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 shape-thinking 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 shape-thinking 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 Value-Thinker

Visual Reasoning Evidence Visual Reasoning Evidence

Output of Function Confounds the output

value of a function with the location of the coordinate point

▪ 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”)

The resulting value from inputting a

value in the function

Speaks about output values

Point on Graph

The coordinated values of the input

and output represented together

▪ Labels points as ordered pairs

▪ Speaks about

points as the result of coordinating an input and output

value

Graph

A collection of geometric points

associated with input values

A collection of coordinates relating

the value of two quantities

“Between f(a) and f(b)”

All points on the curve located between (a, f(a))

and (b, f(b))

Sweeps along entire curve between (a, f(a)) and (b, f(b))

All values between the values f(a) and

f(b)

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.

1Suppose 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)). 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.

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

Nate’s labels for the values 2.5 and 0 on the graph

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 value-thinker. 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 value-thinking 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 real-valued 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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