47 69Now 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?
There are all kinds of approaches to adding thesenumbers, and I’ll bet that if I asked six people 47how they did it, I’d get six different answers. 69
There are all kinds of approaches to adding thesenumbers, and I’ll bet that if I asked six people 47how they did it, I’d get six different answers. 69The interesting thing would be that none of those six people would havedone it the way they were taught at school.
1There’s nothing wrong with the school wayof adding numbers. In fact the school way is 47highly efficient. It’s also a very compactmethod that achieves the answer with the + 69fewest number of internal steps and requires 6the least amount of paper.It’s such a powerful method that we can use it to add dozens of multi-digitnumbers so long as they’re written nicely in columns. That’s how theaccountants and the scientists of the pre-computer age did it. And becausethey had to do it a lot, they stylised it into this short method and shared itwith kids in school.
So why don’t grown-ups ever use the method 47for simple calculations like this? 69
The thing is that few of us have paper and penready when confronted with a calculation. 47That’s one limitation, but I think there’s moreto it than that. Even sitting in front of your 69laptop, looking at this pair of numbers, youcould have wandered off to get a piece ofpaper and a pen, or maybe you could haveactivated the calculator app that comes withyour operating system. But you didn’t! It’sjust toooooo much bother, right?Instead, we scratch our heads a little and put the numbers together in someother way that is certainly less efficient than the school method but whichsomehow suits you better because nothing has to be written down.
1It’s all to do with the subtotals that are requiredby the formal method. You have to add the 9 47and the 7 first to get 16, then tear the 1 off the16 and put it on top of the 4 while putting the 6 + 69down at the bottom and leaving it there for a 6while while we add a 1 and a 4 and a 6 to get a... (pause for breath) what was it that I wasdoing just now?All this doodling around requires a brain that works like a piece ofpaper, and few of us have that kind of brain. We want a method that piles allthe stuff together in one place and then shaves off bits as we go, or adds onbits as we go, so that at any one time there is only one thing for us to lookat.
Now I’m not going to recommend one methodof doing so-called mental arithmetic over 47another because they’re all good. Instead, Iwant to draw your attention to the fact that we 69are failing our kids at school if we don’t letthem know that these ‘mid-air’ methods exist.--YES, show them the formal method because they will certainly need thatwhen they are faced with super-big additions with super-big numbers. Butalso show them the alternatives for simpler additions because the worldabounds in simpler additions. Otherwise we are like gardeners who useonly the biggest tools to do even the simplest pruning jobs, like cutting arose-bud with a combine harvester.
Looking at these numbers we could say, forexample, that 40 and 60 make 100 and that 7 47more makes 107 and 9 more than that makes116. 69...or maybe we could say that 47 and 69 is oneless than 47 and 70, or three less than 50 and69, 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 goodexcept carving 47 lines on the tabletop and carving 69 more and thenadding up all the lines one by one. That shows as much creativity as...as...well, no creativity at all.
Why should there be creativity in a mathslesson? Excuse me? Why shouldn’t there be 47creativity in a maths lesson! Turning the taskinto a journey where the kids themselves decide 69which way to proceed is probably going toappeal to them a bit more than simply rehearsing the same old method amillion 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 howsmart we are too, and make it into a friendly contest. Amazingly (I’vefound), if you stay on really friendly terms with your students, then nomatter whether it’s your turn to show off or their turn to show off, theywant to come round to your side of the table and show you what theythink you should do.
So welcome to the notion of turning a mathsexercise into a decision-making process, and 47seeing which decisions work well and whichones don’t. 69
Here’s an exercise I do with kids once in a while. I pull out a piece ofgraph 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.
By rectilinear I mean that, as I draw the shape, my pencil moveshorizontally or vertically along the blue lines, and comes back to where Istarted.I might get a shape that looks like this:
Now I want to make it very clear that, even as I’m drawing the shape, Ihave no idea what the shape is going to look like, and no idea how manysquares are going to be enclosed in the shape. In fact, even as I type thesewords into my laptop as I create this presentation, I still haven’t taken thetime to count how many squares there are inside this shape (I’m a man; Ican’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 canwork it out. And by showing the student that I trust myself to work itout, I’m showing him/her that it’s easy. But there’s also something morephysiological happening when we do this: We are emanating the bodylanguage of curiosity, whatever that is. The kids pick up the scent andcopy it. They can’t stop themselves.
With any luck the student will spot for him/herself that the region is madeup of rectangles, and that the total number can be found efficiently andpainlessly by separating these rectangles.But how should (s)he do it?
But how should (s)he do it? Or maybe this way?
But how should (s)he do it? Or maybe this way?
It’s not simply about minimising the numbers of boxes. Rather, it’s aboutreleasing as many tens-like numbers as possible from the choices that areon offer.By tens-like, I mean numbers that have zeroes in them.
