The Role of Visual Reasoning in Evaluating Complex Mathematical Statements: A
Comparison of Two Advanced Calculus Students
Erika David, Kyeong Hah Roh, Morgan Sellers, Kody D’Amours
Arizona State University
The purpose of this study is to examine the role of visual reasoning while students evaluate
complex mathematical statements about real-valued functions. We conducted clinical interviews
with nine undergraduate students from mathematics courses at different levels. In the interviews,
we asked these students to evaluate several mathematical statements alone and then using
various graphs. In this paper, we focus on the cases of two students who had completed
Advanced Calculus to highlight the contribution of their visual reasoning about several graphs.
We found that student’s graphical interpretation of “between” in these statements affected their
evaluation of the statements. Even at advanced levels, students’ visual cues dominated their
reasoning about the statements. Our findings indicate that students’ visual reasoning contributes
to their evaluation of mathematical statements and helps to account for differences between
students’ meanings of statements.
Key words: Visual Reasoning, Complex Mathematical Statements, Graphical Interpretations,
Undergraduate Students, Intermediate Value Theorem
Introduction
Mathematical statements, including definitions, theorems, and mathematical claims, which
may be true or false, are a central part of elementary through postsecondary mathematics
curriculum. Mathematics education research has shown that students often struggle with both
understanding mathematical statements (Roh, 2010; Selden & Selden, 1995) and evaluating
whether mathematical statements are true or false (Bubp, 2014; Dawkins & Roh, 2016; Selden &
Selden, 2003). In particular, studies have shown student difficulty with statements involving
multiple conditionals such as “if 𝑝 → 𝑞, then 𝑟 → 𝑠” (Zandieh, Roh, & Knapp, 2014) or multiple
quantifiers such as “∃𝑥∀𝑦 𝑃(𝑥, 𝑦)” (Dubinsky & Yiparaki, 2000). In this paper we focus on
students’ understanding of statements involving both conditional structure and multiple
quantifiers, which we refer to as complex mathematical statements. Although these statements
look complicated, statements with such structure are common in Calculus and Real Analysis. For
example, the Intermediate Value Theorem (IVT) can be stated as a complex mathematical
statement as follows: “Suppose that f is a continuous function on [a, b] with f(a)≠f(b). Then, for
all real numbers N between f(a) and f(b), there exists a real number c in (a, b) such that f(c)=N.”
In order to help students understand these types of statements, instructors may often provide
a graphical illustration to introduce the idea of the statement. Several textbook authors and
researchers have cited the benefits of incorporating visual representations of theorems and/or
proofs in instruction (Alcock & Simpson, 2004; Arcavi, 2003; Davis, 1993; Guzman, 2002;
Hanna & Sidoli, 2007). While research has looked at students’ understanding of multiple
quantifiers (Dawkins & Roh, 2016; Dubinsky & Yiparaki, 2000; Epp, 2003; Roh & Lee, 2011)
and conditional structure (Durand-Guerrier, 2003; Zandieh et al., 2014), few studies have looked
at the role of students’ reasoning about the graphs or images that correspond to a statement about
a real-valued function. Although instructors may consider such explanations clear
representations of the statement, we do not know what sense students make from these graphical
illustrations. While these visual representations have the potential to be valuable to student
understanding, little research has been conducted to examine the role of students’ visual
reasoning in understanding and evaluating complex mathematical statements. This study
addresses the following research question: How do students’ visual reasoning contribute to their
evaluations of complex mathematical statements about real-valued functions?
Theoretical Perspective
Our study is grounded in a constructivist perspective and builds on Moore and Thompson
(2015) and Moore (2016)’s theories of students’ graphical activity. We adopt von Glasersfeld’s
(1995) view that students’ knowledge consists of a set of action schemes that are viable given
their experience, and that we, as researchers, do not have direct access to their knowledge. This
perspective implies that we as researchers can only model student thinking based upon their
actions. Thus, our analysis reflects our best attempt at creating a hypothetical model of student
visual reasoning grounded in evidence found in their words, gestures, and markings on paper.
