Teaching addition to kids.

David Coulson, 2013

Add these two numbers: 47

69

Now tell me how you did it.

Did you add them the way you were shown at school? Or did you try some other, sneaky way?

4769

There are all kinds of approaches to adding these numbers, and I’ll bet that if I asked six people how they did it, I’d get six different answers.

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The interesting thing would be that none of those six people would have done it the way they were taught at school.

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There are all kinds of approaches to adding these numbers, and I’ll bet that if I asked six people how they did it, I’d get six different answers.

47+ 69

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There’s nothing wrong with the school way of adding numbers. In fact the school way is highly efficient. It’s also a very compact method that achieves the answer with the fewest number of internal steps and requires the least amount of paper.

1

6

It’s such a powerful method that we can use it to add dozens of multi-digit numbers so long as they’re written nicely in columns. That’s how the accountants and the scientists of the pre-computer age did it. And because they had to do it a lot, they stylised it into this short method and shared it with kids in school.

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So why don’t grown-ups ever use the method for simple calculations like this?

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Instead, we scratch our heads a little and put the numbers together in some other way that is certainly less efficient than the school method but which somehow suits you better because nothing has to be written down.

The thing is that few of us have paper and pen ready when confronted with a calculation. That’s one limitation, but I think there’s more to it than that. Even sitting in front of your laptop, looking at this pair of numbers, you could have wandered off to get a piece of paper and a pen, or maybe you could have activated the calculator app that comes with your operating system. But you didn’t! It’s just toooooo much bother, right?

47+ 69

All this doodling around requires a brain that works like a piece of paper, and few of us have that kind of brain. We want a method that piles all the stuff together in one place and then shaves off bits as we go, or adds on bits as we go, so that at any one time there is only one thing for us to look at.

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6

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It’s all to do with the subtotals that are required by the formal method. You have to add the 9 and the 7 first to get 16, then tear the 1 off the 16 and put it on top of the 4 while putting the 6 down at the bottom and leaving it there for a while while we add a 1 and a 4 and a 6 to get a ... (pause for breath) what was it that I was doing just now?

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Now I’m not going to recommend one method of doing so-called mental arithmetic over another because they’re all good. Instead, I want to draw your attention to the fact that we are failing our kids at school if we don’t let them know that these ‘mid-air’ methods exist.

--YES, show them the formal method because they will certainly need that when they are faced with super-big additions with super-big numbers. But also show them the alternatives for simpler additions because the world abounds in simpler additions. Otherwise we are like gardeners who use only the biggest tools to do even the simplest pruning jobs, like cutting a rose-bud with a combine harvester.

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Looking at these numbers we could say, for example, that 40 and 60 make 100 and that 7 more makes 107 and 9 more than that makes 116.

...or maybe we could say that 47 and 69 is one less than 47 and 70, or three less than 50 and 69, or maybe it’s like 50 and 70 less 3 and 1.

...or maybe we could ask the kids what they think. Any strategy is good except carving 47 lines on the tabletop and carving 69 more and then adding up all the lines one by one. That shows as much creativity as ...as...well, no creativity at all.

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Why should there be creativity in a maths lesson? Excuse me? Why shouldn’t there be creativity in a maths lesson! Turning the task into a journey where the kids themselves decide which way to proceed is probably going to

appeal to them a bit more than simply rehearsing the same old method a million times. It gives them the chance to show us how smart they are.

--And of course if they get a bit too smart, we can always show them how smart we are too, and make it into a friendly contest. Amazingly (I’ve found), if you stay on really friendly terms with your students, then no matter whether it’s your turn to show off or their turn to show off, they want to come round to your side of the table and show you what they think you should do.

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So welcome to the notion of turning a maths exercise into a decision-making process, and seeing which decisions work well and which ones don’t.

Here’s an exercise I do with kids once in a while. I pull out a piece of graph paper (Oh horror of horrors! A page with lines on it! How old-fashioned! How Victorian!).

I draw (up to) ten lines on it that enclose some kind of rectilinear shape.

I might get a shape that looks like this:

By rectilinear I mean that, as I draw the shape, my pencil moves horizontally or vertically along the blue lines, and comes back to where I started.

Now I want to make it very clear that, even as I’m drawing the shape, I have no idea what the shape is going to look like, and no idea how many squares are going to be enclosed in the shape. In fact, even as I type these words into my laptop as I create this presentation, I still haven’t taken the time to count how many squares there are inside this shape (I’m a man; I can’t add and draw at the same time, even with a university degree).

