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Students sometimes ask whether the $x$ in the expression

$$2x$$ the same kind of thing as the $x$ in the equation

$$2x = 4.$$

In the expression $2x, \;x$ can be any real value.

However, in the equation $2x = 4, \;x$ can only be the value or values (from the some given set of real numbers) that makes the equation true.

Are there some mathematical principles by which we can explain why and how the use of $x$ is different? That is, is there a better explanation than simply telling students that $x$ has a different meaning in these situations? For example, could/should we tell students to interpret the equals sign as a question that asks which value(s) of $x$ make the expressions on both sides of the equation equal?

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    $\begingroup$ @JoeTaxpayer: This is an excellent suggestion. I've made a major revision of the question in trying to capture both what you said and what Frasch was asking. Frasch --- If you don't like what I've done to your question, feel free to completely undo my edits, or revise my edits, as you see fit. $\endgroup$ – Dave L Renfro Jun 9 '19 at 18:12
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    $\begingroup$ Check out my response to "What is a variable?" here and, especially, the paragraph from Velleman at the end. $\endgroup$ – Benjamin Dickman Jun 9 '19 at 18:40
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    $\begingroup$ Related math.stackexchange.com/questions/3244699/meaning-of-an-equation (also by the OP). $\endgroup$ – user5402 Jun 9 '19 at 18:52
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    $\begingroup$ @RustyCore: No! $y=2x$ is not a function but merely an equation. A graph in the $(x,y)$-plane is not the same as the function underlying the graph. $\endgroup$ – user21820 Jun 10 '19 at 16:31
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    $\begingroup$ @user21820 If a function is defined to be a particular type of relation (which it often is) then a function and its graph are the same thing. $\endgroup$ – Jessica B Jun 17 '19 at 21:39
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$2x$ is an expression, a phrase. Compare it to "two ducks". This is neither true nor false. It doesn't have a 'truth value'.

$2x = 4$ is an equation, a statement. Compare it to "two ducks have four legs". This is true (edit: for the ducks, but not necessarily for the $x$).

The meaning of the word "ducks" has not changed. The grammar of what is with that word has.

If you replace "ducks" with "dogs", the "two dogs" phrase makes us think of something else, but doesn't feel much different grammatically than "two ducks". But "two dogs have four legs" would be false. Different values for the variable give different 'truth values'.

(This comparison isn't perfect, because the $4$ turned into $4$ legs some magical way. But I hope it helps you to see that $x$ isn't changing, just the grammar around it.)

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    $\begingroup$ Potential quibble: $2x=4$ may be true or false depending what $x$ is, obviously. There is yet another math-verb: we (attempt to) require of $x$ that this equation/statement be true, without "knowing" what $x$ is otherwise. This kind of thing reasonably confuses students... $\endgroup$ – paul garrett Jun 22 '19 at 22:21
  • $\begingroup$ I was saying the duck thing was true. I wonder if I can make that part clearer. $\endgroup$ – Sue VanHattum Jun 23 '19 at 17:11
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    $\begingroup$ This answer is clear, correct, concise, insightful, informative, and generally a better answer than I would have thought possible upon first reading the question. It clearly comes from a master teacher who has spent serious time thinking deeply about the meaning of their discipline. Since it doesn't look like the OP is selecting anything as the accepted answer, I decided to highlight it with a bounty. $\endgroup$ – Daniel R. Collins Oct 28 '20 at 4:15
  • $\begingroup$ Why thank you. Although I'm a moderator here, my time on stack exchange started with the birth of mese, and I still don't know all the ins and outs. I was just thinking that the points are fun, but of course my real motivation is to educate and share my love of math. (One of my motivations at the start was to be sure that the voices of k12 folks and homeschoolers were not discounted. After joining, another motivation was that a woman's voice be heard.) I don't really get the bounty thing, but it's sweet. $\endgroup$ – Sue VanHattum Oct 28 '20 at 16:46
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Your question is based on a false premise.

