A Foray Into Reasoning And Proof - Part 2
By Dr Audrey Curnock - 8th June 2019
The second half of Dr Audrey Curnock’s session was focused on contextualising proof to everyday life.
We began by comparing proof to the justice system. Initially lawyers construct an argument using the evidence they have. However, any evidence presented for a case isn’t 100% proof. It is ‘proving beyond reasonable doubt’. This means that unless there is evidence to disproof the argument, it is considered valid. But, in Maths, we must be 100% certain that our claim is true.
For exams, students are required to learn Mathematical proofs such as proving that two even numbers will add to give another even number, etc. Naturally, when we present students with information, often we get asked, why? Why is it true?
That’s where proofs come in, they tell us why the argument we are using is sound. Students often find the thought of having to do proofs scary, but when you break it down, majority of the questions are regarding odd (2n +1) and even numbers (2n). So long as they can remember how to represent odd and even numbers algebraically, the rest of it is just playing around with the terms until you get what you want.
One of the examples we worked through (Image 3 below), demonstrates the thought steps required to carry out an algebraic proof. One of the scholars suggested getting students to work through the questions using numbers, then they will be able to present the argument using algebra. This will allow students to see clearly what exactly they are trying to prove. Sometimes students find numeracy simpler as they are familiar with it from an early age, hence it might help them too see the problem numerically before they see it algebraically.
Image 3: Screenshot of example 3
After working through these examples, we discussed how students might find proofs difficult how we can make it more approachable to students. In some disadvantaged areas, students don’t always have access to the vocabulary needed to access the question. This can immediately provide hindrance when attempting to answer reasoning and proof questions. Therefore, when introducing the topic, it might be worthwhile providing a list of key words students will need to be familiar with in order to understand the questions.
Often students find presenting counter-proof easier than proofs simply because they only need to find something to disprove the argument and often this can be using just numbers. However, counter-proof questions are still getting students familiar with answering proof questions and provides a stepping stone to build on. Sometimes students are scared or reluctant to play around with numbers. Something that stuck with me during my time at University was when one of my lecturers said, ‘You should be prepared to do Maths on the back of an envelope!’ We should encourage our students to jot things down, then scribble them out if they don’t think it’s right. Working out doesn’t always have to be neat. I believe that making mistakes and learning from them is the best learning experience any student can have.
To conclude, reasoning and proof are essentially being able to translate English into Mathematics and being able to present an argument using Mathematical language. This is something that needs to be embedded in KS3 to ensure that students have the right skills needed to tackle the GCSE content.
I found Dr Audrey Curnock’s talk very inspiring. It made me pinpoint the finer details I need to focus on when teaching reasoning and proof, and making sure that the content is accessible to all students. Every student has the ability to be good at Maths, they just need to acquire the skills needed to help them do that through learning key vocabulary and mastery of the content.
By Abi Varathanathan