In this study, we reference Moore and Thompson’s (2015) distinction between static shapethinking and emergent shape-thinking as well as Moore’s (2016) constructs of figurative and
operative thought in the context of graphing as a means of situating our constructs that describe
students’ visual reasoning we observed. Moore and Thompson (2015) describe static shapethinking as conceiving of a graph as an object in itself, in which mathematical objects and
actions are subordinate to visual perception. In contrast, they describe emergent shape-thinking
as conceiving of a graph as a trace which emerges from the coordination of two varying
quantities. To develop this distinction, Moore (2016) aligns static shape-thinking with figurative
thought and emergent shape-thinking with operative thought, consistent with Piaget’s (2001) and
Steffe’s (1991) use of the terms. Moore (ibid) explains that figurative thought is dominated by
visual properties of a graph, such as aspects of the shape of the curve which overrides
considerations about relationships of the quantities being represented. In contrast, for students
engaged in operative thought, “figurative elements of their activity are subordinate to that
coordination [of covarying quantities]” (Moore, 2016, p. 3). Students engaged in operative
thought may still perceive visual properties of a graph, but their thought is guided by the
relationships represented in the graph. Furthermore, Moore (ibid) explains that these modes of
thought are not exclusive of the other. Students may engage in both, although one may dominate
the student’s activity, depending on which mode of thought the students’ actions are subordinate.
While these constructs help to distinguish certain student graphing behavior, they do not account
for important nuances of the graphing behavior we observed in this study. Thus, we define the
constructs location-thinking and value-thinking to distinguish between the relevant aspects of the
graphing behavior we observed, as explained in the results section.
Methods
For this study, we conducted two-hour clinical interviews with nine undergraduates at three
different levels: students who had just completed Calculus, an Introduction to Proof course, and
Advanced Calculus, respectively. Due to the complex nature of the interview, we decided to
have one researcher serve as the interviewer with the other three researchers as witnesses, two of
whom were in the interview room, and the third who watched the interview live-streamed from
elsewhere. We recorded the interview with three cameras to capture the entire frame, student
work on the table, and to live-stream the interview to the third witness. All four researchers had
laptops, whose screens the participant could not see, to communicate their current models of the
participant’s thinking in real-time via group chat. This set-up allowed all four researchers to offer
clarifying questions for the interviewer to pose to the participant to test these models.
In the interview, the participants were asked to evaluate four complex mathematical
statements about real-valued functions and provide justification for their evaluation. One of these
statements was the Intermediate Value Theorem (IVT), which we presented in the introduction.
The remaining three statements were variations on IVT with the re-ordering of the quantifiers
(for all, there exists) and/or variables reversed (N, c). For example, another statement read,
“Suppose f is a continuous function on [a, b] where f(a)≠ f(b). Then there exists a real number N
between f(a) and f(b), such that for all real numbers c in (a, b), f(c)=N.” This altered statement
was made from a reversal of both the quantifiers and variables in IVT.
Once the participants had evaluated each of the four statements, the interviewer asked them
to compare the statements and explain whether any of the statements had the same meaning, in
their interpretation. The interviewer later asked students to look at each statement along with six
graphs we created, with the chance of changing their evaluation in order to gain insight into the
effects of students’ reasoning about various graphs on their evaluation. These graphs, which
were intended to represent a spectrum of possible functions and relevant counterexamples,
included: a constant function, a monotone increasing function, a vertical line, a polynomial with
extrema beyond the endpoints of the displayed function, the tangent function, and the sine
function. The participants were also asked to explain how they interpreted various aspects of the
graph and to label relevant points and values on the graphs where appropriate.
After conducting the interviews, our research team closely analyzed the video data for
patterns in student thinking that could explain their statement evaluations. Our data analysis was
consistent with Corbin and Strauss’s (2014) description of grounded theory, in which categories
of student visual reasoning emerged from the data, as we did not begin our study with the
specific intent of investigating visual reasoning. Through analyzing the student interviews,
students’ interpretation of the phrase “between f(a) and f(b)” emerged as highly relevant to their
reasoning about the graph and subsequent evaluation of the given statement.
