# Empirical evidence of the inseparability of mathematics and physics in
expert reasoning about novel 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

Pre-college mathematics modeling instruction often frames mathematics as being
separated from reasoning about the real world—and commonly treats reasoning
mathematically and reasoning about the real-world context as separate stages
of a modeling cycle. In this paper, we present evidence that helps
characterize how experts use mathematics in physics contexts while developing
graphical models. An important finding is that there was essentially no
evidence of experts reasoning in a context-free way with these tasks, but
instead they used physical reasoning—either grounded in the context of the
task or from abstract physical models—to guide their mathematics. The
difference in approach of physics instructors and students may lead to a
mismatch in expectations, and frustration, for both parties. This work
contributes to the body of knowledge about how mathematical reasoning appears
in physics, and can help researchers and instructors recognize the connections
to the physical world in expert mathematical reasoning. This can help
instructors and researchers be better equipped to develop materials and
methods that can help students start building those connections as well.

††preprint: APS/123-QED

##  I Introduction

Instruction about modeling in pre-college mathematics courses
characteristically represents “mathematical” reasoning as distinct from
reasoning about the real world [1, 2, 3, 4]. Emphasizing a separation between
mathematical reasoning and reasoning about the context during instruction may
leave future physics students with the impression that there is physics-less
mathematics associated with solving introductory physics problems. Recent
research reveals that physics students perceive that there is a difference
between the mathematics they do in mathematics courses, and working with
mathematical models in physics [5, 6].

There is growing evidence that upper-division physics and engineering students
do not demonstrate this separation in their practice when they are successful
in their problem solving. There is, in fact, a continuous interplay of
mathematical and physical reasoning [7, 8, 9, 10]. We suggest that the physics
experts who design and teach introductory physics courses may also reason in a
less binary way, with more nuance in the ways that they demonstrate how to
develop and make sense of models—and they may expect that their students are
prepared to follow along. If instructors are modeling in physics in ways that
are inconsistent with students’ pre-requisite mathematics courses, both
students and instructors may be frustrated with the learning outcomes.
Therefore, understanding the interface between expert modeling in introductory
physics and students’ preparation from prior mathematics courses can help
improve the learning and teaching of physics for both students and
instructors. To this end, this work sets out to better understand expert
modeling in the contexts of challenging, introductory-level, graphical tasks.

Initially, we intended to isolate and characterize expert behavior when
reasoning “purely mathematically” in introductory classical physics contexts.
It was quickly evident that when experts modeled in the contexts of these
tasks, physical reasoning was ubiquitous and inseparable from mathematical
reasoning. This finding led to a shift of our research focus from examining
when physics experts engaged in “purely mathematical reasoning” to
characterizing what they actually do. We sought to answer a research question
that represents the core of this paper: “What are some features of the
mathematical reasoning used by physics experts when modeling graphically?”

The results we present contribute to the growing body of knowledge about how
physics reasoning interacts with mathematical reasoning. Examining the
reasoning of physics experts provides researchers with data from one end of
the expert-novice continuum, and may help physics instructors recognize their
own reasoning patterns, assumptions of comprehension, and emergent expertise
in their own students.

##  II Background

There is a growing consensus that quantitative reasoning in pure mathematics
contexts is different from that in physics contexts [11, 12, 13, 14, 15, 16,
17, 18, 10]. Physics education researchers have proposed multiple
characterizations of this difference: Redish described physics as a linguistic
dialect distinct from the language of mathematics [15]; Caballero pointed to
how the operationalization of mathematical tools is often different in physics
than in mathematics [16]; and White Brahmia emphasized the centrality of
quantification to mathematical modeling in physics contexts, where
quantification refers to characterizing a feature of a system with a
quantity—represented by a letter (e.g., $x,v,a$)—that includes a unit of
measure and typically a sign. [19, 20, 21, 22].

Connecting a quantity’s physical meaning to its mathematical representation is
an important part of developing and making sense of mathematical models, and
has been shown to be challenging for many students in introductory physics
contexts and beyond [23, 24, 22, 25, 20]. Research has demonstrated student
difficulties with making sense of the physical meaning of vectors, noting that
many students can make sense of the symbols but struggle with translating
between vector and graphical notation and extrapolating the information to a
vector field [26]. In calculus-based graphical contexts, Bajracharya and
Thompson describe how successful students were able to connect the ideas of
area under the curve and integration, demonstrating both procedural and
conceptual understanding of the underlying mathematics, while also relating
the mathematics to its physical meaning [17]. These findings are aligned with
research in physics education that facility with multiple mathematical
representations is a key element of making sense of the meaning of
mathematical models [27, 28, 29, 30, 31].

Figure 1: Czocher’s redraft of Blum and Leiß’s modeling cycle [32, 9].

A common way to represent the sequence of moves in modeling is as a “modeling
cycle” in which quantification is an early step, and validation—interpreting a
mathematical model within the physical context—is the last (see the example
from Blum and Leiß in Figure 1). Notably, the mathematics space depicted is
accessed between quantification and validation, and does not involve
consideration of the physical context, but instead represents mathematical
manipulation in absence of attention to physical meaning. Similar cycles that
separate mathematical reasoning and reasoning about the real world have been
proposed both in physics and in mathematics education research [32, 33, 34,
35, 36, 3]. While recent education research has moved toward a more blended
perspective of mathematics and physics reasoning, these cyclic representations
commonly frame modeling education in high school mathematics courses and
thereby influence students’ perceptions of mathematical modeling before coming
to college [1, 4].

