The “many meanings” of variables
What is a variable, anyway? Math educators have plenty of complex answers.
In their article “On Developing a Rich Conception of Variable” (part of this volume on undergraduate math education), Maria Trigueros and Sally Jacobs write, “Unlike the concept of function, for example, variable has no precise mathematical definition. It has come to be a catch-all term to cover a variety of uses of letters in expressions and equations.” It appears from a survey of different writing that this point of view — that there is no single definition of variable and each use must be understood in a sort of piecemeal case analysis — is widely accepted.
In this article, I’d like to challenge this notion. In fact, a very simple definition suffices: a variable is just a name that represents a value. Of course, variables are still used with different context and purpose. Instead of introducing variable as a fuzzy word with many definitions, though, educators would do better to nail down a simple definition, and then move on to considering how this unifying idea can be used to communicate and reason in various ways.
Enumerating variable meanings is an obstacle to comprehension and learning.
A typical analysis of the meaning of variables lists several definitions, classifying each use of variables as one of them. For instance:
- Unknowns are variables that stand for a value that isn’t known yet, and should be discovered by solving an equation.
- Generalized numbers are variables that stand for any number to make general statements like (a+ b) + c = a + (b + c).
- Covarying values are variables whose values change together, so that as one changes the other changes with it.
- Parameters are variables whose values identify one from among a family of expressions or functions.
These words definitely get at various conventional ways to use variables. As separate definitions, though, these often come up short. After solving for x, for instance, it changes from an “unknown” to a “known”. Does it really cease to be a variable? Few would be willing to accept that x is a variable for students who did not do their homework, but it is not a variable for those who did! This may seem facetious, yet similar problems arise from the other definitions in the list. They have little to say about what a variable is, and change the subject toward what students should do with them to get an answer. This is ultimately just another instance of the central problem of math education: procedural knowledge without conceptual understanding.
The problem with choosing definitions in terms of what students should do is that as they progress in their level of sophistication, the kinds of things we ask them to do change. If the only way they understood earlier content was based on how to compute answers to certain problem types, then they must now relearn, and come away with the notion that somehow math changed on them. If students are clear on the difference between the definition of a concept and how it’s used in computations in this class, then the latter can change, and they still have their knowledge of the former.
We often want students to use variables in multiple ways even within a single class, unit, or problem! When introducing the standard form ax² + bx + c for quadratics, a, b, and c act as parameters . But if you ask a student for which values of a, b and c the expression ax² + bx + c is the square of a polynomial (and this is actually an important question!), the student must now think of a, b, and c as quantities that vary together, under constraints, and manipulate them algebraically just as they do with unknowns. No single role of variables is enough to solve this problem.
If students need to solve problems like this, it won’t help to tell them that each of these roles is fundamentally a different kind of variable, and try to sort our variables into these buckets. Instead, students must eventually develop a fluid understanding of the ways variables are used in mathematical reasoning, often switching quickly and easily between them. The most interesting problem-solving involves comparing the situation from more than one point of view, where the variables may play different roles, and combining the insights from each.
Variables belong to language, not mathematics.
If defining variables by enumerating the roles they can play in problems is a dead end, surely we must still say something! It’s true, after all, that a variable like x might sometimes be used to generalize over any number, but other times represent a single well-defined number that students should can determine through solving an equation. And what of Trigueros and Jacobs’ claim that variables have no precise mathematical definition?
The key, I believe, is to understand that variables are not mathematical objects at all. They are a notation — a linguistic object instead of a mathematical one — and as such, their definition is simple, but their many senses and uses arise from the context of the communication. We know what a variable is: it’s the name, like x or a, that represents a mathematical object in an expression! You can point to it, and circle it on the page. There is no mystery at all in what x is.
When it comes to using variables, it is not that students have a weak understanding of what they are, but that they lack the communication skills to interpret the surrounding meaning: the qualifying phrases, quantification, and implied setting in what they read.
When we ask a student to solve an equation, we mean: suppose that there exists a value called x, and that this equation is true. What might the value of x be? The “unknown” comes from the existential quantifier on the value of x, not the nature of variables themselves. Because we supposed that the value x exists instead of defining it, we simply do not know — a priori — which it is.
