Subtraction before addition

Why Not? is an interesting book by Ian Ayres and Barry Nalebuff that came out a couple of years ago. It suggests some simple things people might do to be more creative. One of the most interesting suggestions is to turn things on their head. For instance, it is commonplace to see tomato sauce bottles now with the lid at the bottom. Actually, all that has been done is to flip the label. Most of us were putting these bottles upside down for years.

The message of that story is to claim that sometimes thinking about doing things in exactly the opposite way to what you do now may lead to improvements. Of course, George in Seinfeld got this idea first.

A few years ago, while trying to teach my then 3 year-old daughter some mathematics, I hopped on a similar Why Not? moment in reversal. She had always been pretty quick with numbers and could easily count to 20 and beyond. So I set about teaching her how to add. Now the way to do this is to make liberal use of fingers. This is all very well when the numbers you are adding are less than 10 but is a problem after that. Then it requires breaking up the problem; a conceptual advance that is pretty daunting for a child. Of course, you could move on to toes as my son is want to do but that involves taking off shoes.

So I thought, stuff this! How about we start with subtraction first? The idea there was that you could take any number up to ten and subtract any smaller number and still not exhaust your finger options. That dramatically opened up the possibilities and, what is more, I noticed how damn easy it was for a child. They were used to having and then not having and seeing what was left. It was a natural part of their day. When they eat, the food on the plate shrinks. When they paint, the clear bits of the canvas get smaller. Subtraction was a far more natural part of everyday life for children.

What is more, within the same session, I was able to easily move on to subtracting from numbers greater than 10. We didn't even need fingers.

I was very surprised at the results so I tried them on any child, 3 to 6, I could find and later on my son. The same thing occurred. Children had no idea how to add, could easily subtract. What is more, once they understood that comfortably, it was easier to reverse the whole thing and shown them what addition meant. Now I have nothing but my own anecdotal experience here, but it seemed to me that the speed of numeric learning was dramatically improved.

Interestingly, we, of course, have the same 'growth' order in multiplication and division. We teach the thing that requires bigger numbers first (how to multiply) before getting on to the smaller numbers one (how to divide). This is where most difficulty with learning mathematics becomes an issue. Most people have an easy time with addition and subtraction but other operations are far more difficult. The same goes for children. It takes years to really learn these even after the rote learning of times tables (and not division tables!) takes on.

So a few weeks after teaching my 3 year old daughter how to subtract and then add, I decided to move on to divide. Talk about easy! Take a group of 20 objects. If we divide it into 4 groups with the same number of objects in each, how many objects are there? No problem, 5. Do this with even numbers less than 10 and you can illustrate division by two using hands.

Now I am not claiming that she could divide in her head but she could work it out. Division is an even more natural part of a child's life than subtraction. They do it all the time when they learn social behaviour. It is all there and worked out. Not surprisingly, it is also something that they have an abstract understanding of too.

Sadly, learning division did not translate speedily into multiplication. That turns out to be a hard concept. 4 years on and we have got it but not at the same depth as other operations.

The problem with all of this is it is dramatically different from the way these things are taught at school. Students are barely allowed to try division until they have 'got' multiplication. Now maybe a proper scientific study will bear out that that order is the best way for most children. But, for the moment, I am not sure. Any teacher I have talked to about this looks at me blankly and is more willing to attribute this to the skill of my children (which is fine) than an overall problem with the way we do things.

Anyhow, there is no harm in trying this out for yourself. Try subtraction before addition and division before multiplication. I would be really interested in your observations. One day if I dump the whole economics business I may set out to examine this more scientifically. For the moment, my casual observation will have to do.