When we talk about representation in mathematics, it is surprising how often we limit ourselves to looking at formulas. Pictures, diagrams, graphs and tables are also part of how you can represent mathematical ideas and mathematical thinking. Most representations of mathematical ideas are static – understandable, since these representations are mostly drawn on paper. Whole generations of students have responded to the call to “show your work” by drawing something static on paper. It’s such an obvious thing to do that to question it seems silly. Yet I’ll ask the question anyway: in what situations does a snapshot or a static representation materially distort the important mathematical idea?
Let’s look at this through an example. Let’s say my teen has $110 in his account after depositing birthday gifts from various friends and family, and he decides to spend it on a subscription to World of Warcraft, which we’ll assume here costs $14.99 per month. How long will his subscription last? (To simplify things, we’ll also assume that he already has the latest version of the program, which costs money, and that he has already exhausted the free trial period – the situation he’d be in if he had let a previous subscription lapse, and he now reinstates his subscription.)
For adults, the above question usually invokes the need for division. You can do 110/14.99 in your head, or on paper, or using a calculator. The calculator will tell you 7.338, from which you would conclude that he’s good for seven months.
My teenager may have a whole other way of approaching the problem. Since he is very clear he wants World of Warcraft, he has no need to await the answer to the question “how long will it last” before acting. He can just pay the $14.99 each month, and he gets to find out how long it lasts without doing any arithmetic at all. When his balance falls below $14.99 in December when the money is due, the answer to the question is “till now”. Not only is his way easier, it also generalizes much better when there are other money flows coming in or going out, like money coming in at Christmas, or the sudden need to buy an “All Your Base Are Belong To Us” t-shirt from ThinkGeek.
I might argue that my teenager has very good reason to not treat the issue of “how long does my subscription last” as a division problem. It makes more sense to him to treat it as a series of subtractions, $14.99 each month. For him, those subtractions may be punctuated with additions or other subtractions, as other money enters or leaves his account.
Even if no other money flows are involved, the following series of subtractions describes quite well what is happening:
110 – 14.99 = 95.01
95.01 – 14.99 = 80.02
80.02 – 14.99 = 65.03
65.03 – 14.99 = 50.04
50.04 – 14.99 = 35.05
35.05 – 14.99 = 20.06
20.06 – 14.99 = 5.07
– certainly much better than 110/14.99=7.388 describes what is happening – and yet somebody who would write the above series of subtractions as an answer to the question “how long will it last” might not get any credit in a traditional math class, but the person who wrote “7.388” might!
The idea of time is represented in the sequence 110-14.99=95.01 etc above. But how? We are so used to “seeing” time in something that is ultimately static, that we don’t necessarily notice we’re following a convention:
In this comic, we see a progression from left to right, and top to bottom, even though – of course – nothing really moves in the picture. A new picture doesn’t replace the old one, there is no animation. Instead, there is a convention, by which we easily interpret the top left most box as representing the beginning, and the bottom right as the end of something that plays out over time.
The sequence 110-14.99=95.01 etc suggests time and duration, though it doesn’t spell it out explicitly. A simple way to make it explicit is to annotate each line with the name of the month for which the payment applies:
110 – 14.99 = 95.01 May
95.01 – 14.99 = 80.02 June
80.02 – 14.99 = 65.03 July
65.03 – 14.99 = 50.04 Aug
50.04 – 14.99 = 35.05 Sept
35.05 – 14.99 = 20.06 Oct
20.06 – 14.99 = 5.07 Nov
Instead of an annotated sequence, we can also think of this representation as a table, an association of two sets of quantities. Marking events with a “time stamp” is a common and useful way to organize data, and one way to represent a process using a static picture.