Naming Variables

First of all, let me get this out of the way: “Hey, you kids! Get off my lawn!”

In this post, I comment on the notational shifts from what I was trained in back in the 1980s and what textbooks do now.

I was reading Power Puzzles 2 by Philip Carter and Ken Russell when I came across this clue to a word pyramid: “The first known quantity in an algebraic expression.” At first, I thought they meant “unknown” and assumed the answer was “x”, but I couldn’t get that to work with the next (two-letter) clue (“Denoting direction to or towards”). Since “in” seemed like a passable answer to that, I considered “n”, realizing that “i” would also work. I checked the answer in the back of the book, which said “a” (with “at” for the second clue).

This got me thinking about the notational tradition that I’d been trained in, and how things have changed in a few decades.

When I was a lad, I was taught consistent rules about how to name variables. Unknowns to be solved for were named x, y, and z. Parameters were named a, b, and c. Variable indices were named i, j, and k. n was an ordinal placeholder for a sequence. Angles were named with Greek letters: Unknowns were \(\alpha\), \(\beta\), \(\delta\), and \(\gamma\), while general statements used \(\theta\) and \(\phi\). Generic functions (which are not variables) were named f, g, and h.

Judging by modern textbooks, things have apparently changed. For unknowns, modern Algebra I exercises are riddled with any letter of the alphabet that the author chooses to use. Some exercises ignore case, so that M and m are the same thing. The Pythagorean Theorem, for example, might be rendered as \(a^2 + b^2 = c^2\) and \(A^2 + B^2 = C^2\) in the same document. \(f\) can be a variable as readily as a function name, leading to the potential for confusing strings like \(f(f) = f\).

Meanwhile, angles are more often referred to by numbers instead of by Greek letters. My opinion on this: Numerals in diagrams should be reserved for their numeric meaning. \(\angle 1\) is pointlessly confusing. The use of Greek letters doesn’t involve the learning of Greek, and doesn’t even require learning the entire alphabet: Six characters suffice for the purposes of the standard high school mathematics. It’s a (very little) bit of learning at the outset, but it has several clear advantages:

  • It doesn’t mean that diagrams have numerals that have unknown or variable meaning.
  • It reinforces the concept that angle measurements are inherently different from distance/length measurements, a distinction that my students consistently struggle with.
  • \(\pi\) is not imbued with some mystical power, being the only Greek letter students might see.*

I can understand part of the motivation for this. Physics uses meaningful letters in formulas: e is energy; g is gravity. Computer programmers prefer longer variable names that carry the meaning of what they represent, such as MonthlySales and SalesPerCustomer. There’s a point to getting students to be competent at seeing formulas involving all manner of representational names.

However, the key difference between a physics formula or a computer program on the one hand and mathematics on the other is that mathematics is deliberately abstract. Physics and computer programs seek solutions for specific problems. Mathematics provides a tool for writing down those problems.

The first abstraction that students experience is encountered so early in mathematics that we tend to forget about it. Because of the cognitive development of students at the time, it’s not commented on as an abstraction. It occurs when we stop mentioning units: Three butterflies and five butterflies make eight butterflies; three hippos and five hippos make eight hippos; three plus five is eight.

The leap from mere enumeration to abstract mathematics is the generalization that, no matter what we’re aggregating, three things plus five things makes eight things. Three butterflies and five butterflies is not math; three plus five is. Butterflies are a way to concretize the abstract.

That’s not to say that units are not important: Of course they are. But mathematics is indifferent to the specificity of units. When adding, all that matters is that units match: You can’t add meters to feet, or ounces to degrees. When multiplying, that’s not even a requirement, since units are multiplied right along with the values (this may yield unnatural units, like “meter-feet” or “ounce-degrees”, but that’s not a problem with the mathematics, it’s a problem with the application).

