A Problem With Current Conceptions Of Expert Problem Solving

submitted to the Journal of Engineering Education 1
A Problem with Current Conceptions of Expert
Problem Solving
Eric Kuoa
, Michael Hulla
, Ayush Guptaa
, Andrew Elbya,b
Departments of Physicsa
and Curriculum & Instructionb
University of Maryland, College Park
erickuo@umd.edu, mhull12@umd.edu, ayush@umd.edu, elby@umd.edu
Abstract
Background
Current conceptions of expert problem solving depict physical/conceptual reasoning and formal
mathematical reasoning as separate steps: a good problem solver first translates a physical
understanding into mathematics, then performs mathematical/symbolic manipulations, then
interprets the mathematical solution physically. However, other research suggests that blending
conceptual and symbolic reasoning during symbolic manipulations can reflect expertise.
Purpose (Hypothesis)
We explore the hypothesis that blending conceptual and symbolic reasoning (i) indicates
problem-solving expertise more than adherence to “expert” problem-solving steps and (ii) is
something some undergraduates do spontaneously, suggesting it’s a feasible instructional target.
Design/Method
Interviewed students were asked to (1) explain a particular equation and (2) solve a problem
using that equation. In-depth analysis of two students, Alex and Pat, revealed a pattern of
behavior. All 11 interviews were coded to investigate the generalizability of this pattern.
Results
Alex described and used the equation as a computational tool. By contrast, Pat found a shortcut
to solve the problem using a connection he verbalized between the mathematical equation and a
physical process. Coding of 11 interviews confirms a correlation between the shortcut solution
and a conceptual explanation of the equation. Furthermore, Pat’s blended physical/conceptual
and symbolic reasoning is well described by knowledge structures called symbolic forms (Sherin,
2001).
Conclusions
Undergraduate students can and do blend physical/conceptual and symbolic reasoning to solve
mathematics in problem solving. Symbolic Forms provide potential instructional targets for
fostering such blended reasoning. This suggests that researchers should reconsider current
conceptions of problem-solving expertise that do not include such reasoning.

Introduction
We challenge a widely held view in science and engineering education literature about
how to teach problem solving to engineering and physical science students — and more
fundamentally, what constitutes expert quantitative problem solving. Our challenge focuses on
the relation between physical/conceptual reasoning and mathematical manipulations. Most well-
defined descriptions of good quantitative problem solving include two links between physical
reasoning and mathematical manipulations: (1) translating the physical description of what’s
happening into solvable mathematical form, and then (2) after completing the mathematical
manipulations to obtain a symbolic or numerical solution, checking to see if it makes physical
sense (Redish & Smith, 2008; Reif, 2008). Without disputing the importance of (1) and (2), we
argue that an additional feature of expert problem-solving in many cases — and a feasible
instructional target in physics, chemistry, and engineering courses — is a blending of symbolic
and physical/conceptual reasoning (Fauconnier & Turner, 2003; Sherin, 2001) during the
mathematical manipulation phase.
To make our case, we will first review how expertise in quantitative problem solving has
been conceptualized and taught in physics and engineering. Then we will use contrasting case
studies of two students, “Alex” and “Pat,” to illustrate two patterns we documented in how
engineering majors taking a physics course solved a standard textbook physics problem and how
they explained the equation they relied upon to do so. Alex solves the problem by finding
standard physics equations that fit the given situation, solving those equations to obtain a
numerical answer, and reflecting upon that answer — in accord with the common
conceptualizations of good problem solving. Pat, by contrast, solves the problem by blending
mathematical representations with conceptual reasoning about physical processes (which we
refer to as “physical/conceptual reasoning”); his physical interpretation of mathematical objects
influences how he uses the mathematics. After analyzing Alex’s and Pat’s reasoning in detail, we
will show that most of the students we interviewed displayed either Alex-like or Pat-like
reasoning patterns.
Crucially, both Alex and Pat (and most of our other subjects) solved the physics problem
correctly; but Pat’s reasoning, we argue, shows greater expertise. We thereby argue that expert
quantitative problem solving does not necessarily involve an “algorithmic manipulation of
equations” step that is separate from physical/conceptual reasoning. Our contribution is
documenting what that blended reasoning can look like for engineering majors and making the
case that (1) a theoretical construct called symbolic forms provides a good account of Pat’s (and
the Pat-like students’) reasoning and (2) such blending of conceptual reasoning and mathematical
operations is a feasible instructional target in physics and engineering courses, the learning of
which can help students become better quantitative problem solvers.
Literature review: Conceptualizations of expert problem solving
In this section, we will present a common conceptualization of expert quantitative
problem solving in physics and engineering, as well as arguments that begin to challenge that
conceptualization.

