# Expert covariational reasoning resources in physics graphing tasks

Charlotte Zimmerman  Department of Physics, University of Washington  Alexis
Olsho  Department of Physics and Meteorology, US Air Force Academy  Michael
Loverude  Department of Physics, California State University Fullerton
Suzanne White Brahmia  Department of Physics, University of Washington

###### Abstract

Developing and making sense of quantitative models is a core practice of
physics. Covariational reasoning—considering how the changes in one quantity
affect changes in another, related quantity—is an essential part of modeling
quantitatively. Covariational reasoning has been studied widely in mathematics
education research, but has only begun to be used in physics education
research. We present evidence from three studies of 25 individual interviews
with physics experts, in which the experts were asked to reason out loud while
generating graphical models. We analyze the studies through the lens of
covariational reasoning frameworks from mathematics education research, and
determine that the frameworks are useful but do not completely describe the
covariational reasoning of the physics experts we interviewed. From our data,
we identified reasoning patterns that are not described in the mathematics
education research that, together with the mathematics covariational reasoning
frameworks, begin to characterize _physics_ covariational reasoning.

††preprint: APS/123-QED

##  I Introduction

Learning to model the physical world quantitatively is a key objective of
courses in physics, math, and other mathematics-based disciplines [1, 2, 3].
Reasoning quantitatively is at the heart of what it means to “think like a
physicist.” There is a large body of research in physics education that has
examined the role of conceptualizing mathematics in how students reason about
physics [4, 5, 6, 7, 8]. Emergent from this research is evidence that
quantitative reasoning is fundamentally different in a physics setting than in
a purely mathematical one [9, 10, 11, 12, 13, 14].

A central construct for this paper is _covariational reasoning_. Covariational
reasoning is a common subject of mathematics education research [15, 16, 17,
18, 19, 20, 21]. A simple working definition is that covariational reasoning
is the reasoning used when coordinating how two quantities vary. Mathematics
education researchers have effectively operationalized covariational reasoning
by creating a framework of “mental actions” that describes specific ways in
which covariational reasoning can be observed in novice and expert
mathematical reasoning [16, 19, 21]. Covariational reasoning, and the
associated mental actions described by mathematics education research,
resemble ideas widely found in physics; e.g. in the immediate moments after a
switch is closed in a circuit, the voltage increases as time increases.
However, the rate of increase diminishes until the voltage reaches a steady
value equal to the battery voltage.

The work described in this paper began as an effort to determine the extent to
which covariational reasoning, as described by mathematics education
researchers, is a useful lens through which physics reasoning can be studied.
We began by replicating a study of expert mathematics covariation with physics
experts and comparing the two populations’ reasoning. The replicated study
involved novel graphing tasks—that is, the tasks prompted participants to
relate familiar quantities in unfamiliar ways. The lack of familiarity with
the relationship between the quantities was intended to prompt participants to
use covariational reasoning. As our project with physics experts progressed,
it became clear that covariational reasoning as described by mathematics
education research is indeed a useful way to analyze reasoning by expert
physicists, but that _physics_ covariational reasoning encompasses a broader
set of thought processes.

Therefore, we expanded our investigation into two questions:

  * •

Research Question 1: In what ways are the covariational reasoning frameworks
developed by mathematics education researchers productive for describing
expert physicists’ modeling in novel graphing tasks?

  * •

Research Question 2: What patterns do we observe in the covariational
reasoning of physics experts as they solve novel graphing tasks?

We examine these questions across three studies: Study 1, in which we
replicate mathematics education researchers Hobson and Moore’s 2017 study of
expert mathematics covariational reasoning [20] with physics experts; Study 2,
an extension of this work with tasks situated in more advanced physics
contexts; and Study 3, an extension of Studies 1 and 2 with a broader
population. We identify the ways in which the mathematics covariational
reasoning framework is useful in analyzing physics experts’ reasoning, and
propose five emergent reasoning devices and three problem solving modes that
represent a wider set of thought processes related to physics covariational
reasoning. While these devices and modes may not be exclusive to physics, they
were prevalent throughout our experts’ reasoning and seem essential to a
complete description of physics covariation.

##  II Background

In this section, we situate our work in the current literature on
covariational reasoning in mathematics and physics education research. As
covariational reasoning is an essential part of developing, making sense of,
and validating mathematical models [22, 16], we begin by reflecting on
traditional perspectives on the mathematical modeling in physical contexts and
how these perspectives relate to this work. We then discuss covariational
reasoning as defined by mathematics education researchers, and examine work in
physics education research on reasoning about rates of change through a
covariational reasoning lens. Finally, we provide a brief summary of expert
mathematics covariational reasoning as described by Hobson and Moore [20], as
their study forms the jumping off point of our research.

Figure 1: The modeling cycle published by Blum and Leiß [23].

###  II.1 Modeling and Covariational Reasoning

Modeling involves making connections between the real world and mathematics,
as shown in Blum and Leiß’s modeling cycle (reproduced in Fig. 1) [23].
Research in mathematics and physics education demonstrates that attending to
the physical meaning of a quantity is an essential part of modeling across
many contexts [22, 24, 25, 26, 27]. Reasoning about the meaning of quantities
in a particular model is traditionally described as an initial and/or final
step in the modeling cycle. However, a recent study of engineering students
demonstrated that making sense of the meaning of the quantities involved in a
particular model was a continuous activity throughout the modeling process
[28]. These findings are in line with recent work that suggests reasoning
mathematically in physics can be viewed as a blend of conceptual mathematical
ideas and reasoning about the physical meaning of the quantities involved [10,
12, 13, 29, 30, 31, 25, 32]. We take the blended nature of physics and math as
a foundation of our research perspective. Therefore, we suggest that _physics_
covariational reasoning is necessarily grounded in the meaning of the relevant
physical quantities.

In mathematics education research, covariational reasoning is defined broadly
as reasoning about how two or more quantities vary with respect to one another
[19]. It has been studied in the context of mathematics for several decades,
and is an essential tool for reasoning about a relationship between two or
more quantities [15, 33, 34, 16, 35, 36, 18, 20, 19, 37, 38, 39]. Mathematics
education researchers Carlson et al. developed a framework of _mental actions_
(MA) and associated behaviors that effectively operationalizes covariational
reasoning, and has been used by researchers to investigate covariational
reasoning in student and expert statements [16, 19, 20, 21]. This original
framework was based on results from a study in 1998 into undergraduate and
graduate students reasoning about function [16, 40]. The framework was revised
in 2017 [19], and summarized by Jones in 2022 [21]. We find Jones’ simplified
and streamlined framework was most productive for our analysis [21], and a
slightly modified version is shown in Table 1.

Label |  Mental Action [16, 19] |  Brief Description [21] |  Example Behavior
---|---|---|---
MA 1 |  Recognize Dependence |  Identify variables that are dependent |  Labeling axes
MA 1.5 |  Precoordination |  Asynchronous changes in variables |  Articulating that first, one quantity changes, and then the other changes
MA 2 |  Gross Coordination |  General increase/decrease relationship |  Describing that as one quantity increases, another decreases
MA 3 |  Coordination of Values |  Tracking variable’s values |  Plotting points
MA 4 |  Chunky Continuous |  Values changing in discrete chunks |  Articulating that as one quantity doubles, the other triples
MA 5 |  Smooth Continuous |  Continuous, simultaneous changes |  Describing that the quantities vary together, smoothly and continuously
Table 1: A summary of the covariational reasoning mental actions (MA) used in
this study. The framework we provide here is a slightly modified version of
that summarized by Jones in 2022 [16, 19, 21]. We use names and descriptions
from Jones’ summary. Numeric labels and example behaviors were added by the
authors and correspond to the original mental actions from the 2002 framework,
with the addition of 1.5 for “precoordination”, for ease of reference within
this paper.