When we give a formula like \(F = 9C/5 + 32^\circ\), we’re giving a specific formula relating two related sets of data: Degrees Fahrenheit and degrees Celsius. This is not “pure” mathematics: It’s the use of mathematics within science. This is a function where the input is the temperature in Celsius and the output is the temperature in Fahrenheit.

We could write it as a general function: \(f(x) = 9x/5 + 32\). From the view of pure mathematics, it is no more or less significant than any other function. It works the same as \(g(x) = 14x/3 + 18\). It is the abstract rules of mathematics that allow us to convert \(F = 9C/5 + 32^\circ\) into \(C = 5(F – 32^\circ)/9\), just as we can convert \(y = 9x/5 + 32\) into \(x = 5(y – 32)/9\).

There are, naturally, specific facts that belong to mathematics. One of the most famous of these is the aforementioned Pythagorean Theorem, generally written as \(a^2 + b^2 = c^2\). And such facts can rightly be seen as formulas: The Pythagorean Theorem is actually that “the sum of the squares of the side lengths of the legs of a right triangle is always equal to the square of the side length of the hypotenuse” while the standard formula for this is \(a^2 + b^2 = c^2\) (where \(a\) and \(b\) represent the leg lengths and \(c\) represents the hypotenuse length).

It may well be fair to characterize all such mathematics facts (such as the Pythagorean “Theorem” or the area of a rectangle) as formulas. In general, a formula is an applied function.** However, the distinction between “formula” and “function” is philosophically significant, and by permitting the entire alphabet as symbols for functions, this distinction is blurred.

The Algebra I book for my current district (Pearson Common Core) doesn’t even appear to explicitly tie “formula” and “function” together at all. In chapter 2, a formula is defined as “an equation that states a relationship among quantities. Formulas are special types of literal equations” (110). In chapter 4, a function is defined as “a relationship that pairs each input value with exactly one output value” (241). However, what the index lists as a formula appears in the text as a function: “The rule \(V = \frac{4}{3}\pi r^2\) gives the volume \(V\) of a sphere as a function of its radius \(r\)” (250) (this problem also says that \(r\) is the independent variable and \(V\) is the dependent variable, which is another rant).

I understand that we don’t want to foist high-level distinctions on Algebra I students. I’m not suggesting that we try to communicate the interrelationship of functions and formulas at that level, and I’m willing to grimace and accept the casual disregard for multivariate functions. Indeed, my point is that, by blurring the line between acceptable notation for functions and formulas, and by dropping the tradition of using Greek letters in favor of numerals for angle names, we’re muddying the waters before they need to be muddied. When I was a student, a formula looked a certain way; a function looked a certain way. Using letters willy-nilly for generalized, abstract functions is bad practice.

More generally speaking, let mathematics be mathematics: The abstract parts should stay abstract, and that includes using the traditional letter subsets (listed far above) for their traditional mapping.

* There are a few capital Greek letters that also appear in mathematics that a high schooler might encounter, but these are likewise falling by the wayside. \(\Sigma\) and \(\Pi\) are most common in Probabilities and Statistics classes, which generally fall in the “elective” bucket, thus being something only the most ardent mathematics students. \(\Sigma\) also has its place in Calculus classes, which are also only for the mathophile. \(\Delta x\) is being replaced with the less elegant \(x_2 – x_1\), which makes it easier for students to miss the point.

** This is an oversimplification. As most commonly used, a formula is a relationship between interdependent variables while a function returns a single predictable output variable for one or more input variables. However, formulas can be written in function form; a formula with /(n/) variables can usually be written as /(n/) different functions. For instance, the formula relating the area (A) of a rectangle to its side lengths of l and w can be written as \(A = lw\), \(l = A/w\), and \(w = A/l\), all of which are functions. However, we do tend to say things like “the formula for the area of a rectangle”, which refers to the first of these listed functions. Formulas also need not be in strict function form; for instance, the formula relating Fahrenheit to Celsius can be given as \(5F – 160^\circ = 9C\), which is not in the proper form of a function.

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