What problem-solving research says about the role of mathematical manipulations
As a central feature of their professional practice, scientists and engineers apply domain
specific knowledge to solve quantitative problems (Gainsburg, 2006; Redish & Smith, 2008).
For this reason, developing problem-solving expertise in students has become a central concern
of engineering and science education researchers and practitioners. Research has produced many
multi-step, problem-solving procedures, both in engineering (G. L Gray, Costanzo, & Plesha,
2005; Litzinger et al., 2010; Wankat & Oreovicz, 1993; Woods, 2000) and physics (P. Heller,
Keith, & Anderson, 1992; Huffman, 1997; Reif, 2008; Van Heuvelen, 1991). The steps in these
procedures range from vague and open ended to well defined and concrete. An example of a
well-defined procedure comes from Heller, Keith, & Anderson (1992) and is paraphrased here:
1) Visualize the problem – Translate the problem statement into a visual sketch and
identify the known and unknown quantities.
2) Physics Description – Translate the sketch into a physical representation using physics
principles to construct an idealized diagram and to symbolically specify the known and
unknown quantities.
3) Plan a Solution – Translate the physics description into a mathematical representation
using the relevant physics principles in equation form. Work backward from the desired
unknown until you can specify the steps to get from the known to the unknown.
4) Execute the Plan – Using appropriate mathematical methods, execute the planned
solution and write an expression with all the known values on one side of the equation
and the unknown value on the other side. Substitute values given in the problem to obtain
a numerical solution if necessary.
5) Check and Evaluate – Check to see if you have completed the problem. Check the
units, sign of the answer, and magnitude of the answer to see if they are reasonable.
This problem-solving strategy, which resembles those found in many physics and
engineering textbooks and classrooms, illustrates two features common to most well-defined
problem-solving procedures in the literature (Gainsburg, 2006; G. L Gray et al., 2005; P. Heller
et al., 1992; Huffman, 1997; Van Heuvelen, 1991). First, it consists of explicit steps intended to
guide students’ reasoning. Second, the physical/conceptual reasoning happens in separate steps
from the symbolic manipulation of equations; In other words, the interplay or translation
between physical/conceptual ideas and mathematical equations happens before and after the
symbolic manipulations but aren’t “blended” with the manipulations themselves. For instance, in
Heller et al.’s scheme, students translate their physical understanding of the problem into
mathematical equations in the “Plan a Solution” step, and they consider the physical meaning of
their symbolic or numerical answer when they “Check and Evaluate” in the final step. However,
in the “Execute the Plan” step, only mathematical methods are specified in solving the equations.
Some other researchers, by contrast, do not prescribe an explicit method for performing
the mathematical manipulations in problem solving. Although this does not preclude the
blending of physical/conceptual reasoning with mathematical processing, this research typically
does not give examples of how such blending might occur. For example, Reif (2008) splits
problem solving into five general phases: describing the problem, analyzing the problem,
constructing the solution, assessing the solution, and exploiting the solution. “Constructing the
solution” includes assessing the overall problem, choosing an appropriate subproblem (e.g.

solving for a variable that will move you closer to finding the desired unknown variable), and
solving that subproblem. The emphasis in “constructing the solution” is on decision-making,
specifically, identifying which subproblem to solve and what domain-specific knowledge to
deploy while doing so. In contrast to Heller et al. and other authors of well-defined problem-
solving procedures, Reif does not explicitly discuss how students should process the
mathematical equations that they generate. However, in the specific problem-solving examples
Reif provides, the symbolic processing step involves rearranging algebraic equations for
unknown variables and continuing until the desired unknown is expressed in terms of known
variables; as in Heller et al., the symbol processing is a separate step from physical/conceptual
sense-making. So, although Reif’s problem-solving strategy does not preclude blending symbolic
manipulations with conceptual reasoning, he does not give examples of such blending.
Figure 1, from Redish & Smith’s (2008) recent paper in this journal, more generally
illustrates how schemes for defining or encouraging expert quantitative problem solving in the
literature generally incorporate the interplay between formal mathematics and
physical/conceptual reasoning. The central idea is that the physical system is modeled
mathematically, the mathematics is processed to obtain some kind of solution, and then that
solution is interpreted back into the physical system. In this way, the explicit connection between
physical/conceptual reasoning and formal mathematics happens before and after, but not during,
the mathematical manipulations. (Below, we discuss how Redish & Smith add nuance to this
story; our point here is that their diagram captures a common conceptualization of expert
quantitative problem solving and how students should be taught to do it.)
Fig. 1: problem-solving diagram from Redish & Smith (2008).
In brief, accounts of expert quantitative problem solving in the physics and engineering
problem-solving literature generally include a “mathematical symbolic processing” step that is
separate from physical/conceptual reasoning.
What’s wrong with a symbolic processing step?
Despite the lack of explicit alternatives in the science/engineering problem-solving
literature, other pockets of research suggest that processing equations without thinking about
their conceptual meaning during the processing can reflect a lack of expertise. In mathematics
education research, for example, Wertheimer (1959) asked students to solve problems of the

following type: (815+815+815+815+815)/5. Wertheimer argued that students who solved the
problem by computing the sum of the numerator and then dividing by 5 had not understood the
underlying meaning of their calculations and were therefore less sophisticated than students who
jumped to the answer without explicit calculations. Arcavi (1994) suggested the importance of
the ability to reason informally about symbols, called symbol sense. This symbol sense includes
the ability to interpret the conceptual meaning behind symbolic relationships, the ability to
generate expressions from intuitive and conceptual understanding, and the knowledge of when
and how best to exploit one’s intuitive understanding of symbols.
Redish and Smith, writing about problem solving in engineering and physics, also
challenged the view that symbolic processing should be a priori divorced from conceptual
reasoning. Their text adds nuance to the diagram in figure 1:
Note that this model should not be interpreted as a flow chart. There are cross-links
throughout. For example, because of the fact that the equations are physical rather than
purely mathematical, the processing can be affected by physical interpretations (Redish &
Smith, 2008).
Just as Wertheimer showed that students’ conceptual understanding of mathematical operations
influences how they carry out a calculation, Redish and Smith suggest that students’ physical
interpretation of equations in physics and engineering can influence their mathematical symbolic
processing.
By these arguments, teaching students to perform mathematical processing as a separate
step from their conceptual thinking about the meaning of the mathematical objects and
operations can sometimes push them away from problem-solving expertise. Still missing from
this line of argument, however, is a precise description of instructional targets: how can
instruction enable physics and engineering students to blend symbolic processing with
conceptual reasoning when it’s productive to do so?
In this paper, we argue that one way in which freshmen and sophomore engineering
majors can productively blend conceptual and equation-based reasoning during problem solving
is through using a particular class of knowledge elements called symbolic forms (Sherin, 2001).
We describe this construct in more detail below, relating it to our data. Briefly, in a symbolic
form, an intuitive, informal conceptual schema (e.g., “the amount, of some generic stuff, you end
with is the amount you start with plus the amount it changes by”) is combined with an equation
symbol template (e.g.,  =  + ) into a single cognitive unit that the student uses as a whole.
Symbolic forms, we argue, help to explain the patterns we document below in students’ problem
solving and are therefore productive for researchers trying to understand the nature of students’
problem-solving expertise. They are also plausible targets for instruction, partly because they
rely on everyday rather than formal (or discipline-specific) conceptual schemata; therefore,
symbolic forms can be developed while students are still learning difficult physics or engineering
concepts.
Methods and Data Collection
Interview Context

