Problems Identifying Independent and Dependent VariablesKe.docx

Problems Identifying Independent and Dependent Variables
Keith R. Leatham
Brigham Young University
This paper discusses one step from the scientific method—that
of identifying independent and dependent variables—
from both scientific and mathematical perspectives. It begins by
analyzing an episode from a middle school mathematics
classroom that illustrates the need for students and teachers
alike to develop a robust understanding of independent and
dependent variables. It then outlines four rationales (two from
science and two from mathematics) for identifying
independent and dependent variables. Finally, it reports the
results of a textbook analysis that used these rationales to
examine the extent to which typical mathematics textbook
problems support or supplant a sensible view of independent
and dependent variables. The findings indicate that often,
mathematics textbook problems misrepresent the sense-
making aspect of identifying independent and dependent
variables, possibly setting students up to develop misconcep-
tions about this step from the scientific method.
Problems Identifying Independent and
Dependent Variables
The scientific method has had great influence over
science textbooks over the past century (Blachowicz,
2009). Although there is debate about the degree to which
this influence has been positive (Bauer, 1994; Nola &
Sankey, 2007), most scientists agree that viewing the sci-
entific method as a “recipe for doing science” is unhealthy

and counterproductive. Instead, the scientific community
desires that students, scientists, and the general populace
see the work of a scientist as that of exploring and trying
to understand the world around us. Their basic argument is
that “doing science” is exploratory, nonlinear, and cre-
ative. The scientific method should help one to make sense
of scientific work, not remove the need to make sense of
things (Bauer, 1994; Kuhn, 1970; McComas, 1996).
Trying to distill this work into a linear, decontextualized
set of steps trivializes and misrepresents the actual work.
Similar arguments have been made with regard to other
subjects such as mathematics (Hiebert et al., 1997), statis-
tics (Tukey, 1977), and music (Elliott, 1994). In each case,
the argument is that disciplines are made of more than
mere procedures; there are important underlying concepts
and ideas that give meaning to procedures and their
application.
In this paper, I discuss one step from the scientific
method—that of identifying independent and dependent
variables—from both scientific and mathematical perspec-
tives, scientific rationales because of the grounding of the
topic to the scientific method, and mathematical rationales
because of the preponderance of such problems in math-
ematics textbooks. In doing so, I highlight ways procedur-
alized application of these rationales can easily supplant
the underlying conceptual ideas to which they are related.
I also discuss the danger in removing the disciplinary
focus (in this case, mathematical or scientific) when learn-
ing to identify and classify variables. I begin by discussing
an episode from a middle school mathematics classroom
that illustrates the need for teachers and students alike to
develop a robust understanding of independent and depen-
dent variables. I then share four rationales (two from
science and two from mathematics) for identifying inde-

pendent and dependent variables. Finally, I report the
results of a mathematics textbook analysis that used these
rationales to examine the extent to which typical math-
ematics textbook problems support or supplant a sensible
view of independent and dependent variables.
A Classroom Example
Before we discuss in some detail various scientific and
mathematical issues related to determining independent
and dependent variables, consider an episode from a
middle school pre-algebra classroom that illustrates how
these issues can collide.1 The teacher, Ms. Christina2 (a
student teacher in this classroom), was launching a lesson
wherein students were going to input equations into a
calculator and look at the tabular outputs to make deci-
sions about the situations modeled by those equations. In
anticipation of this approach, Ms. Christina asked the stu-
dents to identify the independent and dependent variables
in the equations A = pr2 and C = 2pr.
Now, before proceeding with the episode, consider the
nature of these algebraic “situations.” These equations rep-
resent generalized mathematical relationships, although
not completely generalized, as it is clear in these equations
that the variables stand for particular, contextualized quan-
tities (as opposed to mere abstract quantities). Ms. Chris-
tina’s question indicates that students should be able to
School Science and Mathematics 349
determine the independent and dependent variables here,
possibly by looking at the form of the equations or maybe
by considering the contexts the equations represent.