But I don’t need to know what the answer is because I’m confident I can work it out. And by showing the student that I trust myself to work it out, I’m showing him/her that it’s easy. But there’s also something more physiological happening when we do this: We are emanating the body language of curiosity, whatever that is. The kids pick up the scent and copy it. They can’t stop themselves.

With any luck the student will spot for him/herself that the region is made up of rectangles, and that the total number can be found efficiently and painlessly by separating these rectangles.

But how should (s)he do it?

This way?

But how should (s)he do it?

Or maybe this way?

But how should (s)he do it?

Or maybe this way?

But how should (s)he do it?

It’s not simply about minimising the numbers of boxes. Rather, it’s about releasing as many tens-like numbers as possible from the choices that are on offer.

By tens-like, I mean numbers that have zeroes in them.

70

A quick inspection shows me that there are ten lines over on the right. Ten is an easy number, so I think I’ll carve off that box on the right.

Straight away I know that there are 70 squares in that part. What about the rest? What would you do?

70

Ahah! I think I just roused your curiosity. And that’s a good thing, because if I got you emotionally involved in the task then I should be able to get a kid emotionally involved in the task too.

Exactly how (s)proceeds through the problem is not terribly important. It’s that they have taken control of the problem and are making decisions about it.

70

40

Now maybe my kid is a bit shaky on the times table but can count in fives. (S)he sees that the left side could be sliced off as a group of five, and can count through the lines, saying “5, 10, 15, 20, 25, 30, 35, 40” as (s)he goes because that seems to work for him/her.

We’re already halfway there.

70

40

Now that bit in the middle represents 6x16, and how your student decides to do that depends on how good (s)he is with the times table. That’s a whole ‘nuther story that I’ll describe in another lesson. But if I have one of those kids that can only do tens, fives and ones, I’ll say “Slice the box into tens, fives and ones” and let them grind through the problem that way.

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...or maybe I’ll say “Slice five rows off the left...”

70

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80

16

...or maybe I’ll say “Slice five rows off the left...”

...or maybe something else.

It all depends on what me and the student feel like doing at the time.

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60

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Is it efficient? Maybe so, maybe not. One thing is sure: this is a step closer to how we add in real life, in several respects:

One, there is a mixture of sizes, which is more common in nature than similar-sized things all lined up in a row.

70

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60

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Second, the child can actually see what (s)he is being asked to add, and that is a very important thing for a kid to have. Pulling numbers out of the air is not as satisfying as adding things you can see.

70

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60

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Okay, so in this case I’ve got five numbers to add up. Should I put the subtotals in a neat column and add these in the formal way?

70

40

60

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Maybe so, but in my experience, kids have such appalling handwriting that they can never get the digits to sit properly in columnar fashion.

We can do better than that.

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60

25 11

6070

2511+4

0

Look for numbers that clip together nicely.

70

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60

25

100

11

70

25

100

11

95

Look for numbers that clip together nicely.

Notice that I haven’t bothered to write the numbers in columns. From here you can jostle the numbers together any way you want.

195+11? Or maybe 100+106? Whatever works for you.

100

95

11

Numbers are written wherever there is space, not necessarily in neat columns.

And finally numbers mean something: in this case, they are squares in a page. In another setting that might have been bricks in a brickyard or Lego blocks on the table. (But watch out for things that kids can pick up and fiddle with because ....well that’s exactly what kids will do).

100

95

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This is free-format addition. Numbers are added in pairs that clip together nicely to produce lots of zeroes. Situations where there is a carryover are avoided or at least minimised.

I’ve spent so much more time on this example than I really wanted to.

Rather, I wanted to highlight the benefits of adding numbers to ten, so that kids can recognise which numbers go together nicely, and be able to put them together without much concentration.

8+2 7+3

6+49+1

5+5

8+2 7+3

6+49+1

5+5

The sooner a kid masters this, the sooner (s)he’ll be able to see that 7+5 is 7+3 with 2 more on top,

...and that 9+6 is 9+1 with 5 more on top,

...and that 20-7 must be 13 because it ends in a 3,

...and that 100-33 must be something that ends in a 7.

8+2 7+3

6+49+1

5+5

You can teach number bonds to kids by all kinds of card games. One game I like is Snap, where the matching pairs are numbers that add to ten (so you get to slam your hand down on the pile when you see a 9 go on top of a 1).

But to begin with, just drop the cards on the floor and ask the kid to pick them up in matching pairs (like 7 and 3). It’s simple, it’s tactile, it’s fun. And best of all, it doesn’t feel like maths.

