In the last few sessions in this series we’ve looked at operators that do things like “add 2” or “multiply by 3” or “divide by 10”. Each of these operators takes a number as its input and produces a number for its output. Because of this, these operators can be chained together, as train cars in a train, by connecting the output of one to the input of the next one in the chain. Though these kinds of operators (like “add 2”) are closely related to the arithmetic students learn in elementary school, they aren’t just arithmetic. The thing that’s critical to “add 2”, for me, is that it does not impel the student to find the result of an addition now. Whereas “4+2” looks to almost all students like a command to perform an addition, in which the job isn’t done until you’ve got an answer – by which is almost always meant a number – the “add 2” operator is really just sitting there, waiting patiently for some number to be put on its input. The “add 2” operator is a device, a machine, a service – ready to be invoked; but even if not invoked right now, it can be picked up and held to the light and examined. And when the operator “add 2” is examined, it is clear that it is not at all the same kind of thing as a command to the student to perform an addition now. It is possible to compare two different machines like this, and judge whether they behave the same, or how they behave differently, without ever being confused in thinking you’re doing arithmetic. In comparing machines, you are reasoning about arithmetic. You’re doing algebra. And you’re doing algebra even if you don’t realize you’re doing algebra; thinking about the “add 2” operator is very closely linked to thinking about x+2, and yet it can be introduced and made discoverable to students without any of the formalism and the mystery that surrounds the introduction to algebra when done the traditional way. You might want to look back at the prior post, and imagine the input being “x”, and interpreting what is going on in terms of traditional algebraic notation. To do so, you need to introduce parentheses, order of operations, associative property, distributive property, and – of course – variables. Learning the accepted language of mathematics and formulas is obviously important, but the transition from arithmetic to algebra is notoriously problematic for large groups of middle schoolers – and I have suggested in other places that a key factor of this difficulty is not in the role of “x” but in the new role of “+” and “=” etc.
The operators – “add 2”, “multiply by 3” – all do something, or rather they sit there, ready to do something. And yet, a special case of doing something is doing nothing, and another special case of doing something is undoing something. If you look at the following train of operators, you may notice that the third operators undoes what the second one does, and all three together do the same thing as just the first:
The train of operators a, b, and c has the same effect as just the single operator a. The effect of operator b was entirely undone by operator c. This idea, too, turns out to be very simple for seventh grades to grasp and hold on to. This generation of seventh graders is used to computer applications like Microsoft Word, or Gimp, or any number of other programs in which you can do things in the confidence that these things can later be undone. You can even undo a delete! If you think about what this means, it is pretty clear that delete does more than just throw things away, it must squirrel it away somewhere from which it can later be retrieved, up to a point anyway.
We’ll look in more detail at pairs of operators that undo each other later. Right now, I want to focus on the “do nothing” aspect: if you take two operators that undo each other, and you look for a machine that behaves exactly the same, you get an “empty” machine. The bottom part of the picture above shows this. Boxes b and c together behave just like the empty box in e or the entirely missing box in f. What comes in goes right back out, without modification. The effect of box e is entirely neutral. Nothing happens, other than that the input is connected through to the output. It’s passed on. It’s let through. It’s as neutral as can be. The fancy name for this is the identity operator: what comes out is identical to what went in.
The identity operator is neither an addition nor a multiplication, but you can make additions that are equivalent, as well as multiplications that are equivalent to the identity operator – in other words, they have the net effect of doing nothing.
In the figure above, the identity operator is shown in box a, and the other boxes are intended to be equivalent to it. The question marks stand for particular numbers. What is the number to replace the question mark in box b so that the next effect of box b is to do nothing? Similarly, in box d, what number should we replace the question mark with so that the net effect of box d is to do nothing?
For my seventh graders, this question wasn’t an easy one. Many of them said that the question mark should be zero for all four boxes. They assumed that “+ 0”, “- 0”, “× 0” and “÷ 0” would all leave the number unchanged. When challenged by others in the class, most of these changed their answer for “× 0” to “× 1”. Fewer students knew what to do about the division one, box e. Some of the students figured it out by trying out different numbers for the question mark, and using their calculators to see what it did to a number coming in. Zero has a special role in add and subtract, 1 has a similarly special role in multiplication and division.
We’ll look at pairs of operators that undo each other in the next post.