Reaction to Subtraction
If you are of a certain age, you’ll remember when you learned a rule about how to do subtraction. It was called “Borrow and Payback.” It was a neat little trick to get the right answer to those tricky take away sums. For example, if you were given a random sum (as you often were) 43 – 17, you would start by trying 3 – 7 and realising that this was impossible so you remembered the rule. Stick a one in front of the 3 to make 13 and add 1 to the 1 of 17 to make 2. (See below)
After that, you could get back to normal and find the answer. I assume, no one ever explained why this was the case, but it got you the right answer, which, at that time was more important that understanding how you got the right answer.
If you are younger, you may have come across the concept that is nowadays used throughout most of the English speaking world – the concept of renaming. For the same sum above, one builds upon their knowledge of renaming numbers to find out he answer. We know that 43 is a number containing 4 tens and 3 units. By renaming, we know that this is the same as 3 tens and 13 units. Therefore, we don’t need any tricks. Renaming 43 to “Thirty Thirteen”, we can do the subtraction easily enough.
Both methods have their logic. One can explain the “Borrow and Payback Method” mathematically. In this method you are adding 10 to both numbers in the sum. If I write it a little differently, it might be clearer:
- Start off with the sum: 43 – 17 = ?
- Add ten to both numbers: 43 + 10 is written as 4(13) and 17 + 10 is written as (1 + 1)7
In a way, we have sort of renamed both 43 and 17. However, since this is hard to explain, if one simply learns the rule off by heart, they have a fail proof method for subtracting numbers. The trouble of having to learn things off by heart is that if you rely on this, very soon in the world of maths, you’re going to have a lot of stuff to remember, rather than building on things that you already know.
I’m often challenged on this view and the two main arguments I get are as follows:
- “Weaker” students find the “Borrow and payback” method easier so for their sake, we should let them get on with it.
- That’s all well and good for two digit numbers, what about 1,000,000,000 take away 1?
Starting with the first argument, ignoring that I find it a little disrespectful, I cannot buy an argument that seems to be: “if something is easier to do, it’s better” I feel it’s a very short term view. As subtraction is a very basic concept, if we’re already finding shortcuts by learning a series of rules off by heart, in the long term, these pupils are going to get more and more confused. For example, when they get to long division, they’ll need to learn another rule off by heart, namely: divide, multiply, subtract, bring down. That’s two things to learn off with no obvious logic. Then let’s not forget formulae like area of shapes, calculating speed, distance and time and learning off ways to compute interest rates before they finish primary school. That’s a lot of stuff to learn by rote and it’s only one subject and we’re not even in secondary school yet!
The second argument puzzles me because if we are teaching subtraction properly, it shouldn’t be an issue. I suspect teachers are still using the term “borrow” in their teaching. I’m pretty sure teachers in Ireland still might teach our example above using the following narrative:
43-17. 3 take away 7, you cannot do so you borrow a ten. That leaves 3 tens and 13 units…. and so on.
Using this method, if we try 100 – 8, the teacher might say:
0 take away 8, you cannot do so you borrow a ten… oh, there are no tens so we need to go to the hundreds and borrow there. That leaves zero hundreds and 10 tens and 8 units. Now let’s go again, we borrow a ten and that leaves 9 tens and 18 units.
No wonder the children get confused as the digits expand. Try doing a million take away one using this language! Why not simply rename the number. So:
100 – 8. Zero take away 8 we can’t do so we need to rename the number. 100 is the same as 9 tens and 10 units…
and so it goes on. Picking a bigger number, e.g. 2,002,010 – 123,456, we can rename easily enough if you take it in logical steps.
I think if we spent more time renaming numbers before tackling operations, we’d be doing ourselves a big favour. I also think we need to ban the word “borrow” from subtraction. Even the word “swap” is better. While the above example might seem difficult initially, if you follow simple Place Value rules as you go through the problem, it should be easy enough to deduce where you’re going. At least, there is logic to what’s going on!
If one insists on defending the Borrow and Payback method, at the very least, the teacher must explain what is happening. Using our big number above 2,002,010 and taking 123,456 away from it, the Borrow and Payback can be done quickly but I find it hard to explain why I keep adding one to the bottom digit each time. The first time I’m adding 10 to both numbers, the second time I’m adding 100 to both numbers, the third time 1,000 and so on. Yes, it works but even with an explanation of what I’m doing, it feels a little odd.
Subtraction seems to bring out heavy emotions in the teaching world. It seems to be the first hurdle where students falter in primary school maths. I have witnessed teachers shouting at each other defending their own points of view. There’s suggestions that we should teach both methods to children – a sort of compromise between the old and the new but I’d argue that’s like a doctor today sprinkling vinegar over a sick patient as well as giving him modern drug treatment because back in the day, sprinkling vinegar often worked though they had no idea why! I guess my final point about this is to ask the “borrowers” to teach me their method. I promise I’ll ask “why” throughout and you’d better have a decent answer!