In the blog post Learning and Unlearning Math I suggested that it is natural for kids to assume that “+” in “5 + 7” is a call for action, namely to produce the one number that is the sum of 5 and 7. I also suggested that later in school kids see a formula like “t + 1” and need to come to grips with the fact that in “t + 1” nobody expects them to produce a number, instead something else is called for, and part of the challenge the kid faces is figuring out what is wanted instead.
Imagine a game where two kids each have two dice. In each round, Jana throws both her dice and Joey throws his, and the kids look at all four dice. The round is won by the kid with the highest total of points on their dice. So if Jana throws a 3 and a 2, and Joey throws a 3 and a 1, then Jana wins the round (her total is 5 whereas Joey’s total is 4). If you observe kids playing this game, you find out quickly that the kids often don’t bother to total up the numbers, but take shortcuts in deciding who won the round. In the example we just gave, both Jana and Joey may decide that because each threw a common 3, it’s the other two dice that are decisive: Jane wins because of her 2, which beats Joey’s 1. Even young kids will come up with these kinds of strategies on their own, as a natural shortcut.
If Jana doesn’t add 3 + 2 when comparing 3 + 2 with 3 + 1 but decides in some other way which total is bigger, then Jana has created some wiggle room for herself with respect to the notion that “+” means that you have to stop what you’re doing and produce a number. Maybe what allowed for the wiggle room is that Jana’s goal isn’t to secure a grade or to please a teacher, but simply to agree with Joey who won the round. The problem “3 + 2” gets its entire meaning inside of the larger problem of who won the round. If a kid sees “3 + 2” as merely a step along the way to get somewhere, other roads of getting there may open up naturally.
The phenomenon of holding off on doing a computation until it has become clear to you in which context the result is needed (or not needed, as the case may be) is a very interesting one that we will look at in other situations. The name I’ll give to this phenomenon is deferred computation.
The more complicated or tricky a computation is, the more benefit there can be in delaying the calculation till the last minute. Sometimes, deferring a computation simply gives you more options, and one of those options may be easier or save you time. For example, if I ride 2 miles in 6 minutes, and I want to know my speed, I would divide 2 by 6 to get .33333 miles per minute. If I then want to convert that speed to miles per hour, I would multiply by 60. If I had held off on doing the division of 2 by 6, I could have done the multiplication first, changing 2 / 6 * 60 to 2 * 60 / 6. Alternatively, I could have divided 60 by 6 and multiplied the result (10) by 2. Or, knowing I am going to have to multiply by 60, I leave the 2/6 as a fraction rather than calculating it as a decimal.
In a sequence of additions and subtractions you can often avoid getting into negative numbers by doing the additions and subtractions out of order (much like you can postpone getting a negative bank balance by postponing when you pay each bill.)
So 9 – 12 + 16 – 7 can be done in different ways:
9 – 12 = -3; -3 + 16 = 13; 13 – 7 = 6: final result 6
9 + 16 = 25; 25 – 12 = 13; 13 – 7 = 6: final result 6
16 – 12 = 4; 9 + 4 = 13; 13 – 7 = 6: final result 6
9 + 16 = 25; 12 + 7 = 19; 25 – 19 = 6: final result 6
A child who changes the other of additions and subtractions from the straight left-to-right one is a child who has started to think of adding and subtracting as things you can pick up and hold close to the light, and has stopped thinking of them exclusively as “do this now” commands – in other words, a child who has started to think algebraically.