Research has found little evidence of a clear separation between reasoning
mathematically and reasoning about the real world in quantitative modeling;
moreover, the progression of actual modeling may be messier than the published
cycles would suggest [7, 8]. In a recent study, Czocher conducted interviews
throughout an academic term of four engineering majors enrolled in a
differential equations course [9]. In each interview, the students were
observed solving problems in physics contexts that required generating
mathematical descriptions from a variety of branches of mathematics, including
differential equations. The author described a much finer-grained blending of
mathematical reasoning and physical sense-making than is represented in a
priori “modeling cycles.” Specifically, Czocher notes that “there are
transitions that appear out-of-order. This was largely because three of the
modeling transitions (understanding, simplifying/structuring, and validating)
appeared early and often throughout the students’ modeling processes” [9].
Serbin and Wawro examined junior and senior physics students’ reasoning in
quantum mechanics through the lens of Uhden et al.’s modeling cycle, and
similarly found that successful modeling patterns did not support separating
mathematical reasoning and physics reasoning as predicted in the theories of
how students generate models. The authors state “when students draw on their
mathematical knowledge…it relies on their understanding of how the
mathematical concepts and physics concepts are intertwined” [10].

The results of Czocher, and Serbin and Wawro suggest that modeling cycles that
maintain separate mathematics reasoning and physics reasoning may require
revision to reflect a tighter blend between ideas that are physics-like and
math-like [9, 10]. Typically, conceptual blending is used in physics education
research to describe a fine-grained approach where particular utterances,
equations, or terms are separated into mathematics or physics “input spaces”
by the researcher [37, 38, 13, 39, 40, 41, 42, 43]. These input mental spaces
are described as subconsciously blended into one idea by the speaker. The
blended space may also include utterances that are inseparable by the
researchers. Increasingly, physics education researchers are using the term
more broadly to refer to the blended nature of mathematical structure and
physical meaning [25, 24, 44, 45].

There remains, however, a question in physics about the prevalence of blended
reasoning as compared to “purely mathematical” reasoning—i.e. mathematics free
of contextual meaning—and what moving between these reasoning modes might look
like for physics experts. The research we present here focuses on expert
modeling in the contexts of graphing tasks, adding to current findings
associated with the blended mental space between physics and mathematics in
modeling.

##  III Methods

In this section, we describe the methods used to conduct and analyze the
think-aloud interviews that were part of our investigation into how expert
physicists reason quantitatively during graphing tasks [46].

###  III.1 Data Collection

We report on results from 15 think-aloud, individual interviews that were
conducted with 10 graduate students and five faculty members at an R1
university in the Pacific Northwest. Of the 15 experts, six were female or
non-binary identifying and nine were male identifying. The pool of experts
also spanned a variety of subfields, and included both experimentalists and
theorists. An effort was made to include diversity across race and country of
origin. Participants were solicited by email, and offered a small gift card to
a local coffee shop as a thank you for their time and effort.

The graduate student interview transcripts were divided into two groups of
five for data analysis. The first included early career graduate students who
were enrolled in required graduate courses; the second, late career graduate
students who were no longer taking courses outside their research area and
were fully engaged in research.

One member of the research team conducted the interviews, during which the
participants were asked to solve 3–4 graphing tasks. In the analysis described
in this paper, we focus on three tasks that were identical (or isomorphic)
across the 15-person expert pool. For the first task, experts were given one
of two isomorphic versions: “Electric Charge” or “Gravitation” (Fig. 2). For
the second and third tasks, all experts were given both “Drone” and
“Intensity” (Fig. 3). All of the tasks prompted the participant to watch an
animation and create a graph that related two quantities in the animation.

Figure 2: Stills from the animations associated with (A) Electric Charge and
(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 3: Stills from the animations associated with (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.

Each interview lasted between 25 and 50 minutes, and ended when the
participant expressed that they were finished with all the tasks. At the end
of the interview, the interviewer asked clarifying questions, if applicable.
The interviews were audio recorded; initial transcripts were created
automatically using the Otter.ai software program [47], and were subsequently
hand-corrected.

###  III.2 Data Analysis

|  Code |  Description |  Example
---|---|---|---
|  Symbolizing |  Representing quantity with symbols. |  “This is potential, $V$.”
QUANTITY |  Mathematical Structure |  Choosing a particular structure or discussing why a structure makes sense for a particular quantity. |  “So this is only going to be positive.”
Composite Quantity |  Defining a quantity by others, or making sense of why a quantity is constructed of others. |  “$I$ is gonna be the number sent in times $e$ to the minus $y$ over $y_{0}$, where $y_{0}$ is a characteristic, sort of, thickness of the material.”
Proxy Quantity |  Using one quantity in place of another to make sense of or visualize what is happening. |  “So as I go from distance traveled, I’m basically starting at a fixed $R_{0}$ and I’m going to a smaller $R$ [from] there.”
|  Code |  Description |  Example
TASK CONTEXT |  |  |
Connection to the Task |  Making sense of a quantity or feature of their graph, or drawing a diagram to visualize a quantity in the context of the task. |  “And so when they’re far apart, there’s low gravitational potential energy.”
|  Code |  Description |  Example
MODELS |  Compiled Model |  When participants bring in a previously known model from physics. |  “Okay, so I’m going to sort of invoke Bier’s law.”
Compiled Trend |  When participants refer to a particular function and/or discuss its behavior. |  “I’m going to say this is a graph of something like a minus arctan of $x$.”
Table 1: Codes associated with the code categories Quantity, Task Context, and
Models.

The data were coded using a thematic analysis framework [48]. Thematic
analysis emphasizes consistent communication across the research team to
ensure reliability while recognizing that the research team may bring
perspectives to the work based on prior experience with current literature.
The practice assumes no a priori coding scheme and focuses on grounding the
codes in the data through iterative review and discussion. Our interpretation
of these data is informed by our prior experiences, established theories of
mathematical reasoning [32, 33, 34, 35, 36], mathematics education research
testing the cyclic nature of these theories [9, 10], and research in physics
education that uses conceptual blending to describe mathematical reasoning in
physics contexts [38, 13, 39, 40, 45, 18, 24, 44, 25] as discussed in Section
II.