As the lesson proceeded, it quickly became clear that
Ms. Christina wanted to hear that r was the independent
variable and that A and C were the dependent variables in
their respective equations. After a student had shared his
response that r was the independent variable and A was the
dependent variable in the former equation, Morgan said
that she thought C was the independent variable and r was
the dependent variable in the latter equation. Because this
response was not what Ms. Christina expected, she asked
Morgan to explain her reasoning, in hopes that in so doing,
the “correct” answer would be revealed. We join the lesson
at this point:
Morgan: I did the circumference because the
radius depends on how big or small
the circle is. So I said the circumfer-
ence is independent and the radius is
dependent on the circumference.
Ms. Christina: Okay. Thanks Morgan. Who has the
same thing as Morgan? Who has
something different? Who doesn’t
know? [pause] Who said they have
something different? . . . Kathryn, do
you? Do you want to explain?
Kathryn: I just said the independent would be
the radius and the dependent would be
the circumference.
Ms. Christina: Okay, why?
Kathryn: Because . . . the circumference is
the—. Wait, no, I agree with her
[Morgan].

Ms. Christina: Are you sure? You were going good
there. Do you want to keep explaining
what you were saying?
Kathryn: I was going to say that the circumfer-
ence would change if the circle gets
smaller. But um, you can find the
circumference without the radius, I
think.
Ms. Christina: You can find the circumference
without the radius? How would you
do that?
Kathryn: Um. I don’t know.
.
.
.
Ms. Christina: Brian, what do you think?
Brian: Couldn’t kind of both of them go both
ways? Because like in area, like as the
area gets smaller so does the—. Oh,
never mind.
At this point in the episode, there is a fairly long pause as
Ms. Christina considers where to go next. She asks Abe,
who has not participated thus far, what he thinks and he
indicates that he does not really know what is going
on. Ms. Christina takes this opportunity to explain,
with input from the class, the definitions of circumfer-
ence and radius. Ms. Christina then continues her line of
reasoning:
Ms. Christina: So we’ve got this radius and we are

saying, “What happens as we change
this radius? What does it do to our
circumference?”
Abe: It gets smaller?
Ms. Christina: It gets smaller. So do you see how that,
how it depends on it? And so do you
see why we would switch those?
[Meaning switch Morgan’s responses
around, so r is independent rather than
C] Morgan, does that kind of make
sense?
Morgan: Yeah.
Ms. Christina: Okay, how does that look? [r is now
labeled as the independent variable in
both situations.] Are we okay? Who
agrees with me? I’m getting two
thumbs up from Madison. Who dis-
agrees? Kathryn, do you? Okay. It’s
okay if you do. It’s okay to disagree
with the teacher.
With no further disagreements from the class, Ms. Chris-
tina attempts to move ahead with the lesson. She notices,
however, that Kathryn is still puzzled and is wrestling with
whether she should raise her hand.
Ms. Christina: Kathryn, do you have—?
Kathryn: I have a question about that. In
elementary school we measured a
circle with some string. And so, if you
didn’t have the radius, couldn’t you

also measure the circumference
without the radius?
Ms. Christina: With using that string?
Kathryn: Yes.
Identifying Independent and Dependent Variables
350 Volume 112 (6)
Ms. Christina: Good job. Yeah, you can. We can do it
different ways. Maybe we will do that
sometime. We will be able to figure
that out. But yeah, you are right. There
are different ways that, maybe if you
don’t have the—
Kathryn: Why is the radius the independent?
In this classroom excerpt, Ms. Christina and her stu-
dents seem to be engaged in different, although related,
activities. Ms. Christina is operating under the assumption
that the form of an equation reveals the independent and
dependent variables. The students, on the other hand, seem
to be operating under a different assumption. Their focus
is on whether it is possible to find or control one variable
without needing to know the other. If they can control one
variable without knowing the other, then that variable is
independent. This assumption becomes problematic in this
situation, however, when reasonable arguments are put
forth for being able to determine either variable without
first knowing the other.3