Results
In analyzing the interviews, a distinction emerged in students’ interpretation of the phrase
“between f(a) and f(b),” which stemmed from their meaning for f(a) and f(b). As we analyzed
student meanings for this phrase, students fell into one of two categories in terms of their
interpretation of this phrase, and we labeled each student as a value-thinker or location-thinker,
accordingly. In this categorization, value-thinkers interpret “between f(a) and f(b)” as referring to
output values between the values of f(a) and f(b). In contrast, location-thinkers interpret
“between f(a) and f(b)” as locations between the locations of f(a) (at (a, f(a))) and f(b) (at (b,
f(b))). In other words, value-thinkers distinguish between the value of the output of the function
and the resulting location of the coordinate point on the graph, whereas location-thinkers do not
make this distinction. Students who were labeled as value-thinkers were seen as engaging in
operative thought and emergent shape-thinking, as their visual perception of the graph was
subordinate to their meanings for output value and they coordinated the values of the varying
quantities. Location-thinkers, on the other hand, did not clearly fall into the category figurative
thought, and by extension, static shape-thinking. The visual cues from the graph dominated their
thinking in some aspects of the graph and informed their interpretation of “between f(a) and
f(b).” Although location thinkers engaged in figurative thought, they did not conceive of the
graph statically. Instead, for these students, a graph emerged from the coordination of input
values with locations of the points on the graph. In this way, we consider our constructs of
value-thinking and location-thinking as novel in describing students’ graphical activity. We
summarize the characteristics of each way of thinking in the following table.
Table 1. Comparison of location-thinking and value-thinking characteristics
Location-Thinker
Visual Reasoning
Output of
Function
Point on Graph
Confounds the output
value of a function with
the location of the
coordinate point
Graph
A collection of
geometric points
associated with input
values
“Between f(a)
and f(b)”
All points on the curve
located between (a, f(a))
and (b, f(b))
Evidence
▪ Marks output on
the curve
▪ Speaks about
points as the
resulting output of
the function (e.g.
“an input maps to a
point on the curve”)
Sweeps along entire
curve between (a,
f(a)) and (b, f(b))
Value-Thinker
Visual Reasoning
The resulting value
from inputting a
value in the function
The coordinated
values of the input
and output
represented together
A collection of
coordinates relating
the value of two
quantities
All values between
the values f(a) and
f(b)
Evidence
Speaks about
output values
▪ Labels points as
ordered pairs
▪ Speaks about
points as the result
of coordinating an
input and output
value
Marks the interval
of values between
f(a) and f(b) on the
output axis
Based on our analysis of the interview data, four students were identified as location-thinkers
and five students were classified as value-thinkers. In this paper, we choose to highlight the cases
of two students, Jay, a value-thinker, and Nate, a location-thinker, to illustrate the difference in
these visual reasoning and their subsequent impact on the student’s understanding and evaluation
of the given statements. Both Jay and Nate had recently completed Advanced Calculus and
earned an A in the course. Additionally, both students showed evidence of having strong
meanings for multiple quantifiers and conditional structure. However, their evaluation of the
truth values of these statements differed. Jay correctly evaluated each of the four statements and
offered justifications that were consistent with the mathematical community. Nate paraphrased
one of the statements1 as, “for every single c there is a point on the curve that it maps to,”
indicating that he correctly interpreted the multiple quantifiers in the statement. While Jay
correctly evaluated this statement as false, Nate evaluated it as true. Jay and Nate’s evaluations
of these statements serve as a contrast due to the significance of visual reasoning in their
evaluations of the statements.
Jay: A value-thinker
The excerpt of Jay’s interview below provides evidence for Jay’s interpretation of “between”
as value-thinking. In this excerpt, Jay has just looked at the IVT and evaluated the statement as
true. He used the graph he drew (Figure 1, left) to explain why this statement, in his
interpretation, is true.
