/ 6 July 2011

Don’t be fooled by tricks

Throughout my columns, I make the point that children need to experience mathematics as a meaningful, sense-making, problem-solving activity. When people experience mathematics in this way, they feel empowered and capable.

By contrast, when mathematics is experienced as the memorisation of facts, rules, formulas and procedures needed to determine the answers to questions, then the subject becomes tough. When mathematics is reduced to rules, formulas and procedures without understanding, it becomes the domain of the magician. In this column I want to look at some of the magical mathematical tricks that children are exposed to.

Go to the internet (YouTube in particular) and search for Chinese multiplication, Russian Peasant multiplication and grid multiplication. There are more words you can search for, but this is enough to get you started. What you will find are numerous videos of “tricks” that you can use to do arithmetic. What you will find is magic.

The magic works. You can use the magic to get the right answer. Look at the following example I found among the YouTube video clips. One presenter was explaining a method for multiplying by 11.

Check. The answer is correct. The “method” works.
Let us try another sum: 57 X 11:

  • Write the 5 and the 7 of 57 with a space between them: 5 7
  • Add the 5 and the 7 of 57: 5 + 7 = 12 and write the answer into the space: 5 12 7

But wait, 57 X 11 is not equal to 5127.

If you watch the video again, you will learn that we can only write one number in the “space” – in this case the 2 of the 12 and what we should do is add the 1 of the 12 on to the 5 and so the answer becomes:
(5 + 1) 2 7 6 2 7.

A method that, at first, looks easy becomes more complicated very quickly.

So what if we want to multiply 245 by 11? Does the method work now? Not as it is. But the method can be adapted:

  • Write the 2 and the 5 of 245 with two spaces between them: 2 5
  • Add the 4 and the 5 of 245 and write the answer into the right-hand space: 2 9 5
  • Add the 2 and the 4 of 245 and write the answer into the left hand space: 2 6 9 5

Now try 478 X 11 yourself. What adaptations are needed for the method to work? What about 1 232 X 11? Can we use the method for 25 X 111?

The problem I have is not so much that the method cannot be adapted to three, four- and five-digit numbers multiplied by 11. The problem is that the method works only when multiplying by 11.

There is another method, involving fingers, that can be used for multiplying single-digit numbers by 9 — this one cannot, however, be adapted for multiplying two, three and four-digit numbers by 9. Imagine having to remember a different trick for each multiplication sum you want to perform. If that is what it means to do mathematics then of course mathematics becomes tough.

The other problem I have is that these sorts of trick are finding their way into school textbooks.

I find this problematic because it reduces mathematics to the memorisation of tricks and their related conditions without any attempt to make sense of what is happening.

Much more subtle than this is the very notion so much of the mathematics that children learn is, in fact, experienced as tricks. To the mathematician, doing column addition makes sense; there is a good understanding of place value, of how addition works and of what is actually being done. To children who are presented with the “column method” without any attempt to justify it and/or to understand it, the method is as much a trick as the method used to multiply by 11 presented above is a trick to you. That “it works” is not a good enough reason for teaching it without understanding.

That leaves me with one question for this column: Can the multiplication by 11 trick be understood? Can we explain why it works? Of course we can. Think about what it means to multiply 25 by 11:
25 X 11 = 25 X (10 + 1)
= 25 X 10 + 25 X 1
= (20 + 5) X 10 + (20 + 5) X 1
= 20 X 10 + 5 X 10 + 20 X 1 + 5 X 1
= 200 + 50 + 20 +5
= 2 hundreds + 5 tens + 2 tens + 5
= 2 hundreds + (5 + 2) tens + 5
= 2 hundreds + 7 tens + 5
= 2 7 5

I am not suggesting that we need nine lines to multiply 15 by 11.

We don’t — in simple mental arithmetic terms: 25 X 11 = 25 X 10 + 25 = 250 + 25 = 275!

The point I want to make is that the trick is hardly spectacular; it is simply doing what we would do anyway but hiding the thinking. That is the nature of magic — hiding what you are actually doing to create an illusion — an illusion that leaves the audience amazed.

When we teach mathematics, do we want to amaze the audience or do we want them to know what they are doing?

Aarnout Brombacher is a private maths consultant. For more information go to www.brombacher.co.za