Physics education researchers describe student reasoning about change using
the language of proportional reasoning [41, 42, 43, 9] and scaling [44, 45,
46]. In the context of covariational reasoning, we argue that proportional
reasoning is a _linear version_ of covariation and that scaling could be
described in the language of the mathematics covariational reasoning framework
as Mental Action 4: Chunky Continuous, in which one might ask “if I double
this, what happens to that?” Physics education research has already begun to
benefit from integrating the language of covariational reasoning in
investigations into mathematical modeling in physics [29, 47, 48, 49, 50].
Incorporating the language of covariation has thus far been useful in physics
education research, and we suggest this may be because it allows for
descriptions of reasoning about change across a wide variety of continuous,
functional relationships between two or more quantities [21].

Covariation in mathematics is centered on explicitly coordinating the change
in one quantity with the change in another, i.e. “as time increases, the
distance will also increase.” We will refer to this as “direct” covariational
reasoning. While direct covariational reasoning certainly appears often in
physics, there is some preliminary evidence that covariation and reasoning
about quantities more generally may have additional, distinct features in
physics [51, 25]. In particular, early findings in our prior work suggest that
physics covariational reasoning may involve a broader tool set than reasoning
focused on the specific functional relationship between two quantities [51,
48], perhaps due to the blended nature of physics reasoning. In this paper, we
will use the mental actions as described in Table 1 as a launching point for
our investigation into physics covariational reasoning.

###  II.2 Expert Mathematics Covariational Reasoning: Hobson and Moore’s 2017
Study

The first study described in this paper is a replication of an study conducted
by Hobson and Moore in 2017 [20]. That study examined how expert
mathematicians use covariational reasoning by asking several mathematics
graduate students to complete novel graphing tasks. The authors considered
graduate students to represent experts in the context of the introductory
calculus problems, as graduate students are expected to “have sufficient
experience engaging in tasks involving representing and reasoning about
quantities” [20]. The graduate students were given animations developed by the
researchers of three graphing tasks identical to those shown in Figure 1, with
place names adjusted to the local region.

All three tasks asked participants to relate a particular quantity (e.g., the
height of a Ferris wheel cart) to the total distance traveled of an object
moving at constant speed. This was an intentional choice by the researchers,
who noted the facility with time-based motion tasks that many experts in math-
based disciplines have acquired. The research demonstrated that mathematics
graduate students were likely to:

  * (1)

Directly covary the quantities asked for in the task (in some cases, with
prompting from the interviewer), and

  * (2)

Equally divide the domain of the task, or large sections of the motion, into
small sections to compare the change in the two quantities and then map these
equal segments to their graphs [20].

In this paper, we report the results from our replication study and compare
our observations to those provided by Hobson and Moore. We then discuss the
results from the extension studies that examined a broader set of physics
contexts and included a larger population. We report on the patterns we
observed across all the studies as physics experts reasoned covariationally.

##  III Methods

|  Study 1: Replication |  Study 2: Extension |  Study 3: Faculty
---|---|---|---
Subjects |  10 Graduate Students |  5 Graduate Students from Study 1 |  5 R1 Faculty
|  |  5 New Graduate Students |
Tasks |  Going Around Tacoma |  Gravitation / Electric Charge |  Gravitation
|  Ferris Wheel |  |  Ferris Wheel
|  Square Track |  |
|  |  Drone |  Drone
|  |  Intensity |  Intensity
Table 2: A summary of the differences between the populations and tasks across
the three studies.

In this section, we describe the methods, population, and interview tasks used
to conduct the series of studies. We discuss the two analytical approaches we
used in our coding process—grounded theory and thematic analysis—to account
for the difference between Study 1, during which we were informed only by our
literary review, and Studies 2 and 3, during which our analysis was influenced
by our experiences from Study 1. Finally, we describe how we sought patterns
in the coded data to address our two research questions.

###  III.1 Study Design

This work was situated at a large, R1 university in the Pacific Northwest. The
population across the three studies included 20 experts, nine of whom were
female or non-binary identifying and 11 of whom self-identified as male. The
pool of experts spanned from first-year graduate students through tenured
professors with over 30 years of experience. The population also spanned a
variety of physics subfields, including both experimentalists and theorists.
An effort was made to include diversity across race and country of origin. We
categorize the graduate students into two groups: those who were early in
their graduate career and were still enrolled in required courses, and those
late in their graduate career who were fully engaged in research. We entered
the research study assuming all of our participants to be experts in the
physics contexts of the tasks across all three studies. We discuss how the
populations compare in the findings.

Study 1 was a replication of the Hobson and Moore 2017 investigation into
expert covariational reasoning that was conducted with mathematics graduate
students (see Sec. II.2). In order to study expert physics covariational
reasoning, we repeated their study with a population of physics graduate
students. Study 2 was an extension of Study 1, using new tasks situated
outside of constant motion contexts. Constant motion contexts are so
commonplace for physics graduate students that they are likely part of their
everyday thinking. We hoped to elicit a wider variety of reasoning by asking
the graduate students to engage with less-familiar contexts in Study 2.

Study 3 was conducted with five tenure-track or tenured physics professors.
The intent was to see if the patterns we noticed in graduate students’
reasoning were present in faculty members’ reasoning, and if any new patterns
might emerge. Study 3 used a subset of tasks from the Studies 1 and 2 (see
Table 2).

In all three studies, participants were solicited by email and offered a gift
card to a local coffee shop as a thank you for their time and effort. An
interview involved one interviewer and one participant. During each interview,
the participant was presented with a handout of three to four graphing tasks.
Participants had access to a computer that allowed them to interact with short
animations associated with each prompt. Each task prompted the participant to
create a graph to represent the relationship between two quantities. The
participants were invited to move between questions at their own pace, and
play the animations as desired. The interviews lasted between 25 and 50
minutes, depending on how long the participants chose to work on the tasks.
The participants were encouraged to think aloud while they created their
graph, and the interviewer asked follow up questions after the participant
expressed they were finished with all of the tasks. The interviews were audio
recorded; initial transcripts were created automatically using the Otter.ai
software program [52], and were subsequently hand corrected.

The following sections discuss details particular to each study, including a
brief description of the tasks. A summary comparison of the populations and
tasks used in the three studies can be found in Table 2.

Figure 2: Stills from the animations associated with tasks administered in
Study 1: (A) Going Around Tacoma, (B) Ferris Wheel, and (C) Square Track. All
tasks asked participants to create a graph that related a particular quantity
(distance to Tacoma, height of the cart, and distance from the cart to the
wall, respectively) to the total distance traveled of an object in constant
motion. These graphing tasks are identical those in the original study
conducted by Hobson and Moore [20], with the exception that we changed the
place names in Going Around Tacoma. Figure 3: Stills from the animations
associated with the first task administered in Study 2. Participants either
saw (A) Electric Charge or (B) Gravitation. The tasks prompt participants to
create a graph that relates either the electric or gravitational potential and
the total distance traveled of the probe or spaceship, as it moves at constant
speed from start to finish. Figure 4: Stills from the animations associated
with the second and third tasks administered in Study 2: (A) Drone, in which
participants were prompted to create a graph that relates the height to the
angle of the drone, flying in the arc shown, and (B) Intensity, in which
participants were prompted to create a graph that relates the intensity
measured by the probe at the side of the liquid column to the total distance
traveled of the probe as it moves from the bottom of the column to the top.

###  III.2 Study 1: Replication Study

In Study 1, we followed the procedure described in Hobson and Moore’s 2017
paper. We used the same tasks as those they administered with minor
modifications of the place names to reflect the local area (Fig. 2). We
included both written descriptions and animations of each task’s context. We
recruited 10 physics graduate students to participate as our experts,
consistent with the original study [20].

Each task prompts the participant to create a graph that relates a particular
quantity to the total distance traveled by an object (Fig. 2). In the first
task, Going Around Tacoma, participants are asked to relate the distance from
Tacoma and the total distance traveled by the car. In the second task, Ferris
Wheel, participants are asked to relate the height of the Ferris wheel cart
and the total distance traveled by the cart. In the third task, Square Track,
participants are asked to relate the distance of the go-cart from the wall to
the total distance traveled by the go-cart.