1
Suppose that f is a continuous function on [a, b] such that f(a)≠f(b). Then, for all real numbers c
in (a, b), there exists a real number N between f(a) and f(b) such that f(c)=N.
… So this right here is y equals N, this line is y equals N (points to horizontal line which
he labeled N). If f crosses this line (points to horizontal line he just drew), then there
exists a real number c in the open interval such that f(c) is N. Because if I cross these
lines... lines say here, this (marks c on x-axis) is my value of c, right? Because that's
where f(c) is N. Okay, so the only way for like this not to be true is if you can draw a
continuous graph through it or a continuous graph from a to b with f(a) being this (marks
f(a) below the line drawn), f(b) being this (marks f(b) above the line drawn), such that
you skip N.
Figure 1. Jay’s hand-drawn explanation for why the IVT is true (left) and Jay’s possible values
of N (right)
Both Jay’s graph and his explanation above of why IVT is true reveal how he made sense of
the phrase “between f(a) and f(b)” in the statement. Jay drew a horizontal line between these
output values that intersected the y-axis, which indicated that he attended to the values of the
output, rather than the location of the ordered pair point in space. Later, Jay described that the
horizontal line he drew in his diagram is arbitrary, indicating that he imagined all of the possible
horizontal lines passing between the horizontal line through f(a) and the horizontal line through
f(b). For Jay, the phrase “between f(a) and f(b)” was connected to this image that he shared. Not
only did Jay consider the values of f(a) and f(b), he visualized the graphical meaning of IVT,
which is that a continuous function will always intersect any horizontal line drawn between the
output values of f(a) and f(b). Further evidence of Jay’s visual reasoning for “between” was
found later in the interview, when Jay evaluated another statement and provided graph, and
marked off the relevant interval for N along the y-axis (Figure 1, right). Jay’s interval of possible
N values between f(a) and f(b) in Figure 1 indicates he again considered the values between the
value of f(a) and f(b) rather than the spatial locations of these points.
Nate: A location-thinker
In contrast with Jay’s value-thinking, Nate’s reasoning was labeled as location-thinking.
Evidence for Nate’s thinking can be seen in the following excerpt. Nate had already evaluated
Statement 1 (which is false)2 as true, and below explained why it is true using the same provided
graph that Jay used above (which, from our perspective, is a counterexample to Statement 1).
Nate: So for all these c’s (sweeps pen along x-axis between a and b) you can see that it
mapped to a point on the curve. For every single c there is a point on the curve that it
maps to…So after that c, N would be here (marks c around 1 on x-axis, N at ordered pair
location of (c, f(c)). So it maps to that. And this c would be in here (marks c around 2 on
x-axis). And this c would be like, N right here (marks corresponding N’s for each c on
graph).
Interviewer: Okay. Lets say we picked a point over here (points to a point beyond (a,
f(a)), (b, f(b)), rightmost marked point on Figure 2). Could we say that the output would
be between f(a) and f(b)?
Nate: I would not say its between f(a) and f(b). Even though the, yeah. This is the, the
actual numbers 2.5 and 0 (marks the points 2.5, 0). This would be… If you are looking at
numbers 2.5, 0, this would be in between that interval. But it’s in between that number
interval. But it’s not in between the functional interval in this case. So f(a). The interval
will refer, refers to all these points between f(a) and f(b) (sweeps pen along curve). All
points of the function. That's what I am interpreting.
In the transcript above, when asked about the far right point the interviewer selected, if the
output would be between f(a) and f(b), Nate explained, “it’s in between that number interval,
yeah, but it’s not in between that functional interval in this case.” The “it” Nate referred to here
is the point in question, which he confounded with the output of the function at that point. Nate
further clarified that “the interval refers to all the points between f(a) and f(b)” and illustrated
what points he is talking about by sweeping his pen along the curve between the points (a, f(a))
and (b, f(b)).