I’ve sometimes heard educators do worse by suggesting that x could have multiple values, or none! This is unfortunate, since it tries to change what is really a familiar situation of not knowing into a more complex definition. If there’s not enough information in the equation to determine the value of x, one can just say that we do not know its value. Perhaps it could be either 3 or 5, for instance, but we cannot tell which. Perhaps the equation is contradictory and so cannot be true at all. But it’s plainly nonsense to say that x has both values at once, or none at all. Moving the complexity to the definition like this prevents students from applying what they already know about not knowing. A common error, for instance, is for a student to believe they have the answer from solving an equation, and not consider whether each possible value of x is consistent with other information they have besides the equation.
There are other kinds of quantification, as well. When we state the distributive property, we mean: for all numbers x, y, and z, this equation is true. The fact that x, y, and z this time generalize over all numbers — something which many authors called a definition of variable — instead arises from the universal quantifier that says so! On the other hand, when we write that F = ma in physics, or y = x² as the equation of a line, we mean: the relevant situations — accelerating masses under a force, or pairs of numbers (x, y) — are those for which this equation holds. This is bounded quantification. In general, the crucial phenomenon of covariation arises from bounded quantification of values, not from a third meaning of variable.
As you can see, the variation in the meaning of variables is actually variation in the stated or implied context of the communication. More often than not, that context involves some kind of quantification, and it’s the meaning of the quantifier, not the concept of variable, that changes.
Teaching language comprehension and communication is not easy.
Correcting this misunderstanding doesn’t make the task easy. Certainly, no one would claim that teaching students language comprehension is an easy task. Neither does it mean we can throw up our hands and leave language arts teachers to the job. Mathematics is its own language, both in the formal language of algebraic expressions themselves, and in the heavy use of semi-formal logical phrases and qualifiers attached to English or other natural languages. Nested and alternating quantifiers, for instance, add significant complexity to mathematics, and are far more rare in other common uses of language. Addressing these students’ language comprehension difficulties is still part of teaching mathematics.
Unfortunately, there are incidental challenges, as well. In current curricula and resources, these implied phrases and qualifiers are left out for students far too early. Students cannot be blamed for missing the details of communication when those details are not stated! Authors or teachers are too familiar with common types or patterns of math problems, and don’t always take the time to express the structure of what they are asking. Perhaps they don’t consciously understand it themselves; many of us get by with an intuitive understanding of language that we picked up second-hand from our own teachers and textbooks, but have never analyzed ourselves.
Consider the notation f(x) = 3x+5. In writing this, we usually mean: “Let the function f be defined such that for all numbers x, f(x) = 3x+5.” I find that most math educators struggle to express this meaning. A substantial number believe that “f(x)” is itself the full formal name of the function, and f just an abbreviation! This leads to ad hoc rules about why f(t) = 3t+5 is the same function, why the equal sign makes sense even though 3x+5 is a number, why it’s okay to write f ∘ g but not f(x) ∘ g(x), and so on. Understanding of the implied quantifiers makes all of this clear, but as many students have discovered, math gets bewilderingly complex when you’re compensating for incorrect fundamentals.
(I say usually above because one could write the same equation in a context where x is defined elsewhere, such as “If x is prime, then f(x) = 3x+5”. Then, different quantifiers would be inferred from that context. In short, inferred context is not compositional: the meaning of a statement is quite ambiguous without full context.)
We can teach fluid comprehension of variables.
Then a proper understanding of variables depends on interpreting complex, multi-level, and sometimes implicit use of mathematical language. This is not trivial, but it’s important to get right anyway. Mathematics is a language for communication, and however hard we try, we cannot divorce mathematics from the usual concerns of human communication, such as reading comprehension, and drawing inferences of intent and context.
The question arises of how math educators might help students build the skill of communicating and understanding the use of variables in mathematics, including the various kinds of quantification and implied phrases. Though I do not claim to have all the answers, I can share some thoughts in this direction. I’m also intensely interested in the ideas of others.
Teachers should model variables as a communication technique.
When variables are only introduced as part of new procedural learning such as solving equations, it’s not surprising that students connect their meaning to those procedural skills. But successful students, and professional mathematicians as well, often start with defining variables so that they will have vocabulary to reason with, before they know what they’ll be doing with them. Teachers can model this by introducing variables purely for didactic reasons in class.