From fall 2008 through fall 2010, we interviewed 11 engineering majors taking the
University of Maryland’s first-semester, calculus-based, introductory physics course. The course,
geared toward engineering majors, covers mechanics. Students were recruited by email and
through an in-class announcement. All the students who expressed an interest were interviewed
(except for 2-3 students who could not be interviewed due to scheduling issues).
The two subjects on whom we focused our analysis, Alex and Pat (pseudonyms chosen to
protect the identity of students), were interviewed two months into the course in fall 2008. By
that time, the course had covered kinematics, including objects moving under the influence of
gravity (the topic of the interview prompts discussed below). We chose Alex and Pat because
they were among the first students we interviewed and because the strong differences between
their responses motivated us to seek an explanation for the differences. In addition, from our
experiences teaching undergraduate engineering majors, Alex’s and Pat’s reasoning patterns
struck us as mirroring common student behaviors — an assumption we later tested by coding the
responses of all our interview subjects, as described below.
Interview protocols
The semi-structured interviews were designed to probe engineering students’ approaches
to using equations while solving quantitative physics problems, specifically what mathematical
and conceptual tools they bring to bear and which epistemological stances they take toward the
nature of the knowledge they invoked (e.g., is it integrated or piecemeal?). To that end, we had
students think aloud while solving specific problems. We also asked them to explain the meaning
of both familiar and unfamiliar equations and to discuss more generally how they know when
they “understand” an equation. The complete protocol can be found online
(http://umdperg.pbworks.com/AlexPat). Our analysis in this paper focuses on the first two
prompts.
Prompt 1: Explain the velocity equation
The interviewer shows the student the equation v = v0 + at and asks, Here’s an equation you’ve
probably seen in physics class. How would you explain this equation to a friend from class?
Prompt 2: Two Balls Problem
(a) Suppose you are standing with two tennis balls on the balcony of a fourth floor apartment.
You throw one ball down with an initial speed of 2 meters per second; at the same moment, you
just let go of the other ball, i.e., just let it fall. I would like you to think aloud while figuring out
what is the difference in the speed of the two balls after 5 seconds – is it less than, more than, or
equal to 2 meters per second? (Acceleration due to gravity is 10 m/s2
.) [If the student brings it
up, the interviewer says to neglect air resistance]
(b) [If student solved part (a) by doing numerical calculations] Could you have solved that
without explicitly calculating the final values?

Analysis phase 1: Alex and Pat
We began with fine-grained qualitative analysis of Alex’s and Pat’s approaches to using
and explaining the velocity equation, v = v0 + at, with the goal of characterizing how the two
subjects were conceptualizing the equation and its role in solving the Two Balls Problem.
Phase 1a: Two balls problem
To start, we looked at video and corresponding transcript of the two subjects’ responses
to the two balls problem (prompt 2). In trying to characterize how they were thinking about the
velocity equation while using it to solve the problem at hand, we attended not just to the literal
meaning of their statements but also to other markers of their thinking suggested by the discourse
and framing analysis literature (Gee, 1999; Tannen, 1993), including word choice, pauses, pitch
and register, body language, and facial expressions. Of course, we also analyzed the solutions
they were writing while thinking aloud. After formulating possible characterizations, we went
line by line through the data, looking for confirmatory and/or disconfirmatory evidence for each
one. In this way we refined and narrowed down the plausible characterizations of how they were
thinking about the equation (Miles & Huberman, 1994). To further hold ourselves accountable,
we provide the full transcripts of the interviews online (http://umdperg.pbworks.com/AlexPat) so
that readers can check their own interpretation against ours.
Phase 1b: The velocity equation
With Alex and Pat, we had reached consensus in phase 1a about how they were thinking
about the velocity equation. We then analyzed their response to prompt 1, where they explain the
velocity equation as if to a friend. Using the same analytical tools as in phase 1a, we tried to
characterize how they were thinking about the equation in the context of explaining it. We then
compared our characterizations to what we had found in phase 1a. As discussed below, the
alignment was strong; our characterization of Alex’s and Pat’s use of the equation while solving
the Two Balls Problem matched our respective characterization of how they were thinking about
the equation while explaining it. By providing detailed analysis below, and also the complete
transcript, we give readers the opportunity to check if we unconsciously “matched up” our
characterizations in phase 1a and 1b when it wasn’t warranted.
Analysis phase 2: Checking the pattern
As described below, Alex treated the velocity equation as a plug-and-chug “gizmo” while
explaining it (in prompt 1) and did not see a certain shortcut while solving the Two Balls
Problem (in prompt 2). By contrast, Pat invoked physical/conceptual reasoning while explaining
the equation and used a shortcut to solve the Two Balls Problem. To check if this correlation
between “gizmo” vs. conceptual responses to prompt 1 and “the long way” vs. shortcut responses
to prompt 2 was idiosyncratic to Alex and Pat, two researchers (the first and second author)
independently coded all subjects’ responses to prompt 1 as containing conceptual reasoning or
not, and responses to prompt 2 as noticing the shortcut or not. Inter-rater reliability was 93%.

