Copyright 1997 Carl Wenning
In recent years, physics education researchers and cognitive psychologists have turned their attention to the question of how individuals solve basic physics problems. The presenter summarizes the surprizing results of a multiple case study in which three experts and three novices were observed as they solved kinematics problems using a "think aloud" protocol. Follow-up interviews and content analysis led the researcher to concluded that expert problem solvers do not always follow the most efficient routines, nor do they always use the most effective methods for teaching basic problem-solving skills to students.
The class began on time at 10:25 AM with the instructor asking
the students if they had any questions about the kinematics homework
problems that they were supposed to have attempted the night before.
A discussion dealing with three homework problems (and one example
problem) ensued for the next 45 minutes. While the instructor
was solving these problem, students observed intently. The majority
of the students listened while the instructor talked and worked
on the board, yet about one third of the students ceaselessly
recorded in notebooks everything that the instructor wrote.
In each case of problem solving, the treatment by the instructor
was consistent and methodical. The instructor began with a statement
of the problem. Next, he drew a picture. Thirdly, he stated what
was known or given as part of the problem. Fourth, he identified
a principle by which the problem could be solved. Fifth, he stated
the relevant equation that related the knowns and unknowns. Sixth
he restated the knowns and unknowns. He then solved the equation
for the required unknown, inserted the knowns, and carried out
the arithmetic calculation. The instructor then made reference
to checking the answer for reasonableness. The instructor's approach
to problem seemed clear and, yet, something seemed to be missing.
During the problem-solving session there were 19 questions asked
by students. The questions, interestingly enough, were more frequently
metacognitive questions ("How do you know when to...?"
and "What do you do if...?" and "How do you go
about...?") than any other variety.
Beginning at 11:10 AM, the instructor moved on to a 20-minute
lecture about Newton's first and second laws. He did not provide
many significant real-life examples of the first law, and the
second law was treated entirely at a theoretical level. During
this time, all students appeared to be diligently taking notes.
At the outset of the lecture portion of the class the instructor
dealt momentarily with the alternative conception that moving
things need a constant force to keep them in motion.
At the end of this session, and near the end of the class, the
instructor worked another example problem. He assigned 16 exercises
for homework at the end of the hour. Eight of the exercises were
questions, six were "standard" problems, and two were
"challenge" problems. The students diligently recorded
the list of required homework problems and promptly left the classroom
at the end of the period. Another typical introductory physics
class had come and gone.
What do we leave students with at the end of a series of such
introductory physics lessons? Are students better able to solve
physics problems now that they have seen a few examples? Do they
have a metacognitive understanding of this simple problem-solving
process that is so frequently tendered with almost every class?
Do such courses leave the students with the perception that the
scientific process is nothing more than searching for the right
equation? How important are concrete examples to true student
understanding of physical phenomena?
These are only a few of the questions that might arise from intently
watching and seriously reflecting on what happens in many introductory
physics classes. To focus on all these questions would be too
great a task in the limited space available for this article and,
so, a more narrow view will be centered on the difficulties associated
with teaching the general problem-solving paradigm so frequently
taught in didactic introductory-level physics courses -- find
the knowns and unknowns, state the relationship between them,
and solve for the unknown.
In recent years physics education researchers and cognitive psychologists
have turned their attention to the question of how individuals
solve physics problems. Recent research has focused on two areas
as they pertain to physics problem solving: (a) the overall plan
of attack used to solve problems, and (b) the identification and
use of heuristics in problem solving. The researchers generally
approach a study of the first focus area by comparing and contrasting
the performance of novices (generally defined to be students in
introductory physics classes) with that of experts (generally
defined to be physics teachers). Studies in the area of problem
solving frequently utilize qualitative approaches and involve
a relatively small number of subjects. "Think-aloud"
protocols are normally used in these efforts. Computer models
are generally associated with the heuristic aspect of problem
solving and will not be dealt with in this article.