However, in the equation $2x=4$, $x$ can only be the value or values (from the some given set of real numbers) that makes the equation true.

No! In the equation "$2x=4$", $x$ is merely a variable, and the equation is meaningless without any further context. You can ask many different questions about that equation, such as what real $x$ satisfies it, or what is its graph in the (cartesian) $(x,y)$-plane (it is a vertical line), or what are the free variables in it (just $x$), and so on. Furthermore, that equation may be just one part of a larger expression such as "$∀x∈\mathbb{R}\ ( 2x = 4 ∨ ∃y∈\mathbb{R}\ ( (x-2)·y = 1 ) )$", in which certainly $x$ is not only "the value or values that makes the equation true".


"$2x=4$" is an equation, but equations do not imply anything about the symbols in them. "What real $x$ satisfies the equation $2x=4$?" is a question with a well-defined answer. If you do not specify "real", then it is not only not a well-defined question, but also a common example of imprecision. Namely, "Solve $2x=4$." is not a precise question.

An equation is a (mathematical) statement of the form "$A = B$" where $A,B$ are terms (also called expressions). A statement has a truth-value once all its free variables have been bound (i.e. their values are specified). Namely, "$2x=4$" has a truth-value once you specify what $x$ is. Hence it is meaningful to ask what real $x$ satisfies that equation (i.e. what value[s] for $x$ makes that equation true).

"$2x$" and "$4$" are simply terms, and "$2x$" is a compound term built from the terms "$2$" and "$x$", where multiplication has been represented by juxtaposition. Yes, every variable is also a term, and as with statements, terms are meaningless until you specify the values of all its free variables.

So the $x$ in "$2x$" and "$2x=4$" are actually exactly the same kind of variable, and mean exactly the same thing. Just as you can ask a question about the equation "$2x=4$" (e.g. what real $x$ satisfies it?), you can likewise ask a question about the term "$2x$" (e.g. what real $x$ makes it equal to $4$?). No difference.

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    $\begingroup$ I'd have to disagree: the $x$ in $2x$ could be the default name for the input to a function $f(x)=2x$... Sure, all these things are related, but they are not quite the same. $\endgroup$ – paul garrett Jun 22 '19 at 22:22
  • $\begingroup$ @paulgarrett: Please carefully read my comments about that starting here. $\endgroup$ – user21820 Jun 23 '19 at 15:13
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    $\begingroup$ Well, yes, I agree that it is possible to make arguments that "they are the same", as well as make arguments that "they are different". Debate team and all that. But a more sincere answer, including not only denotation but connotation, and intelligible to "intellectually naive/honest" students, is that they are subtly different, I think. After all, what kind of "x" is it that allows itself to be replaced by a matrix, in the Cayley-Hamilton theorem? $\endgroup$ – paul garrett Jun 23 '19 at 15:27
  • $\begingroup$ @paulgarrett: I have an answer to that question, if you are interested, but it is in my opinion unrelated to this question. You can find me in this chat-room. $\endgroup$ – user21820 Jun 23 '19 at 15:31
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I have read much of what is here, so, I'd wager the answer I'm about to offer will not be welcomed by some.

However, I think it might be useful to students to give the following answer:

  • In $2x$ we have variable $x$ appearing in the expression $2x$.
  • In $2x = 4$ we have variable $x$ being constrained by the equation $2x=4$. We could also call $2x=4$ a condition.

Really, $x$ is the same thing in both cases. It is a label for a quantity we do not know.

In both cases we assume $x$ can be multiplied by $2$. There is often a natural context for possible choices of $x$ given the source of the question. Especially if this is from an applied problem it may make sense to assume $x$ is a real variable. But, I think (depending on the student's attention span) expanding the point of context may be unwise in the first pass.

At least this is one reasonable interpretation before we find more abstract algebraic ways to think about $x$. Those abstract algebraic interpretations can wait for the next grade since this is just a question for the 10-year-old crowd.