Without going into detail here, we note that Alex’s reasoning was typical of one set of
students while Pat’s was typical of another set. We noticed no gender or other demographic
differences between those two sets of students.
Results of analysis phase 1a: The Two Balls Problem
We anticipated that some students would find something like the following “shortcut”
solution: according to the kinematics equation, v = v0 + at, since both balls gain the same amount
of speed over 5 seconds (with the gain given by the mathematical term at), their final difference
in speeds equals their initial difference in speeds, in this case 2 meters per second. We wanted to
see if students would notice this “shortcut” solution, either on their own or in response to the
follow-up prompt about whether the problem could have been solved without calculations.
Alex’s and Pat’s reasoning patterns were typical of students who did not and did notice the
shortcut, respectively.
How Alex uses the velocity equation while solving the Two Balls Problem
Alex solves the problem procedurally
Alex starts by drawing a diagram of the two balls and labeling their speeds (figure 2). She
makes a mistake in labeling this picture, confusing acceleration for initial velocity, as she
initially writes “9.8 m/s” next to the dropped ball. After deciding to use the velocity equation to
solve this problem, Alex pauses, because she doesn’t have a value for a. She realizes that a
should be 9.8 and writes this value in her diagram. The interviewer interjects and says that she
could use 10 if she wanted, but that she is also free to use 9.8 if that makes her feel more
comfortable. Alex says that using 10 is probably easier, crosses out “9.8” for the acceleration and
for the velocity of the dropped ball, and replaces those values with “10.” She then explicitly
solves for the velocities of the thrown and dropped balls after five seconds and writes down the
difference. Figure 2 shows all of her work.
Fig. 2: Alex’s work in the Two Balls Problem

After working out the speeds of the two balls to be 50 m/s and 52 m/s, Alex explains her thought
process:
A31 A: …Ok, so after I plug this into the velocity equation, I use the acceleration and
the initial velocity that’s given, multiply the acceleration by the time that we’re
looking at, five seconds, and then once I know the velocities after five seconds of
each of them, I subtract one from the other and get two. So the question asks “is it
more than, less than, or equal to two?” so I would say equal to two.
In this segment, Alex’s behavior is almost script-like, following a set of steps similar to
the well-defined problem-solving procedures described above, or even the steps most textbooks
recommend (e.g., Giancoli (2008), Young and Freedman (2003)), to solve a routine physics
problem: draw a picture of the situation, write down the known values, choose the equation that
can take your known values, and calculate the desired unknowns to answer the question. Alex
executed this procedure smoothly, with the only pause coming when Alex wondered whether to
use 9.8 or 10 for the acceleration.
Alex exhibits hints of conceptual reasoning, but not integrated with the velocity equation
After Alex gives her answer, the interviewer asks if someone could have answered the
question without working out the numbers and explicitly solving for the velocities of both balls.
Her first answer is yes, but when asked to elaborate, she seems unsure:
A35 A: Well, I’d have to think about it, since you’re dropping one and throwing one. If
you’re, I mean I guess if you think you’re throwing one 2 m/s and the other has 0
velocity since you’re just dropping it, its only accelerating due to gravity, you can
just say that since you know one is going at 2 m/s, it’s going to get there 2 m/s
faster, so 5 seconds faster, it would get there 2 seconds… er… it’s going 2 m/s
faster, I guess.
A36 I: OK. So, they would say that you threw one, so this was getting 2 m/s faster. So
what happens 5 seconds later?
A37 A: Uh…it’s going…uh…I don’t know. (laughs)
As the interviewer followed up, Alex continued to sound less and less sure about her
answer. Finally, she changed her mind:
A51 I: So you’re saying that they need not have actually plugged in the numbers? Is that
what I’m hearing?
A51 A: No, I think you’d have to plug in the numbers because…uh…I mean you just
would to be sure. I guess you, I don’t think you can just guess about it.
In line A35, Alex expects that the Two Balls Problem can be answered without an
explicit calculation. The least charitable interpretation of this exchange is that she never came up
with a well-formed conceptual explanation for how to solve the problem without calculations, as
evidenced by her mixing up the units (“...so 5 seconds faster, it would get there 2 seconds, err,

it’s going 2 meters er second faster...”). A more generous interpretation is that she is trying to
express the following conceptual argument: since both balls are accelerating due to gravity only,
both balls will gain the same amount of speed, so the thrown ball will be traveling “2 meters per
second faster” (line A35). Either way, she backs off this line of reasoning in lines A37 and A51,
possibly because she feels she is on the spot, trying to answer the interviewer’s questions. Our
main point, however, is that any conceptual reasoning in line A35 is not well integrated with
Alex’s mathematical, symbolic reasoning. Evidence for this lack of integration comes from (i)
the lack of explicit mention of the equation or implicit reliance on its structure in line A35, and
(ii) her view in line A51 of the calculation as a way to “be sure” of non-calculation-based
reasoning, which is more of a “guess” than something reliably connected to the calculation in
some way.
In summary, the procedural way in which Alex solves the problem, along with the lack of
connection between the equation and conceptual reasoning, evidenced by her follow-up
comments, points us toward the tentative conclusion that Alex, in this context, is viewing the
velocity equation as a plug-and-chug tool — a “gizmo” — for grinding out a final velocity given
an initial velocity, an acceleration, and a time. We will put this initial interpretation to the test
below when we analyze Alex’s response to prompt 1, where she explains the velocity equation to
a friend. But first, to emphasize the contrast between Alex and Pat, we present Pat’s solution to
the Two Balls Problem. He solves the problem without explicitly calculating the final velocity of
either ball.
How Pat uses the velocity equation while solving the Two Balls Problem
Pat solves the problem without plugging in numbers
Pat also turns to the velocity equation. However, he uses it much differently from the way
Alex did:
P41 P: …Well, the first thing I would think of is the equations. The velocity, I suppose,
is the same equation as that other one [the velocity equation he had just explained in
prompt 1], and I’m trying to think of calculus as well and what the differences do.
So the acceleration is a constant and that means that velocity is linearly related to
time and they’re both at the same…so the first difference is the same. I think it’s
equal to two meters per second.
Later, asked how he got this answer by the interviewer, Pat elaborates on his solution a
little more:
P45 P: So the first differences are the same.
P46 I: Mhm.
P47 P: And if the first differences are the same then the initial difference between the
two speeds should not change.
When asked, Pat explained that the term “first differences” comes from his high school algebra
class, where sets of data points would be analyzed by taking “delta y over delta x,” which is