###  III.3 Study 2: Extension Study

The extension study was designed upon reflection of the results from the
replication study (Study 1). The interviewers noticed that the physics
graduate students were so well acquainted with motion-based tasks that few of
the graduate students engaged in direct covariational reasoning—that is, they
were unlikely to compare the change in one quantity to the change in another,
and instead assigned functional relationships between the two quantities [51].
The similarities and differences in the patterns from our interviews are
compared to those reported by Hobson and Moore thoroughly in Section IV.1.

In order to better understand how physics experts use covariational reasoning,
a new set of three tasks were developed using physics contexts outside of
constant motion. A summary of the tasks can be found in Figures 3 and 4. We
designed two isomorphic versions of the first task, Going Around Tacoma. One
version, Electric Charge, asks the participants to relate the electric
potential measured by a probe and the total distance traveled as the probe
moves around a charge (Fig. 3A). The other version, Gravitation, asks
participants to relate the gravitational potential of the system and the total
distance traveled of a rocket ship as it moves around a planet called
Rigamarole (Fig. 3B). Half of the graduate students were given Electric
Charge, and the other half were given Gravitation.

The second task, Drone, was designed to relate two familiar quantities—height
and angle—in a novel way (Fig. 4A). The task asks participants to relate the
height of a drone to the angle the drone makes with the horizontal axis, with
the constraint that the drone is designed to tilt such that its axis is always
aligned with its velocity vector. Finally, the third task, Intensity, depicts
a laser shining through a column of liquid and asks participants to relate the
intensity measured by a probe and the total distance traveled by the probe as
it moves upward along the column of liquid (Fig. 4B). All of the graduate
students saw both Drone and Intensity.

Five graduate students from the first study elected to participate in the
extension study, and five additional graduate students were recruited to
ensure there was no bias from participating in the first round.

###  III.4 Study 3: Faculty Study

The final study probed whether the patterns that emerged amongst graduate
students could be observed in faculty reasoning and whether any new patterns
emerged in a population with more extensive physics experience. The tasks for
this study were chosen based on the researchers’ experience with the tasks in
the first two studies. Ferris Wheel, Gravitation, Drone, and Intensity had
been most successful at eliciting covariational reasoning.

Five faculty members were recruited individually by email based on their
strong engagement with teaching. We considered engagement with teaching as a
measure of those faculty that may be most likely to be able and willing to
think aloud about their reasoning.

###  III.5 Data Analysis

Figure 5: An example time on task chart for a faculty member from Study 3. The
horizontal axis represents normalized time, or the fraction of time on task.
The symbol represents the time the category was coded, and the band represents
the duration that the interviewee was engaged with that kind of reasoning.

The interviews were analyzed through an iterative, qualitative coding process.
For Study 1, the research team’s analysis was informed by prior literature on
covariational reasoning in mathematics education research, as well as prior
literature on proportional reasoning and scaling in physics education
research. The aim of Study 1 was to investigate the similarities and
differences in the ways the physics graduate students reasoned
covariationally, as compared to the covariational reasoning of mathematics
graduate students reported in Hobson and Moore’s study. However, we wanted to
ensure that the results rose out of the data itself. We chose to set aside the
mathematics covariational reasoning framework and employed a modified grounded
theory approach [53] to seek patterns in the data. A grounded theory approach
provided a framework to acknowledge the prior knowledge the research team
brought to the analysis and to reflect the effort made to seek emerging codes
from the data.

Studies 2 and 3 were conducted after Study 1 was analyzed. A thematic analysis
framework [54] was used to account for the ways in which the results of Study
1 impacted the research teams’ interpretation of the data. Thematic analysis
provides a structure for the research team to seek emergent patterns while
keeping a record of how prior experiences with the subject may be influencing
the choices they make. To ensure the codes were grounded in the data, research
team members met regularly during the coding process to discuss how the data
supported the particular codes that emerged. Studies 2 and 3 were coded
together using the same codebook.

For each set of studies, the lead researcher cyclically coded the transcripts
for emergent patterns related to quantitative reasoning. Between coding
cycles, the lead researcher conferred with the research team to establish
inter-rater reliability and wide-spread agreement with the coding scheme. We
established code categories by independently clustering codes, and then
discussing our clusters as a group until consensus was reached. The resulting
code categories that are pertinent to the discussion in this paper can be
found in the Appendix.

The codes and code categories allowed us to compare small pieces of reasoning
to the mathematics covariational reasoning framework, and begin to pull apart
the similarities and differences between the reasoning of mathematics and
physics experts. For example, when we compared the emergent code categories
from Study 1 to the framework in Table 1, we found some important connections
between the codes we identified and the mental actions in the mathematics
covariational reasoning framework (see Section IV.1). As a result, we used
these mental actions as codes in Studies 2 and 3.

It was also evident that there were larger patterns across the experts and
tasks for all three studies. Therefore, we sought patterns between code
categories to characterize the ways in which physics experts engaged with
covariational reasoning as part of the wider group of strategies they used to
solve the tasks. We approached this pattern analysis differently for Study 1
and for Studies 2 and 3 to account for our prior experience; our process for
each is described in detail in the following sections.

####  III.5.1 Study 1 Data Analysis

To seek patterns between code categories for Study 1, two members of the
research team performed a card-sorting task [55, 56] to characterize larger
patterns in expert reasoning. They separately grouped sections of coded
transcript based on the ways participants expressed their reasoning. The two
researchers compared their findings and compiled a list of reasoning patterns
that were observed across all participants. Each pattern is associated with
one or more code categories.

####  III.5.2 Studies 2 and 3 Data Analysis

Studies 2 and 3 were led by the same researcher who led Study 1. In order to
mitigate confirmation bias while interpreting the patterns in our code
categories, we opted out of using a card sorting task and instead created a
timeline (see Figure 5) for each participant. The timelines are plots that
show the code categories assigned to a participant over time. Each plot shows
the fraction of total time on task along the horizontal axis, and a mark for
each code category at the each moment it was assigned. The tails associated
with each mark show for how long the participant was engaged in that code
category. Each code category is plotted with a different shaped mark and
color. Therefore, this method allowed the research team to find patterns
between colors in the diagrams and then draw conclusions about how code
categories may be interacting, grounding the analysis in the data.

The research team sought patterns in the code categories by looking for
categories that occur at the same time and categories that rarely appear
together. As an example, consider the patterns in Figure 5. The code category
“Quantity,” which includes symbolizing and relating quantities to one another,
often appears at the same time as the code category “Physical World,” which
includes making sense of the physical context of the task. In contrast, the
code category “Function,” which includes assigning functional forms, often
appears next to—but not at the same time as—the code category “Mental
Actions,” which includes the mental actions from the mathematics covariational
reasoning framework.

Once initial patterns were identified across the 15 interviews in Studies 2
and 3, we went back into the transcripts to examine how the participants
described their thought processes and what behaviors were observed by the
interviewer at those moments where code categories coincided or were apart. We
then iteratively went back and forth between the timelines and the transcripts
to characterize the patterns we observed.

During this time, some new patterns emerged, some were revised and clarified,
and others were set aside. For example, the code category “Graphing,” which
describes behaviors associated with drawing a graph, often appears at the same
time as “Mental Actions” in Figure 5. The transcript reveals, however, that
this is because the mathematical mental actions describe reasoning that is
closely tied to graphical behaviors (e.g. identifying that two quantities are
related is tightly associated with the behavior of labeling the axes of a
graph). This relationship is well documented in mathematics education
research, and taken as a premise of the mathematical mental actions.
Therefore, we did not further characterize this relationship.

##  IV Findings

Three main patterns emerged:

  1. A.

Physics experts engaged with covariational reasoning in some ways that align
with the mathematics covariational reasoning framework (towards RQ1).