One member of the research team coded the data, using participants’ written
work and notes from the interviewer to confirm the interpretation of the audio
transcripts. Between coding cycles, the researcher conferred with the research
team to establish wide-spread agreement.

Once the codebook was stable, the same iterative process was followed to seek
themes amongst the codes. Several code categories emerged, three of which we
share here:

  * (1)

Quantity, which refers to when participants generate, relate, symbolize, or
discuss the mathematical structure of quantities which are inherently situated
in the real world.

  * (2)

Task Context, which refers to when participants made explicit reference to the
physical context of the task.

  * (3)

Models, which refers to the ways in which participants use previously known
physical models or apply their knowledge of particular functions common in
physics.

Table 1 shows the codes in these categories, along with a short description
and example of each code.

The research team took care to allow the codes to emerge from the data. Upon
first glance, a code for “physical reasoning” seemed appropriate, however it
quickly became clear that the “physical reasoning” code was ubiquitous across
the data, rendering it meaningless. The lead coder also felt they could not
distinguish between “physical” and “mathematical” reasoning throughout the
data, even after lengthy discussion with the research team. Therefore, we
instead developed the code Connection to the Task, and established the
boundary that this code only included instances in which experts explicitly
relate their reasoning to the task at hand. This distinction removed some of
the subjective interpretation associated with deciding what kind of reasoning
is physical, and grounded the coding in the statements of the experts. A
consequence of this decision is that reasoning about an abstract physical
model is not included in the Task Context category. For example, one expert
stated:

> “If I think about $v_{x}$ over $v_{y}$, $v_{x}$ is a constant and $v_{y}$ is
> linear in $t$.”

This statement is categorized as Quantity, since the expert is symbolizing
velocity quantities, but not Task Context since they are reasoning using the
kinematic equations in abstract terms. Instead, we applied a separate code,
Compiled Models, to indicate they were using a previously-known model [46]. In
contrast, later in their reasoning this expert stated:

> “As we get towards the top…as my height increases, my angle—does it change
> faster?”

It is clear the expert is considering the context of the task at hand when
they talk about “the top” and the statement is categorized as Task Context.

We created timeline charts to seek patterns between code categories. The
timeline charts visually represented segments of transcript for when a
participant’s reasoning was assigned a particular code category, and the
duration for which they were coded in that category. Figure 4 shows six
representative timelines from our data. We sought patterns in which categories
frequently appeared together and frequently appeared apart. After identifying
segments of transcript in which a given pattern appeared (e.g., context coded
with quantity), we returned to the transcripts for further analysis. We
categorized the transcript segments into two groups: (1) those in which the
text supported the pattern and (2) those in which the pattern was refuted. For
example, for a pattern that appears together in the timelines, we collected
all transcript pieces where the code categories coincided, and separately
collected all transcript pieces where the code categories had not. We examined
these sections of text carefully to find patterns in the ways that experts
reasoned in order to characterize, or dismiss, the relationship identified in
the timeline.

Finally, we used quantitative measures to compare the frequency of categories
appearing separately or together. For example, we calculated the time on task
for all utterances where Quantity had been coded (on its own, or with any
other category), and the time on task where Quantity and Task Context had been
coded together.

##  IV Findings

Figure 4: A visual representation of when participant statements were assigned
the code categories Quantity, Task Context, and Models. The length of the bars
corresponds to the time the participant was engaged in that line of reasoning.
The horizontal axis represents time, normalized to the length of each
interview. Therefore, it can be thought of as a percentage of their total time
on task. The highlighted regions are where Quantity appears without Task
Context. Vertical lines represent when participants switched tasks.

Our research question for this study focuses on the relationship between
physics and mathematics reasoning: “What are some features of the mathematical
reasoning used by physics experts when modeling graphically?” Our data
revealed that there was almost no time during which physics experts reasoned
without reference to the physical world. This is in contrast to what is
suggested by the presence of “purely mathematical reasoning” in common
modeling cycles. Instead, the distinction between how experts reasoned
physically and mathematically lay in whether they reasoned with reference to
the physical world abstractly or with reference explicitly to the context of
task at hand.

We identify two major features of mathematical reasoning in physics from this
work. (1) When physics experts reasoned mathematically, they generally did so
with explicitly reference to the context of the task. (2) During the
relatively infrequent occasions when experts did not attend explicitly to the
task context, physics experts were likely to use other familiar physics models
as a starting point to find an approximate, analytical solution.

In the time segments during which physics experts did not consider the task
context, we observed a behavior that we have named “mathematical riffing”:
experts were seen to algebraically manipulate a model to see if it could be
adjusted to fit the novel context of the task. We found that early career
graduate students spent a larger percent of time-on-task riffing, as they were
less likely to reject an unproductive model and thus spent longer trying to
make the unproductive model work. In this section, we describe each of these
findings in detail.

###  IV.1 Blended Mathematics and Physics

A dynamic interplay between behaviors categorized by Quantity codes and Task
Context codes emerged across all experts. The timeline charts (Figure 4)
indicate that behaviors coded as Quantity and Task Context appeared together
frequently. The Quantity and Task Context categories were assigned at the same
time when the research team could not separate the 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.”

The speaker associated their symbolic expression with what it implies is
happening in the physical system—the negative sign is required for the model
to describe an attractive gravitational field that the spaceship can “fall in”
to.