In this section, I discuss various rationales (such as
those used in the preceding classroom episode) one might
use in identifying independent and dependent variables. I
describe and contrast scientific and mathematical ration-
ales. A review of the literature revealed no treatises to date
on such rationales, nor on students’ understanding of inde-
pendent and dependent variables. Thus, these rationales
are derived from a conceptual analysis of independent and
dependent variables and the ways they are commonly used
in science and mathematics. Therefore, I present a frame-
work of rationales that proved useful in the textbook
problem analysis that follows and that can be tested, devel-
oped, and refined through applying it in future research.
Scientific Rationales
When conducting formal experiments, scientists often
seek to identify the variables involved in a phenomenon.
They also seek to identify the effect of these variables on
each other. Such work entails isolating, controlling for,
and measuring varying quantities in order to posit relation-
ships among them. For example, a study of crickets might
lead one to wonder at the variation in their chirp frequen-
cies. The wondering might lead to initial data collection
and exploration in various contexts. Eventually, one might
hypothesize that chirp frequencies are related to the time
of day, the time of year, the ambient temperature, or maybe
the size or age of the cricket. Eventually, the following
conjecture might be made: Cricket chirp frequency is
related to ambient temperature. An experiment could then
be designed wherein one places crickets in environments
of varying temperature and records their chirp frequency.
Other variables such as age, sex, or time of day or of year
could also be held constant or controlled in order to ascer-
tain their possible relation to chirp frequency. In this

example, there are three types of variables—independent,
dependent, and constant. The constant variables are the
ones we hold constant (e.g., age, sex, time of day); the
independent variable is the one over which we have control
and which we choose to vary (in this case, ambient tem-
perature); the dependent variable (in this case, chirp fre-
quency) is the one over which we do not have direct
control and whose variance we seek to measure as we
change the independent variable.
When viewed in the context of an experiment, the
process of identifying the independent and dependent vari-
ables in a situation involves considering the context and
the purpose of the experiment. It also involves considering
what you can or do control. Identifying independent and
dependent variables in a situation void of experimental
context or causation means very little. A robust under-
standing of independent and dependent variables from the
scientific standpoint recognizes it is up to the individual
who is exploring a relationship between variables to
choose (not so much determine) the independent and
dependent variables. Although context (including issues of
causality and control) certainly helps one to make this
decision and also helps one to determine just how reason-
able that decision is, in many situations, context is insuf-
ficient; purpose is preeminent.4 The upshot of this
requirement for understanding and decision making is that
it makes little sense and is actually misleading from a
scientific standpoint to ask someone to identify the inde-
pendent and dependent variable in a decontextualized situ-
ation, as the answer is almost always, “It depends on what
your are doing and the questions you are asking.”
Mathematical Rationales
One significant purpose of mathematics is to describe,
generalize, and abstract relationships between varying

quantities. At its most fundamental level, mathematics
seeks to describe relationships. Theorems, formulas, and
algorithms all have at their core the desire to describe what
always happens under given circumstances. The more this
description can be abstracted from the context and
described in terms of the underlying mathematical struc-
tures and relationships, the better.
In mathematics, we have many different ways of repre-
senting relationships between variables. For example (the
rule of four): verbal, symbolic, graphic, numeric. There
are certain meanings or conventions associated with each
of these representations. In the symbolic representation, y
= 2x + 1 can be read as the equation of a line where x is the
independent variable and y is the dependent variable. We
do this by convention, however, as the equation y = 2x + 1
actually only defines a relationship between x and y. There
are at least two conventions at play when we assume that x
is the independent variable here: (1) In general, we use the
variable x to be the independent and y to be the dependent.
The choice of letter actually carries meaning; and (2) the
equation is “solved for y,” meaning, y is isolated from the
x and constant terms. A (somewhat implicit) convention is
that when we have equations solved in this manner, we
intend people to read the equation like this: Given an x
value, what is y? In other words, the form of the equation
somehow tells us which variable is independent and which
is dependent.

