In this series, we’ve been playing with the idea of operators as some kind of action that alters the state of some device or some system, perhaps invoked at the press of a key or a button. We’ve emphasized that it is useful to think of operators as not necessarily operating on numbers. For example, the window on the screen in which you read this blog post has a “minimize” button that invokes a “minimize” operator, which alters the state (and the shape and size) of the window; this operator does more than just the obvious change of the size of the window, it also squirrels away the old size and position somewhere, so that it can be restored to its original size and position later.
All the same, we can get a lot of use from looking at operators that operate on single numbers. In the last post we looked in some detail at what happens if you press the “2” key on a calculator when entering a big number. Now we are going to turn to some operators based on the familiar arithmetic operations of add/subtract/multiply/divide, and see if we can shed some new and interesting light on those familiar operations.
Yet there is something that is useful to do first, and that is to picture the relationship between the state, the operator, and the button that invokes the operator. This is done in the figure above. A button is pictured in red, and when it is pushed, the operator is invoked. The operator takes the current state and produces a new state. In the example we used in the previous post, the state is the number in the display of the calculator, the button is the “2” key, and when the current state is 33, pressing the “2” will invoke an operator that changes the state to “332”.
The whole thing pictured above is called a “state machine”, and many devices can be modeled or pictured as a state machine. Of course, most will have multiple buttons invoking multiple operators. Nor does the state of a state machine have to be a single number. In fact, a typical 4-function calculator has a state that keeps at least two numbers, only one of which is shown in the display. For example, if you enter “2+5=”, the display will show, in turn, 2, 5 and 7. At the time that it shows “5”, it clearly hasn’t forgotten about the number two, nor has it forgotten that the “+” has been pressed. How it remembers this, in what form it remembers this, we can’t really tell without knowing more about the way a particular calculator has been designed.
With this picture, we can now focus on the operator part of it, ignoring the state part:
So how can we fashion some operators of this type from the familiar arithmetic operations? Here’s a simple way:
This box takes the number that comes in, and adds 2 to it, and that gives us the number that comes out. So, if “7” goes in, then “9” comes out. If “3” goes in, “5” comes out. Pretty simple, right? I’ve got lots of anecdotal evidence that seventh graders think this is very simple.
These same seventh graders think it pretty obvious what happens when you string two such operators together, as in the picture above, on the left. They can trace that when “5′ goes in, a “7” goes out of the top box, which then goes in the bottom box, and “10” will come out. The “net effect” is that “5” goes in and “10” comes out. If they are then shown the single box on the right, and asked what it should have in it to produce an identical result, they will quickly and easily come up with “+5”. In subsequent posts, I’ll lay out more of the evidence that the students actually think this through from arbitrary numbers going in. What I’m suggesting here is that kids do in fact do the same kind of thinking that we usually express in formulas like “(x+2)+3” and see it as equivalent to “x+5”, but without formulas and without “x”es and without parentheses. The key thing, as far as I am concerned, is that these kids seem to be able to think of the “+2” box as something they can pick up and hold to the light, separately from doing any addition. They are not only able to do addition, they are able to think of a machine that will do addition, and think of the action of that machine even if they don’t know the number that goes in. These are the same kids, by and large, who display no understanding of “x+2”.
The notion of stringing operators together by connecting the output of one to the input of the other is known as composition of operators. We’ll show more of that in subsequent posts, including composition of multiplication and addition, and we’ll have a chance to show the dreaded distributive property in a new light.