  2. B.

Physics experts used a set of five distinct “reasoning devices” that were not
well described by the mathematics covariational reasoning framework (towards
RQ2).

  3. C.

Physics experts broadly engaged in three different modeling modes when they
were solving covariational graphing tasks (towards RQ2).

During our analysis, we also found empirical evidence of the intertwined
nature of physical sense-making and mathematical modeling. This integrated way
of reasoning about math and physics was an essential part of how all experts
engaged with the tasks. In this section, we first discuss the relationship
between math and physics, as it underlies the rest of our analysis. We will
then discuss the main findings in the subsections that follow.

While we did not intend to investigate the relationship between the meaning of
the physical quantities and conceptual mathematics, it was a consistent
finding that was integral to how the experts reasoned covariationally. As the
example timeline in Figure 5 shows, the “Physical World” and “Quantity” code
categories often appeared at the same time. Here, our code category “Physical
World” represented when the experts were relating their reasoning back to the
context of the task. “Quantity” referred to when experts symbolized a
quantity, discussed its mathematical structure, or related one quantity to
another. In particular, we found that 78% of the time that Quantity was coded,
Physical World was as well. While this makes sense because they are reasoning
about covariation in a physics context, we note that mathematicians rarely do
this.

The majority of the time these codes coincided, it was because the research
team could not separate the two ideas within a single utterance. For example,
one expert working on Gravitation stated:

“It’s got to be negative one over R. Because I want it to fall in.”

We stated in Sec. II that we take the blended nature of physics and
mathematics as a foundation of our research perspective and our evidence
supports this perspective. We suggest that the rest of the findings presented
here should be interpreted as all involving a blend between physics and
mathematical reasoning. This finding warrants a more thorough discussion than
is appropriate here, and we are preparing a subsequent report to describe our
findings in detail.

###  IV.1 Seeking Mathematics Mental Actions in Physics Expert Reasoning

Research Question 1 states, “In what ways is the covariational reasoning
framework, as defined by mathematics education researchers, productive for
describing how expert physicists reasoned while solving novel graphing tasks?”
We found that the mathematics covariational reasoning framework (Table 1) was
helpful to describe the physics experts’ reasoning in all three studies.

In Study 1, we made an effort to set aside the mathematics covariational
reasoning framework to seek emergent codes from the data. Once the data were
fully coded, we were able to draw connections between the codes and some of
the mental actions described in the mathematics covariational reasoning
framework. We found that MA 1 (Relating Quantity), MA 2 (Gross Coordination),
MA 3 (Coordination of Values), and MA 4 (Chunky Continuous) were all present
in the codebook (see the Appendix for more details about the codes
themselves). However, we note that we were unable to relate any of the codes
that emerged from our analysis to MA 5 (Smooth Continuous).

In Studies 2 and 3, we used MA 1-4 as codes under the code category “Mental
Actions.” These codes were very productive towards characterizing physics
experts’ reasoning. However, there were other code categories which we
consider to be related to the broader definition of covariational
reasoning—how the changes in one quantity are related to changes in another
quantity—but are not well described by the mathematics covariational reasoning
framework.

Research Question 2 states, “What patterns do we observe in the reasoning of
physics experts as they solve novel graphing tasks?” We identified five
“reasoning devices”—thought processes and behaviors that experts used in while
reasoning covariationally—and three larger modes that physics experts engaged
in to create graphical models. These reasoning devices and modes are distinct
from MA 1-5, but are all representative of covariational reasoning more
generally. The following two sections define the devices and modes, and
discuss how they are related.

###  IV.2 Reasoning Devices

During the card-sorting task for Study 1, we identified five patterns of
expert reasoning that are only partially described by the mathematics
covariational reasoning framework. These patterns were consistent across
populations and tasks during Studies 2 and 3, and they encompass many of the
code categories that were different than the mental actions. We call these
“reasoning devices,” to illustrate that they are a combination of how the
experts described their own reasoning while talking out loud and the graphing
behaviors we observed the participants perform. In the following subsections,
we describe each of these reasoning devices and the associated mathematics
mental action (MA 1-4), if any. A summary of the reasoning devices can be
found in Table 3.

Reasoning Device |  Related MA |  Description |  Example
---|---|---|---
Proxy Quantity |  MA 1 |  A quantity used to replace another to make the task easier to think about. |  “It’s going at a constant rate. So time maps one-to-one to, uh, distance traveled.”
Regions of Consistent Behavior |  MA 2 |  Separating a domain into sections that are modeled by the same function. |  “So there’s a moment when they’re doing that loop around Tacoma—looks like half a circle. So that’s going to be a constant distance.”
Physically Significant Points |  MA 3 |  Plotting only those points that hold physical significance. |  “So it’s starting and finishing the same distance away…[plots end points]”
Neighborhood Analysis |  MA 4 |  Examining the rate of change of one quantity with respect to another over a “small chunk” around a physically significant point. |  “So it’s, it’s angular velocity has to be faster when it’s in the narrow part of the peak…[and] I have more, more, change in the angle for smaller change in the height.”
Compiled Models |  [none] |  The strong association between a particular functional form and a physical context. |  “And as you get closer the potential is one over R. So I’m literally just plotting one over R.”
Table 3: The reasoning devices identified from the physics experts’
descriptions of their reasoning as they thought out loud while solving novel
graphing tasks. Each reasoning device is associated with—but not the same as—a
covariational reasoning mental action identified by mathematics education
research (described in Table 1).

####  IV.2.1 Proxy Quantities

We define a “proxy quantity” as a quantity that is not present in the prompt
of the task, but brought in by the interviewee to replace another. We observed
that it was often done to make the task easier to think about. The use of
proxy quantities is characterized by an interplay between MA 1 and codes
related to defining, relating, and symbolizing quantities 111The related code
categories are “Considering Quantity” in Study 1, and “MA 1” and “Quantity” in
Studies 2 and 3, as defined in the Appendix for the interested reader. All
physics experts were observed to use proxy quantities.

During our interviews, we found use of proxy quantity occurred most often when
participants substituted time for total distance during constant motion tasks.
For example, a graduate student working on Going Around Tacoma said:

“It looks like they’re going at a constant speed, so total distance is just
going to map to time.”

They then proceeded to solve the three tasks in Study 1 by thinking out loud
about time instead of total distance while simultaneously labelling the
horizontal axis as total distance.

We note that proxy quantities were also observed between quantities that were
not related to motion. For example, one graduate student found it easier to
think about the number of particles than the intensity during the Intensity
task:

“I want to say the number of particles is proportional to the intensity… I
don’t know if there’s a square in there or not.”

They proceeded to solve the task relating number of particles to the depth of
the liquid, and then translated the number of particles to intensity, and
depth of the liquid to the total distance traveled of the probe.

####  IV.2.2 Regions of Consistent Behavior

The use of “regions of consistent behavior” is defined as dividing a task
based on what parts of the domain make sense to be modeled by the same
function. We observed that the boundaries between regions were often
identified by an abrupt change in motion, and experts tended to focus on
modeling one section at a time. This reasoning device is characterized by an
interplay between MA 2 and problem approaches such as breaking down the
problem into functionally distinct parts 222The related code categories are
“Approach to the Problem” and “Trends of Change” in Study 1, and “MA 2” and
“Graphing” in Studies 2 and 3, as defined in the Appendix for the interested
reader. All physics experts were observed to use regions of consistent
behavior.

When dividing the tasks into sections in Study 1, all graduate students
divided Going Around Tacoma into three sections: traveling from Portland to
the semi-circular path, going around the semi-circular path, and traveling to
Tacoma. For example, one graduate student says upon watching the animation:

“So just from the picture, I’m just going to split it up into each [of the]
segments just because it’s easy to find the distance from two lines, and then
just from half of a circle.”