However, mathematics generally eschews these conven-
tions. There is nothing sacred about x—any placeholder
will do. And one reason for introducing function and func-
tion notation is because y = 2x + 1 does not provide
sufficient information to determine the independent and
dependent variables. One could as easily write this rela-
tionship as p = 2k + 1 or x y= −( )
1
2
1 (okay, perhaps not as
easily in the latter case, but certainly as valid). When we
write the equation as f (x) = 2x + 1, however, the notation
itself defines the independent and dependent variables.
Our decision to write the relationship as a function has
indicated our choice of independent and dependent
variables.
A robust understanding of independent and dependent
variables in mathematics includes an understanding of the
difference between a relation and a function. A relation
carries no meaning when it comes to identifying indepen-
dent and dependent variables; it simply communicates a
relationship between variables. A function, on the other
hand, carries all the meaning when it comes to identifying
independent and dependent variables; by definition, in y =
f (x), y is the dependent variable and x is independent. In
addition, certain mathematical conventions for given rep-
resentations (whether advocated or not by the mathemati-
cal community) often complicate matters. Thus, a robust
understanding of independent and dependent variables in
mathematics includes an understanding of the affor-
dances and constraints of such conventions for various
representations.5

Comparing Scientific and Mathematical Rationales
Thus far, I have argued that the primary rationales for
identifying independent and dependent variables in science
and in mathematics differ. In science, the main rationales
are control and causation; in mathematics, the main ration-
ales are convention and function (Table 1). In mathematical
and scientific contexts, both logic and choice play a role in
applying these rationales. For example, consider again our
previous example of the experiment involving number of
cricket6 chirps per minute (N) and temperature in degree
Fahrenheit (T). According to Dolbear (1897), there is
indeed a relationship between these two variables, which
can be written as follows: N = 4 (T - 50) + 40. And this
relationship is a causal one—the temperature influences or
causes the change in number of cricket chirps per minute.
So changes in T cause changes in N.
So how does one determine the independent and depen-
dent variables in this situation? From a “control” stand-
point, the only thing you have control over is the
temperature—you can alter the temperature and measure
the number of chirps per minute. The alternative experi-
ment is ludicrous. In addition, although one could cer-
tainly attempt to change (i.e., control) the number of
chirps per minute by some means other than temperature
(biologically, socially, medicinally), we certainly do not
expect such changes to influence the ambient temperature,
or even the temperature of the cricket. There is no reason
to believe that N causes T, regardless of whether one could
control N. Thus, when considering the scientific rationales
of control and causation, it is reasonable and defendable
to define T as the independent variable and N as the
dependent.

This situation looks somewhat different, however, when
viewed with the mathematical rationales, for there is
clearly a relationship between T and N. It is certainly a
meaningful question to ask, “Given this number of chirps
per minute, what is the approximate temperature?”
In actuality, that was Dolbear’s (1897) point in presenting
his findings: “The rate of chirp seems to be entirely
Table 1
Descriptions of Scientific and Mathematical Rationales for
Determining Inde-
pendent and Dependent Variables
Rationale Description
Scientific
control
Whether one has the ability to directly control
the values of a given variable.
Causation Whether variation in the values of a given
variable cause changes in another variable.
Mathematical
convention
Whether there are conventional notations or
practices that communicate which variable
should be thought of as independent.
Function Whether the mathematical relation at hand is
defined more specifically as a function.

352 Volume 112 (6)
determined by the temperature and this to such a degree
that one may easily compute the temperature when the
number of chirps per minute is known” (p. 971). In fact,
Dolbear’s original formula is not solved for N as earlier,
but rather for T: T N= + −( )50 1
4
40 . One particularly
valuable result of this experiment is that one can think of
the dependent variable from the experiment (N) as the
independent variable. (In other words, Dolbear found a
relationship between variables.) Once that relationship is
represented in symbolic form, either variable can be
chosen as the dependent variable, so appealing to the func-
tion rationale does not determine dependence. By contrast,
by convention (solving for T), which Dolbear used more
than 100 years ago, the intention of the formula is for N to
be independent and T to be dependent. This example thus
illustrates how the scientific and mathematical rationales
can lead to quite different determinations.7
Textbook Analysis
Analysis of the classroom episode presented earlier left
me wondering how one “should” identify the independent
and dependent variables in a given situation. My answer to
that question is the conceptual descriptions for scientific
and mathematical rationales just outlined (and summa-
rized in Table 1). It might be tempting for the reader to

