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There are two kinds of answers I'm looking for:

  1. What do students think a variable is?
  2. What do YOU, the teacher, think a variable is?

I'm also interested in why you think a variable is what you think it is.

More Context: Historically, there has been a great deal of confusion surrounding the ill defined notion of "variable". The word has been used so loosely even in modern treatments of algebra and logic, that it is not only difficult to identify "which definition is 'right'" but also how one is to develop a student's understanding of variable towards "the 'right' definition".

Perhaps it is impossible to give a final definition of the term variable, but I do believe there is no better place to start than by asking educators and students just what they think variables are.

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  • $\begingroup$ One totally illogical but extremely widespread notion among my students is that variables have units like miles per hour, but constants don't. Don't ask me why they think this -- they can't seem to explain it, but it comes up over and over. Another silly hang-up is that in calculus, students insist on treating any letter the same way they would treat a variable, even if they've been told that the letter represents a constant. For instance, if I tell them that $y=x^2+b^2$, where $b$ is a constant, they say $dy/dx=2x+2b$. $\endgroup$ – Ben Crowell Mar 4 '16 at 4:24
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    $\begingroup$ It would be interesting, maybe even necessary to answer the question, to compare the meanings of the words variable, argument, unknown and indeterminate. $\endgroup$ – Benoît Kloeckner Mar 4 '16 at 12:53
  • $\begingroup$ Variables do exist in mathematics, but what we usually call a "variable" is in most cases a constant. See also this. $\endgroup$ – dtldarek Mar 4 '16 at 22:53
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    $\begingroup$ Questions about understanding of variables should have their own tag. $\endgroup$ – Dag Oskar Madsen Mar 5 '16 at 10:07
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This is a very difficult question to answer; I recommend as a first place to look:

Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. The ideas of algebra, K-12, 8, 19. Link (no paywall).

As Usiskin writes (emphasis in original):

My thesis is that the purposes we have for teaching algebra, the conceptions we have of the subject, and the uses of variables are inextricably linked. Purposes for algebra are determined by, or are related to, different conceptions of algebra, which correlate with the different relative importance given to various uses of variables (p. 9).

The paper explores your question further, and contains a few nice examples. One more excerpt:

enter image description here

Really, the question at the heart of all of this is how one conceives of algebra writ large. To this latter end, there is an interesting trio of sources tackling a similar conception as follows:

Usiskin, from the aforelinked:

enter image description here

Chazan, using similar language in a later publication:

enter image description here

And even in a blog post (underline added) describing math classes with a single word:

enter image description here


What is my answer to the question? First, here is an excerpt from my personal communication with D.J. Velleman (author of the popular How to Prove It textbook):

On variables: In proofs, I think of variables as standing for fixed but unspecified objects. For example, suppose you're proving that A is a subset of the union of A and B. You think of A and B as being fixed but unspecified sets. The first sentence of the proof is "Let x be an arbitrary element of A." You're now thinking of x as being a fixed but unspecified element of A. You proceed to prove that x is in A U B. Then you say that since x was arbitrary, this is true for all x in A, so A is a subset of A U B.

I find that students don't always understand variables this way. I've had students tell me that in this proof x stands for all elements of A simultaneously, or something like that. And when proofs get complicated, with multiple variables that are introduced in a particular order, with some depending on others (for example, delta = epsilon/3), I think this way of thinking about things is hopeless.

I've been wondering if students have trouble with the meaning of variables in proofs because the way we use variables in earlier courses is different. For example, in calculus we might say something like "Let y = x^2. Then dy/dx = 2x." Here x and y don't stand for fixed numbers. In fact, the whole point is to think of x and y as changing. So if variables don't mean the same thing in calculus as they do in proofs, then what do they mean in calculus?

My own answer, perhaps somewhat appropriately, is that the meaning of variable does, in fact, vary according to its context (e.g., across different classes such as Velleman's example of proofs versus calculus, and within a single class such as Usiskin's approaches to algebra). Rather than trying to pin down a single definition, my own feeling is that we should explore these different meanings and challenge students to articulate the ramifications of adopting different conceptions at different times.