This observation was consistent across Studies 2 and 3 for the analogous
tasks, Gravitation and Electric Charge. Every graduate student in Study 1 also
divided Square Track into two sections: going towards the wall and going away
from the wall. They all recognized that the turning behavior at the midpoints
to the wall did not affect the distance of the cart to the wall.

Experts did not necessarily consider the sections “in order.” Rather, they
were likely to model each section independently, in whichever order made the
most sense to them. For example, many experts began Going Around Tacoma,
Gravitation or Potential by first drawing the middle section.

####  IV.2.3 Physically Significant Points

The use of “physically significant points” is defined as choosing and plotting
a small number points that hold physical meaning. For these tasks, this
physical meaning might represent bounds of a quantity (the maximum or minimum)
or important inflection points (e.g., the sides of the Ferris wheel). We
observed that experts often chose physically significant points as a first
step towards modeling the tasks. This reasoning device is characterized by an
interplay between MA 3 and behaviors such as labeling points on a graph or
diagram 333The related code categories are “Approach to the Problem” in Study
1, and “MA 3” and “Graphing” in Studies 2 and 3, as defined in the Appendix
for the interested reader. All physics experts were observed to use physically
significant points.

For example, one graduate student began Electric Charge by choosing some
points:

“So I start at some small positive, go to some bigger positive, stay there,
and then go back to my initial point.”

They plot four points to represent this story, and then complete their graph
saying, “And it’ll go like one over R in between.” Many of the interviewees
were not sure how to begin with the tasks in Studies 2 and 3 that were less
familiar, and nearly all of them chose to start by plotting points.

####  IV.2.4 Neighborhood Analysis

“Neighborhood analysis” is defined by creating a smooth graph by choosing
physically significant points and drawing small line segments centered around
those points that represent the slope of the graph. This was typically
followed by connecting these small line segments with a smooth curve and a
verbalization of the second derivative. Neighborhood analysis was used when
participants did not have a ready-to-apply model from prior physics
experience. It is characterized by an interplay between MA 4 and graphing
behaviors such as sketching slopes 444The related code categories are “Trends
of Change” in Study 1, and “MA 4” and “Graphing” in Studies 2 and 3, as
defined in the Appendix for the interested reader. A majority of physics
experts used neighborhood analysis.

For example, a graduate student checking the curve they drew for Drone stated:

“Because like, the angle is changing really quickly near the middle. You kind
of think the derivative should be bigger there. And it sort of is, like, if I
look at two, you know, two points around the middle. Like it’s changing a lot
faster than say two points here [gestures to the far left of their graph].”

This graduate student observed that the angle appears to change faster at the
apex of the curve than on the sides, and used that as justification for the
slope tangent to the curve at corresponding points on the graph.

####  IV.2.5 Compiled Models

Finally, we define the use of “compiled models” as accessing an already-known
model for a particular physical context that carries with it a common
functional relationship between two or more quantities. The “$1/R^{2}$” model
for conservative forces is a familiar example to all physicists. Accessing a
compiled model provides the reasoner with a wide variety of resources about
how two quantities relate to one another. We observed experts use compiled
models frequently for tasks that had familiar physical contexts. It is a code
category that emerged from data analysis during Study 1, and was included as a
code in Studies 2 and 3 (for more information, please see the Appendix).
Compiled models is not associated with a mental action besides MA 1 (which we
consider trivial in this context), because it involves the application of a
model and often does not engage direct covariational reasoning. All experts
were observed to use a variety of compiled models.

For example, upon reading the Ferris Wheel task and watching the animation,
one graduate student stated:

“I feel like this is where my like, understanding of trig functions really
comes in handy. Because I know this is a circle. And so the height goes like a
trig function.”

This graduate student then quickly generated a graph using the initial
position of the cart to determine the phase of the sine function.

We observed a number of compiled models in various contexts: in constant
motion contexts, a linear function that relates distance and time ($d\propto
t$); in scattering light contexts, an exponential function that relates
intensity and depth ($I\propto e^{-y/y_{0}}$); and in potential energy
contexts, an inverse function that relates potential energy and distance from
the source ($U\propto 1/r$).

These associations manifested verbally (“So potential goes like $1/R$”),
symbolically, and graphically, suggesting that they hold deep meaning to the
user. They were often used in conjunction with proxy quantities to form a
final model for the task, or were used to justify the use of a proxy quantity.
Because compiled models rely on familiar mathematical models, they are one way
that expert physicists were able to arrive at an answer to the tasks without
engaging with direct covariational reasoning besides MA 1.

These five reasoning devices often simplified the covariational reasoning
tasks, reduced cognitive load, and reduced direct use of covariational
reasoning (i.e. explicitly comparing the change of one quantity to another).
However, we suggest that these devices appear to be in part habitual, and the
avoidance of direct covariational reasoning was likely unintentional.

In addition to simplifying the tasks as part of using reasoning devices, we
also observed physics experts simplify tasks explicitly using arguments about
symmetry and limiting cases. This result adds to other research in physics
education on simplification techniques and limiting cases [61]. Recent work
has reported that students who were prompted to evaluate the validity of
proposed mathematical expressions for problems in introductory physics used a
variety of strategies to do so including the canonically taught strategies of
limiting/special cases, unit checking, and considering reasonableness of
numbers [50]. In a recent study of expert techniques when problem solving
across scientific fields, 100% of those interviewed were observed to consider
what assumptions and simplifications they could make [62]. Therefore, we focus
our analysis on thought processes associated more closely with direct
covariational reasoning.

###  IV.3 Modes for Graphical Modeling

Our third main finding is that physics experts were likely to engage in one of
three modes when graphically modeling the relationship between two quantities.
These modes are larger patterns of reasoning that describe the integrated ways
in which experts were engaged with covariational reasoning as described by the
mathematics education research frameworks (Table 1) and the reasoning devices
described in Sec. IV.2 (Table 3). This relationship is shown in Figure 6. In
this section, we describe how these modes were identified, how they are
defined, and how they relate to one another.

Figure 6: The modes identified in our data are representative of a combination
of covariational reasoning mental actions as described by mathematics
education research and the reasoning devices we identified in Section IV.2.

During our examination of the participants timelines in Studies 2 and 3, we
sought patterns between the code categories. A striking pattern emerged: there
was a consistent interplay between the code categories “Function” and “Mental
Action,” as shown in Fig. 7. The code category “Function” describes how
experts reasoned using functional forms (i.e. linear, quadratic, exponential,
etc.); it includes applying a Compiled Model and using Neighborhood Analysis
(as described in Table 3). The code category “Mental Actions” consists of MA
1-4 from the mathematics covariational reasoning framework 555The code
categories are described in more detail in the Appendix for the interested
reader. The interviewees tended to go between these two code categories while
modeling—the categories rarely appeared at the same time, and physics experts
were likely to use one and then the other back-to-back. The data clustered to
three modes of interplay between Function and Mental Actions, which we call
Function Knowing, Choosing, and Generating.

Figure 7: Four examples of the interplay between Function and Mental Actions
across Studies 2 and 3. These examples were chosen because they are
illustrative of the trends we saw for both faculty and graduate students. The
horizontal axis represents time on task, and is normalized across
participants.

Here we use function to refer to the functional form of a relationship between
two quantities that describes how two quantities change together. This
relationship can be expressed both graphically and symbolically. For example,
a quadratic function can be represented symbolically $y\propto x^{2}$ and as a
parabolic shape in which the dependent quantity varies smoothly in proportion
to the independent quantity squared. For the purposes of this work, we
consider both of these to be “reasoning about function.” Across tasks and
expert populations, we observed physicists use the phrase “goes as” or “goes
like” to refer to such a relationship (ex: “the height goes like a sine
function”). We view the use of the phrase “goes as” or “goes like” to be a
clue that someone is engaged in reasoning about a functional relationship
between two quantities [64].

