The road to flexible maths is paved with inflexibility

We’re not embarrassed about the fact that at Michaela, pupils learn capitals, dates, quotations and definitions by heart. We expect it because it’s a mistake to treat novices like experts. It’s not because we think that’s History is inherently and ultimately about memorising dates, or Geography about capitals. But doing so is an important and unavoidable step to getting the point of being a critical Historian or Geographer.

For a long time, I didn’t think we had an equivalent argument in Mathematics. My experience of number talks this year has made me realise we most certainly do.

Number talks are the darlings of the anti-traditionalists. Here is Jo Boaler speaking about number talks.

Lovely. I see a lot of merit in what Boaler is advocating. I certainly want my pupils to solve problems flexibly, without resorting to pen and paper unnecessarily, having chosen from a variety of methods, and be able to have a sense of whether their answer is realistic.

And, at the end of the first year at Michaela, this is something I’m pleased to say my kids are pretty good at.

However.

I have rarely had less productive experiences that simply projecting a number talk and telling the pupils to have a go.

In fact, what has got them to the point of being able to have productive number talks in class is doing a series of things that Boaler et al. would absolutely despise.

It is a heck of a lot of inflexible work that has enabled the flexibility to happen.

1. Learning by heart

Number sense has been used as an argument for why pupils don’t need to learn their number bonds, times tables and other number facts. If they’ve got number sense, they can work out 6 x 7 by doubling 3 x 7, can’t they?

Yes, they can. I still don’t want them to. This is for two reasons.

Reason one: Working memory

Firstly, you always have to start from some sort of number fact to get to where you need to get to. The difference is just quite how many steps it takes them to get there. I prefer that number of steps to be lower. It’s faster, and reduces working memory overload. Let’s say we’re calculating 12 x 7.

They could know it straight away.

They could double 6 x 7, if they know that fact. One extra step. One extra burden on working memory.

They could double 3 x 7 twice, if they know that fact. Two extra steps. Two extra burdens on working memory.

They could start with 2 x 7, if they know that’s 14, then add 7, then double and double again. Three extra steps. Three extra burdens on working memory.

How far are you willing to go? How much pressure are you happy to put on your pupils’ working memory – especially the weakest ones? I’d like it to be as little as possible.

This isn’t something a maths teacher can opt out of making a choice on. By failing to teach number facts, you are making a choice. You are making a choice to give children extra overload on their memory.

Reason two: Pattern spotting, generalising and checking

Secondly, knowing the facts by heart makes it easier to spot, remember, generalise and check the rules of arithmetic that help them to work flexibly. Pupils who just know that 4 x 6 and 2 x 12 are both 24 find my explanation of the strategy of doubling one factor and halving the other much easier to understand. When I move onto examples like 2.5 x 18, they’re more confident in telling me it’s definitely the same as 5 x 9. Something @danicquinn and I have spoken about is how when we’re calculating something complex we’ll often check “oh yeah, it’s OK to use this strategy” by trying it with some easy numbers. The little check propels you along, making you confident you’re doing the right thing. Pretty tough to do that if you don’t have the knowledge of simple facts in your long term memory.

 

2. Drill, baby, drill

How do I get the kids to learn the number facts by heart? Drill, drill, drill. With time pressure.

Jo Boaler recommends a two player dice game called “How close to 100?” as a method of developing fluency without fear. In the time it’s taken pupils to practice one multiplication in the dice game, they could have done 100 on a plain old worksheet.

But don’t they have maths anxiety? Don’t they fear maths? I don’t see it.

Pupils love seeing themselves improve. They love knowing stuff by heart. They love feeling successful and masterful.

3. Naming the steps

There seems to be a belief that to encourage multiple ways of solving a problem, we should keep things a bit fuzzy, a bit vague, a bit ethereal. A feeling that strategies for solving a calculation like 3.75 x 14 should emerge organically from their conceptual understanding. A belief that telling them “here are four strategies you can use to solve multiplication problems” is too rigid, and is in direct opposition to the flexible thinking we have as our end goal. I disagree.

It was Doug Lemov’s Teach Like a Champion that convinced me I should not only tell children specific methods for manipulating calculations, but name them. Teaching an incredibly nuanced art – more so than arithmetic. Still, having named techniques like Cold Call has been invaluable in my development as a practitioner. As a novice, it helps you pin down something nebulous. But more importantly, long term, it gives a common vocabulary. It means my line manager can observe me and say “give more wait time when you’re insisting on right is right during pepper” and I know exactly what he means. That’s powerful.

So I’ve done the same with maths techniques. When I project a question like 3.52/22 on the board, I’ll say something like “shall we roll the divisor here? Partition the dividend? Or decompose and divide twice?”.  You might not know what I mean by “roll the divisor”, but Mohammed and Abdoulai and Sijan do, and that’s what matters. Rather than me spending time explaining in a heavily laboured, abstract way what I mean by those terms each time, we can actually have a class discussion focussed on which is best and why.

Yes, it’s an oversimplification. Yes, the names can sound a bit crude. But once pupils have got the named ones down, it’s much easier for them to add tweaked versions or combined versions of the ones  to their schemas. So when I project 3.3 recurring times 12, they might say “Miss, it’s like doubling and halving, except I’m going to triple and divide by 3”.

 

4. Telling them when their method is bad

I worry that number talks can encourage an attitude of “every method that gets the right answer is equally great”. No. There are bad, laborious methods of doing calculations. Fudging that is not good for the children long term. It leaves them with a sense that maths is more complex than it actually is. It discourages them from learning new strategies and number facts. The teacher, as the expert in the room, has the responsibility to show children easier, better ways of doing things.

Should you make them feel bad personally for using that method? Of course not. Praise the effort. Acknowledge it gives the right answer. But be clear that quicker and easier is better. After all, if maths is about so much more than arithmetic – a core argument of the anti-traditionalist teachers (which of course is true) – we want to minimise the effort expended on that arithmetic.

5. Gimmicks, carrots and sticks

Finally, I’m happy to use some gamification, silly props, humour, and competition as a spoonful of sugar to help the medicine go down. Extrinsic motivators? Yes. But the medicine needs to go down. Building habits is hard. Work is hard. The reality is it’s much easier to be intrinsically motivated at something you’re good at. So let’s do what it takes to get them good at maths. If that means a silly pair of glasses and some rockstar music, I’m more than happy to oblige.

Children can have the rich discussions about complex arithmetic that Boaler wishes for. The inflexibility of drill, learning by heart, named steps, best methods and a bit of carrot and stick is not in opposition to flexible knowledge. That inflexibility is the very way we get flexibility.

 

 

 

4 thoughts on “The road to flexible maths is paved with inflexibility

  1. Lee MacArthur

    Thank you for this nice discourse on the topic. I have hopes one day that the elementary and middle school will produce more than 10 to 20 percent who actually know their multiplication tables by heart. Too many of my students struggle with this and it really does slow them down tremendously when working on say area, or completing the distributive property. So I am teaching multiplication in my High school math classes to help remedy this.

    Reply
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