assume that the primary cause of the independent/
dependent variable dilemma discussed in the episode was
the inexperience and somewhat misguided mathematical
conception of the student teacher (i.e., unquestioned reli-
ance on mathematical convention). I had similar thoughts
as I analyzed the situation. But the student teacher’s under-
standing of independent and dependent variables origi-
nated somewhere, likely in her own middle school
mathematics experiences, and mathematics textbooks both
reflect and influence the nature of classroom mathematics.
Because of this important relationship, many researchers
have used textbook analyses as one lens to explore the
mathematics students are learning (e.g., da Ponte &
Marques, 2007; Son, 2005; Vincent & Stacey, 2008; Zhu
& Fan, 2006). How do mathematics textbooks address the
identification of independent and dependent variables? In
particular, what is the nature of textbook problems asso-
ciated with this identification? I conclude this paper with
the results of a small textbook analysis study designed to
begin to answer this latter question.
Analysis began with an ad hoc review of textbooks from
publishers with multiple mathematics textbook series
(Glencoe McGraw-Hill and McDougal Littell) in order to
get a sense of where in curricula series the concept of
independent and dependent variables seemed to be placed.8
We discovered it tended to be discussed in Algebra I texts,
but was sometimes located in pre-algebra texts, and some-
times in Algebra II and precalculus texts. With this infor-
mation in mind, we then scoured our curriculum library for
the latest edition of every Algebra I (or roughly equivalent)
textbook we could find, as well as a sampling of the other
textbooks just mentioned. Our intent was to locate a sub-
stantial number of problems associated with independent

and dependent variables so as to analyze their nature, not to
analyze the textbooks per se. As such, the textbooks were
purposefully selected in an attempt to create a collection of
typical problems. In the end, our sample consisted of a
collection of 23 textbooks (Table 2). We then reviewed each
text looking for any discussion of variables and, in particu-
lar, independent and dependent variables. We scanned each
chapter and analyzed the table of contents and index. We
sought to identify every instance where the textbook asked
students to identify independent and dependent variable.
Our search of 23 textbooks yielded a total of 73 problems
from 10 of these texts (Table 2). Thus, a number of these
textbooks contained no problems related to identifying
independent and dependent variables.9
We then analyzed each of these 73 problems according
to the scientific and mathematical rationales described pre-
viously (Table 1). For each rationale, we answered yes or
no to the question of whether one could reasonably deter-
mine the independent and dependent variables by appeal-
ing to that rationale. As an example of how this analysis
played out, consider the problem from the classroom
episode previously discussed. In a textbook, the problem
would be stated as follows:
Identify the independent and dependent variables: C =
2pr
We now consider each rationale:
1. Control: No—One can control either r or C (as so
eloquently argued by Kathryn in the classroom episode).
2. Causation: No—Varying either variable causes the
other to change.
3. Convention: Yes—The representation is symbolic

and solved for C, so one could appeal to convention to
argue that C is the dependent variable.
4. Function: No—The equation is presented as a rela-
tion, but not as a function.
Each problem can thus receive yes or no for each of the
four rationales, yielding a total of 16 permutations or
“types” of problems, as determined by they type of ratio-
nale one might use in order to solve the problem (Table 3).
The sections that follow consider several of the most inter-
esting subsets of these permutations.
No Reasonable Rationales
As can be seen in Table 3, almost half of the problems
received “No” in all four categories. This means that for
these 32 problems, the problem statements and situations
were insufficient to determine the independent and depen-
dent variables based on any of the four rationales. Let us
take a closer look at several NNNN problems. For
example, consider the following problem:
(A) Name the two variables involved. Explain which
you would list first and represent on the x-axis. Which
would you list second and represent on the y-axis?
Height and weight of players on a soccer team.
(Answer: [height, weight])

According to the textbook, there is a correct answer to
problem A. So, how could students determine that answer?
Table 2
Collection of Textbooks and the Number of “Identify the
Independent and Dependent Variables” Problems in Each
Title Series Textbook Publisher Number of “Identify”
Problems (n = 73)
Algebra I Glencoe/McGraw-Hill 12
Algebra: Concepts and
Applications
Glencoe/McGraw-Hill 2
MathMatters 3 MathMatters: An Integrated Program
Contemporary Mathematics
in Context: Course 1
Core-Plus Mathematics Project (CPMP) Glencoe/McGraw-Hill 8
Grade 8 Core-Plus Mathematics Project (CPMP)
Algebra I Holt 25
Algebra II Holt 3
Algebra I McDougal Littell 3
Algebra II McDougal Littell 1
Integrated Mathematics 1:
Algebra
McDougal Littell 0
Book 1 Mathematics Thematics McDougal Littell 0

