## Friday, 31 January 2020

### Rules First, Exceptions Later

I know that we as modern, innovative maths teachers like to say that maths isn't about rules and that it's about relationships and reasoning and all that stuff. But there are some mathematical processes that students have to be able to carry out in a fairly standard way and for the sake of this discussion, we're going to call them rules.

Just so you know what I'm talking about it, here are a few examples of processes that involve rules:
-finding the area of a sector
-finding the volume of a prism
-solving equations of the form 5(x+3)=7.

When we teach these processes, I think the most obvious way to do it is to start off with easy examples that students can work out without having to use a formal rule, and then extrapolate their reasoning to apply to more complex situations, which is used to develop the rule that is then eventually used. But it's occurred to me that this intuitive approach may actually be inefficient and even detrimental to long term ability to recall these processes.

Let's take adding fractions as our first example. I'm guessing you probably teach this in the same order I've always taught it in:
1) adding fractions that have the same denominators
2) adding fractions with similar denominators (like 5/8 +3/4)
3) adding fractions with denominators that share common factors (like 5/6 + 3/8)
4) adding fractions with denominators that are coprime (like 3/8 + 5/7)

The logic behind this sequencing is obvious: number 1) requires a one-step process, number 2) requires a two- or three-step process, and number 3) requires a three- or four-step process, so this suggests a kind of hierarchy. Number 4) involves applying a rule that you have to use when no other option is available, so it makes sense it goes last.

Students often intuitively understand the rule for number 1), so they can master it within seconds. They are then pretty accepting of the rule for number 2), as it's easy to see that writing 3/4 as 6/8 will allow you to apply the rule from number 1). This then logically leads to a rule for number 3). Number 4) is technically the same rule as number 3) because you are still finding a common multiple, but in practice your approach to these questions is slightly different.

In summary, the rules students need to learn are:
1) Since the denominators are the same, just add the numerators.
2) The denominators are kind of the same, so change one fraction so that it matches the other, and then apply rule 1).
3) The denominators are similar, so identify what the lowest common multiple (or any common multiple) of the two denominators is, and then change both fractions so that the denominators match, and then apply rule 1).
4) Spend a while trying to find a common multiple using your times tables facts, realise you can't think of one, multiply the two numbers together to create a common multiple, then change both fractions to make that denominator. Then apply rule 1).

This is quite a lot of rules to remember just for one process.

Now consider an alternative (still referring to the original four cases):
1) Since the denominators are the same, just add the numerators.
2) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).
3) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).
4) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).

In this version, there's only two rules to remember. Doesn't that sound so much nicer? The standard rule that works for case 4) works for cases 2) and 3) too, so why not just use this rule for all three?

Now I know a lot of you are feeling very upset about the fact that I am endorsing the idea of students adding 3/4 and 5/8 by making an unnecessarily large denominator of 32. You may also point out that this way would involve a lot more simplifying at the end. But hear me out!

I'm suggesting that we treat fractions with dissimilar denominators as the rule and the fractions with similar denominators as the exceptions. We teach the students the rule, we make sure they're super-duper confident with this process, let them feel really successful in doing this, give them lots of drills in this so that they're well practised and then teach them the exceptions. Say to the class, "of course with question e) there is a slightly easier way to do it, can anyone spot it?" and then discuss this and identify a few more exceptions like this and then do some practice with these. Of course, it's not really an exception because the other rule will still work, but it's exceptional in that there is another method you could use.

Let me give you another example: finding the area of a sector. I've always started out by finding the area of a semicircle, because students know straight away that this involves halving the area of the whole circle. They'll also get how to do quarter circles straight away. Then I would throw in a 120 degree angle or a 60 degree angle, and with a bit of thought they'd be able to tell me that you need to divide the area by 3 or 6. And then I'll put up a 24 degree angle and watch them struggle. They might eventually do 360 divided by 24 to get 15 and then tell me we need to divide by 15. And then finally I'll go to a 37 degree angle and they wouldn't be able to do it. At that point I would tell them, well the fraction that's shaded is 37 out of 360, so we do blah blah blah.

Now consider an alternative: start off with the 37 degree angle. Ask the class what fraction of the circle is shaded. Eventually they'll get that it's 37 out of 360. Then simply show them the correct calculation. And that's it. No faff. If they come across a question with a 24 degree angle, they can still do the same calculation, there's absolutely no need to work out that you can divide the circle area by 15. And even if you have a semicircle, the same calculation will work, and it won't exactly take much longer that halving (especially if it's a calculator question). Feel free to teach semicircles and quarter circles and even third circles as exceptions afterwards, as a little bonus shortcut that students will grasp easily. But I don't really see how starting with that will help students understand the process of finding the area of the 37 degree sector.

Another example: finding the volume of a prism. Isn't it weird that we learn how to find the volume of a cuboid separately from finding the volume of a prism, despite a cuboid being a prism and the prism rule working perfectly well for a cuboid? Why not just teach how to find the volume of a prism, and throw in cuboids, triangular prisms, trapezoidal prisms and compound prisms altogether (assuming they are all fluent in finding the area of rectangles, triangles, trapezia and compound shapes)? I would love to add cylinders to that list too but I found out embarrassingly recently that a cylinder is not, as I always believed it to be, a prism. Who knew?

One final example: solving equations that look like 4(x+2) = 8. Now there are two obvious methods to solving this equation. You could divide both sides by 4 or you could expand the bracket first. Dividing by 4 is a nicer method in this case and gets you the answer faster. However, what about the equation 4(x+2)=7? In this case, it would be unwise to divide by 4 first, so you should really expand the bracket. But there are no examples where expanding the bracket first doesn't work, so why don't we just teach our students to always expand first? (I have a feeling a lot of you do teach them this actually!) There are a lot more numbers that don't divide nicely by 4 than numbers that do (although Hilbert might disagree...) so most of the time the dividing method won't work, but the expanding method always will. It's nice to show the dividing shortcut as a little aside, once the students are really secure in the process of solving other equations with brackets. But I believe fluency in the rule should take precedence over the ability to identify exceptions.

I hope I've managed to explain my point well enough in this post. It would have been much easier if I could scribble stuff onto a whiteboard at times! Maybe that's the push I need to finally start the NotJustSums YouTube channel...

Oh and I should mention that this post was inspired by things I heard in Craig Barton's podcast which is excellent. My favourite episodes have been his interviews with Dani Quinn and with Tom Francome, and I strongly recommend that you check them out.

Emma x x x

## Tuesday, 14 January 2020

### 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!