called the “first difference.” So, “first difference” connects at least roughly to the notion of
slope.
A few moments later, Pat states that “there’s a couple of methods of attacking” the
problem if he get stuck. Pat then further discusses different ways to solve the Two Balls
Problem:
P61 P: So if I started from thinking about the equations and I’m not quite sure whether
the velocities are changing at the same rate, then like sometimes I’ll use several
[solution methods] and see if they’re consistent. Then I could switch to thinking
about the derivatives of the velocity and I’ll think, ok, so the initial conditions are
off by 2 and then the velocities are changing at the same rate so that should mean
they stay at 2…
Pat does not follow a more structured, computational problem-solving approach that
describes Alex’s solution to the Two Balls Problem. Instead, he finds a shortcut past a plug-and-
chug calculation by reasoning conceptually. Pat starts his solution by explicitly saying he’s
thinking of the equations in line P41 (Later on in the interview he makes it clear that he was
thinking of the velocity equation.). He says that the constant acceleration “means that velocity is
linearly related to time,” and this leads him to say that the “first difference” is the same. Since
“first differences” is similar to the idea of slope, this aligns with his reasoning in line P61, where
he offers a similar argument in terms of derivatives.
Pat’s solution does not tell us the details of how he thinks of the relation between
acceleration and velocity, nor do we know the ideas Pat associates with ”first differences.”
Nonetheless, we can draw some inferences about how he is thinking about the velocity equation
in these moments. He is not viewing it as a mere gizmo for plugging numbers into. Instead, he’s
viewing the equation as connected to conceptual reasoning, as expressing physical meaning
about how the initial and final velocities relate. Specifically, his reasoning relies in part on an
intuitive schema, the idea that if two things undergo the same change, the difference between
those things doesn’t change. But his reasoning also relies on mathematical formalism; he
explicitly refers to the equations in lines P41 and P61, and he exploits the linearity of the relation
between the final velocity and time. So, although we lack sufficient evidence to fill in the details,
we can infer, at least tentatively, that Pat’s reasoning combines formal mathematics with
physical/conceptual reasoning in these moments.
Results of analysis phase 1b: The velocity equation
So far, we have tentatively concluded that in the context of the Two Balls Problem, Alex
views the equation as a tool for plugging numbers to generate numerical solutions while Pat
views the velocity equation (v = v0 + at) as expressing conceptual meaning. To test these
interpretations, we now analyze Pat and Alex’s response to prompt 1, which asked how they
would explain the velocity equation to a friend.
Alex explains the velocity equation: A computational tool

Alex initially seems puzzled by this question but eventually answers:
A10 A: Umm…Ok…well…umm…I guess, first of all, well, it’s the equation for
velocity. Umm…well, I would…I would tell them that it’s uh… I mean, it’s the
integral of acceleration, the derivative of {furrows brow} position, right? So, that’s
how they could figure it out…I don’t know. I don’t really {laughs}, I’m not too
sure what else I would say about it. You can find the velocity. Like, I guess it’s
interesting because you can find the velocity at any time if you have the initial
velocity, the acceleration, and time…
Alex’s explanation here has two main parts. First, the velocity equation is defined
through its relation to other kinematic equations; it’s the integral of acceleration and the
derivative of the position equation. Second, the equation can be used to calculate the velocity at
some time if you know the other values in the equation.
The interviewer then asks if that is what she would have said on an exam. She says “no”
and responded on how she would answer on an exam:
A14 A: Um…well, it depends on what it was asking…’cause I feel like your question’s
kind of vague, but, I mean, I would probably just say ‘it’s the velocity equation’
{nods and laughs}. I mean, if it was a more specific question, I could probably like,
elaborate, I guess.
Finally, Alex is asked to explain the equation to a 12-year old who knows math but
doesn’t really know physics.
A16 A: Well, these two sums will tell you how fast something is going. If you know
how fast it’s going when it first starts and after it first starts moving and you know
its speed when it first starts moving, and you know a certain point in time. You’re
looking at a certain point in time at which the object is moving, and you know how
fast it’s changing its speed, you can find how fast it’s moving at that time, or you
can find out the acceleration from it if you know how fast it’s going at that time.
The interviewer asks Alex if she would feel comfortable giving this explanation on an
exam. Alex says no, but she’s not sure because she has never had to answer this kind of
question before:
A20 A: Well, I’ve never taken physics before until now. So, um…like, the test, we’ve
had one test, and there just wouldn’t be a question on there that asks, like, to explain
the equation. You know, it would ask you to assume that you understand what it is
and then use it to figure out something more complicated, so I guess that’s what I
mean. If it was on there, sure I’d feel comfortable with that answer, but I’ve just
never really seen it.
In explaining the equation to a 12-year old (line A16), Alex reiterates the computational
utility of the equation but in a way that explains the meaning of the variables: “if you know how

fast it’s going when it first starts...and you know a certain point in time... you can find out... how
fast it’s moving at that time.”
In summary, Alex’s explanations of the velocity equation display a view of the equation
as a computational tool. She sees the individual variables as representing physical entities and
she sees the equation as coming from somewhere (“the derivative of position”), but the equation
itself is a gizmo into which you plug numbers to calculate the unknown quantity. This is the
same view of the equation she displayed when using it to solve the Two Balls Problem, as
discussed above.
Pat connects the equation to a physical process
When asked to explain the equation to a friend from class, Pat starts by looking at the
units and meaning of the variables:
P2 P: Well, I think the first thing you’d need to go over would be the definitions of
each variable and what each one means, and I guess to get the intuition part, I’m not
quite sure if I would start with dimensional analysis or try to explain each term
before that. Because I mean if you look at it from the unit side, it’s clear that
acceleration times time is a velocity, but it might be easier if you think about, you
start from an initial velocity and then the acceleration for a certain period of time
increases that or decreases that velocity.
Pat then talks about how the signs make things more difficult, because when he sees the
plus sign in the equation it naturally biases him to think that something is increasing. The
interviewer then asks Pat what he meant when he mentioned “the intuition part” in line P2:
P9 I: So right when you started you said something about “well, then from the intuitive
side.”
P10 P: Yeah, the problem is dimensions are just numbers really, or units, and it doesn’t
really explain what’s going on in the motion.
…
P15 I: Ok, so how would you explain it intuitively?
P16 P: I would say that an acceleration is the change in velocity, so you start from the
velocity you have in the beginning and you find out how the acceleration affects
that velocity. Then that would be the significance of each term.
For Pat, it’s not sufficient to explain the equation only by defining its variables and
focusing on units; “the intuition part” is also present and important. Specifically, for Pat, the
velocity equation schematizes a physical process of an object starting at some velocity and then
speeding up or slowing down due to some acceleration. He treats the at term in the equation
(“acceleration times time”) as a velocity term, not just because of its units, but because of its
conceptual interpretation as the increase or decrease in velocity due to the acceleration: “...you
start from an initial velocity and then the acceleration for a certain period of time increases that
or decreases that velocity” (line P16).

