Can YOU Subtract ’12’ From ‘9’ Using ‘Number Bonds’?

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..by Anura Guruge

If so, as the incomparable Kipling said: “You’re a better man than I am, Gunga Din!”.

I guess unlike me, an old man pushing 60, who learned his 3 Rs in the 1950s (in a third world country), you must know all about ‘Number Bonds‘. Just this year, looking at Teischan’s homework I became vaguely aware that they were doing this stuff called ‘number bonds’. Initially all I had seen was the two dangling ball efforts to deconstruct an integer and I had no problems with that since I do think that kids should appreciate how a number comes to be what it is. Now to be fair, ‘number bonds’ is part of the new, ‘new math‘ – the so called ‘Singapore Math‘ (and in case you don’t know, ‘Singapore‘ is a tiny Asian country, really best known for its infamous capture by the damn Japs during WW II, when it, like my home country was a British colony). Singapore Math is being taught all over the place, not just in Alton.

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Then yesterday Deanna showed me this homework that Teischan was struggling with. It took my breath away. I never realized that they were going to use the two dangling balls to do arithmetic operations. So, have a look at this.

They had done the top 2 in class, on the blackboard. She had supposedly transcribed the method and answer from the board onto her sheet. Number 2 was wrong and we asume she copied it wrong.

Though I had never seen subtraction done this way, I could work out how they got ’14’ for #1 and the (correct answer) ’12’ for #2.

Then I noticed that we had a problem! Number 3 and 4 (’17.’ & ’18’ on the sheet) were very different to the other three, and the two they had done in class. Can you spot the difference? Yes, the ‘ones’ number of the second operand is BIGGER than that of the first operand. To use the same technique as for one & two, you would have to use a NEGATIVE NUMBER, in this case ‘-4′ which when added to the ’10’ will give you the right answer ‘6’! But even I, with my high expectations of kids, do not really expect 6 and 7 year olds to be that conversant with negative numbers. IF you don’t use negative numbers, then you have to use a DIFFERENT technique to handle numbers 3 & 4!

I had no idea what that different technique would be. So I did what I always do when I am stuck. I Googled. I found this excellent video tutorial, with exactly the right example, at ‘onlinemathlearning.com‘. Here it is. You have to watch it.

Click to access page. It is the 1st video of the three.

Notice the BIG ‘No!’. I was mortified.

There is an exception to the method. This is for 6 and 7 year olds.

I have two issues with using this strange, two dangling ball approach for teaching kids subtraction.

1/ This method does NOT ELIMINATE the need to do subtraction! Ah? Kids still have to do subtraction with this approach. So what is the gain. I would be all in favor if this method eliminated the need to subtract and said kids could do subtraction by just adding numbers. Now that isn’t as crazy as it may sound to the uninitiated. Logarithms. Now that is real math. We (as kids who didn’t have calculators) used logarithms because ‘logs’ allowed you to do complex multiplication and division using just addition and subtraction. That is neat and useful. You eliminate a complicated process with an easier, better mastered technique. Not so with the two dangling balls. You still have to do the damn operation — in this case subtraction. Plus, how do they teach subtraction. They count the difference between the two numbers. If so, why bother with the two dangling balls. Just count the difference to begin with!

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2/ Having an exception to deal with a common occurrence is beyond unacceptable. The abiding, (to some of us sensually stimulating) beauty of maths is its predictability, its uniformity. You can’t have a so called ‘easy method’ that has exceptions to deal with common occurrences. This is plain crazy.

Yes, I am the first to admit that I am an old fashioned and stuck in my ways. But, I see no problems with the way we learned our arithmetic, algebra, geometry and trigonometry. We had no electronic calculators or even mechanical ones. We learned things by rote and repetition, over and over and over again.

This was the dedication in one of my recent books.

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P.S.: I collect old logarithmic/trigonometric tables (i.e., the  so called ‘log’ books) and old slide rulers. Send me pictures and quote me a price. Yes, every once in awhile, late at night, when I feel that I am due a treat, and have a few dollars stashed away, I log onto eBay and see what they have. Got a real beauty of a slide rule, cheap, very cheap, a couple of months ago — making use of the eBay, ‘make an offer’ feature.