Figure 8: This figure depicts the modes we observed experts engage in when
modeling, and the interplay between the mathematics mental actions and
reasoning devices for each mode. The modes are shown to lie along a spectrum
from Applying to Generating Models.

####  IV.3.1 Defining Function Knowing, Choosing, and Generating

In this section, we define the three modes, Function Knowing, Choosing, and
Generating, and present evidence from across the three studies. We note that
the distinctions between Function Knowing, Choosing and Generating are not
hard and fast; there were many instances where the research team held
discussions about whether a particular portion of transcript was better
characterized as one or another. We suggest that these modes describe a
spectrum of approaches to modeling from recognizing a well-modeled physical
situation towards generating a model where one doesn’t exist. This is
represented by the arrow in Fig. 8, which goes from Applying Models to
Generating Models.

Reasoning Mode |  Item |  Example
---|---|---
Function Knowing Quickly associating a physical context with a known mathematical model. |  Gravitation |  “I want to talk about the gravitational potential energy of the entire system. Okay, so gravitational potential energy goes as $1/R$, one over the total radius to the system.”
Ferris Wheel |  “So, the total distance traveled is proportional to the time because it’s moving at constant speed.”
Function Choosing Plotting physically significant points and using the trend between them based on the physical context to choose a function. |  Intensity |  “And so we expect that at the maximum distance traveled, the intensity will be greatest. And as light is absorbed, it will get less great…I have no idea. I would expect, and it depends if its in the linear regime, how exactly it will go. But naively, I’d expect that it’d be roughly linear.”
Intensity |  “So at the top, you’ll get the maximum intensity here and it will drop—it will drop according to some some curve that depended on this. It could be an exponent—and likely an exponential. It really would—should be, uh—should be an exponential fall because the differential probability of having less light drives the thing.”
Function Generating Connecting physically significant points with a smooth curve using the rate of change as a guide. |  Drone |  “The top—basically, the height changes the most slowly, and then the bottom…and, well, as you increase the angles, the height goes—changes faster. So you should have a similar trajectory as you would see in the picture. But how exactly it is, is a different question. I think you can’t solve this question quantitatively, but qualitatively you can draw a curve like this.”
Table 4: The ways in which experts engaged with mathematical modeling while
using covariational reasoning during graphing tasks.

Function Knowing is characterized by accessing a compiled model and applying
it to the context using important, physically relevant points (such as those
described by the initial conditions of the task). Function Knowing often
involves the use of proxy quantities to make sense of the accessed model in
the context of the task. It is described by an interplay between MA 1 and the
use of compiled models (Fig. 8).

For example, upon seeing the Ferris Wheel task, many of our interviewees
immediately identified the context as circular motion, and that they would
need to use a trigonometric function. Most then reasoned about how to “map”
between time and total distance to make sense of their initially accessed
model that related height and time.

Function Choosing is characterized by using the general trend between
physically significant points to choose between a handful of compiled models.
It is described by an interplay between MA 2 and Compiled Models (Fig. 8).

For example, during Intensity, many experts reasoned that the lowest intensity
was at zero distance traveled and the highest at maximum distance traveled.
Then interviewees chose how best to connect the points by using the general
trend between them and by reasoning about which functions made sense in the
physical context—e.g. linear or exponential.

Function Generating is characterized by using Neighborhood Analysis:
determining the rate of change at a small number of physically significant
points and then plotting the points along with small line segments that
represent the slope of the curve at those points. The points are then
connected with a smooth curve using the line segments as a guide. Some
participants would name the function that looked like the curve they drew, and
others simply ended the task with the graph. It is described by an interplay
between MA 4 and Neighborhood Analysis (Fig. 8).

Examples of each pattern are shown in Table 4. These examples also help to
demonstrate the distinctions between the three approaches. In the examples of
Function Knowing, both speakers have a model ready at hand, and spend their
time deciding how to apply it. In the examples of Function Choosing, the
speakers use the general trend between points to choose between a handful of
possible models. Finally, in the example for Function Generating, the speaker
does not identify a possible function and instead is guided by their
understanding of the first and second derivative. We note these approaches are
emergent from the data and are therefore necessarily reliant on the ways the
experts chose to talk about how they were reasoning. However, they do give us
some insights into common approaches expert physicists took when solving
graphical covariational reasoning tasks.

Across all three studies, Function Generating was consistently the last-resort
option: experts were likely to begin with Function Knowing or Choosing and
finally engage in Function Generating only if Function Knowing or Choosing had
failed. Figure 8 illustrates the spectrum we observed from applying models
with Function Knowing to developing new models with Function Generating. The
arrow from Applying to Generating Models is inspired by Tall’s 2008 diagram of
proceptual reasoning where it represents increasing sophistication [65]. In
Figure 8, we suggest the arrow is illustrative of increasing cognitive load
and the observed reasoning progression.

####  IV.3.2 Evidence of Function Knowing, Choosing and Generating from the
Studies

We predominately observed Function Knowing and Choosing during Study 1. For
Going Around Tacoma and Square Track, every expert associated the constant
motion context with a linear relationship between the distance asked for in
the task and the total distance traveled. Therefore, the time spent on these
tasks was focused on connecting physically significant points with straight
lines.

During Ferris Wheel, the majority of the graduate students held strong
associations between the circular motion context and trigonometric functions.
They quickly identified that they needed a sine function, and used initial
conditions to identify the phase and the amplitude:

“So our height is just going to be kind of going sinusoidal…[and the cart]
started out at highest point [draws curve].”

Function Generating was used during Ferris Wheel by three of the ten graduate
students in Study 1. These participants did not demonstrate an activated
association between circular motion and trigonometric functions, and instead
examined the rate of change at particular points in order to determine the
graph. For example, one graduate student stated:

“So those — kind of — be some kind of curvature to- to this.

So towards the halfway point, when it’s almost, when it’s halfway to the
bottom, the speed should be the greatest. And then otherwise, it should be
like — should kind of have a lower slope….

So low to high slope back to low slope.”

This student drew small line segments to guide their reasoning, and then
connected them with a smooth line to form their final graph (Fig. 9).

Figure 9: A graduate student’s work from Ferris Wheel that demonstrates
Function Generating reasoning; the point and line segment they used are
highlighted in green.

Studies 2 and 3 were designed to probe expert reasoning outside of constant
motion (and circular motion) contexts. Two of the tasks, Gravitation (and
Electric Charge) and Intensity required physics knowledge to solve, and
therefore necessarily required either Function Knowing or Choosing. However,
these contexts were less familiar to most participants and we observed that
graduate students were likely to toss out several different models while
engaged in Function Choosing. For example, one graduate student stated:

“Not going to use it, but intensity is proportional to the magnitude of the
field squared.

Yeah, I’m not going to use that. Because…no, I mean, never mind, I can’t use
that. That is what that’s what intensity is. But I’m not going to calculate….

Should I consider the situation…exponential from? I guess that’s just me
reaching for a model. That happens a lot. Exponential wouldn’t make sense.
That’s like skin depth penetration. That would be like a beam hitting a
conductor.”

This line of reasoning was ultimately productive for the graduate student, as
they were able to get some initial compiled models out of the way by realizing
that the models did not apply to the question being asked. However, they also
distracted themselves from the correct answer (exponential) with other content
knowledge.

This example is provided to show the range of Function Choosing—for some, it
was a process of choosing between compiled models that had physical meaning in
the context of the task. For others, it looked like coming up with several
possible models from physics that are related to the quantities involved, and
working through which were relevant to the prompt. Faculty were more likely to
engage in the former approach, and early career graduate students were more
likely to use the latter.

Finally, Drone prompted participants to engage in Function Generating. This
may reflect the fact that deriving an analytical solution is challenging and,
as one faculty member said, “messy.” Nearly every expert solved Drone by
engaging with Function Generating, either as a check on an initially drawn
curve or to develop a curve.

Figure 10: The time participants in Study 2 and 3 spent Function Knowing,
Function Choosing, or Function Generating as a percentage of time spent
engaged in function reasoning.