A clear and concise definition of problem solving must be given
if the problem statement is to be meaningful. A review of secondary
sources shows that there are a number of definitions of the word
"problem," but the definition that is most apropos to
this project is a characterization -- work associated with those
tasks found at the end of chapters of introductory physics text
books. Typically, these tasks involve a statement of information
and/or circumstances, and an additional variable or variables
are determined on the basis of the information provided. These
tasks tend to be very specific and the work and goal well defined.
Problem solving then is the process of attaining the goal of any
specified problem.
Context
Studies of novice and expert physics problem solvers have suggest
that there are two distinct and contrasting patterns of problem
solving among experts and novices. These variations have led to
the formulation of two major models for problem solving. According
to Larkin et al. (1980), expert problem solving is typified by
the KD model, the so-called knowledge-development approach. Novice
problem solving is typified by the ME model, the so-called means-end
approach. In the ME model the student typically works "backward"
from the unknown to the given information. Under this scenario
the novice problem solver (NPS) essentially writes an equation
and then associates each term in the equation with a value from
the problem. If there are additional unknowns, the problem solver
moves on to the next equation. In the KD model the expert proceeds
in the opposite direction, working forward from the given information.
Under this second scenario, the expert problem solver (EPS) associates
each of the knowns with each term of the equation as the equation
is set up. That is, novices move from equations to variables,
while the experts move from the variables to the equation.
The research in the area of physics problem solving accelerated
rapidly in the early 1980s and is now the focus of much attention
in the research literature. There are a number of questions left
unresolved, including those given by Maloney (1994), "What
knowledge do novices typically use when faced with physics problems?"
and "How is the knowledge that a novice possesses organized
in memory?" and "How do alternative conceptions affect
novices' representations?" However important these questions,
the basis of this research still depends upon the answer to the
question, "How do problem-solving approaches differ between
novice and experts?"
Method
In case studies, the researcher is the primary research instrument.
When this is the case, validity and reliability concerns can arise.
The human investigator may misinterpret or hear only certain comments.
Guba and Lincoln (1981), as well as Merriam (1991), concede that
this is a problem with case study work. Yin (1994, p. 56) lists
six attributes that an investigator must possess to minimize problems
with validity and reliability associated with the use of the human
research instrument.
The researcher believes that he exhibited these personal characteristics,
though "no devices exist for assessing case study skills."
(Yin, 1994, p. 56)
Five kinematics physics problems were written for this project.
The five questions ranged from simple one-step problems with a
single output variable, to more complex two-step problems where
more than one output variable was requested. These problems used
in this study can be found in Appendix A.
Three faculty members and four students were then self-selected
to participate in this study. All faculty members were male; one
of four physics students was female. Though this may at first
appear to be too large a sample for a case study, "any finding
or conclusion in a case study is likely to be much more convincing
and accurate if it is based on several different sources of information."
(Yin, 1994, p. 92) The problem-solving skills of these individuals
were examined through observation, interview, and content analysis.
Such use of multiple data sources also enhances validity and reliability
via triangulation.
All volunteer faculty members participating in this study had
experience teaching introductory physics courses for non-majors.
All students were volunteers who were currently enrolled in an
introductory, algebra-based physics course for non-majors at a
middle-sized midwestern university. Students were informed that
a wide range of problem-solving abilities were needed, and that
excellence in problem solving was not a prerequisite for participating
in the study. (The female student was subsequently dropped from
the study due to an apparent lack of ability to solve even rudimentary
algebraic equations.)
Three data collection strategies were used in this project. Participants
first solved the five physics problems using a "think aloud"
protocol. The researcher listened to the problem solvers, recording
pertinent details dealing with the solution of the problems. He
later coded these comments for analysis. Following problem solving,
the researcher collected the written work which would be used
in content analysis, and then commenced the interview process.
In follow-up interviews, the faculty members were asked three
questions common to all study participants, and two additional
questions reserved to expert problem solvers. The students were
asked the same three common questions and three additional student-specific
questions. The questions can be found in Appendix B.
Appendix C shows the coding plan for problem solver statements
made while working on the problems using a think aloud protocol.
The coding plan consists of steps in a theoretical scheme of problem
solving enunciated by Heller, Keith, and Anderson (1992), and
modified and extended slightly for this study. Each step of the
problem-solving process is operationally defined with descriptors.
