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- 1. Teaching addition to kids. David Coulson, 2013

2. Add these two numbers: 47 69 3. 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? 4. There are all kinds of approaches to adding thesenumbers, and Ill bet that if I asked six people47how they did it, Id get six different answers.69 5. There are all kinds of approaches to adding thesenumbers, and Ill bet that if I asked six people 47how they did it, Id 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. 6. 1Theres nothing wrong with the school wayof adding numbers. In fact the school way is 47highly efficient. Its also a very compactmethod that achieves the answer with the + 69fewest number of internal steps and requires6the least amount of paper.Its such a powerful method that we can use it to add dozens of multi-digitnumbers so long as theyre written nicely in columns. Thats 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. 7. So why dont grown-ups ever use the method 47for simple calculations like this? 69 8. The thing is that few of us have paper and penready when confronted with a calculation. 47Thats one limitation, but I think theres moreto it than that. Even sitting in front of your69laptop, 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 didnt! Itsjust 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. 9. 1Its all to do with the subtotals that are requiredby the formal method. You have to add the 947and 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 a6while 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. 10. Now Im not going to recommend one methodof doing so-called mental arithmetic over 47another because theyre all good. Instead, Iwant to draw your attention to the fact that we 69are failing our kids at school if we dont 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. 11. 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 its 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. 12. Why should there be creativity in a mathslesson? Excuse me? Why shouldnt there be47creativity in a maths lesson! Turning the taskinto a journey where the kids themselves decide69which 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 (Ivefound), if you stay on really friendly terms with your students, then nomatter whether its 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. 13. So welcome to the notion of turning a mathsexercise into a decision-making process, and 47seeing which decisions work well and whichones dont.69 14. Heres 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. 15. 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: 16. Now I want to make it very clear that, even as Im 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 havent taken thetime to count how many squares there are inside this shape (Im a man; Icant add and draw at the same time, even with a university degree). 17. But I dont need to know what the answer is because Im confident I canwork it out. And by showing the student that I trust myself to work itout, Im showing him/her that its easy. But theres 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 cant stop themselves. 18. 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? 19. But how should (s)he do it? This way? 20. But how should (s)he do it?Or maybethis way? 21. But how should (s)he do it? Or maybe this way? 22. Its not simply about minimising the numbers of boxes. Rather, its 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. 23. A quick inspection shows me that there are ten lines over on the right.Ten is an easy number, so I think Ill 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 24. Ahah! I think I just roused your curiosity. And thats 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. Itsthat they have taken control of the problem and are making decisions aboutit. 70 25. 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.Were already halfway there. 7040 26. 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. Thats awhole nuther story that Ill describe in another lesson. But if I have one ofthose kids that can only do tens, fives and ones, Ill say Slice the box intotens, fives and ones and let them grind through the problem that way.6070 40 306 27. ...or maybe Ill say Slice five rows off the left... 80 70 4016 28. ...or maybe Ill 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. 6070 4025 11 29. 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 704025 11 30. 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 704025 11 31. Okay, so in this case Ive got five numbers to add up. Should I put thesubtotals in a neat column and add these in the formal way?607040 2511 32. 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 70402511 33. Look for numbers that clip together nicely.10060 704025 11 34. Look for numbers that clip together nicely.10095 7025 11 35. Notice that I havent 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 36. 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 that95might 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 thats exactlywhat kids will do). 11 37. Ive 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 tog