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  • $\begingroup$ What is your personal answer to the question "What is a variable?" What answers have you heard from your students to the question "What is a variable?" $\endgroup$ – John Mar 3 '16 at 20:19
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    $\begingroup$ Also, thank you for the many references. I have an insatiable desire to understand how different people view these fundamental concepts of mathematics as it directly impacts their use of mathematics as a tool for seeing and solving problems in everyday life. $\endgroup$ – John Mar 3 '16 at 20:26
  • $\begingroup$ What different meanings of variable have you explored thus far? What ramifications have you noticed to adopting one of those meanings rather than another? $\endgroup$ – John Mar 3 '16 at 20:32
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    $\begingroup$ My principle reason for pressing you for your answer to these question is that while a reference is wonderful for exploring someone else's ideas, I can not question a reference like I can you. :) Well I guess I could question a reference, but it wouldn't have much to say back I don't think. $\endgroup$ – John Mar 3 '16 at 20:34
  • $\begingroup$ @JohnMeuser Ha, well, perhaps start with the various sources (and, of course, those that have cited them subsequently, e.g., Usiskin) and we'll see how others answer! The reason I happen to have these sources ready is because I am teaching a course called Mathematics for Teaching: Algebra this very semester, and Usiskin and Chazan are among our readings (and sources for discussed meanings of 'variable'). That said, I'll have to let the post above speak for me [at least for now...]. $\endgroup$ – Benjamin Dickman Mar 3 '16 at 20:39
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I teach people (informally) how to make iOS apps. A lot of the people I teach are not people who were good math students. Of course in programming variables are important and anyone with a basic understanding of algebra picks up the concept pretty quickly. The term "variable" is pretty scary for even for people who did ok in algebra because it's a math term and that frightens a lot of people. I lose a lot of people when I bring up variables and constants.

The way I usually explain it in terms of computers helps a little bit. Basically, in CS to make the computer do something you need to write code. The code is usually long. I show them that they can store all that code in a simple variable so they have to write less code, and use that variable when I want to refer to that long piece of code. This helps them understand a little better the benefit of a variable. I explain it as a little basket that can hold any "object". So visually, I draw some baskets on a piece of paper, say the basket has a name, draw an animal in basket (dog, cat) and tell that they can change the animal for that basket. Then I contrast it with the concept of constants. You can use the concept of a bus as the variable and the number of passengers as a bus, and that in a lot of cases, you won't know the number of passengers or that it can change.

In CS, a variable is literally a place in the computer's memory that holds information (not exactly but conceptually this works as an explanation). Rathe than treat is an an unknown as it is usually presented, present it as a container for things. You can address visual and kinesthetic learners this way.

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    $\begingroup$ One of the hurdles in teaching (beginning) programming is precisely that by then students have the idea that it is called "variable" but is really fixed but possibly unknown tatooed on their brain... and in programming (at least bog procedural one), it means something else altogether... and when they finally wrap their mind around that, you tell them about functional programming. Neverending fun. $\endgroup$ – vonbrand Mar 5 '16 at 16:18
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    $\begingroup$ @vonbrand Amen brother. It's a never ending process of learning, but that's life I guess. $\endgroup$ – noobsmcgoobs Mar 5 '16 at 22:27
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Ok, let me take my comment seriously and have a try at a personal answer to the second question (I really don't know what students think a variable is, and I feel it is a question that needs difficult research to be answered more than anecdotally).


Short version

Constants and variables are names (often a single letter) given to mathematical object for various reasons. Constants are mostly a convenience, while different kind of variables are used to denote object whose identity is not completely specified. I distinguish (at least) three kind of variables: placeholders, arguments and unknowns.

Placeholders are variables used to structure a general reasoning around a fixed but unspecified object. Arguments are used in functions, as a way to express what part of the expression of the image is the object taken as entry by the function. unknowns denote putative objects, that may or may not exist, in order to express an equation (or more generally a set of constraints) we want to study (e.g. to determine whether said equation has solutions).


Here is the long version, hopefully useful by its examples (which are addressed to students, for use by teachers). We should distinguish at least the following terms: variable, which will be the most generic term, and three specific kind of variables: placeholder, unknown, argument (the terminology might not be completely standard, but that is about what fulfills my needs). I won't speak about the indeterminants, which are still another beast and somewhat more confusing (at least to me), and that I would not consider as variables.