Algebra University of Chicago School
Mathematics Project (UCSMP)
McGraw-Hill/Wright Group 0
Pre-Transition Mathematics University of Chicago School
Pre-Transition Mathematics University of Chicago School
Precalculus: Enhanced with
Graphing Unilities
Pearson 0
Algebra I Center for Mathematics Education
(CME)
Pearson 0
Grade 7 Connected Mathematics Project (CMP) Pearson 8
Grade 8 Connected Mathematics Project (CMP) Pearson 0
Algebra I Prentice Hall 6
Grade 8/7 Saxon Math Saxon Publishers 0
Functions Modeling Change: A
Preparation for Calculus
Wiley 5

Table 3
Summary of Categorization of the 73 Problems According to the
16 Possible
Permutations
Control Causation Convention Function Total
(n = 73)
N N N N 32
N N N Y 4
N N Y N 7
N N Y Y 2
N Y N N 11
N Y N Y 0
N Y Y N 5
N Y Y Y 0
Y N N N 3
Y N N Y 1
Y N Y N 0
Y N Y Y 0
Y Y N N 7
Y Y N Y 0
Y Y Y N 0
Y Y Y Y 1
354 Volume 112 (6)
Again, let us consider the four rationales. First, consider
causation. Although height and weight are likely corre-
lated (remember, these are middle school students answer-
ing this question, and no other data are provided), changes
in an individual’s height do not necessarily cause changes

in their weight, nor do changes in their weight cause
changes in their height. It could be argued here, however,
that the one is much more likely than the other. That is, it
makes sense that if you get taller, your weight will
increase; whereas if you gain weight, you may not grow
taller at all. Thus, it is likely that the textbook problem
expected students to use “causation” in order to determine
the solution, although changes in neither variable actually
cause changes in the other.
Considering the control rationale illustrates how confus-
ing such scenarios might be for students. We have no direct
control over our weight or our height. But of these variables,
the only one that is even reasonable to try to control is
weight. Managing to change our weight, however, does not
cause changes in our height. Thus, the one variable we
might try to control does not cause change in the other. The
other variable, height, we have no control over, yet changes
in height are likely to accompany changes in weight.
There are no verbal conventions to appeal to in this
situation, but notice the use of conventions in the state-
ment of the problem. Students are not actually asked to
identify the independent and dependent variables, but
instead are asked to identify which variable they would list
first (in an ordered pair) and place on the x-axis. Thus,
although convention does not help in determining whether
height or weight is the independent variable, an under-
standing of convention is necessary in order to answer the
question as intended. Finally, the situation implies a rela-
tionship between height and weight but not necessarily a
function. None of the four rationales considered here
would be adequate for merely determining the indepen-
dent and dependent variables in this problem (as
requested), but one could certainly justify a sensible
choice were one asked to do so.

Next, consider this pair of problems from two different
textbooks:
(B) Identify the independent and dependent variables:
The faster Ron walks, the quicker he gets home.
(C) Name the two variables involved. Explain which
you would list first and represent on the x-axis. Which
would you list second and represent on the y-axis?
Driving speed and time required for a trip.
Both problems B and C are situated in the context of the
relationship between distance, rate, and time, with the
main variables in question being speed and time. Now,
does one of these variables either control or cause the
other? For either rationale, one can reasonably argue either
direction. I can certainly determine my rate based on how
much time I have (I’ve got to get home by my curfew, so
I’ll pick up my pace) or I can determine my time based on
my rate (I’m driving 60 mph, so it will take me a certain
amount of time to get to my destination), and such changes
in one variable do cause a change in the other. So, on what
basis can one possibly determine the independent and
dependent variables in these situations? By the way, the
textbook for problem B gives the answer of (speed, time).
The textbook for problem C says that either (speed, time)
or (time, speed) are acceptable answers.
Causation and Control
Another interesting subset of the 73 problems are the
seven problems that were coded as YYNN (Table 3)—that
is, problems that could reasonably be determined by con-
sidering either control or causation. Here is an example:

