Different Kinds of Variables

Question

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?

Answer 1

$2x$ 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.)

Answer 2

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".

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"$2x=4$" 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.

Answer 3

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.

Answer 4

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.)

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.)