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I would encourage you not to teach your students that an equals sign represents a question or implies that an action should be taken. Many students already struggle to interpret the equals sign as relational (indicating equality or balance) instead of operational (indicating an action to be taken). The Importance of Equal Sign Understanding in Middle Grades by Knuth et al. discusses the difference, and notes that students with a relational view of the equals sign perform better at equation-solving than students with an operational view.

In addition, when students are presented with an identity, such as $\sin^2t+\cos^2t=1$ which is true for all real numbers $t$, do we want our students to see the equals sign and assume their goal is to attempt to solve the equation for $t$?

I, too, have been searching for ways to help my students understand what seem to be contradictory uses of "variables" in mathematics.

Susanna S. Epp suggests that we define variables as placeholders, and this is what I do now. I would argue that we absolutely should tell students that $x$ has different meanings in different contexts, and that it is the context we are given and the questions we are asked about the expression, equality, number sentence, mathematical phrase, etc that give meaning to $x$. The context and the questions indicate how we should use or manipulate the expression, equality, number sentence, etc that we are given. In one context the variable could be a placeholder for finitely many numbers that make the equality (or inequality) true. In another context, like the Pythagorean identity above, the variable could be a placeholder for any real number. The variable could represent a fixed quantity, or could represent a quantity that changes. Variables can also be placeholders for angles and points and matrices and a myriad of other mathematical objects that are not real numbers.

The values that can be represented by $x$ in the single equation $(x^2+5)(x+1)(x-.5)=0$ depend on context; are we looking for all real numbers, all complex numbers, or perhaps all integer values that satisfy the equality?

Questions and context such as

  • Find all real numbers for which the equality is true.
  • Show the equality holds for all real numbers.
  • Consider the points $P(x,y)$ with integers $x$ and $y$ defined by $x^2+y^2=4$.

will tell us what the placeholders are saving places for.

I have seen attempts to differentiate between and define symbols used for variables, parameters, and constants. I like this treatment from Calculus: Newton, Leibniz, and Robinson Meet Technology which uses multiple cylinders to illustrate the difference between a constant ($\pi$) which doesn't change between cylinders, a parameter (the radius $r$) which is fixed in a given cylinder, but changes with different cylinders, and a variable $x$ representing the depth of water in the cylinder, which can take on different values even with a specific cylinder. Again, context matters.

(I really notice which students have a robust understanding of variable and symbol use when we start using $\pi$ to represent a permutation, and not the ratio of circumference to diameter they are used to.)

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I think perhaps the definitive discussion of this topic was written by Zalman Usiskin in "Conceptions of School Algebra and Uses of Variables", originally published as a chapter in the 1988 NCTM Yearbook, The Ideas of Algebra, K-12 (A. F. Coxford and A. P. Shulte, eds), and later reprinted in the 1999 book Algebraic Thinking, Grades K–12: Readings from NCTM’s School-Based Journals and Other Publications (B. Moses, ed).

Usiskin opens with an enumeration of different cases:

Consider these equations, all of which have the same form -- the product of two numbers equals a third:

  1. $A=LW$
  2. $40=5x$
  3. $\sin x = \cos x \cdot \tan x$
  4. $1 = n \cdot (1/n)$
  5. $y = kx$

Each of these has a different feel. We usually call (1) a formula, (2) an equation (or open sentence) to solve, (3) an identity, (4) a property, and (5) an equation of a function of direct variation (not to be solved). These different names reflect different uses to which the idea of variable is put. In (1), $A$, $L$, and $W$ stand for the quantities area, length, and width and have the feel of knowns. In (2), we tend to think of $x$ as unknown. In (3), $x$ is an argument of a function. Equation (4), unlike the others, generalizes an arithmetic pattern, and $n$ identifies an instance of the pattern. In (5), $x$ is again an argument of a function, $y$ the value, and $k$ a constant (or parameter, depending on how it is used). Only with (5) is there the feel of “variability,” from which the term variable arose. Even so, no such feel is present if we think of that equation as representing the line with slope $k$ containing the origin.