First, the common between these objects.

They all are about giving a name to mathematical objects, but with slightly different goals and convention. That name is usually a Latin or Greek letter, but it may in principle be any symbol or word (e.g. $\square$). They thus relate to constants, which are names given to well-defined, fixed mathematical objects. For example the golden ratio is defined as $\frac{1+\sqrt{5}}{2}$, and writing $\phi$ for it is just a shortcut. The number $\pi$ has conveniently been given a one-letter name to avoid having to constantly use periphrases such as "the ratio of any circle's perimeter to its diameter" as well as recalling that said ratio does not depend on the chosen circle.

What distinguish variables from constants is that their value is not fixed, or unclear, or unspecified. The reason why we use a letter is not mere convenience, it is a way to speak about mathematical object before we know much about them. Before going further, let us take an example: $\sqrt{2}$. This looks like a bad example, as it is a constant, whose name has a quite different form from a $x$ or a $y$. But thinking about it, we name it as soon as we are able to define it (i.e. as soon as we can prove or accept that there is a unique positive number whose square is $2$), and then we can start studying it. For example, if we want to give the first decimal digits of $\sqrt{2}$, we will use its definition and thus will need its existence, and we will use the name we have created before starting the computation. That is an example why we may need to give names to quantities or objects we don't yet know perfectly.

The difference between the three words to be defined lies in the type of unspecification we deal with. What is confusing is that these different situations are something mathematician are so used to that they don't need to really think about them: they have build an intuitive understanding of how to use them, to the point they don't feel the need to use different words. And that is how we tend to call variables very different things that look the same, but which are better distinguished at first.

The main point: differences between different kind of variables

placeholders

This is probably what most mathematically educated people think of when told about variables. A placeholder is a name given to an object of some specified type (we ask that it belongs to some set, for example "let $x$ be a real number"), possibly with some specified property (e.g. "assume $x$ is positive"), but whose identity is not specified either to ensure that the ensuing reasoning can be applicable generally, or because its precise value cannot be determined without specifying completely previously introduced placeholders. Placeholder thus enable us to do one reasoning, and apply it at once to all possible values in the specified set with the specified properties.

For example, assume we want to prove that the square of an even integer is even. We would let $n$ be an even integer, then apply the definition of $n$ to state that there exists an integer $k$ such that $n=2k$, then compute $n^2=4k^2=2\times(2k^2)$. Observing that $2k^2$ is an integer, we deduce from the definition that $n^2$ is even. In this proof, $n$ is a placeholder that ensures generality, while $k$ is a placeholder whose value depends on $n$ (and is unique given $n$, in this case).

Arguments

When we define a function, we specify (a domain, a target set, and) an assignation to each element of the domain of an element of the target set. We can for example define a function $f$ by stipulating that it maps each given real number to the sum of the square of the given number, minus three times the given number, and two. We see that this is not a very comfortable way of phrasing things. What we do is we use an argument: we choose any symbol (that does not provoke collision with previously used symbols), say $x$, and use it instead of "given number" above. The example above can then be written $f : x\mapsto x^2-3x+2$; or we say that $f$ is the function defined by $f(x)=x^2-3x+2$.

An important point is that the chosen symbol must appear both on the domain side and on the target side, so that we know what it means; for example, assume that there is a parameter $t$ in our context, which is known to be an unspecified real number. Then we could want to study the function $g$ defined by $g(x)=x^2-tx+2$, and then we have to distinguish clearly between the roles of $x$ and $t$.

Unknowns

An unknown appears notably when one wants to study an equation: it is a name for a putative object of a specified nature (i.e. belonging to a specified set) which satisfies a given constraint or set of constraints (often equations). It is important to stress that the object is putative, in that it may or may not exist, depending whether the equation (or set of constraints) admits a solution or not. But we need to name the thing just to phrase the equation, thus before determining whether it exists or not.

For example, consider the question: "which real numbers have their square equal to themselves plus one?" It is rephrased as "find the solutions of the equation $x^2=x+1$ in the real unknown $x$". (We should always specify the set of allowed values of the unknown, but this is often neglected).