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, becauseif I got you emotionally involved in the task then I should be able to get akid emotionally involved in the task too.Exactly how (s)proceeds through the problem is not terribly important. It’sthat they have taken control of the problem and are making decisions aboutit. 70
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 cancount through the lines, saying ‚5, 10, 15, 20, 25, 30, 35, 40‛ as (s)he goesbecause 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 decidesto do that depends on how good (s)he is with the times table. That’s awhole ‘nuther story that I’ll describe in another lesson. But if I have one ofthose kids that can only do tens, fives and ones, I’ll say ‚Slice the box intotens, fives and ones‛ and let them grind through the problem that way. 60 70 40 30 6
...or maybe I’ll say ‚Slice five rows off the left...‛ 80 70 40 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. 60 70 40 25 11
Is it efficient? Maybe so, maybe not. One thing is sure: this is a step closerto how we add in real life, in several respects:One, there is a mixture of sizes, which is more common in nature thansimilar-sized things all lined up in a row. 60 70 40 25 11
Second, the child can actually see what (s)he is being asked to add, and thatis a very important thing for a kid to have. Pulling numbers out of the air isnot as satisfying as adding things you can see. 60 70 40 25 11
Okay, so in this case I’ve got five numbers to add up. Should I put thesubtotals in a neat column and add these in the formal way? 60 70 40 25 11
Maybe so, but in my experience, kids have such appalling handwritingthat they can never get the digits to sit properly in columnar fashion.We can do better than that. 60 70 40 25 11
Look for numbers that clip together nicely. 100 60 70 40 25 11
Look for numbers that clip together nicely. 100 95 70 25 11
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
This is free-format addition. Numbers are added in pairs that clip togethernicely to produce lots of zeroes. Situations where there is a carryover areavoided or at least minimised.Numbers are written wherever there is space, notnecessarily in neat columns. 100And finally numbers mean something: in thiscase, they are squares in a page. In another setting that 95might have been bricks in a brickyard or Lego blockson the table. (But watch out for things that kids canpick up and fiddle with because ....well that’s exactlywhat kids will do). 11
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 toput them together without much concentration. 5+5
The sooner a kid masters this, the sooner (s)he’ll be able to see that 7+5 is7+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. 5+5
You can teach number bonds to kids by all kinds of card games. One gameI like is Snap, where the matching pairs are numbers that add to ten (soyou get to slam your hand down on the pile when you see a 9 go on top ofa 1).But to begin with, just drop the cards on the floor and ask the kid to pickthem 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. 5+5
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 4Would you start at the left and plough through the numbers like weeds ina field? That would be heroic and hard work and really intellectual and allthose other things that maths is supposed to be ...and a damned stupid wayof doing it.
Maybe instead you could bunch them into tens, like so: 10 101 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
Maybe instead you could bunch them into tens, like so.Cross out the pairs as you go.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 2 410 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 withthree 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 itrequires you to do is memorise three pairs of numbers that add to fifteen: 9+6 7+8 10+5
Watch what happens when I add a long list of numbersby grouping to fifteens instead of tens: 5 2 6 7 9 6 7 5 3 2 3 8 3 8 9 8 3 6 7 4 3 15 15
Big numbers knock out big numbers. 15 15 15 5 2 6 7 7 5 3 2 3 8 3 8 8 3 7 4 3 15 15
Eventually all the big numbers are gone. 15 15 15 5 2 6 5 3 2 3 3 3 4 3 15 15
I’ve created a card game that practises additions to 15, and the kidsconsistently enjoy playing it, to the point where they share the gamewith their friends at school.I don’t have space in here to share the rules of the game, but I’m sureyou could come up with some interesting ideas of your own.The point is to turn maths into something worth exploring, not a choreto be done (like washing the dishes). I hate washing dishes because itsthe same damn thing every time, and there’s no imagination in it.Sound like the maths you did at school? Remember that the next timeyou’re hunched over the dish rack.
Fifteens are clearly useful when it comes to adding long lists ofnumbers. But fifteen’s importance goes beyond that. Along with ten, itserves as a useful stepping stone when adding simple pairs ofnumbers. 10 15 20
Like the tens-bonds I described a while ago, pairings like 9+6 and 7+8help you to jump quickly to the sums of things like 19+6, 17+18 and soon.They also shed light on the numbers adjacent to them, like 9+5 and 9+6and 7+7 and 7+9 and so on. 10 15 20
The point is to use a few memorised sums to shed light on other sumsthat 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. 10 15 20
So to finish, I want to summarise what I think about addition: There’s more than one way to add. Some methods suit big numbersand other methods suit little numbers. Some are best done on paper andsome are best done in your head. Learn several. Memorise a few simple combinations, like numbers that add to tenand numbers that add to fifteen. Teach addition as the addition of objects rather than the addition ofnumbers. Kids need to see (or imagine seeing) what they’re adding. Be prepared to let go the formal, numbers-in-columns methods if akid has lousy handwriting. Be innovative. Explore. Go back to the rootsof arithmetic and rebuild it in your own way. In other words, be a kidwhen you teach kids.