The article is far too long and thorough to successfully summarize here, but here is a brief summary that I wrote in my own book, Secondary Mathematics for Teachers and Mathematicians: A View from Above (see p. 114 for context):

Usiskin (1999) distinguishes between multiple distinct notions of “variable” in school algebra. He describes the use of variables as pattern generalizers, unknowns (to be solved for), arguments (to be substituted in to functions), parameters, and referent-free symbols (“marks on paper”). Usiskin also calls attention to “the question of the role of functions and the timing of their introduction”:

It is clear that these two issues relate to the very purposes for teaching and learning algebra, to the goals of algebra instruction, to the conceptions we have of this body of subject matter. What is not as obvious is that they relate to the ways in which variables are used... 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 related. 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. (pp.8–9, emphasis in original) Usiskin identifies four distinct “conceptions of Algebra”, each corresponding to a different use of variables. Algebra is, in his analysis, (1) a generalization of arithmetic, (2) the study of procedures for solving certain kinds of problems, (3) the study of relationships among quantities, and (4) the study of structures. When we (for example) study the graph of a polynomial function or inquire after its zeros, we are primarily attending to uses (2) and (3); in that context, a variable stands for an unspecified element of a replacement set. On the other hand, when we factor polynomials we are primarily focusing on (4). In that context, variables stand for indeterminates; that is, “marks on paper” that are to be manipulated without attending to what they stand for.

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For the function $f(x)=2x$, the letter $x$ is a variable; it can have any value in the domain of $f$. For the polynomial $p(x)=2x$, $x$ is an indeterminate that is just a placeholder. We can write the polynomial $p$ without it : $p=(0,2,0,0,\ldots)$.

$2x=4$ is an (algebraic) equation and $x$ is the unknown to be found.

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    $\begingroup$ @user21820 "Let $f: \mathbb{R}\longrightarrow\mathbb{R}$ by $f(x)=2x$" is often shortened to "let $f(x)=2x$". $\endgroup$ – user5402 Jun 11 '19 at 14:18
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    $\begingroup$ @user21820 I can accept y=2x as equation (of two variables). Equation evaluates to true or false when variables are replaced with actual values, like (2,4) turns the above equation true. But f(x) = 2x has just one independent variable, and the dependency is clearly identified with f(x) expression. Semantically this defines a function. I do agree that typical mathematical pedagogy is defective, though. ;-) $\endgroup$ – Rusty Core Jun 12 '19 at 21:39
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    $\begingroup$ @user21820 Funny that you wrote in the chat what to me sound the same as I wrote: "How do we refer to the output of f on some input x? We use the notation "f(x)" to denote exactly "the output of f on input x". " $\endgroup$ – Rusty Core Jun 12 '19 at 21:58
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    $\begingroup$ @user21820 This type of language shortcuts is everywhere. For example we say "the area of a triangle" even though a triangle is just three segments. We use L'Hôpital's rule on a sequence, we use the same function $f$ to denote $f:E\longrightarrow F$ and $f:\mathcal{P}(E)\longrightarrow \mathcal{P}(F)$, etc. $\endgroup$ – user5402 Jun 13 '19 at 13:56
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    $\begingroup$ I removed the comments discussing things other than the answer and left the productive discussion here. $\endgroup$ – Chris Cunningham Jun 14 '19 at 12:15
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About a week ago I ordered 7 or 8 books from amazon (I do this 2 to 4 times a year, depending on how much I can afford to spend), and yesterday Wooton’s book arrived (bibliographic details below). This is the book that, in my answer to Where can I find primary sources from the New Math movement in the 60s?, I said I don’t believe I’ve ever looked at before. At some later time I might archive some information about Wooton’s book in that earlier answer. However, as I was reading through his book the past couple of days, I came across some comments that I thought would fit well in this thread.