These kinds of statements were consistent throughout the data; we found that
about 3/4 of the time, Quantity codes were assigned for the same utterances as
Task Context codes (Fig. 5). We interpret this to mean that, for experts
engaged in graphing tasks, the majority of their reasoning about symbolic
representations is grounded in the physical context of the task they are
working on.

Figure 5: Percentage of time spent reasoning in a blended way, which we
measure as Quantity and Task Context coded together, abstractly, which we call
mathematical riffing and is measured as Quantity coded without Task Context,
and other reasoning, which is measured as when Quantity is coded without Task
Context, but the transcript reveals that the participants are not riffing.

We also observed that when the experts were making sense of quantities, the
mechanism for how the quantities might affect one another was an essential
tool [49, 50]. For example, during the Intensity task, many experts reasoned
about the physical interaction between the light and the liquid to help
develop their model. Several experts noted that they couldn’t imagine a glass
of water changing the intensity in a measurable way:

> “I don’t know, it’s hard to imagine a laser light being that much affected
> by water, but I guess so. So as soon as you went infinitely far down into a
> glass, there would be so much light scattered before reaching the bottom
> that there would be no light.”

Many used the ocean as an example, preferring to think about a body of water
large enough that they could imagine the light diminishing. Of those physics
experts who convinced themselves the light may be affected measurably, most
then wrestled with whether a linear or exponential function would be more
appropriate. This debate was either because they were considering the
attenuation of the light due to the liquid, or because of self-described
inexperience with the context.

Experts were also observed to use reasoning about the physical mechanism to
validate their models. For example, Gravitation prompted many physics experts
to invoke braking as a mechanism to explain a conflict between their
assumptions and the animation. To begin the task, physics experts uniformly
produced an already known model of potential energy as a function of distance
from the source (either graphically or symbolically), and then used it to
create new graphs of potential energy as a function of total distance
traveled. However, they had expectations of how kinetic energy and potential
energy would change together based on a constant total energy model. One
faculty member stated,

> “It’s going to speed up as it gets zooming into this thing. So I’m going to
> swing it down here and swing it up there because there’s more kinetic energy
> down here [referring to the points closest to the planet Rigamarole].”

To make sense of this with the animation, which shows a spaceship moving at
constant speed, several experts described braking as a mechanism to explain
the motion:

> “There’s probably some energy source the shuttle is using when it’s flying
> away, is my guess. And when it’s braking. ’Cause I don’t know how else it
> will take that path.”

These examples illustrate how the physics experts were engaged not just in the
technical definition of the quantity—its unit, sign and physical meaning—but
also the _physical implication_ when it changes. The quantities bring to mind
both particular functions or contexts and a sense of physicality; the physics
experts expressed ideas about what quantities made sense to change and what
mechanism would cause the change 111For more information about how the physics
experts reasoned about rates of change and functions in particular, please see
our accompanying publication [46]..

###  IV.2 Mathematical Riffing

When transcript portions are categorized as Quantity but not categorized as
Task Context, we typically see physics experts setting aside the context of
the task to reason algebraically. However, the algebraic reasoning we observed
is not “purely mathematical,” that is, devoid of contextual or physical
meaning. For example, one expert working on Drone stated:

> “I think all I know is the…$x$ is going to be some $v_{0}t$…And $v_{0}$ is
> related to $\theta_{0}$. So let’s write $x$ of $t$.”

This expert makes a series of algebraic moves based on their assumption that
the problem can be modeled with a constant horizontal velocity. Their
algebraic reasoning carries with it implicit physical meaning—$v_{0}$
represents velocity, and is introduced as an already established fact (despite
this being its first occurrence in the interview). This quote is
representative of what we observed across the expert pool when we reexamined
sections of transcript categorized as Quantity and _not_ Task Context: the
reasoning seemed to be anchored in already familiar physical models. These
moments are outlined in Figure 4. The timeline charts illustrate that physics
experts are likely to engage with physical models before, after, and often
during these periods of mathematical reasoning (as shown by the prevalence of
green squares, indicating the category Models, near outlined sections of the
timelines).

We name this behavior “mathematical riffing”: the physics experts are playing
around with familiar models to generate a new expression. We use this language
as an analogy to the practice of riffing by jazz musicians. To the untrained
ear, jazz improvisation may appear to be random. However, playing
improvisational jazz is deeply informed by the musician’s experience and
knowledge. Phrases such as “What happens if…” or “Let’s see what we can do
with this…” were strong indicators that an expert was engaged in mathematical
riffing. During this process, it was common for physics experts to refer back
regularly to the task to check that what they were doing made sense—in the
timeline charts, this is seen by the Task Context code category appearing in
between, or at the ends of, outlined sections. These checks were used to make
sense of the model they were generating and to confirm, or reject, an emerging
model as productive towards solving the task. To exemplify mathematical
riffing, we provide a transcript excerpt from an interview with one of the
faculty members whom we call Ernest.

Figure 6: Ernest’s written work for the Drone item.

While working on the Drone task, Ernest recognized that the parabolic shape
might mean the motion of the drone could be modeled using a quadratic
function. He stated:

> “So [the height of the drone] is going to be minus some constant times $x$
> squared, which we can work out.
>
> “This is just assuming it’s a quadratic, actually, I don’t know that for
> sure.
>
> “Well, we said no air resistance, so we do know that for sure.”

Ernest wrote $h=h_{o}-cx^{2}$ on his paper to define the height, $h$, by the
horizontal position of the drone, $x$, and then checked his expression against
the context of the task to make sure it made sense (see Fig. 6). The reference
to the absence of air resistance suggests the model Ernest is using may be
linked to a model of constant acceleration in his mind, but he did not state
so explicitly. Later, he worked on defining the angle, $\theta$:

> “The tangent of this angle is going to be given by $dh$ by $dx$.
>
> “Okay, which is minus $c$ times $x$.
>
> “And that’s in some complicated relation to $h$, which is given by $h_{0}$
> minus $h$, square root, thereof and some factor, which I could work out. So,
> okay, so this is some messy formula, I’m not going to come up with this very
> well.
>
> “So we’ll just draw a smooth curve, which goes between these two, and we’ll
> call it a day at that point, okay.”