(D) Identify the independent and dependent variables.
In warm climates, the average amount of electricity
used rises as the daily average temperature increases
and falls as the daily average temperature decreases.
(Answer: [temperature, amount of electricity used])
In this problem, it is fairly straightforward that the daily
temperature is heating homes, which causes the use of
air-conditioning to cool off the homes, which in turn causes
increased electricity use. So it is reasonable to determine
the solution given by the textbook based on causality (as
electricity use clearly does not cause changes in daily
temperature). Neither convention nor function comes into
play here, but it is interesting to consider the issue of
control. One cannot control daily average temperature, but
one can control electricity use. The latter, however, cer-
tainly does not cause a change in the former. Thus, in this
problem, the variable one does not have control over is the
one that causes the change in the variable one does have
control over. Appealing to either control or causation as a
rationale results in different responses to the question.
Contrast problem D with problem E, also from the set of
problems coded as YYNN—that is, problems that could be
determined by either causality or control:
(E) Identify the independent and dependent variables.
An employee receives two vacation days for every
month worked.

In this instance (and all YYNN instances), cause and
control issues coincide. One can only control the number
of months worked and changes in that variable also cause
the variable of vacation days to change. Thus, for problems
like problem E, consideration of either the control or the
causation rationales results in the same determination of
independent and dependent variables; whereas for prob-
lems like problem D, consideration of either of these
rationales results in different determinations.
Convention and Function
Finally, consider the set of problems for which either
convention or function are reasonable rationales, but
neither causation nor control are (NNNY, NNYY, NNYN).
There are 13 problems in this subset. Here are three
examples of NNYN problems:
(F) Copy and complete: In the equation y = x + 5, x is
the ___ variable and y is the ___ variable.
(G) Identify the independent and dependent variables
(Figure 1):
(H) A convenience store has been keeping track of its
popcorn sales (Figure 2). Make a coordinate graph of
the data in the table. Which variable did you put on the
x-axis? Why?
In each of these problems, convention (symbolic,
graphic, then numeric) is really the only way one could
determine the answer to the given problem. Imagine how
confusing such problems might be to students who are
unaware of the conventions, or who are trying to
make sense of these situations by appealing to other
rationales.

Conclusion
The quality of the contexts of these “Identify the inde-
pendent and dependent variable” problems varies signifi-
cantly. Such variation potentially sets students up to
develop misconceptions about independent and dependent
variables. On the one hand, students are given contexts
from which one might reasonably conclude that one vari-
able clearly must depend on the other, and that the alter-
native is unrealistic (e.g., problem D). On the other hand,
students are asked to determine the independent and
dependent variables in mere mathematical relationships
(e.g., problem F). Given this range of contexts for these
problems and the fact that almost all problems had a “right
answer” in the text, one could infer the intended learning
outcome of these textbooks with regard to independent
and dependent variables: Given any situation, one can
objectively determine the independent and dependent vari-
ables. When students are asked to make sense of such
situations, however, as illustrated in the classroom
example, they are able to see through this fallacy.
The terms “independent” and “dependent” are relatively
straightforward for students to understand; they easily
grasp the idea that the dependent variable “depends” on
the other. What is not so clear to students—and I posit
because this idea is unclear in some textbooks, textbook
problems, and for many teachers—is just what it means for
a variable to “depend” on another. Is the issue “cause and
effect”? Is the issue “control”? Or is the issue merely “can
be determined by”? These variations are complicated, and
in fact, are quite useful dilemmas in building a rationale
Figure 1. The graph provided for problem G.

