Functions: Standard Notation and Schools, Continued
In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. In the previous post, we started to explore function notation, which in middle school and high school shows up as , and examined its use. This post is a continuation of the previous post.
Let’s continue to look at functions as little boxes that take an input and produce an output, and play with that a bit. The most obvious way to play with it is to connect the output of one box to the input of another. An example is shown below:
Two simple function boxes, one which adds three to whatever comes in, and one which doubles whichever comes in. Together, they take a number that comes in at a and produce a number that comes out at c. We could hide this entire apparatus (i.e. the combined +3 box and the double box) in a single box, as follows:
Since we know what the smaller function boxes do, we can figure out specifically what the new function box does. We could try some different numbers at a, and follow them through the boxes at b and then at c. The value 1 at a will result in the value 8 at c. The value 10 at a will result in the value 26 at c. If we collect a whole set of value pairs like this, we can graph them. We would discover that the graph is linear.
When we don’t know much about the smaller boxes, we need a different approach. In the picture above, we have two function boxes, labeled and , respectively. Also notice that we’ve labeled the inputs and outputs again, this time as . With regard to the function box , we’d say that x is the input and y is the output. In the standard notation, we write . Similarly, we write . Combining both, we’d get . Yes, in the standard notation, the is shown before the .
We should note that in this standard notation, we need to give a name to the value coming in to the function box. The function , in standard usage, is pronounced “ef of ex” ( f of x) rather than plain “ef” (f). This appears to be because in standard notation, the name of the variable x is important; for example, if we say , the name of the variable x shows up again inside of the expression .
Let’s look again at the function box labeled “+ 3” above. Notice that it doesn’t contain any variable. Though we might call the number coming in “x” (or “a” or anything else), the function box doesn’t use “x”. It just says “+ 3”. In contrast, in standard notation, we’d talk about the independent variable and the dependent variable , and would write . It seems like the price we pay for using a “normal” looking expression such as is that we have to commit to the use of a particular variable, here . And yet, a function defined as is the same function in all respects as the function defined as . The notion, so beloved in secondary school, that x is always the independent variable and y is always the dependent variable, this gets in the way completely once we look at functions as things that can be combined (composited) easily.
I’m by no means the first one to notice that the “x” in f(x) could just as easily be “y” or “z” or “t”. The development of lambda calculus in the 1930s gave us a careful and precise model for function definition and function invocation, complete with a system of notation. This system of notation, involving the Greek letter lambda (λ) has become standard in certain branches of mathematics and computer science. It makes a clear and precise distinction between bound variables and free variables and elucidate how substitution works, to enough precision so that computers can do it automatically.
All the same, I’d say that lambda calculus is overkill for secondary school, even if introduced only for the notation, e.g. . I think there are easier ways to make clear through notational means that the bound variable (also called dummy variable) doesn’t matter. In fact, we’ve already seen examples of it.
Above are shown five identical function boxes, but with different notation. Box (a) shows the action of the box as an expression, x + 3, and labels the input as x. The suggestion is that the “x” in the x + 3 expression matches the number on the input. Box (b) shows the action of the box as an equation, y = x + 3, and labels both input and output, with x and y, respectively. Box (c) shows the action of the box as a function using lambda notation, and the input is not labeled. Here, the suggestion is that the label on the input has no bearing on the notation of the function in the function box. Box (d) shows the function in typical high school notation, using the function label “f” (so the function now has a name, f, even if that name is not used anywhere else.) Box (e) simply says “+ 3”, suggesting that whatever number is on the input gets three added to it.
Though we might have esthetic preferences for one of the boxes above over the others, it is the use of composition that will really drive up the reasons to prefer one over the other. Remember that all these 6 boxes are identical inside, and differ only in the labels. They all add three to the number going in. I could connect two of these identical boxes, output to input, and achieve the net effect of adding 6 to the number going in:
In situation (a), we see that the two identical boxes need to be given different labels, since the number going into the bottom box is not x, is not the same number as the number going into the top box. In situation (b), we also need to use different labels for identical boxes, since neither the number going into the bottom box nor the number coming out of the bottom box is the same as those for the box above. In situation (c), we can indeed use the same labels, since the lambda notation doesn’t presume anything about the name or value of the number coming in. In situation (d), we could call both the functions f, but can’t consider both boxes defining instances of the function f. In situation (e), as in (c), we can use the same box with the same label in both places, and have the notation work consistently.
So, for my money, the notation used in situation (e) gives us all the power and grace of the lambda notation while being much simpler for use at the middle school level. None of the more traditional school notations for functions has the same power and grace once we start to use functions in composition.
My experience with students at the middle school level suggests that the boxes and situations (e) give no problems. However, we need to examine this approach with examples other than adding constants to get a good feel for how expressive this notation really is. This will be the subject of our next post.