This analysis supports and refines our conclusion from analysis phase 1a, where we
found that Pat’s reasoning connected mathematical formalism to an intuitive schema rooted in a
physical/conceptual understanding. We now see that that he views the velocity equation as not
just connected to but actually expressing an intuitive schema, the idea that the final velocity is
the velocity you start with plus the change (increase or decrease) in velocity due to the
acceleration.
Summary of differences between Pat and Alex’s view of the velocity equation
Looking across the first two prompts in the interviews (“explain the velocity equation”
and “solve the Two Balls Problem”), we see a key difference in how Alex and Pat connect
physical/conceptual reasoning to the equation. Alex’s connection is at the level of assigning a
physical meaning to the individual variables in the equation, but the equation as a whole is
treated as a problem-solving tool. By contrast, to Pat, the equation as a whole also expresses
intuitive, physical meaning, the idea that the velocity you start with plus the velocity you gain (or
lose) is the velocity you end up with. We have also argued that this difference between Alex’s
and Pat’s conceptualizations of the equation explains many of the differences in their responses
to those two prompts.
Recall from our methods section that we decided to focus on Alex and Pat in part because
their reasoning patterns seemed typical to us given our experiences teaching introductory physics
courses for engineers. The above analysis allows us to refine this instructional intuition into a
testable question.
Results of analysis phase 2: Are Alex and Pat idiosyncratic?
Given our phase 1 analysis, our intuition that Alex and Pat both exemplify “typical”
reasoning patterns becomes operationalized as follows. We expect such students to be “Pat-like”
as follows: (i) when explaining the velocity equation, they will include physical/conceptual
reasoning that goes beyond just explaining the physical meaning of the individual variables, and
(ii) on the Two Balls Problem, they will discover a shortcut by tying mathematical formalism to
physical/conceptual reasoning, either in their initial solution or in response to the follow-up
question about whether the problem could have been solved without calculations. By contrast,
we expect other students to be “Alex-like” in that (i) when explaining the velocity equation, they
will not include conceptual reasoning that goes beyond just explaining the physical meaning of
the individual variables, and (ii) on the Two Balls Problem, they will not clearly articulate the
conceptual shortcut.
Of the 11 students interviewed, 7 students, including Alex and Pat, went through prompts
1 and 2, as described in this paper, at the beginning of their interviews and were coded in our
analysis below. Of the 4 students omitted from our coding, 3 received a different version of the
Two Balls Problem and 1 went through a different interview protocol.
There were two codes for each student; we coded each subject’s explanation of the
velocity equation for whether it included physical/conceptual reasoning that went beyond
interpreting the individual variables, and we coded the Two Balls Problem for whether the
student found a conceptual shortcut using the velocity equation. Note that an important feature of

the shortcut is the connection between the mathematical formalism and physical/conceptual
reasoning. Before coding, we decided that an algebraic shortcut, with no accompanying mention
of its conceptual/intuitive meaning, would be coded as Alex-like, and that use of purely
conceptual reasoning to solve the Two Balls Problem, not tied to the velocity equation, would be
coded as neither Pat-like nor Alex-like. However, no students’ solutions to the Two Balls
Problem fell into these two special codings.
Before discussion, the two raters agreed on 13/14 codes (93%). This single disagreement
was resolved after discussion. Figure 3 shows the results.
Two Balls Problem:
Found conceptual shortcut
(Pat-like)
Two Balls Problem:
No conceptual shortcut
(Alex-like)
Explaining equation:
Includes conceptual schema
(Pat-like)
4 1
Explaining equation:
No conceptual schema
(Alex-like)
0 2
Figure 3: Results of phase 2 coding. All but one student was either consistently Pat-like or
consistently Alex-like across the two interview tasks.
Readers may be surprised that more students were Pat-like than Alex-like. We find that students
who volunteer for interviews tend to be better-than-average students. Also, we observed no
gender patterns; of the two Alex-like students, one was female, and of the four Pat-like students,
one was female.
Although our sample size is too small to support robust claims of consistency in how
students understand and use equations, the data support our instructional intuition that the pattern
we see in Alex’s and Pat’s responses is not completely idiosyncratic. A future round of
interviews will enable us to perform statistical analyses. However, the argument we’re making in
this paper does not rely upon the ubiquity of Pat-like reasoning; it relies merely upon the
existence of such reasoning among (some) good students in traditionally taught introductory
courses, as discussed below.
Discussion: Symbolic forms
In this section, we discuss a previously documented cognitive structure called a symbolic
form. The activation of this cognitive element, we argue, helps to explain Pat’s reasoning (and
Pat-like reasoning more generally), while the non-activation of this cognitive element helps to
explain Alex-like reasoning. We will then argue that symbolic forms-based reasoning is a
concrete, feasible target of instruction.
What is a symbolic form?