William Wooton (1919-1988), SMSG. The Making of a Curriculum, Yale University Press, 1965, x + 182 pages.

REVIEWED BY:

Harry Merrill Gehman (1898−1981), Science (NS) 150 #3693 (8 October 1965), p. 202.

Bryan Thwaites (1923−___), Mathematical Gazette 50 #374 (December 1966), pp. 403−404.

Robert Marion Todd (1928−2015), Arithmetic Teacher 14 #3 (March 1967), p. 232.

5-paragraph excerpt from pp. 26−29:

A detailed description, here, of the outline or of the philosophy underlying the 9th grade text would involve mathematical considerations beyond the scope of this work. To show the nature of the dialogue involved, however, it seems worthwhile to look at one of the more fundamental problems the group faced. Most persons who, at one time or another, have been exposed to a course in elementary algebra have little difficulty recalling that central to the subject is the use of letters of the alphabet as symbols. In particular, the letter $x$ comes easily to mind. It is around the use of such symbols that some of the “modern” controversy centers.” Traditionally, such symbols have been referred to as “unknowns,” “literal numbers,” “general numbers,” and “variables,” in some cases depending on context, in others depending on the whim of the user. The student has been told that these symbols are unknown numbers and that, in some cases, he can add these numbers $(x+x = 2x)$ while, in other cases, he cannot (e.g., $x+y).$ He has been instructed in ways of “finding the unknown,” as, for example: if $x+2 = 5,$ then $x=3.$ But, on the other hand, he has been told that he is not to try to “find $x$” when writing such things as $x(x+2) = x^2 + 2x.$ This is due to the fact that the usual manipulations in which he becomes involved when working with such symbol groups produce $\text{O} = \text{O},$ which, though true, does not seem to tell him anything he wants to know about $x.$ Worse, should he inadvertently apply his ingenuity to $x = x+1,$ he arrives at the mystifying $\text{O} = 1,$ which not only does not tell him anything about $x,$ but causes him to call into question the sensibilities of anyone who finds interest in a discipline that deals with such absurdities.

It is one of the concerns of those seeking to revise the mathematics curriculum to make the meaning of such symbols clear to students, and to place their use on a sound logical foundation. Granting this, however, the best way to establish such a foundation is a matter of much controversy (hence, one reason for the lengthy discussions of the 9th grade subgroup). Present-day logicians, in their inquiries into the foundations of mathematics, have had occasion to use the notion of what they call a “placeholder,” and it was this viewpoint that UICSM had adopted. Briefly: a number is an abstraction. Nobody has ever heard, felt, or seen a number, but the body proper of mathematics stems from the fact that the human mind can conceive of such abstractions. Furthermore, in discussing numbers, symbols are used which are called, in some cases, numerals, and in others, pronumerals, placeholders, or variables. The logicians, having found it necessary to work at varying levels of abstraction, have come to view a symbol such as “$2$” not as a number, nor even as the name of a number, but rather as a representation of the name of a number. Another representation of the same name of the same number is “two.” A number has infinitely many names; for example, another name for the number whose name can be represented by “$2$” is the name represented by the symbolism “$1+1.$” A symbol such as “$x,$” then, can be viewed as holding a place in an expression such as “$x+5$” for a representation of a name of a number, hence the name “placeholder.”

A step from the logicians are those who believe that the distinction between a name and its representation is unnecessary in all but the deepest discussions of foundations, and that it is sufficient to distinguish between the number itself and its name. Thus, the symbol “$2$” is the name of a number, just as the word “Tom” is the name of a person. At this level of abstraction, however, there are various ways of viewing symbols such as “$x.$” In one view, when one is considering the expression “$x+5,$” the symbol “$x$” is conceived of as holding a place for the name of a number and is thus, conceptually, a “placeholder.” In another view, the symbol “$x$” is used in the expression “$x+5$as the name of a number and is called a “variable.” Inherent in both viewpoints, of course, is the agreement that there is some specified set of numbers with which the symbol “$x$” is associated.