Across Studies 2 and 3, we found the graduate students were most likely to
engage in Function Choosing, and faculty were most likely to engage in
Function Knowing or Function Generating (see Fig. 10).

##  V Discussion

This work seeks to characterize some ways that physics experts reason
covariationally while solving novel graphing tasks. Research Question 1 asks
“In what ways are the covariational reasoning frameworks developed by
mathematics education researchers productive for describing expert physicists’
modeling in novel graphing tasks?” We found that some of the ways mathematics
covariational reasoning has been described were consistent with physics
experts’ reasoning, and some were not. Research Question 2 asks “What patterns
do we observe in the reasoning of physics experts as they solve novel graphing
tasks?” We found five reasoning devices and three modeling modes that were
used consistently by all experts while reasoning covariationally. In this
section, we describe how these findings relate to current literature and
present a hypothesis for how _physics_ covariational reasoning is distinct
from that in mathematics.

###  V.1 Comparing Mathematics and Physics Expert Reasoning

Our original study—Study 1—was designed as a replication study of Hobson and
Moore’s 2017 investigation into the covariational reasoning of expert
mathematicians. In this section we examine Research Question 1 by considering
how our findings are similar to and different from those of Hobson and Moore.

Hobson and Moore reported that mathematics graduate students:

  * (1)

Directly covaried the quantities asked for in the task (in some cases, with
prompting from the interviewer), and

  * (2)

Equally divided the domain of the task, or large sections of the motion, into
small sections to compare the change in the two quantities and then mapped
these equal segments to their graphs [20].

In comparison to (1), we observed that the physics graduate students from
Study 1 all used time as a proxy quantity for total distance. Hobson and Moore
report that, after prompting from the interviewer, the graduate students they
interviewed articulated a direct covariational relationship between the two
distances in their equivalent of Going Around Tacoma. For example, they report
that one student stated, “for every one mile you move towards the total
distance towards [Portland] you’re still getting that one mile closer to
[Tacoma]” [20]. In contrast, the physics graduate students working on Going
Around Tacoma simply stated that because the objects moved at constant speed,
distance “maps” to time. One of the graduate students went so far as to create
a graph with respect to time. Upon realizing their mistake, they simply
crossed out “time” and wrote “total distance” on the horizontal axis. No
physics graduate students drew a direct connection between the two distance
quantities while thinking aloud or during questioning by the interviewer. We
suggest that this reflects how ingrained constant motion contexts are for
physics experts, and perhaps also an indication of how ubiquitous proxy
quantities are in reasoning about how quantities are related in physics.

We also found that physics graduate students were highly likely to section the
domain of a task only if there were distinct regions of consistent behavior.
Within a region of consistent behavior, the graduate students either applied
an already-known model or used the rate of change to generate a graph. Rather
than relate equal portions of change to gather points on the graph and justify
a non-linear increase in one quantity for an equal increase in another, the
physics graduate students considered nearly instantaneous rates of change
centered on particular points and assumed a smooth connection between them.

We recognize that Hobson and Moore were not attempting to characterize all of
expert mathematics covariational reasoning; nor are we attempting to describe
all of expert physics covariational reasoning. Rather, we take this
replication study as an opportunity to consider how practices in the two
fields may differ when reasoning about change. Recognizing the differences in
how experts reason covariationally may be informative for instructors to
reflect on their own reasoning about function and rates of change as it
compares to what kinds of instruction their students may have received.

###  V.2 The Emergent Reasoning Devices and their Relationship to the
Mathematics Mental Actions

During our analysis across all three studies, we identified several reasoning
devices that accompany covariational reasoning used by physics experts. We see
these devices as closely related to many of the direct covariational reasoning
mental actions defined by mathematics education researchers [16, 19] (towards
Research Question 1). There are also ways that these devices are distinct, and
we suggest these devices contribute to patterns of expert _physics_
covariational reasoning when solving novel graphing tasks (towards Research
Question 2).

Proxy quantities were consistently used by physics experts across most of the
tasks. In particular, we saw that proxy quantities allowed expert physicists
to use models they already knew efficiently and “map” the familiar model to
the contexts and quantities the task asked for. We propose proxy quantities as
an extension of MA 1 that is particular to physics covariational reasoning, as
it is a tool that we observed physics experts use to replace one quantity for
another in order to make the task easier to think about.

This reasoning device is reminiscent of “simultaneous covariation” in
mathematics education research [18], which has been characterized by
considering how two quantities change independently with respect to time and
then comparing the independent changes of one to coincident changes in the
other. While a small number of physics experts described their use of proxy
quantities as “parameterization,” most simply talked about “replacing”
distance with time. Further, we observed physics experts engaged in using
proxy quantities that were not time, namely, radius for total distance in
Gravitation and Electric Charge.

We also observed experts sectioning tasks into regions of consistent behavior,
and using neighborhood analysis to examine the rate of change at particular,
physically significant points. We see sectioning as an extension of MA 2
particular to physics covariational reasoning, as the experts were seeking
sections where the relationship in the trend was constant (i.e. articulating
that one quantity increases while the other stays the same for a section of
the domain). Similarly, we suggest that neighborhood analysis could be an
extension of MA 4 for physics contexts. Thompson and Carlson’s 2017 framework
defines MA 4 as comparing discrete change in two quantities. The next step (MA
5) is smooth continuous variation, in which the person views the quantities
changing smoothly together. We suggest that Neighborhood Analysis sits between
these two mental actions; physics experts had in mind that the quantities
changed smoothly, but used small, nearly infinitesimal, chunks of discrete
change modeled at particular points to guide their smooth curve.

Finally, we suggest there are many differences in the goals of physics and
mathematics that might lead to these different approaches in covariational
reasoning. As mathematical modeling is often one step towards a larger goal in
physics, a degree of speed and approximation is expected. We observed many
experts use simplification methods—making assumptions about the task, only
focusing on particular points, using limiting cases to constrain the context,
and using symmetry to get the second half of a task immediately—to great
effect seemingly in order to minimize the effort they needed to extend. It
could be that the epistemological approach of finding a model that is, as
Albert Einstein said, ““As simple as possible, but not simpler,” is a dominant
feature of modeling in physics.

###  V.3 Modes Associated with Modeling using Physics Covariational Reasoning

An important pattern in how experts were reasoning about function while
mathematizing during the graphing tasks emerged from the data. We observed
three categories of approaches to reasoning covariationally, both symbolically
and graphically, which we call Function Knowing, Function Choosing, and
Function Generating. In this section, we propose a hypothesis for how these
approaches are used and describe how they represent some important ways that
expert physicists reason covariationally.

We observed that experts were likely to begin with Function Knowing (if
accessible), then work through Function Choosing, if more than one compiled
model surfaced. Function Generating was consistently the last-resort option,
and only attempted once Function Knowing or Choosing was eliminated. We
summarize how Function Knowing, Choosing, and Generating relate in Figure 8.
From our findings, we described the arrow as descriptive of the observed
reasoning progression. We also suggest that it may be related to increasing
experience generating models, as we saw that faculty members were more likely
to engage in Function Generating than either group of graduate students (see
Fig. 10).

In our model, Function Knowing, Choosing, and Generating are characterized by
an interplay between the direct covariational mental actions defined by
mathematics education researchers [16, 19] ( Table 1) and the reasoning
devices described in Table 3. The cyclic representation between mental actions
and the devices compiled models and neighborhood analysis represent the ways
in which Function Knowing, Choosing and Generating are intimately connected to
mental actions and go beyond how the mental actions are described in
mathematics education research (see Fig. 6 and 8). In particular, Function
Choosing is an interplay between using MA 1 (relating quantity) and accessing
a compiled model. Function Knowing is an interplay between using MA 2 (gross
coordination) and accessing one or more compiled models. Function Generating
is an interplay between MA 4 (chunky continuous) and neighborhood analysis.
Function Choosing, Knowing, and Generating are all founded on reasoning about
quantities in the context of the physical world and their change with respect
to one another. Thus, we propose that Function Choosing, Knowing, and
Generating are three patterns associated with expert _physics_ covariational
reasoning.