For instance, a problem solver can be said to be visualizing the
problem if he or she draws a sketch, identifies the known variables
and constraints, restates the question, or identifies the general
approach to solving the problem. While problem solvers were working
problem number one (and all subsequent problems), the researcher
recorded statements for later coding. The results of the coding
can be found in Table 1.
This table shows the logical approaches used by expert and novice
problem solvers. If a problem solver uses what is theoretically
the most efficient scheme for solving the problem, then his solution
should consist of five sequential steps: 1, 2, 3, 4, and 5. If
expert problems solvers depart substantially and consistently
from this model, it might lead the researcher to conclude one
of two things: either these particular EPSs are inefficient, or
the model proposed by Heller et al. is simply wrong.
The data tabulated in Table 1 shows that EPSs do not generally
follow the same paths to a solution as the theoretical model.
In all three cases, the EPSs chose different routes to solve the
problem. These paths were 123, 231, and 213. NPSs #1 and #2 took
similar mixed routes, while NPS #3 departed from the general problem
solving model when he failed to include step two. Among the six
problem solvers, this was the only person to neglect this step,
leading possibly to the long, convoluted solution to the problem
as indicted by the twelve steps. Interestingly enough, five of
the six problems solvers made the effort to mentally check their
answers for apparent correctness.
The overall impression gained by the researcher while observing
the problem solvers was that the problem-solving procedures utilized
by novice problem solvers (NPS) are very unstructured and inefficient.
Problems are not systematically approached, knowns are rarely
written down in equation form (for instance, = 1 m/s2), starting
equations are rarely written down, equations are not solved for
unknown variables before inserting the knowns, work is done without
units, solving algebraic equations appears to be a problem for
most, etc. Students, in many cases, quite randomly choose equations
to solve for the unknown. They, not infrequently, expected a calculator
to "solve" the problem for them. One student in particular
regularly multiplied and divided numbers in a random fashion looking
for solutions that "looked right." This procedure might
work on a multiple-choice test -- something that is normally used
at the introductory level -- but not in this research project
where students had to derive precise answers of their own. In
general, the time required for expert problem solvers (EPS) to
solve problems was one third that required by NPSs.
It is clear from the interview process that in the area of kinematics,
students tend to follow the same general procedures as the experts
when it comes to problem solving: search for knowns and unknowns,
establishing or finding a relationship between the knowns and
unknowns, and then solve for the unknown. The general procedure
for problem solving is shown in Figure 1. In some cases the students
would check their answers to see if they made sense; this was
normally the case with experts. Checking the answer generally
took the form of looking at the magnitude and sign of the solved
variable. The students interviewed seemed to be clear on the overall
process. When they did have trouble, it was in selecting the appropriate
equation to relate the known and unknown variables through the
most direct route. In this procedure two faculty members were
very efficient; however, one expert problem solver almost invariably
started the problem solving process with the same kinematics equation,
no matter what the original given quantities were.
Two students were unable to explain clearly the "black box
"procedure for selecting the appropriate kinematics equation
to relate the variables (see Figure 1). For instance, "I
look to fit all the information into a model" and "I
see what formula gives me the information I need." The result
of this uncertainty was clearly evident as these two students
randomly selected one equation after another in an effort to "plug
and chug" their way through the problem set. One student
was clear about the procedure, "The equation I would select
would be that which has one unknown variable -- the one you are
looking for. Alternatively, using a formula with two unknowns
where one of the unknowns can be obtained with the use of another
formula." All problem solvers, novices and experts alike,
appeared to use the means-ends approach to solve the five physics
programs provided.
The physics teachers explained how they taught kinematics problem
solving in their introductory courses. In all cases teachers indicated
that they made use of examples almost exclusively. In one case
an instructor noted that from time to time he would attempt to
clarify the process; in another case an instructor indicated that
he would never use a metacognitive approach. In his words, "...I
do not discuss general strategies....I'm not sure some students
at this level can conceptualize general strategies. Strategies
are drawn by example." Another instructor noted, "I
don't think that there is any particular procedure that you can
describe to the students for them to become more expert. In special
areas I point out what they have to do to recognize the unknown,
the data, and what sort of formula for them to use. Students often
randomly search for formulas. I warn them against this."