Ernest determined a symbolic representation for both $h$ and $\theta$, and
attempted to find an expression of $h$ in terms of $\theta$. However, he
quickly changed tack when he realized the expression would be complicated and
possibly unhelpful towards completing the task. During this time, it was clear
that the symbols held physical meaning for Ernest, even when he was
disconnected from the task itself.

Figure 7: Percentage of time spent reasoning: in a blended way, as measured by
Task Context and Quantity coded together; mathematically riffing, as measured
by when Quantity is coded without Task Context; and Other, as measured by when
Quantity is coded without Task Context but the transcript reveals that the
reasoning is not fully aligned with our definition of mathematical riffing.
Percentages were calculated as percent time on task for each participant, and
averaged by population to produce the charts shown.

The majority of the physics experts engaged in mathematical riffing at some
time during the interview, suggesting it is a useful tool and a mark of
expertise. We observed that productive riffing is characterized by adjusting
established physical models to describe a novel context. Equally important to
productive riffing is the flexibility to change tack quickly when a model does
not appear to work well for a given task.

We also observed unproductive riffing, which is characterized by continuing to
persevere with a model that does not get the reasoner closer to a solution. As
an example, we provide a short excerpt from an early career graduate student,
Peyton. While working on Drone, Peyton spent a long period of time attempting
to derive an analytical expression:

> “…So right now I have $y$ is equal to $x\tan{\theta}$, which is equal to
> $x_{0}$ plus $v_{x}t\tan{\theta}$. I don’t want to eliminate $\theta$, but
> I’d like to eliminate time…
>
> “If I put $y$ over $\tan{\theta}$ into here…then…have something that’s
> quadratic in both. I don’t really like that…”

Peyton continued in this manner for a large percentage of their time on task,
as shown by the dashed boxes in Figure 4. Throughout their time reasoning
algebraically, they seemed unconvinced that they were on the correct path,
regularly making statements like “This looks messy…” and “That must be wrong.”
Despite their concern, they continued working with the same model for the
majority of their time spent solving Drone.

We observed a difference in prevalence of mathematical riffing across
populations. Faculty and late career graduate students were less likely to
spend much time riffing, as shown in Figure 7, which we attribute to the fact
that they were quick to abandon unproductive models. The early career graduate
students percentages are dominated by Peyton, who engaged with mathematical
riffing far more than their peers and often in unproductive ways. However,
even when we removed Peyton from the data set, a trend is apparent: the
average percent time on task riffing for the other four early career graduate
students was 30%. A full timeline chart of each participant can be found in
the Appendix for comparison.

##  V Discussion

In this study we focus on how physics experts engaged with mathematical and
physical reasoning during a series of graphing tasks. The tasks in these
interviews were designed to probe graphical reasoning, and therefore did not
prompt participants to generate symbolic expressions. However, many experts
spontaneously chose to develop symbolic representations to help themselves
reason. We found that experts engaged in physical sensemaking whether
reasoning symbolically or not; physics experts continually maintained a
connection to the physical world throughout their interviews. There was little
to no evidence that the physics experts were ever reasoning in a purely
mathematical space, that is, one in which symbols hold no physical meaning.
This finding suggests that mathematical reasoning in the contexts of the
chosen tasks almost always contains some reference to the real world for
physics experts.

It was common for the physics experts to reason with specific functions to
help them reason through the graphing tasks, which may feel to them like they
are doing “pure mathematics.” However, we found there was always physical
reasoning embedded in their mathematical reasoning—the variables carried
physical meaning and experts had expectations about how they would behave as
another quantity changed. These periods of algebraic reasoning, what we are
calling mathematical riffing, are characterized by physics experts
manipulating already familiar physical models, seeing what happens, and either
noticing when the model is not fruitful for them and abandoning the line of
reasoning, or building on the model. Essential to mathematical riffing is that
the expert bases their reasoning on expected common functions in physics and
on familiar physics models. The expert manipulates these models in a way that
may appear random or inexpert-like to a novice, but is informed by an
understanding of what makes sense for the context. This finding suggests that
physics experts associate physical meaning with algebraic variables in ways
that mathematicians typically don’t, and that it is an essential part of
mathematical thinking in physics.

We also found that the physical mechanism by which quantities changed was
integral to how physics experts reasoned about the meaning of their
representations. Research has shown that using the physical mechanism of a
system to make sense of why a quantity may change is an important aspect of
conceptual physics reasoning [52, 49, 53, 54]. Euler, Rådahl, and Gregorcic
describe how mechanistic reasoning is an important part of students developing
explanatory models; research in physics education has demonstrated the value
of explanatory models in modeling instruction [55, 56]. This study provides
further evidence that mechanistic reasoning also informs the development of
mathematical models by physics experts.

Finally, we observed differences in the effectiveness of mathematical riffing
across our expert populations. Figure 7 shows that the early career graduate
students in our sample spent more time mathematically riffing than more
experienced experts, because they were less likely to reject an unproductive
line of reasoning quickly. This emergent pattern is aligned with Bing and
Redish’s description of a journeyman: “that level where students have
developed sufficient skills that they can no longer be considered novices but
where they have not yet had sufficient experience with sophisticated problem
solving (and research) to be considered experts” [57]. Bing and Redish
compared novice and journeymen reasoning, and showed that journeymen more
quickly recognized their approach was unproductive and were more likely to
change tack than novices. Our data represent another section of the spectrum,
from journeyman to expert. Together, these results demonstrate that rejecting
unproductive models is a hallmark of expertise, and that this skill may be
slow to develop across many years of physics experience.

