were shown how proof is contextualised into everyday life by looking at the justice system where the lawyers and jury decide whether the case can be proven one way or another. These real-world examples can show comparison of different types of proof. However, it is only in mathematics that there can be 100% proof.

 We also learned that there are 4 levels of proof. 

1. Naïve empiricism
2. Crucial experiment
3. Generic example
4. Thought experiment

Students are usually able to understand the first 2 levels, but it important to encourage them to go further to develop logical proofs in the higher levels of proof. It is vital that we encourage the students to justify and explain answers as this is the precursor to proof.

Practical suggestions for maths teachers to promote proof:

- Make sure pupils understand vocabulary and keywords as this can lead to better understanding.
- Encourage students when starting proof as they can be reluctant to begin.
- Get students to critique their neighbours’ answers and ask for justification.

It is clear that developing the skills students need to be able to reason mathematically is vital for the promotion of further understanding of mathematical concepts. It is important that all teachers encourage students to justify answers and methods as this will provide the basis for developing proof.

By Kathryn Jones.