A Foray Into Reasoning And Proof -
By Dr Audrey Curnock - 8th June 2019
I recently attended the Alumni CPD event at Aston Conference in Birmingham. One of the talks was by Dr Audrey Curnock on Reasoning and Proof. Our aim was to begin to understand how to develop reasoning and proof across the curriculum. To essentially see proof as a tangible element of the syllabus rather than an abstract and difficult topic.
The first half of the session involved working in small groups to work through and discuss various reasoning problems. We had to identify the skills needed to answer the question and what aspect of the problem students will find challenging and what the misconceptions were. The first problem (Image 1 below) illustrates a calculator question, where students simply had to put the numbers into their calculator to get the answer. However, in my group we identified that although it looks simple, it was quite easy to forget to square root their answer or forget to include the cubed in the denominator. Or even forget to put their final answer in standard form. When you’re trying to answer the question quickly, especially in test conditions, it is easy to overlook the small details. Therefore, this made me realise the importance of teaching our students to work through the questions calmly at their own pace to ensure that they have fully understood and answered the questions correctly.
For part b, we discussed various different thoughts the student might have on the problem. 5% is small, surely this means this won’t affect the final answer? If both w and d are increased, so the overall answer must increase too?
This problem emphasised the need for reasoning itself. It’s important for students to identify that although d is only increased by 5%, it is still cubed. So, overall a smaller number divided by a bigger number will give us a smaller answer.
Image 1: Screenshot of reasoning exercise 1.
The second question we discussed was regarding circle theorems (Image 2 below). The skills needed to answer part a of this question involved being familiar with the circle theorems, remembering them and being able to recognise the theorems. In part b, Dylan has gone about solving the problem algebraically, however this is a circle question involving triangles. So, how could y be 200° if the angles in a triangle only add up to 180°?
This is a good example of how students should be able to reason with themselves the possibilities of the value of y using their understanding that y mustn’t be greater than 180° if it was to be an angle in a triangle. This highlights for a need to link geometry and algebra so that students can see the crosslink between different topics and be confident applying their knowledge to unfamiliar situations. Sometimes students are not able to think logically, simply because they don’t know the facts. Thus, it is imperative that students master the topics to become more confident with applying their knowledge.
Image 2: Screenshot of reasoning exercise 2.
So, what is reasoning?
Reasoning is being able to think logically. It is a long-term process and can’t be developed overnight. In school, we teach Mathematics in layers, slowly building up their knowledge over the years so that they are able to develop their reasoning skills as well as building up their Mathematical knowledge. However, how well the student retains this information and is able to apply it is ultimately down to how much time they are prepared to spend to fully understand the topic. The wonder in Mathematics is that there are very little facts to remember, most of the time you are applying the knowledge you know. Knowing a little fact can go a long way in Maths!
By Abi Varathanathan