Given the small sample size and the potential for embarrassment while problem
solving in an interview, we emphasize that further research is needed here
before we strongly claim that there is an “early-career” effect. We cannot
ignore the potential influence of imposter syndrome—the participants were
aware the study was focused on expert reasoning, and this may have had an
impact on how they felt during the interview.

In summary, we interpret our findings that, in the context of challenging,
novel, classical physics graphing tasks, physics experts continually maintain
a tie to the real world, even when reasoning mathematically. This finding is
aligned with those of Czocher and Serbin and Wawro [9, 10]. Despite the
differences in methodology and context—both Czocher and Serbin and Wawro
examined tasks that prompted symbolic answers, interviewed undergraduate
students in engineering and physics respectively, and used a priori coding
schemes based on established modeling cycles—the results are strikingly
similar. These complementary findings do not support the prevalence of
frameworks that separate mathematical reasoning and reasoning about the real
world; in the contexts encountered by the vast majority of students who take
physics courses, rather, the findings support descriptions of modeling that
incorporate a continuous interaction between mathematical reasoning and
physics reasoning. Future research might consider revising theoretical
frameworks of modeling to support expert behaviors of using physical models as
a basis for reasoning mathematically.

We encourage instructors to reflect on their thinking in the context of
modeling by noticing how and when physical ideas are incorporated, often
implicitly, when “doing math” in physics contexts. Even considerations such as
why particular arithmetic operations make sense in a model often have physical
foundations (i.e., summation in conservation laws, subtraction and division in
rates of change). Acknowledging this subtle expert knowledge might help
students recognize what is familiar, and what is new about the modeling that
they are learning to do in physics.

##  VI Conclusion

In this paper, we present empirical evidence of the emergent reasoning of
physics experts while they develop graphical models. While there is a spectrum
of reasoning from more math-like to more physics-like, there is scant evidence
in our data of purely mathematical reasoning. The majority of the time,
physics experts’ reasoning involved interwoven mathematical and physical
reasoning. When experts reasoned mathematically without considering the
context of the task at hand, they engaged in mathematical riffing—using
abstract physical models as a starting point, adjusting the mathematics to
work towards a novel expression, and quickly rejecting models that were not
productive. Considering whether models were consistent with physical
mechanisms played an important role in their decision-making.

This study was limited by the context of the tasks, which were all prompting
for graphical representations (although several experts spent a good fraction
of their time generating and engaging in symbolic representations) of some
aspect of constant speed motion in classical physics contexts. The narrow
scope of the contexts correspondingly limits the generalizability of the
results; we do not claim that our findings characterize physics reasoning in
every context.

Additional research can help span the space of expert mathematical reasoning
in physics. For example, we can imagine enriching these findings by
replicating this study in more abstract contexts, such as quantum or
statistical mechanics. While it was difficult to generate prompts that are
actually novel for experts, our findings suggest that the process of examining
experts generating models is potentially fruitful for future research.

Basing instruction on an assumption a mathematical model is isolated from its
real world, or physical, context is not consistent with the way that physics
experts appear to model. Instruction that is implicitly based on mathematical
procedures isolated from physical meaning might serve to increase the gap
between students and their instructions, who largely don’t reason this way. We
encourage instructors to reflect on their own experiences of “doing math” in
physics, and to be aware of implicit assumptions about deeper physical
meanings that are nearly always embedded in the mathematical reasoning of
introductory physics. In addition, instructors, specifically when rejecting a
model, may find it productive to consider ways to make their decision-making
process explicit to the students in their physics courses. In general,
instructional interventions that help bridge gaps between what experts may
consider foundational mathematical reasoning in physics and what novices are
prepared for in their prerequisite courses will enrich the learning of
physics. Our findings can help the ongoing and future efforts to close these
gaps.

##  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. This work was partially supported by the
National Science Foundation under grants No. DUE-1832836, DUE- 1832880,
DUE-1833050, DGE-1762114, DUE-2214765, and DUE-2214283. 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.