Time Total Bags Sold
6:00 A.M. 0
7:00 A.M. 3
8:00 A.M. 15
9:00 A.M. 20
10:00 A.M. 26
11:00 A.M. 30
noon 45
1:00 P.M. 58
2:00 P.M. 58
3:00 P.M. 62
4:00 P.M. 74
5:00 P.M. 83
6:00 P.M. 88
7:00 P.M. 92
Figure 2. The table provided for problem H.
356 Volume 112 (6)
for the development of a function; for once one has a
function y = f (x), one knows by definition that y depends
on x. However, if one is merely looking at a relationship
between y and x, one is left with insufficient information to
determine which is the independent and which is the
dependent variable. The correct response really is, “It
depends on what you want.”
Most textbook problems asking for the identification of
independent and dependent variables seem to be sending

mixed messages and implicitly impeding students from
developing a robust understanding of independent and
dependent variables. These messages include: (1) one can
determine the independent and dependent variables not
just from functions, but from any relation, often, it seems,
by using implicit conventions; (2) given a context, one
can determine the independent and dependent variables;
and (3) if one determines what causes or controls, one
can determine the independent and dependent variables.
Messages such as these not only confound students’
understanding, they serve to perpetuate the belief that
mathematics is mysterious and nonsensical.
What message should we send? The scientific and
mathematical rationales discussed in this paper might
provide a good starting point. In particular, mathematics
textbooks should make it clear how they use independent
and dependent variables, and should discuss the differ-
ence between mathematical and scientific uses of the
terms. Students need explicit opportunities to make sense
of the meaning of “dependence” and that the variation in
situations, all the way from real-world cause-and-effect
situations to mathematical functions, need to be com-
pared and contrasted. I fear we are mistaken if we think
conceptually problematic problems like the ones dis-
cussed in this paper will simply take care of themselves.
Further research could help to support this conceptual
analysis by analyzing the same kinds of problems in text-
books from disciplines such as science, statistics, and
economics; by expanding the analysis to study how the
entire curriculum (the problems and the text) addresses
the topic; by documenting students’ current understand-
ings of identifying independent and dependent variables;
and by developing and testing curricular materials that
are designed to help students make these important
connections. I was unable to locate any extant research

literature on students’ or teachers’ understanding of
independent and dependent variables.
This paper illustrates one small corner of the curriculum
where typical textbook problems may be thwarting rather
than supporting students’ attempts to make sense of the
content. Conceptual analysis of curricular holes such as this
one have the potential to contribute to our ever-increasing
commitment to viewing science and mathematics as a
sense-making activity and to deepen our understanding of
how to enact such a view in classroom teaching and learn-
ing. Our students’ understanding depends on it.
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Author’s Notes
1 An earlier analysis of this episode initially motivated
this paper (see Peterson & Leatham, 2009 for a related
discussion of the episode).
2 All names are pseudonyms.
3 Certainly, there are aspects of this episode that can be
attributed to the inexperience of the student teacher. The
phenomenon of significant differences between teachers’
and students’ mathematics, however, is common to all
teaching situations and is, in fact, a hallmark of what
makes teaching “problematic” (Simon, 1995; Steffe &
D’Ambrosio, 1995).

4 Hill’s (1965) classic causality criteria illustrate the
problematic nature of determining causation and the pre-
eminence of purpose in the process.
5 In the numeric representation, numbers are often orga-
nized into tables, and by convention, the independent vari-
able is often listed first, the dependent variable second.
This convention is in line with the conventional ordered
pair (independent, dependent). In the graphic representa-
tion, the independent variable is often placed on the hori-
zontal axis, the dependent variable on the vertical axis.
6 Snowy tree crickets, to be exact.
7 The rationales are likely not exhaustive. Further analy-
sis (in science and mathematics, as well as of other disci-
plines such as statistics) may reveal yet further refinement
to the framework of rationales. For example, in science,
one ideally looks for cause and effect. In statistics, on the
other hand, one is more concerned with an exploration of
correlation than necessarily determining cause and effect.
Because of this focus on correlation, statisticians often
give preference to terms like “explanatory” and
“response” variables (Yates, Moore, & Starnes, 2008).
8 Many thanks to Jillian Busath for her help in data
collection and to Annalisse Daly for her help in refining
the problem analysis.
9 We make no judgment as to whether such textbooks
should or should not have problems asking students to
identify independent and dependent variables. Our con-
ceptual analysis focused on the nature of the problems that
do exist, wherever they were.

358 Volume 112 (6)
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