Sherin proposed the existence of a knowledge structure called a symbolic form (Sherin,
2001). Each symbolic form connects a symbol template with a conceptual schema.
A symbol template is the broad structure of a mathematical expression. For example, the
symbol template for Newton’s 2nd law (F = ma) is represented by  = ; and the symbol
template for the first law of thermodynamics, ∆E = Q + W, is  =  + . Each symbol
template is not unique to a single equation. For example, the symbol template  +  + 
describes both the expression x0 + v0t + 1/2 at2
and the expression P0 + 1/2ρv2
+ ρgh.
A conceptual schema is an informal idea or meaning that can be represented in a
mathematical equation or expression. For example, one conceptual schema is the idea that a
whole consists of many parts. Another conceptual schema, applicable to intuitive reasoning
about a game of tug-of-war or to a marriage between a spendthrift and a miser, is the idea of
opposing influences.
A symbolic form connects a symbol template to a conceptual schema. For example, the
parts-of-a-whole symbolic form connects the symbol template of “ =  +  +  + ...” to the
conceptual schema of something consisting of many parts. The box on the left side of the
equation takes on the meaning of “whole” and the boxes on the right side of the equation take on
the meaning of “part.” The fact that these “parts” are added up to be the “whole” completes this
blend between the symbol template and the conceptual schema. When using a symbolic form in
her reasoning, the student is reasoning neither purely symbolically nor purely conceptually; the
conceptual/intuitive and symbolic reasoning are blended into a unified reasoning that has “dual
citizenship” in the world of informal reasoning and world of mathematical formalism (Sherin,
2001).
What counts as evidence of symbolic forms-based reasoning? The simultaneous
presence of both a symbol template and a conceptual schema in the student’s reasoning is not
enough. The symbol template and conceptual schema have to be bound together. In practice,
good evidence for this can consist of the student’s articulating a conceptual schema to explain
the meaning of an equation; by doing so, the student shows that she views the conceptual schema
as attached to the symbolic formalism in some way.
For example, a student talking about the equation x = x0 + v0t + 1/2 at2
might explain the
three terms on the right side as the initial displacement, the additional displacement due to
moving at the initial velocity over some time, and further displacement due to the acceleration
changing that initial velocity. Thinking of these three terms as individual displacements that add
up to overall displacement indicates the activation of the parts-of-a-whole symbolic form in the
student’s reasoning in that moment.
This is not to say that activation of a symbolic form necessarily leads to an articulated
connection between the conceptual schema and symbolic formalism. In theory, such a
connection could be implicit in what appears to be a student’s purely symbolic or purely
conceptual reasoning. However, such reasoning cannot provide evidence supporting the
activation of a symbolic form.
The “Base + Change” Symbolic Form
Sherin documented a particular symbolic form, called Base + Change, what we will
argue was present in Pat’s — but not Alex’s — reasoning when explaining and using the velocity
equation, v = v0 + at. The symbol template for Base + Change is:  =  +  . The conceptual
schema is different from the one associated with parts-of-a-whole. Instead, it is that “the final

amount is the initial amount plus the change to that initial amount.” In the context of the
velocity equation, the “amount” that is being changed is the velocity of an object.
Now that we have introduced the language of symbolic forms in general, and the Base +
Change symbolic form in particular, we can revisit Alex and Pat to check for the absence or
presence of that knowledge element in their reasoning.
Looking back at Alex
There is no evidence that Base + Change is present in Alex’s explanation and use of the
velocity equation. As discussed above, her physical/conceptual reasoning was confined to
interpreting the meaning of the individual variables. When solving the Two Balls Problem, she
treats the equation as a problem-solving tool and doesn’t discuss its intuitive meaning. When
later asked if the Two Balls Problem could be solved without calculations, Alex briefly thinks
conceptually but does not use the velocity equation in her reasoning. This too is consistent with
the absence of Base + Change; if the velocity equation has no conceptual interpretation, it makes
no sense for Alex to try to integrate that equation into her conceptual solution.
Crucially, we are not claiming that Alex doesn’t “have” or “know” the Base + Change
symbolic form. In other interviews, we found that students who displayed Alex-like reasoning
about the velocity equation showed evidence of “having” Base + Change when solving more
everyday problems about money. Alex herself shows evidence of blending symbolic and
conceptual reasoning later on in the interview. Nonetheless, for whatever reason — and we
suspect part of the reason has to do with how introductory physics courses are typically taught —
Alex does not activate whatever symbolic forms she may possess, when responding to the first
two interview prompts. We believe that, in different contexts, Alex would use the Base +
Change symbolic form in her reasoning. For example in a problem of thinking about bank
accounts, we predict that Alex might reason about the amount of money in the account as some
amount she started with plus her weekly salary, and that she could express this understanding in
a symbolic equation. Below, we will discuss an intervention being piloted where at least one
student was cued into using Base + Change by asking about money. In the literature, Sherin
found that students asked to write an equation for the final height of a sand pile given the initial
height, rate of growth, and time could activate intuitive resources, which they could have also
used in physics contexts but didn’t, for generating the equation (Sherin, 2001). What we are
claiming is that, for whatever reason, Alex is not tying this particular symbolic form to the
velocity equation in this context – and that this makes her reasoning appear less expert than
Pat’s.
Looking back at Pat
By contrast, one of the things Pat said when asked to explain the velocity equation was,
“you start from the velocity you have in the beginning and you find out how the acceleration
affects that velocity. Then that would be the significance of each term” (line P16). This
explanation, and his follow-up clarifications, indicate that he is attaching a “final amount equals
initial amount plus change” conceptual schema to the velocity equation. As discussed above, this
constitutes good evidence that Base + Change is activated in his reasoning: he sees v0 as the
base and at as the change.