There are situations where the different kind of variables are difficult to separate.

Take for example the following problem: "determine which pairs of real numbers are the sum and the product of two real numbers". To solve the problem, we can first reformulate it with variables: "let $a$ and $b$ be real numbers, and consider the equation system \begin{cases} a =x+y \\ b = xy \end{cases} in the real unknowns $x,y$." Here $a,b$ are placeholders and $x,y$ are unknowns. But then we work a little, and arrive at the conclusion that "the above system has a solution if and only if $a^2-4b\ge0$." So the answer to the original question is best phrased as an equation on the placeholders. This is ok in the phrasing above, because $a^2-4b\ge0$ is an assertion rather than an equation to be solved. But the difference is thin and can be overlooked by the reader, or the writer may phrase things more loosely. This is probably the kind of situation where extreme confusion can happen, with teachers struggling to understand what is unclear to the students, because teachers have a much better intuitive understanding of these matters, but an understanding that they only rarely phrase explicitly.

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Suggestion: Instead of using single letters for variables, you might consider using meaningful words or phrases.

Instead of s=d/t, for example, write speed = distance / time.

Computer programmers always use this technique. They are required to consistently assign meaningful names to variables so that another programmer reading their code can more easily understand what it is supposed to do. It also helps in debugging your own code.

Maybe teaching some rudimentary programming or spreadsheet skills would help. Of course, you have to somehow initially avoid things in programming languages like x=x+1. Maybe some language that uses a different symbol for this kind of assignment of values. I don't think spreadsheets do this -- not sure. It's been a while.

You can start abbreviating the words as students get used to the idea of a variable.

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I am both a teacher and a student so I will answer 1. and 2. at once:

  • an $S$-variable $x$ is a possible quantity taken from set $S$. For example, suppose $x$ is a real variable. If $x^2+1=0$ then no such $x$ exists, whereas if $x^2-1=0$ then $x=\pm 1$. In contrast, if $x$ is a complex variable and $x^2+1=0$ then $x = \pm i$.
  • an indeterminant $x$ is a symbol which is used as a place-holder for expressions. For example, $a+bx+cx^2 = (a,b,c,0,..)$ where $1 = (1,0,0,..)$, $x=(0,1,0,..)$ and $x^2=(0,0,1,0,..)$. We can capture the algebra of polynomials either via expressions with $x$ or with sequences with only finitely many nontrivial terms. In this context, $x$ is not an arbitrary element of some set $S$, it is the sequence $(0,1,0,...)$.

My example of an indeterminant is by no means general! You could think of indeterminants $x,y$ for which $xy \neq yx$ etc... In any event, an indeterminant can be made into a variable by limiting $x$ to assume the value inside some given set (through the appropriate valuation homomorphism).

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  • $\begingroup$ I find your example for a variable quite troubling. If $x$ is a real variable, it is a possible quantity taken from $\mathbb{R}$. Right. Now you say "If $x^2+1=0$ then no such $x$ exists"; but didn't you assume $x$ exists by introducing it as a real variable? So what exactly is the status if $x$ in "$x^2+1=0$"? I may sound picky, but what I want to point out is that your answer seems an example of conflation of variable and unknown, and I guess many student's misunderstanding come from such conflation. $\endgroup$ – Benoît Kloeckner Mar 8 '16 at 12:05
  • $\begingroup$ @BenoîtKloeckner I said "possible" quantity. So, no, I do not assume $x$ exists within $\mathbb{R}$. So, you would say what I define here is not a variable, but rather an "unknown". Your definition of variable presupposes that $x$ is a quantity in $S$. (not trying to be argumentative here, it is certainly possible that my terms could use refinement!) $\endgroup$ – James S. Cook Mar 8 '16 at 14:27
  • $\begingroup$ @BenoîtKloeckner you include "uknown" in your term variable in the answer you posted. It seems we agree ? $\endgroup$ – James S. Cook Mar 10 '16 at 11:47
  • $\begingroup$ in fact I changed terminology between my comment and my answer. I still think that there are several flavors of variables which benefit of not being conflated (here placeholder and unknown). $\endgroup$ – Benoît Kloeckner Mar 11 '16 at 12:48

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