## References

  * Usiskin [2011] Z. Usiskin, Mathematical Modeling in the School Curriculum (2011).
  * Usiskin [2015] Z. Usiskin, Mathematical Modeling and Pure Mathematics, Mathematics Teaching in the Middle School 20, 476 (2015).
  * Martin [2016] C. Martin, Mathematical Modeling: Part 1 of Series, Colorado Mathematics Teacher , 26 (2016).
  * Greefrath  _et al._ [2022] G. Greefrath, H. S. Siller, K. Vorhölter, and G. Kaiser, Mathematical modelling and discrete mathematics: opportunities for modern mathematics teaching, ZDM - Mathematics Education 54, 865 (2022).
  * Bajracharya  _et al._ [2023] R. R. Bajracharya, V. L. Sealey, and J. R. Thompson, Student Understanding of the Sign of Negative Definite Integrals in Mathematics and Physics, International Journal of Research in Undergraduate Mathematics Education 10.1007/s40753-022-00202-y (2023).
  * Taylor and Loverude [2023] H. Taylor and M. Loverude, “I forget about physics when I go to math”, in  _The Learning and Teaching of Calculus Across Disciplines Conference Proceedings_ (2023).
  * Ferri [2006] R. B. Ferri, Theoretical and empirical differentiations of phases in the modelling process, ZDM — Mathematics Education 38 (2006).
  * Ärlebäck [2009] J. B. Ärlebäck, On the use of realistic Fermi problems for introducing mathematical modeling in school, The Mathematics Enthusiast 6, 331 (2009).
  * Czocher [2016] J. A. Czocher, Introducing Modeling Transition Diagrams as a Tool to Connect Mathematical Modeling to Mathematical Thinking, Mathematical Thinking and Learning 18, 77 (2016).
  * Serbin and Wawro [2022] K. S. Serbin and M. Wawro, The Inextricability of Students’ Mathematical and Physical Reasoning in Quantum Mechanics Problems, International Journal of Research in Undergraduate Mathematics Education 10.1007/s40753-022-00174-z (2022).
  * Sherin [2001] B. L. Sherin, How Students Understand Physics Equations, Cognition and Instruction 19, 479 (2001).
  * Kuo  _et al._ [2013] E. Kuo, M. M. Hull, A. Gupta, and A. Elby, How students blend conceptual and formal mathematical reasoning in solving physics problems, Science Education 97, 32 (2013).
  * Hu and Rebello [2013] D. Hu and N. S. Rebello, Using conceptual blending to describe how students use mathematical integrals in physics, Physical Review Special Topics - Physics Education Research 9, 1 (2013).
  * White Brahmia [2014] S. White Brahmia,  _Mathematization in introductory physics_ , Ph.D. thesis, Rutgers University (2014).
  * Redish and Kuo [2015] E. F. Redish and E. Kuo, Language of Physics, Language of Math: Disciplinary Culture and Dynamic Epistemology, Science and Education 24, 561 (2015).
  * Caballero  _et al._ [2015] M. D. Caballero, B. R. Wilcox, S. J. Pollock, and L. Doughty, Unpacking students’ use of mathematics in upper-division physics: where do we go from here?, European Journal of Physics 36 (2015).
  * Bajracharya and Thompson [2016] R. R. Bajracharya and J. R. Thompson, Analytical derivation: An epistemic game for solving mathematically based physics problems, Physical Review Physics Education Research 12, 10.1103/PhysRevPhysEducRes.12.010124 (2016).
  * Van Den Eynde  _et al._ [2019] S. Van Den Eynde, P. Van Kampen, W. Van Dooren, and M. De Cock, Translating between graphs and equations: The influence of context, direction of translation, and function type, Physical Review Physics Education Research 15, 10.1103/PhysRevPhysEducRes.15.020113 (2019).
  * Thompson [2011] P. W. Thompson, Quantitative reasoning and mathematical modeling, in  _New perspectives and directions for collaborative research in mathematical education_ , Vol. 1, edited by L. L. Hatfield, S. Chamberlain, and S. Belbase (University of Wyoming, Laramie, WY, 2011) Chap. Quantitati, pp. 33–57.
  * Czocher and Hardison [2021] J. A. Czocher and H. L. Hardison, Attending to Quantities Through the Modelling Space, in  _Mathematical Modelling Education in East and West_ (Springer International Publishing AG, 2021) pp. 263–272.
  * White Brahmia and Boudreaux [2017] S. White Brahmia and A. Boudreaux, Signed quantities: mathematics based majors struggle to make meaning, in  _Proceedings of the 20th Annual Conference on Research in Undergraduate Mathematics Education_ (2017).
  * White Brahmia [2019] S. White Brahmia, Quantification and its importance to modeling in introductory physics, European Journal of Physics 40, 10.1088/1361-6404/ab1a5a (2019).
  * Torigoe and Gladding [2011] E. T. Torigoe and G. E. Gladding, Connecting symbolic difficulties with failure in physics, American Journal of Physics 10.1119/1.3487941 (2011).
  * Olsho  _et al._ [2019] A. Olsho, S. White Brahmia, A. Boudreaux, and T. I. Smith, The physics inventory of auantitative reasoning: assessing student reasoning about sign, in  _Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education_ (2019) pp. 992–997.
  * White Brahmia  _et al._ [2021] S. White Brahmia, A. Olsho, T. I. Smith, A. Boudreaux, P. Eaton, and C. Zimmerman, Physics Inventory of Quantitative Literacy: A tool for assessing mathematical reasoning in introductory physics, Physical Review Physics Education Research 17, 10.1103/PhysRevPhysEducRes.17.020129 (2021).
  * Bollen  _et al._ [2017] L. Bollen, P. Van Kampen, C. Baily, M. Kelly, and M. De Cock, Student difficulties regarding symbolic and graphical representations of vector fields, Physical Review Physics Education Research 13 (2017).
  * Hestenes [1997] D. Hestenes, Modeling methodology for physics teachers, AIP Conference Proceedings 399, 935 (1997).
  * Van Heuvelen and Zou [2001] A. Van Heuvelen and X. Zou, Multiple representations of work–energy processes, American Journal of Physics 69, 184 (2001).
  * Kohl and Finkelstein [2006] P. B. Kohl and N. D. Finkelstein, Effect of instructional environment on physics students’ representational skills, Physical Review Special Topics - Physics Education Research 2, 10.1103/PhysRevSTPER.2.010102 (2006).
