## Operators, Functions, and Properties – part 1

I intend to write some things on operators in mathematics, beginning in this post.  Functions and properties will be brought in as appropriate.   The idea of operators is absolutely basic and central to computers and computing; they are also important in mathematics, but aren’t usually put front and center in K-12 education.  In mathematics education, operators and functions don’t usually come to the fore until college level math, though you can easily find them foreshadowed in high school math or earlier.  Earlier still, kids learn about four particular operations: add, subtract, multiply and divide, and kids typically know that a calculator that can perform these four operations (but not much else) is called a four-function calculator.   So one of the things we can look at is the question whether a function is the same as an operation, and if not, how they are different.  Or whether an operation is the same as an operator, and if not, how they are different.  More important, I think, is to look for what is interesting and relevant about these ideas that is worth introducing earlier in school.  How, and how early, can kids get a handle on the idea that the “+” in 3+5 isn’t just a command to do something with 3 and 5 and get us a new number, 8, but that “plus’ is something that can be looked at independently of the particular numbers 3 and 5 – that “plus” can be held up to the light and looked at and talked about.

As our starting point, I’d like us to imagine a small device with a single red button, and a screen that shows a single digit.  When you receive it, the device shows “3” on the screen, and nothing further happens until you push the button.

The screen then shows “4” on the screen, and the “4” stays there until your patience runs out and you push the button again.  After playing with this device for several days, you’ve observed that the screen cycles through the single digits as follows, and it has kept up its repetitive behavior for that entire time:

Can we tell what the device does and how it works?  We have seen its outside, we’ve been able to shake it and rattle it, and listen for any signs of a clock or a rodent inside.  Probably more importantly, we’ve played with it and observed its behavior, which we characterized in the diagram above.

Just as important, but trickier: you probably have formed a model for what’s inside of the device.  The model may be detailed enough so that a device could be built from your model, or it may be a simple sketch or a simple mental image with lots of details lacking.  It is easier to predict that you have in fact formed some kind of a model of what the device is like than to predict what your model looks like.  There are many different models that fit the behavior observed.

For right now, I’d like to focus on the button and on what it does.  You might say that it makes a “4” appear on the screen, but that is only true in certain situations.  Since the device was showing a “3” when you got it, the first time you pushed the button, it did make a “4” appear.  Also, later, when another “3” is showing, when you pushed the button again a “4” appeared.  Yet in other situations, e.g. when a “0” is showing, pushing the button does not make a “4” appear.

So the description of what pushing the button does is not as straightforward as saying it makes a “4” appear – and yet clearly the button does something, regardless of what particular digit is currently showing, and there are different ways to characterize what it does.  We can talk about the button as doing something as part of the device, as an operator acting on the current state of the device.  How we talk about the operator, and what language we use for that, will depend on our model for the device.  In future posts, I will show three different models which yet all show the same outside behavior, each described in a different language.  And our view of the button as an operator will be colored accordingly.

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