## Representations – Formulas and Some Alternatives

There are systems of notation for mathematical expressions that are in wide use.  One of them is so widely used and so well-known that we often think of it as the only one, the real one, the true one and the best one.

If I write $\pi r^2$, then people the world over will agree that this means the multiplication of three quantities: the quantity indicated by $\pi$, the quantity indicated by $r$ and again the quantity indicated by $r$ .   Within the same general scheme, we can yet find alternative ways to render this mathematical expression: $r^2 \pi$ or $\pi r r$ .

Certainly, $\pi r^2$ stands as the standard – some of you may even consider the alternatives I gave as incorrect.  After all, you might say, the constant must come before the variable, and exponents should be used rather than repeated multiplication.  True, as far as it goes – but I don’t think it goes very far.  Among several equal renderings $pi r^2$, $r r \pi$, $r^2 \pi$, $r \pi r$, some are “more equal than others”, in the wonderful phrasing of George Orwell.  By writing the formula in the standard way, $\pi r^2$, others will recognize it immediately, and remember it as the formula for the area of a circle.  The other variants require more effort to recognize as the area of a circle.

Yet the historical fact of the spread of a particular system of notation needn’t blind us to the virtues of other kinds of representations, even if we don’t typically encounter them in math class in secondary schools.  It isn’t actually hard to imagine an alternate history in which a whole different system of representation would have come down to us, a system we then would think of as the true one.  As recent as a hundred years ago, we would still encounter peoples in jungles or in remote places like the highlands of New Guinea who had lived in isolation, with strange languages and strange cultures and strange civilizations.  Star Trek also got us interested in boldly going to seek new civilizations.  Such new civilizations may well know how to get the area of a circle, but there is no reason to assume their way to represent this would be $\pi r^2$.

Let’s imagine a civilization where the formula for the area of a circle would routinely be represented as follows:

Let’s be clear: I’m not asking you to look at this as a picture for the formula, I’m asking you to consider a civilization in which this would be the formula.  In this civilization, the basic building blocks for formulas are boxes where something goes in and something comes out.  If what happens in a box is simple enough, you simply state it (“multiply”), if what happens is more involved, you draw the box with smaller boxes inside that indicate what happens to produce the quantity that comes out.  You could also imagine that when people in this civilization jot down something quickly on scratch paper, they would cut corners and streamline it some, perhaps coming up with something like this:

The representation with boxes, as well as the streamlined version with arrows, shows quantities being obtained from other quantities through some orderly process.  It’s the same “orderliness” that underlies the notation $\pi r^2$ but note how it is expressed in a completely different way.

Here is yet another way to represent these ideas, this time in a more verbal way, but still quite precise.

The area of a circle =  pi times s,
where pi is a constant, which is often given as 3.14 or 22/7, though each is an approximation,
where s is the area of a square that has r as its side:   s = r times r,
where r is the radius of the circle,
where the radius of the circle is the distance from the edge of the circle to its center.

This style of representation, based on “where clauses”, may seem cumbersome at first glance, yet it contains a lot of information.  It also maps onto a picture of the situation really well:

We could spend a lot of time on each of these representations, and showing how come each is in fact a legitimate and precise way to render the idea of the area of the circle.  In addition, we could look at the advantages and disadvantages of each.  Each pushes something into the foreground, relegating something else into the background.  Some may be more suitable for learners, some may be more suitable for experienced folks.  Some may be more suitable on a piece of scratch paper or a blackboard, some may be more suitable using a standard keyboard.

Of all these representational approaches, quite different from each other, one approach is the one we happen to have inherited.  The result is that, for most of us, we think of $\pi r^2$ as the true representation, better than all the other ones.

In my experience with sixth and seventh graders, the representation with the boxes appears natural for them.  You show it, they start using it on their own.  The boxes don’t seem to occur to them as something that’s hard to learn – it seems to occur to them as something that doesn’t require learning at all.  It is interesting to watch them play with these and discover notions like “nesting” (boxes within boxes) and never having to stop to talk about parentheses or order of operations.  Still, I am not suggesting we change which representations we teach, and in what grades.  Rather, I’m suggesting that freeing yourself up from the single standard representation yields important benefits when working with students.  It is good when math students show a fluidity and flexibility in moving in and out of various representations, and coming up with their own.  Sometimes, such non-standard use of representations is discouraged in the classroom rather than celebrated.