# The upside down subtraction bug

There are some issues with Maths that pop up across all agegroups, and the upside down subtraction bug is one of them. If you now have a mental picture of some sort of 6-legged brightly coloured ladybird creature hanging under a branch, then that’s not quite what I mean. I’m talking about the commonest error people make in traditional “Column Subtraction”.

The one where they say that 573 – 254 = 321.

It goes like this:     500 – 200 = 300, then 70 – 50 = 20, then lastly 3 – 4 = 1.

Most kids, when you point out their error, say “Oh yes”, and try again, doing the borrowing correctly and getting the correct answer of 319. But I just have a sinking feeling that the bug will reappear the very next time they subtract… nothing is solved. And I hate not solving a problem….

Working as a maths tutor in a primary school gives me a new perspective on how maths is learned and taught. The language that is used is pretty consistent – the teachers all sing pretty much from the same hymnsheet, and a lot of the kids make really good progress. And this is how this sum “should” be done….

3 – 4 YOU CAN’T so go next door. 7 becomes 6 and 3 becomes 13. 13-4 is…. 13, 12, 11, 10, 9. NINE. 6-5=1. 5-2=3. The answer is 319.

This makes me uncomfortable, it’s simply not true that 3 – 4 is impossible, and it set me wondering if kids’ later dislike of negative numbers partly stems from this rather strange approach to subtraction. What if the poor things really believe that you can’t do 3-4? I remember being 6 and if an adult told me I couldn’t do something, then it was TRUE.

So that’s one reason that traditional subtraction bothers me, but the other reason is that it bears no relationship with the mental method that is taught in primary, the whole concept of counting on. Using this method, 573-254 becomes:

+6                 +40              +100            +100          +70              +3

254 —–> 260 ——> 300 ——> 400 —–> 500 —-> 570 ——> 573

the answer is 100+100+70+40+6+3 = 219  and that is actually VERY difficult. So it’s only really suitable if the 2 numbers are a small number of steps apart.

I like written methods that flow seamlessly into mental methods, because kids feel SOO good if they can do maths mentally, but a lot of them do need to jot as a first step. So the other day, I tried out this with Year 6:

573-254 goes like this:

500 – 200 = 300

70 – 50 = 20

3 – 4 = -1

THEN

300 + 20 – 1 = 319

Several of them experienced genuine panic and distress – why were we working left to right? Surely this was wrong? 3-4 I REALLY can’t. They also believed that they needed to learn a whole load of new subtraction facts like 4-9=-5. It took a while for some of them to learn to cheat (ie do 9-4=5, so 4-9=-5) and yet cheating is pretty much the upside down subtraction bug, used properly…

What was surprising, after I had done this with 4 groups over the morning, was that the kids who had the biggest panics were the ones who ended up fastest and most confident with the method.

Like so many alternative methods, I don’t expect it will suit them all… but it opened their minds a bit, and empowered one or two of them to do some wonderful, fast, mental work.