In solving the Two Balls Problem, we have previously argued that Pat is connecting a
conceptual schema with formal mathematics. Now, we make the case that his reasoning is
specifically rooted in the Base + Change symbolic form. Pat’s initial solution, as discussed
above, comes from thinking of the formal equation (lines P41 and P61) and relies on the
conceptual idea that since the initial (base) difference in velocities is 2 meters per second, and
since both balls undergo the same change in velocity, it follows that their final difference in
velocities is still 2 meters per second. Although it’s unclear whether he is applying Base +
Change to the individual velocities of each ball or directly to the difference in those velocities,
it’s clear that he is connecting the equation — its general structure and its linearity to the
intuitive idea of “final equals initial plus (linear) change.” Along with Pat’s explicit explanation
of the velocity equation with Base + Change, this is evidence that the symbol template and the
conceptual schema are not only simultaneously present but somewhat bound in his reasoning.
However, our key point is not the specific form of Pat’s blended reasoning, but rather that
Pat productively blends physical/conceptual reasoning with formal mathematics in problem
solving. Outside of these two prompts, Pat uses blended conceptual and symbolic reasoning —
which we sometimes identify as symbolic forms-based reasoning - throughout the interview to
address problems and explain phenomena.
Discussion: Revisiting what counts as good problem-solving
We now argue that Pat’s (and other Pat-like students’) productive use of symbolic forms
gives us reason to question the common view of what constitutes good problem-solving in
physics and engineering, and hence, of what strategies instructors should nurture in their
students.
Most problem-solving procedures don’t include Pat-like blending
As discussed in the literature review, most problem-solving procedures include a symbol
manipulation step that is separate from the steps involving conceptual reasoning. In these
procedures, conceptual reasoning “meets” equation-based reasoning at two points: in translating
the physical situation into equations, immediately before the symbol manipulation step, and in
checking the answer for plausibility, immediately after the symbol manipulation step (Gainsburg,
2006; Giancoli, 2008; G. L Gray et al., 2005; P. Heller et al., 1992; Huffman, 1997; Van
Heuvelen, 1991; Young & Freedman, 2003). However, as Wertheimer and Redish & Smith
argue, blending symbolic manipulations with conceptual reasoning is a part of problem-solving
expertise, as it allows for more adaptive, flexible solutions — and more opportunities to learn
while solving the problem1
— than is afforded by step-by-step problem-solving procedures.
Our contribution to this debate is to illustrate in some detail what the contrast looks like
between students who are and are not incorporating such blending into their reasoning. Pat’s
blended reasoning was demonstrably productive for him, and Redish & Smith would say it
displayed more expertise than Alex’s procedural reasoning. For these reasons, we urge both
1
While addressing a hydrostatic pressure problem later in the interview, Pat’s blended
reasoning enables him to reach a new conceptual insight: the difference in pressure between two
points under water does not depend at all on the pressure at the top surface.

researchers and instructors to revisit the standard problem-solving procedures discussed above.
Instead of encouraging students to treat symbol manipulation as a separate step, we should help
students learn to be more Pat-like, looking for the meaning expressed by an equation and
blending conceptual with symbolic reasoning.
Symbolic forms as an instructional target
Our analysis also helps us refine a vague instructional suggestion (“teach students to
blend!”) into something more concrete. Symbolic forms-based reasoning, we argue, is a
productive and feasible target of instruction, and techniques for helping students engage in such
reasoning in the physics classroom exist.
For instance, Redish & Hammer (2009) discuss an instructional strategy that helps
students “see the sense in the equation” and engaging students in that particular sense-making
game is a frequent part of lecture. In recent interviews, we’ve been testing an intervention that
specifically tries to cue the use of the Base + Change symbolic form. When a student can’t find
the shortcut to the Two Balls Problem, we switch gears and ask the following prompts about
money:
• $0 is the amount of money you have now, and you earn r dollars per day for d days, how
much money do you have after d days, assuming you don’t spend any? [We predicted
that physics and engineering students will be able to answer this question
unproblematically.]
• How would you explain that equation to a 3rd grader? [The goal is to cue the “final
equals initial plus change” conceptual schema in relation to their equation.]
• Now suppose a car starts with speed s and gains speed at rate r for time t. How fast is it
going after time t?
• Explain that velocity equation to a 3rd grader. [The goal is for the conceptual schema,
cued by the money problem, to carry over to the physics-related problem.]
Although this intervention is still being piloted, at least one student in an interview has been cued
into activating the Base + Change symbolic form. Furthermore, when asked to go back to the
Two Balls Problem, this student was able to find the Pat-like conceptual shortcut.
This is not to say that we expect use of Base + Change to automatically lead to finding
the Pat-like, conceptual shortcut. For example, a student could be activating symbolic forms-
based reasoning in doing a more Alex-like, procedural solution. For this one student and for Pat,
we claim only that the use of symbolic forms-based reasoning plays an important role in their
problem-solving shortcut, not that that reasoning is the cause of it.
Although this intervention is still being pilot tested, our preliminary results counter the
view that symbolic forms-based reasoning is something that cannot be taught and that it’s
something only “good” students can do. Other research supports this conclusion. For instance,
Iszak (2004) documented eighth graders constructing and using Base + Change after interacting
in a rich learning environment for several hours. For these reasons, it is plausible that Alex-like
students’ failure to blend conceptual and symbolic reasoning reflects not a lack of ability, but a
lack of scaffolding. As emphasized above, some standard problem solving procedures taught in
textbooks encourage students to view symbol manipulation as a separate step from conceptual
reasoning.

Conclusion
Analyzing Pat and Alex’s explanations and uses of a standard kinematic equation, v = v0
+ at, we attributed part of the difference in reasoning patterns to the presence vs. absence of a
knowledge element called a symbolic form. A symbolic form is a blend of symbolic and
conceptual knowledge. Pat’s use of that blend enabled him to give an intuitive explanation of the
velocity equation and to quickly find a shortcut to the Two Balls Problem. Alex’s explanation
and problem solving were more procedural. So, as Redish & Smith, Wertheimer, and others
argue, Pat’s solution to the Two Balls Problem shows more expertise. But Alex’s solution aligns
more closely with the standard problem-solving procedures advocated by researchers and taught
to students, according to which the symbol processing step is separate from conceptual
reasoning.
We have used this result to argue that blending conceptual and symbolic reasoning —
specifically, symbolic forms-based reasoning — can be an instructional target, valued by both
researchers and instructors. We have offered plausibility arguments that symbolic forms are a
desirable and feasible instructional target.
Given our arguments, a researcher might take away the message that we advocate
replacing the “plug-and-chug” step in a problem solving procedure with a “symbolic forms-
based reasoning” step. This is not what we are suggesting. As Reif (2008) argues, good problem
solving involves decision making, not just following a set procedure. By this argument, a good
problem solver doesn’t always plug-and-chug or always activate symbolic forms. Instead, she
has all of those tools in her toolbox, and selects which tools to use based on the details of the
problem (Reif, 2008; Reif & J. I. Heller, 1982; Schoenfeld, 1992).
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