Notes on Notation – Variables

Variables have often been seen as the big bugaboo of the middle school math curriculum and beyond.  They have become inextricably identified with that 7-letter curse word “algebra”.  I submit most teachers have encountered some variation on student Jesse who says in utter frustration: “How can a possibly equal b?  a is already equal to a!”  Yet before we dismiss Jesse as confused, it would be useful to note that Jesse clearly got what was important in school up till now: that you keep all your letters straight.

The introduction of letters for numbers in school marks the end of the transition of kids from arithmetic to algebra, not the beginning.  This transition, from the doing of arithmetic to the thinking about arithmetic (both numbers and operations) as a domain with patterns and regularities, involves many mathematical ideas, and that doesn’t wait till middle school, and it doesn’t wait for formal algebra with formal variables.

And yet the introduction of letters as variables, letters for numbers, does mark a big hurdle for many students.  The last shred of pretense that what we’re doing is arithmetic, that we’re out to produce a number, is now out the window.

If we look at the notational conventions for variables, it seems that the most obvious aspect of it is also the most important: variables look different from numbers.  For the old Greeks, this was never obvious, for their way of writing numbers re-used their letters!  (They used nine letters to indicate 1-9, nine different letters to indicate 10-90, and so on.)  For them, letter variables wouldn’t have been clearly distinct from numbers.  To conclude, as some have done, that this was single-handedly what kept the Greeks from inventing algebra strikes me as rather silly – they could have  found any of another number of small variations that would have dealt with the difficulty.  But for us, letters aren’t confused with numbers, so hence letters for variables.

However, there is an additional convention that variables in mathematics are always single letters.  You often see this in physics, too, so that you see d = \frac{1}{2}a t ^ 2 instead of distance = \frac{1}{2} acceleration \times time^2 .  The latter version favors ease of reading over compactness of writing: sometimes I get the impression that the entirety of mathematics still carries around the traces of old economic trade-offs: the Greeks wrote their mathematics with a stick in the sand; much of school work over the last several centuries was done with a marker on a small slate: a board literally made out of slate – easily written on, and easily erased.  In those centuries, paper was very expensive, and not wasted on small matters such as calculating sums.

If you look at modern programming languages for computers and websites, every single one allows variables to be more than a single letter.  Variables tend to be chosen so that they are meaningful for the reader, so ‘distance’ rather than ‘d’.  Of course, in movies it would be much less imposing to have a scientist with wild hair say “energy is mass times the square of the speed of light” instead of E = mc^2 , but I’d settle gladly for a mathematics that didn’t sound like magical incantations that have to be repeated just right.

Einstein on variables

Einstein on variables

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3 Responses to Notes on Notation – Variables

  1. José says:

    You say that:
    “And yet the introduction of letters as variables, letters for numbers, does mark a big hurdle for many students. The last shred of pretense that what we’re doing is arithmetic, that we’re out to produce a number, is now out the window.”

    How is the transition to variables a hurdle? Maybe a very, very small hurdle, but nothing of great difficulty. If anything, the use of variables makes arithmetic steps much easier. The variables are like a name for a number when the value is not immediately needed (until later). Variables make many things in arithmetic easier to understand.

  2. Bert Speelpenning says:

    Thanks for your comment. You raise an important issue.

    There is a distinction between what we think ought to be hard or easy, on the one hand, and what children can be observed to have difficulty with.
    Those are not the same. Many things we think ought to be easy are consistently stumbled over, and many things we think ought to be hard are taken in stride by generation after generation of kids.

    As you read more of this blog you’ll see me drag in examples of both kinds. The examples come from fairly extensive personal observations as well as reports from many teachers.

    The connection between algebra and the ability to deal with numbers whose value is not immediately needed – that’s a great point. I’m referring to that in the blog as “deferred computation” and I’ve got many entries on just that.

  3. Pingback: Notes on Notation - Parentheses « Learning and Unlearning Math

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