###  V.4 Comparing Faculty and Early Career Graduates

In Function Knowing, Choosing, and Generating, early career graduate students
were more likely to stay in the Function Choosing space. Faculty were more
likely to use Function Generating overall, but were also the most likely to
use Function Knowing of the three populations.

Bing and Redish describe a “journeyman” level that is situated between expert
and novice. Journeymen are characterized as “having sufficient skills that
they can no longer be considered novices, but not yet had sufficient
experience with sophisticated problem solving (and research) to be considered
experts” [66]. We suggest that, for early career graduate students, rigorous,
mathematically complex problem solving is the norm. It might be that they are
so continually thrust into a mindset of complex textbook-like physics problems
that they are less likely to solve the tasks we provided with a generative
frame of mind. As early graduate students haven’t had much experience in
research, our data suggest that engaging in the theoretical aspects of
research may foster comfort with modeling quickly and making simplifying
assumptions. It may also be that “sophisticated problem solving” looks
different from the kind of highly complex tasks early career graduate students
are asked to engage in during coursework. We suggest that our data for early
career graduate students are better described by less experience in research
and being in a “test-taking” mindset of coursework in graduate school, but
more research is needed to fully investigate these ideas.

##  VI Conclusion

The mathematics education research community has demonstrated the importance
of covariational reasoning for understanding concepts in pre-calculus and
calculus. Recognition of the importance of covariational reasoning led to the
development of a framework of covariation mental actions, which effectively
operationalizes covariational reasoning. Mathematics education researchers
have found that the covariational reasoning frameworks in mathematics are
helpful when investigating student reasoning.

Though prior work suggests that physics experts’ covariational reasoning
“looks different” than that described by mathematics education researchers, it
also suggests that covariational reasoning is essential for developing and
understanding quantitative models in physics. The studies described in this
paper were motivated by a desire to understand how physics experts use
covariational reasoning. We examine the reasoning of physics experts engaged
in graphing tasks through the lens of the mathematics covariation mental
actions framework. Our investigation indicates that the covariational
reasoning mental actions described by mathematics education researchers are
part of productive quantitative modeling for physics experts across varied
physics contexts.

However, we also find that physics experts use a number of other devices, not
described by the mathematics covariational reasoning framework, to coordinate
two varying quantities. These devices are used alongside the mental actions
described in mathematics education research to form patterns in physics
experts’ reasoning that we identify as modeling modes. We observed these modes
across a range of physics contexts and in a range of physics experts. The
modes rely not only on direct covariation, but also on facility with the
underlying mathematics and physics content. In addition, we find that for
physics experts, direct covariational reasoning is intertwined with
sensemaking about the physical world and the relevant physics quantities.
These results suggest that the covariational reasoning frameworks developed by
mathematics education researchers do not fully operationalize _physics_
covariational reasoning.

Covariational reasoning has been defined by mathematics education researchers
as “the cognitive activities involved in coordinating two varying quantities
while attending to the ways in which they change in relation to each other”
[16]. We suggest that this definition is productive for physics covariational
reasoning as well, with the understanding that for physics experts there exist
a number of “cognitive activities” beyond the mental actions described by
mathematics education researchers. We contend that the set of devices and
modeling modes described in this paper, along with the mental actions
described by mathematics education researchers, characterizes physics
covariational reasoning.

The work described in this paper provides a foundation for operationalizing
physics covariational reasoning. As a next step, we are using the data
collected for these studies to investigate how physics experts use sensemaking
about the physical world to guide their use of mathematics during
covariational reasoning. We are also working to characterize the understanding
of the mathematical foundations and physics quantities that seem to be
necessary for expert-like physics covariational reasoning. These findings will
be synthesized into a framework of covariational reasoning in physics (CoRP).
Our hope is that the CoRP framework will not only operationalize physics
covariational reasoning but will also be productive for characterizing physics
students covariational reasoning.

##  VII Acknowledgements

We would like to thank the faculty and students at the University of
Washington who have made this work possible—their participation and support
has been integral to this project. We would also like to thank Peter Shaffer
and John Goldak for their thoughtful reviews during the preparation of this
material. This work was partially supported by the National Science Foundation
under grants No. DUE-1832836, DUE- 1832880, DUE-1833050, DGE-1762114. Some of
the work described in this paper was performed while the second author held an
NRC Research Associateship award at Air Force Research Laboratory.

*

## Appendix A

Here we present the codebooks for Study 1 and for Studies 2 and 3 for the
interested reader. In the codebook for Studies 2 and 3, an effort has been
made to connect the code categories for Studies 2 and 3 back to those for
Study 1 to aid the reader.

Code Category |  Description |  Example
---|---|---
Mapping Math to Physics |  The epistemic game of attempting to fit a particular mathematical function to a physical scenario [10]. |  “And then total distance starts to increase, so we’re going to go up [draws a line increasing linearly].”
Considering Quantity |  Identifying, defining, and representing quantities either given in the problem statement or devised in the process of problem solving. |  “So distance from Tacoma’s on the y axis and then total distance, its going to be on the x axis.”
Approach to the Problem |  Identifying the “best way” to start a task, unprompted. This includes but is not limited to breaking down the task and identifying important points. |  “So there’s a moment when they’re doing that loop around Tacoma looks like half a circle. So that’s going to be a constant distance. So I’m going to start off with that.”
Trends of Change |  Relating the change in one quantity to the change in another; most commonly by discussing the rate of change or slope of a graph. |  “When it’s halfway to the bottom, the speed should be the greatest.”
Graphical Structure |  Finding a larger graphical trend between two quantities; this includes linking together known rates of change with a smooth curve, and seeking graphical features such as extrema. |  “So if I want it to be slow, fast, slow, I think it looks like…[draws a curve connecting tangent lines].”
Compiled Models |  Using an already-known relationship between two quantities that is strongly associated with a physical context to model a particular situation. This new term is further discussed in Section IV.2. |  “I’m just like — oh, trig function — its a circle.”
Table 5: Study 1 code categories that emerged during data analysis. Code Category |  Description |  Example
---|---|---
Physical World |  Analogous to Mapping Math to Physics from Study 1; we limit this to when interviewees specifically discuss the physical world in the context of the task. |  “This point here [gesturing to the top of the Ferris wheel], that’s the maximum height.”
Quantity |  Analogous to Considering Quantity from Study 1; additionally includes explicit reference to the use of a quantity not named in the task to make the task easier to reason about. |  “Distance traveled…uh..well, [is the] same thing as time basically.”
Mental Actions |  The use of the mental actions as described by mathematics education research in Table 1. |  “So you get basically some some curve like this. Slowly, slowly changing in the bottom.”
Graphing |  Analogous to Graphical Structure from Study 1; includes behaviors associated with drawing a graph, such as connecting points, assuming a smooth curve, and distinguishing between independent and dependent quantities. |  “Okay, total distance traveled and the electric potential it measures. So, I want to do distance traveled on the x-axis… and the y will be electric potential.”
Function |  Analogous to Compiled Relationships from Study 1; includes both compiled relationships and the identification of a particular relationship as a particular functional form (parent function). |  “So the trajectory looks like is a quadratic—It’s sort of a parabola.”
Table 6: Studies 2 and 3 code categories that emerged during data analysis.
These codes were developed using thematic analysis [54] such that they
describe emergent patterns from the data. While an effort was made to set
aside the codebook from Study 1, many similar themes arose. We connect these
code categories to those of Study 1 in the descriptions for the ease of the
reader.

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  * Note [1] The related code categories are “Considering Quantity” in Study 1, and “MA 1” and “Quantity” in Studies 2 and 3, as defined in the Appendix for the interested reader.
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