Although there is almost universal agreement on the importance of distinguishing between a number and its name, the best way, mathematically, and pedagogically, to view the use of a symbol such as “$x$” is the center of much controversy. In view of the eminence of many proponents of each point of view, it would appear that, for the present, the way in which such symbols should be handled depends on the spirit in which the subject matter of algebra as a whole is handled, and this is chiefly a matter of who is doing the handling. It should be apparent, however, that the question is not a trivial one, since, in a sense, the way in which it is resolved determines to a greater or lesser extent the spirit in which the subject of algebra is taught. The question is essentially one of the “level of abstraction” and, barring gross misrepresentations, the argument reduces as much to a pedagogical matter as it does to a mathematical one. The problem of finding the level of abstraction appropriate to the cognitive readiness of the student is a very real one, and the definitive answer, if such exists, has not yet been found.

To return to the matter at hand, the question of how to treat symbols was one (though not the only one) of the causes for the lengthy pre-discussions held by the 9th grade group before beginning detailed outlines of chapters, and their final decision was not free from critical attack by proponents of alternative viewpoints, as was to be expected.

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    $\begingroup$ This would be improved by at least summarizing what you think the take-away from this long quote is. I'm not sure it really responds to the question in a meaningful way. $\endgroup$ – Daniel R. Collins Oct 26 '20 at 5:52
  • $\begingroup$ @Daniel R. Collins: I'll revise this sometime in the next couple of days. I was busy with other stuff yesterday (as always seems to be the case lately), and I used up more time than I should have typing in the excerpt (and then carefully proofreading it) and looking up the reviews (especially the reviewer name info. stuff), so I didn't try to do anything more than just throw it out for others to read. I suppose the introductory stuff can be put in a comment, and I'll try to say something about the "Present-day logicians, in their inquiries $\ldots$" part in reference to the OP's question. $\endgroup$ – Dave L Renfro Oct 26 '20 at 17:32
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With older students (university age, or maybe late school) I'd be inclined to try and show them that they probably don't really know what they are asking, rather than actually trying to give a definitive answer (which is much harder to do properly).

Write $y-2x=3$ next to a pair of axes. Write it again separately with $y+x=1$ underneath. Now ask if the $x$s are the same.

I would expect most students to think of the first instance as a line to plot, while the second is some equations to solve. While these are in some sense the same, I think a student who asks the opening question probably means these to be considered as different. Using the same equation in both cases helps to show that the issue isn't so much adding $=4$ to $2x$ as how you choose to think about it.

Telling the students to interpret the equals sign as asking a question is great, so long as they understand in which contexts they should think of it that way and when they should think of it a different way. (That students generally don't know this is, I think, probably one of the big reasons they find maths confusing.)

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    $\begingroup$ Does the downvoter want to explain what's wrong with this as an answer? $\endgroup$ – Jessica B Jun 18 '19 at 6:18
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    $\begingroup$ Oh, I see now that it was meant to be a system of two equations to solve... Maybe the same point can be made by writing once $y-2x=3$ and then something like $5-2x=3$. $\endgroup$ – Michael Bächtold Jun 18 '19 at 7:14
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    $\begingroup$ I don't quite understand that: Solving a system of two linear equations with two variables is ok, while a solving a single linear equation with one unknown would confuse students. $\endgroup$ – Michael Bächtold Jun 18 '19 at 16:09
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    $\begingroup$ @MichaelBächtold Don't expect the students to think. They solve the two using a fixed algorithm. Using it with only one variable in one equation requires modification of the algorithm. $\endgroup$ – Jessica B Jun 18 '19 at 17:08
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    $\begingroup$ "Don't expect the students to think. They solve the two using a fixed algorithm." — Who taught them this fixed algorithm? Who taught them not to think? Rhetorical questions. $\endgroup$ – Rusty Core Jun 19 '19 at 5:46

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