Now consider what happens when you add long lists of numbers:

1 2 5 7 9 6 4 5 3 2 3 5 3 8 9 8 4 6 7 4 3 1 5 2 8 3 8 6 9 7 3 1 2 4

Would you start at the left and plough through the numbers like weeds in a field? That would be heroic and hard work and really intellectual and all those other things that maths is supposed to be ...and a damned stupid way of doing it.

Maybe instead you could bunch them into tens, like so:

1 2 5 7 9 6 4 5 3 2 3 5 3 8 9 8 4 6 7 4 3 1 5 2 8 3 8 6 9 7 3 1 2 4

10

10

10

1 2 5 7 9 6 4 5 3 2 3 5 3 8 9 8 4 6 7 4 3 1 5 2 8 3 8 6 9 7 3 1 2 4

Maybe instead you could bunch them into tens, like so.

Cross out the pairs as you go.

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

1 2 5 7 9 6 4 5 3 2 3 5 3 8 9 8 4 6 7 4 3 1 5 2 8 3 8 6 9 7 3 1 2 4

...and then combine the little numbers into tens.

10

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

1 2 5 7 9 6 4 5 3 2 3 5 3 8 9 8 4 6 7 4 3 1 5 2 8 3 8 6 9 7 3 1 2 4

It’s a bit like a treasure hunt, isn’t it!

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

1 2 5 7 9 6 4 5 3 2 3 5 3 8 9 8 4 6 7 4 3 1 5 2 8 3 8 6 9 7 3 1 2 4

So now it’s just 16 tens and 3.

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

Now, it won’t always be this convenient. Sometimes you’ll get stuck with three or four biggish digits that won’t go together nicely, like two 9s, a 6 and a 5.

So I’ve come up with an idea that gets around that quite nicely, and all it requires you to do is memorise three pairs of numbers that add to fifteen:

9+6 7+8

10+5

15 15

5 2 6 7 9 6 7 5 3 2 3 8 3 8 9 8 3 6 7 4 3

Watch what happens when I add a long list of numbers by grouping to fifteens instead of tens:

15 15

5 2 6 7 7 5 3 2 3 8 3 8 8 3 7 4 3

15

15

15

Big numbers knock out big numbers.

Eventually all the big numbers are gone.

15 15

5 2 6 5 3 2 3 3 3 4 3

15

15

15

Make some 10s, then.

15 15

5 2 6 5 3 2 3 3 3 4 3

15

15

1510

10

15 15

2 3 2 3 3 3 3

15

15

1510

10

15

...and if it helps, you can bunch repeated numbers into a single number.

15 15

2 2

15

15

1510

10

15

Look! There are two 2s.

15 15

15

15

1510

10

Pair the 15s together to make 30s.

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15

10

10

The rest is easy.

30 30

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4

I’ve created a card game that practises additions to 15, and the kids consistently enjoy playing it, to the point where they share the game with their friends at school.

I don’t have space in here to share the rules of the game, but I’m sure you could come up with some interesting ideas of your own.

The point is to turn maths into something worth exploring, not a chore to be done (like washing the dishes). I hate washing dishes because its the same damn thing every time, and there’s no imagination in it. Sound like the maths you did at school? Remember that the next time you’re hunched over the dish rack.

Fifteens are clearly useful when it comes to adding long lists of numbers. But fifteen’s importance goes beyond that. Along with ten, it serves as a useful stepping stone when adding simple pairs of numbers.

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Like the tens-bonds I described a while ago, pairings like 9+6 and 7+8 help you to jump quickly to the sums of things like 19+6, 17+18 and so on.

They also shed light on the numbers adjacent to them, like 9+5 and 9+6 and 7+7 and 7+9 and so on.

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The point is to use a few memorised sums to shed light on other sums that are similar to them in some sense.

We already do this with 2+2: - knowing 2+2 helps you calculate

12+2.- knowing 2+2 helps you calculate

20+20.

Give a kid a few more ‘lighthouse numbers’ and his/her skill will grow.

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So to finish, I want to summarise what I think about addition:

There’s more than one way to add. Some methods suit big numbers and other methods suit little numbers. Some are best done on paper and some are best done in your head. Learn several.

Memorise a few simple combinations, like numbers that add to ten and numbers that add to fifteen.

Teach addition as the addition of objects rather than the addition of numbers. Kids need to see (or imagine seeing) what they’re adding.

Be prepared to let go the formal, numbers-in-columns methods if a kid has lousy handwriting. Be innovative. Explore. Go back to the roots of arithmetic and rebuild it in your own way. In other words, be a kid when you teach kids.

[END]

dtcoulson@gmail.com