For a simple example, consider the usual presentation of the so-called “distance formula” (essentially the Pythagorean theorem, but stated using Cartesian coordinates instead of right triangles):
Many younger students find this rather daunting as an expression, because there are so many levels of nesting. It can also be written like this:
The two new variables here are non-essential, since the expression can just as well be written as it usually is, without them. Adding these variables, though, gives names to interesting quantities in the original expression, making the top-level expression shorter and clearer. This is a communication technique, and nothing more. Nevertheless, it’s an interesting conversation to have with students: Is the first or second form clearer? Easier to apply? Better for specific purposes?
Students should define their own variables.
We can also encourage students, as well, to make up their own variables by choosing an unclaimed symbol and explaining which quantity is represents. This dispels the myth that the variables are inherent in the problem itself, when in fact any interesting mathematical situation is likely to have dozens of different quantities involved. The choice of which to name as variables, and which to leave as unnamed and implied, is a communication decision, and the resulting variables are a working vocabulary which can make problem-solving easier or harder. (As an advanced example, consider the decision of a physicist to work with momentum instead of mass and velocity.)
Students can define variables with informal language, or by writing equations that define them, or of course preferably both! They can justify their choices and explain why they feel one quantity is more informative or important than another. And they should do so before they have solved a problem.
(It is interesting to note that in computer science, it’s typical to start out writing a complex definition by defining variables — for a moment ignoring that the word means something different there, because the difference isn’t relevant now — to capture simple but interesting quantities first. This organization and communication technique is taught explicitly under the name top-down decomposition. Mathematics students, though, are often expected to pick up skills like this on their own, without direct instruction.)
Quantification should be taught explicitly.
Logical quantification is sometimes viewed as an advanced topic in either philosophy or mathematical logic, and reserved for classes like [pre-]calculus or even the university level. But we can see here that quantification is implicit in much of the basic middle school curriculum! Therefore, we cannot get away with not teaching it.
There is no need to do so with a great deal of formalism. Instead, teachers should understand that they are teaching language comprehension, and that students have brains hard-wired for intuitive understanding of language. Instruction should focus on statements in plain language, not logical connectives! What is the difference between saying “For all birds, there is a nest they live in.” and “There is a nest that all birds live in.”? Can the first be true without the second? Can the second be true without the first?
Students should also practice writing out the quantification in their math problems explicitly. As I did above, students should write out whether a statement claims that such a value exists, or that it’s true for all values, or perhaps that it defines certain associations between values (such as all pairs of x and y that are part of a curve or region of the plane). Again, the focus must be on communicating, so it helps to ask them to write for a specific audience, such as a hypothetical student (the foil) who expresses some incorrect understanding, and to convince that student.
Background and Credits
My thoughts here originally stem from my project of the last eight years, teaching an enrichment curriculum about creative expression with mathematics. The curriculum attempts to answer the question: what if algebra were taught as a language for creative self-expression? The implications are exciting:
- The emphasis shifts from calculating answers to expressing abstract ideas, and use computers to do any calculation so students can see the consequences of their descriptions.
- Students no longer seek to answer unmotivated questions posed by others, but instead find ways to express their own ideas. They experience mathematics as a creative medium, which gives them autonomy and empowers them.
In effect, this turns the entire middle-school mathematics curriculum inside-out. Even without stretching, one easily reaches around 70% of middle school math concepts (for instance, as listed in the Common Core math standards, or other standards documents). And yet, most of these ideas are seen from a radically different perspective. It makes memorized procedural knowledge quite irrelevant, to the extent that some teachers express skepticism that it is even real mathematics in the first place. But students understand ideas like variables, expressions, functions, operators, and so on with a kind of concreteness that is missing from a typical math education.
I often find that the way ideas are taught in conventional mathematics is incompatible with what I’m doing, because the ideas are too tied to traditional math problems and procedures, rather than the big shared ideas.
For this article in particular, I’m also grateful to Maria Trigueros and Sally Jacobs, whose writing on the subject helped focus my thoughts. Though I treated their central claim as a foil to some extent, they also address the importance of a fluid understanding of the uses of variables in mathematics, and how this causes problems for students who conceptual knowledge is too limited. Henri Picciotto, who has authored a large number of resources on the teaching of algebra, was also helpful in responding to my earlier rants on Twitter! Thanks to you all.