Nate’s
labels for
the values
2.5 and 0
on the
graph
A point that the
interviewer marked and
asked Nate about in order
to further probe his
meaning for “between”
Nate’s labels for points
along the curve
corresponding to various
c values
Figure 2. Nate’s work on provided graph with his possible N values labeled on the curve, outside
the range of values between f(a) and f(b).
Nate’s response indicates that his perception of the graph guided his interpretation of and
evaluation of Statement 1. Nate labeled f(a) and f(b) not on the y-axis, but at the location of the
point (a, f(a)) and (b, f(b)), which indicates that he conceived of the output of the function as the
point itself. These two labeled points became visual boundaries for Nate, between which lay all
2
Statement 1: Suppose that f is a continuous function on [a, b] such that f(a)≠f(b). Then, for all
real numbers c in (a, b), there exists a real number N such that f(c)=N.
of the relevant N values. In other words, he interpreted the relevant values of N “between f(a)
and f(b)” as all of the points on the path of the curve between the starting point that he labeled as
f(a) and ending at the point in space that he labeled f(b). Nate’s response to the question of the
curve extending beyond the point (b, f(b)), but including values that are numerically between f(a)
and f(b), is further evidence that Nate conceived of “between f(a) and f(b)” as all of the points
along the path between (a, f(a)) and (b, f(b)). While Nate said the point in question was not
between f(a) and f(b), he acknowledged that this point was numerically between 2.5 and 0 (the
values of f(a) and f(b)). The fact that Nate did not ignore the numerical value of the outputs, and
even acknowledged the confusion surrounding this meaning indicates that he recognized
multiple possibilities for the meaning of between. Nate’s interpretation of “between” in terms of
the location of the points, prompted by his visuospatial perception of the graph, reveals the
power of his visual perception to override his numerical evaluations.
Together, both Jay’s and Nate’s meanings for “between” and their subsequent evaluations of
the given statements highlight the role of visual reasoning in evaluating such statements. Since
both participants understood the quantifiers and conditional structure of Statement 1 the same
way, but had different truth-value evaluations, the difference in their visual reasoning became the
focus of our analysis. For Jay, the phrase “between f(a) and f(b)” was visually connected to
horizontal lines between the values of f(a) and f(b). Jay interpreted the outputs of the function as
values, distinct from the coordinate points on the graph and thus was classified as a valuethinker. For Nate, “between f(a) and f(b)” referred to every point along the path of the graph
between the location f(a) (at (a, f(a))) and the location f(b) (at (b, f(b))). Since Nate did not
distinguish between the outputs of the function and the locations of the coordinate point on the
curve, Nate was labeled a location-thinker. In summary, Nate and Jay, both successful Advanced
Calculus students consistently interpreted “between” differently, which led to their different
understandings, and subsequent opposing truth-value evaluations of Statement 1.
Discussion
Based on our findings that almost half of the students whom we interviewed were identified
as location thinkers, visual reasoning about mathematical statements plays a significant role in
students’ understanding. These results highlight and explain some important aspects of students’
graphical activity that were not previously accounted for by current theories and studies on visual
reasoning (Moore & Thompson, 2015; Moore, 2016). Thus, the use of our constructs of valuethinking and location-thinking could progress the depth of analysis in the field of student
graphical activity, especially with regard to ideas from Calculus and Analysis. As illustrated
through the cases of Nate and Jay, different interpretations for “between,” both rooted in visual
reasoning, contributed to different evaluations of a complex mathematical statement about realvalued functions. While in our study, value-thinking helped students to understand versions of
IVT, in other contexts, such as Geometry, location-thinking may be preferable. Ideally, students
should possess the ability to think in both ways, as well as the discernment for when to use each.
Overcoming various perceptual cues found in graphs, beyond conceiving of the graph as a static
shape, is a nontrivial achievement, even for advanced students. Teachers utilizing such
representations should support students in overcoming adherence to visual cues. In the
classroom, instructors should be aware of the various ways in which students may interpret
information from a visual representation. Further research on this topic may include developing
instructional tasks to address this topic, whether in the context of the IVT or other statements
commonly associated with graphs, such as the formal limit definition of continuity at a point.
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