Within any Key Stage 3 or GCSE Maths classroom, I am an expert. I am certainly not saying that I have the best subject knowledge (more on this to come!) in the world, or even in my department, but I don’t have to struggle to solve equations or calculate the mean from a frequency table, and I can detect immediately whether a problem will require the use of trigonometry or Pythagoras’ Theorem. Obviously, with the advent of the new GCSE, some of the more unusual problems may occasionally cause me to pause and consider the most efficient method, but even then, my expertise means that I can come a solution with relative ease. This surely, is important, for a teacher. I’ve never been particularly convinced by the claims that ‘as long as you’re one step ahead of the students, you’ll be fine’. I don’t just want to be one step ahead – I want to be multiple steps ahead so that I can challenge and probe and I know exactly where the learning is headed.
My expertise, as is the case for any teacher, has been developed after years of practice. Now, obviously I was reasonably lucky at school in that I enjoyed Maths and it came relatively easily, but the cause of my expertise is simply the sheer quantity of Maths that I have done. Thinking back to my own years at school, although I could replicate procedures/solve problems/explain concepts to others, I was far less of an expert than I am now. When I reflect back, while I never struggled with factorising quadratic expressions, for example, the procedure was not automatic in the same way that it is for me now. More importantly, when I reflect more deeply, I remember that learning Maths was difficult.
Yes, I enjoyed Maths, and yes, I was successful at Maths, but the process of learning new concepts/methods is something that I would have found difficult – because it is difficult! This is something that I am trying to work on conveying to my students at the moment. In the past, I would have prefaced any introduction to a topic with how successful they were going to be and how easy they would find it. ‘Angles in parallel lines? You are going to love this! This is so straightforward; I promise you are all going to be great at this!’ Regardless of the topic, they would hear a similar introduction. On occasion, I might begin with ‘today, we’re going to look at Circle Theorems. Now, people may say this is hard, but all of you are going to find this super easy and be amazing at it – let’s go!’.
This was obviously done with the best of intentions – it seemed important to promote the idea that all students would experience high levels of success and it would boost their self-esteem from the beginning. Except, it didn’t. And it doesn’t. Even with the best teacher explanation and resources and careful scaffolding in place, learning is (or at least, should be!) hard work. Every time that I told a student that they were going to find something easy and they didn’t (which was inevitable – most of the students that I teach are novice learners of Maths), they would blame themselves. ‘If this is so easy, and I still can’t get it, then it must be me that’s the problem’.
In recent months, I have started teaching myself Further Maths A Level – I only studied Maths at A Level and then a non-Maths related subject at university, so the vast majority of the content has been brand new to me. I am learning about matrices and complex numbers and sequences and all sorts of wonderful mathematics, and this is fascinating and I am so excited about everything that I am learning. It’s been really hard work, as you might imagine. Although I can now find the roots of polynomial equations with relative ease, trying to learn it was really tough! To add some context, my partner is currently completing his PhD in Bayesian statistics – so his mathematical expertise is far greater than my own. When I asked him for help recently on a particularly challenging problem, his immediate response was ‘oh, well obviously you need to…’. To him, it was obvious, because he had acquired that expertise. To me, a relative novice, it was far from obvious, and now I felt frustrated in the fact that I hadn’t known how to do it.
I felt similarly frustrated when a colleague recently emailed around a difficult problem. I had sat down with my pen and paper, and was just getting stuck in, when another colleague emailed around the solution. Although I still went on to obtain the correct answer, I’d felt far less satisfied by achieving it than I would have otherwise.
This has been a massive revelation to me in terms of how I relate to my students in the classroom. Firstly, although I will still introduce topics to my students by telling them how much I love/they will love/how fascinating it is, I am far more honest about the fact that it will be hard, hard work. They will need to focus entirely and want to achieve and be prepared to make mistakes – and then, after time, they will find it easy and forgotten that they ever had reason to struggle. I am also promoting the fact that hard work, when you are successful at it, is far more enjoyable than not having to think at all.
I am also far more thoughtful about how I give students answers to problems – in the past, I would stop the class at particular points to read out answers to ‘the questions you should have completed so far’. Imagine that you are the student working on question 5 – it’s a difficult question involving multiple properties of angles in polygons, but you’re confident you know what you need to be doing. You’re working through, quite happily, when all of a sudden you are interrupted: ‘Question 5 is 25°’. There’s little point in continuing with that question now – the satisfaction that you would have experienced when you had reached the solution was taken away from you. As a result of this, I now will typically have answers to problems/exercises available to all students – these will be on the back of a worksheet/printed separately, and this means that they can check their work when they are ready. I don’t know if I’ve been particularly lucky with the students that I am teaching currently, but the vast majority don’t want to cheat.
Despite our difference statuses as novice and expert, we achieve that same ‘YES!’ feeling when we make it to the end – and that ‘YES’ feeling is all the more worthwhile when it has taken considerable effort to get there. Yes, Maths can be easy and straightforward – but learning it often isn’t.