I Was Wrong

That's right, I'm still here!

It's been a long time since my last confession.

I've been wanting to get back into blog writing for a while now but for some reason no particular topic really seemed interesting enough to write about (and let's face it, I've set the bar pretty low with some of my previous topics - "is zero a factor of zero?", anyone?) However, this week I've been inspired to crack open the ol' blog and do something with my spare time that doesn't involve my Nintendo Switch.

The reason for this sudden burst of inspiration? Reading Craig Barton's wonderful book, How I Wish I'd Taught Maths. If you've not heard of Mr Barton, you're missing out. He's a UK Maths teacher whose resources website has got me through many a Tuesday afternoon double lesson. His book is part memoir and part how-to guide, with lots of lovely academic references to back up his claims. I'd definitely recommend getting yourself a copy.

Mr Barton (why does it feel wrong to use his first name, like he's my teacher or something?) talks in his introductory chapters about how his opinion of what good maths teaching looks like has changed dramatically over recent years, and I've experienced something similar. When I was in my PGCE training year back in 2010/11, there was a belief held by many cutting edge maths teachers that maths lessons should be all about exploring, discovering, and open-ended problem solving. The suggestion that a traditional "chalk and talk" lesson which was teacher-led and full of repetitive exercises may be effective was met with haughty derision by us (wise beyond our years) trainees and may have even elicited an "OK boomer" type eye roll. We knew that the way forward was rich tasks from NRICH and a hands-on activity involving spaghetti and marshmallows. We quoted Piaget and Skemp as if they were old friends and bragged to our colleagues "oh I never use textbooks". I think we were rebelling against our own education, which might have involved those SMP booklet thingies or OHP slides covered in hundreds of near-identical quadratic equations. And the fact that we were really rather successful under that system was irrelevant; we knew that we could do better, and better must mean different. So clutching a copy of Jo Boaler's latest with our paper-cut (bloody Tarsias!) fingers, we marched into our classrooms determined to teach our students nothing and yet have them learn everything and when things didn't quite go the way we planned (although the marshmallow came unstuck from the ceiling eventually), we blamed the Primary school teaches for raising our students to have fixed mindsets and no independent learning skills. We pulled off Outstanding (capital O) lessons and wowed our observers whilst managing to cover remarkably little content. Those were the glory years, the halcyon days of group work and bits of string and "what do you notice?". But somewhere along the road I think we all started to realise that not only was this approach not anywhere near as much fun as we thought it would be, it also just didn't seem to work.

I'm in my ninth year of teaching now and my pedagogical beliefs have definitely changed. Here is a summary of how:

I used to think:
You couldn't tell students anything explicitly, you had to let them discover it for themselves, otherwise it wouldn't be meaningful.

Now I think:
Mathematical concepts don't become meaningful when you discover them, they become meaningful when you use them to solve a problem. Think about the circle theorems. Discovering the "angle at the centre is twice the angle at the circumference subtended by the same chord" rule by drawing a lot of random lines inside circles is very unlikely to happen without significant scaffolding (is it really discovering if you were given a guided tour?) and students may well discover something completely different (or "discover" something that's not even true!) along the way. And would they even know they've discovered it when they discover it? Would they even see its value? How about instead, we give students a circle with some angles in it and ask them to find the missing angle. They will try to apply the rules they already know, but will eventually get stuck. We can then say, "hmmm, you need a rule that's going to help you solve this easily. Luckily, I have just the thing!" and then the circle theorem saves the day and makes a difficult looking question dead easy. Much more meaningful, in my opinion.

I used to think:
You must show students the proof of rules or theorems before you let them use them. This includes basic rules like the laws of indices, up to more complex ones like differentiation.

Now I think:
Proofs should only ever be shown after students are fully comfortable with using and applying the rule or theorem. When I was shown proofs of Pythagoras' theorem as an adult, I thought they were amazing and interesting and beautiful, and I thought, why didn't my year 9 teacher show me this? So then I decided I would always show my students these proofs when introducing Pythagoras. It turns out, the only reason I found these proofs so fascinating is because I had already been using Pythagoras for eight years and was pretty intellectually invested in it. Why should students who have never heard of or used the theorem to solve any problems care about how the theorem is derived or why it works? I'm even starting to question my approach to teaching differentiation: I usually demonstrate graphically using a concrete first principles approach to find the derivatives of x² and x³. Along the way I generally lose the interest of 50% of my class, and the other 50% finish the lesson believing that differentiation is the most difficult topic EVER and they should drop A Level Maths before it's too late. Wouldn't it be better to teach them the quick little "lower the power" rule, get them used to it, and then show them where it all comes from? And then the sudden appearance of the 3x² out of nowhere may be met with a chorus of satisfied "ahhhhh"s rather than a sea of bewildered faces.

I used to think:
Slower methods are better than quick ones if it means that students understand how and why the method works.

Now I think:
The best method is whichever one has the lowest cognitive burden, as this frees up valuable thinking space for other aspects of the question at hand. Problem solving questions can sometimes involve five different mathematical processes to be carried out, wrapped in a "real life" situation that involves unpicking. The problem solving aspect of the question takes up a considerable amount of brain power, so being able to carry out the mathematical processes efficiently is really beneficial. I would say, however, that for students with certain SEND needs, sometimes the memorability of the method is more important than the cognitive burden, which means that slower methods can sometimes be favourable. But for students with a decent long term memory, the quickest method is usually the best.

I could probably list another ten beliefs that have changed, but I've written an awful lot today and I want to save some material for future posts. Stay tuned!

Have your beliefs about teaching changed since your training year? Let me know in the comments.

Emma x x x