  * Rosengrant  _et al._ [2006] D. Rosengrant, E. Etkina, and A. Van Heuvelen, An Overview of Recent Research on Multiple Representations, in  _Physics Education Research Conference Proceedings_ (2006).
  * Brewe [2008] E. Brewe, Modeling theory applied: Modeling Instruction in introductory physics, American Journal of Physics 76, 1155 (2008).
  * Blum and Leiß [2007] W. Blum and D. Leiß, 5.1 - How do Students and Teachers Deal with Modelling Problems?, in  _Mathematical Modelling_, edited by C. Haines, P. Galbraith, W. Blum, and S. Khan (Woodhead Publishing, 2007) pp. 222–231.
  * Redish [2005] E. F. Redish, Problem solving and the use of math in physics courses, in  _World View on Physics Education in 2005: Focusing on Change Conference Proceedings_ (Dehli, 2005).
  * Redish and Smith [2008] E. F. Redish and K. A. Smith, Looking Beyond Content: Skill Development for Engineers, Journal of Engineering Education 97, 295 (2008).
  * Uhden  _et al._ [2012] O. Uhden, R. Karam, M. Pietrocola, and G. Pospiech, Modelling Mathematical Reasoning in Physics Education, Science and Education 21, 485 (2012).
  * Wilcox  _et al._ [2013] B. R. Wilcox, M. D. Caballero, D. A. Rehn, and S. J. Pollock, Analytic framework for students’ use of mathematics in upper-division physics, Physical Review Special Topics - Physics Education Research 9, 10.1103/PhysRevSTPER.9.020119 (2013).
  * Fauconnier and Turner [2003] G. Fauconnier and M. Turner, Conceptual Blending, Form and Meaning, Recherches en Communication 19, 10.14428/rec.v19i19.48413 (2003).
  * Bing and Redish [2007] T. J. Bing and E. F. Redish, The cognitive blending of mathematics and physics knowledge, in  _Physics Education Research Conference 2006 Conference Proceedings_, Vol. 883 (2007) pp. 26–29.
  * Huynh and Sayre [2018] T. Huynh and E. C. Sayre, Blending mathematical and physical negative-ness, Proceedings of International Conference of the Learning Sciences, ICLS 2, 957 (2018).
  * Taylor and Loverude [2018] H. Taylor and M. E. Loverude, “So it’s the same equation.“: A blending analysis of student reasoning with functions in kinematics, in  _Physics Education Research Conference 2018 Conference Proceedings_, Vol. 2018 (2018).
  * Schermerhorn [2018] B. P. Schermerhorn,  _Investigating Student Understanding of Vector Calculus in Upper-Division Electricity and Magnetism: Construction and Determination of Differential Element in Non-Cartesian Coordinate Systems_ , Ph.D. thesis, University of Maine (2018).
  * Van Den Eynde  _et al._ [2020] S. Van Den Eynde, B. P. Schermerhorn, J. Deprez, M. Goedhart, J. R. Thompson, and M. De Cock, Dynamic conceptual blending analysis to model student reasoning processes while integrating mathematics and physics: A case study in the context of the heat equation, Physical Review Physics Education Research 16, 10114 (2020).
  * Odden [2021] T. O. B. Odden, How conceptual blends support sensemaking: A case study from introductory physics, Science Education 105, 989 (2021).
  * Eichenlaub and Redish [2019] M. Eichenlaub and E. F. Redish, Blending Physical Knowledge with Mathematical Form in Physics Problem Solving, in  _Mathematics in Physics Education_ (Springer International Publishing, 2019) pp. 127–151.
  * Loverude [2018] M. E. Loverude, Mathematization and the ‘Boas course’, in  _Physics Education Research Conference 2017 Conference Proceedings_ (2018) pp. 15–18.
  * Zimmerman  _et al._ [2023] C. Zimmerman, A. Olsho, M. Loverude, and S. White Brahmia, Expert covariational reasoning resources in physics graphing tasks, Physical Review Physics Education Research, in review (2023).
  * 201 [2019] Otter (2019).
  * Nowell  _et al._ [2017] L. S. Nowell, J. M. Norris, D. E. White, and N. J. Moules, Thematic Analysis: Striving to Meet the Trustworthiness Criteria, International Journal of Qualitative Methods 16, 10.1177/1609406917733847 (2017).
  * Russ  _et al._ [2008] R. S. Russ, R. E. Scherr, D. Hammer, and J. Mikeska, Recognizing mechanistic reasoning in student scientific inquiry: A framework for discourse analysis developed from philosophy of science, Science Education 92, 499 (2008).
  * Russ  _et al._ [2009] R. S. Russ, J. E. Coffey, D. Hammer, and P. Hutchison, Making classroom assessment more accountable to scientific reasoning: A case for attending to mechanistic thinking, Science Education 93, 875 (2009).
  * Note [1] For more information about how the physics experts reasoned about rates of change and functions in particular, please see our accompanying publication [46].
  * Hammer and Elby [2003] D. Hammer and A. Elby, Tapping Epistemological Resources for Learning Physics, The Journal of the Learning Sciences 12, 53 (2003).
  * Richards [2013] A. J. Richards,  _How students combine resources to build understanding of complex topics_ , Ph.D. thesis, Rutgers, The State University of New Jersey (2013).
  * Jones [2015] D. C. Jones,  _Cognitive resources of physics experts_ , Ph.D. thesis, Rutgers, The State University of New Jersey, New Brunswick, NJ (2015).
  * Etkina  _et al._ [2006] E. Etkina, A. Warren, and M. Gentile, The Role of Models in Physics Instruction, The Physics Teacher 44, 34 (2006).
  * Euler  _et al._ [2019] E. Euler, E. Rådahl, and B. Gregorcic, Embodiment in physics learning: A social-semiotic look, Physical Review Physics Education Research 15, 10.1103/physrevphyseducres.15.010134 (2019).
  * Bing and Redish [2012] T. J. Bing and E. F. Redish, Epistemic complexity and the journeyman-expert transition, Physical Review Special Topics - Physics Education Research 8, 10.1103/PhysRevSTPER.8.010105 (2012).

*

## Appendix A

Figure 8: A visual representation of when participant statements were assigned
the “Quantity,” “Physical World,” and “Models” code categories. The length of
the bars corresponds to the time the participant was engaged in that line of
reasoning. The horizontal axis represents time, normalized to the length of
each interview. Therefore, it can be thought of as a percentage of their total
time on task. The highlighted regions are where “Quantity” appears without
“Physical World.”