The students interviewed mentioned that they did make use of
examples to learn how to do kinematics problem solving. In all
three cases the students reported reading over the example, and
sometimes working the example, in an effort to comprehend the
general procedure. They did not indicate using examples as templates
for solving problems except in one instance. This student reportedly
resorts to using examples like templates to find one variable
in a two-step problem in which the desired variable is not immediately
obtainable directly from an equation.
When queried, student expressed the opinion that they had learned
general problem-solving strategies prior to taking the physics
class mentioned in this study. One student attributed his physics
problem-solving skill to a high school classmate; another to life
experiences; and yet another to related coursework in business
classes. Students generally felt that their problem-solving skills
were enhanced by taking the physics course, and this helped them
to gain a broader perspective on the problem-solving process.
Subsequent to the follow-up interviews, the written work of problem
solving was collected for content analysis. The procedures used
by problem solvers were coded on the basis of equations used to
find intermediate or final unknowns following the work of Simon
and Simon (1978). The equations referred to are those appearing
on the problem sheet shown in Appendix A. The first equation is
labeled 1, the second 4, the third 5, the fourth 7, and the fifth
8. This numbering sequence was chosen to remain consistent with
previous research on kinematics problem solving. The coding procedure
is a "short-hand" that indicates how problem solvers
approached problems. For instance, if a problem solver found the
average velocity, , using equation 5, then the approach was coded
( ). If the instantaneous velocity, , was found from equation
5, then the approach was coded ( ).
Table 2 shows the results of coding the mathematical steps used
by EPSs and NPSs. The designations running horizontally along
the top numerically distinguish EPSs and NPSs. The numbers running
vertically along the left side of the table indicate problem number.
Each cell contains the equation-based problem solving approach.
False starts have not been included in this table, nor have unsuccessful
attempts to solve problems. If a cell in the table is blank, it
is an indication that the problems solver was unable to find the
correct solution.
From an inspection of the approaches outlined in this table,
it is clear that not all expert problem solvers determine unknowns
in the same fashion or with the same efficiency (efficiency being
defined as working toward the answer by taking the most direct
route -- using the fewest number of steps and equations to solve
for an unknown). Admittedly, there are several ways to solve each
of these problems, with some routes being different but equally
efficient. This can be seen in the solution of problem 5 by expert
problem solvers.
Differences in problem-solving efficiencies were notable among
EPSs attacking problem 4. For example, compare the procedure of
EPS #2 with those used by EPS #1 and EPS #3. EPS #2 used a solution
procedure that was less efficient than that used by other EPSs.
EPS #2 solved for the product of and from equation 4, and
then divided this product by while the other EPSs solved equation
4 directly. This appears to have do with EPS #2's propensity for
beginning most problems with a statement of equation 7, and then
searching for variables to insert into the equation -- not always
the most efficient procedure.
Interestingly, some NPSs exhibited what appears to be greater
insight in solving some problems than EPSs. For instance, note
how all NPSs solved problem 1 in a much more direct fashion than
any EPS, not solving for acceleration ( ) in order to find .
Though the table does not show it, NPSs took a significant number
of dead-end approaches to solving the problems.
The findings of this research project do not lend support to
the claim that expert problem solvers tend to use a KE approach
and novice problems solvers an ME approach -- at least in the
area of kinematics. Both NPSs and EPSs used the same technique
of searching for an equation among a group of equations that contains
the end variable. They then worked from this end using any means
necessary. One might argue that their is no alternative to the
solution of kinematics problems, but the contrasting solution
of problem 1 by EPSs and NPSs would seem to indicate that the
students interviewed have used a more "insightful" KE
approach than did the EPSs.
It would appear that the general procedure for solving kinematics
problems (find the knowns and unknowns, state the relationship
between them, and solve for the unknown) are clear to the students
studied. It is also clear that these students have learned the
general procedure by watching instructors solve example problems,
by looking at examples in the course textbook, and through shared
experiences. What students are not consistently clear about is
how to select the appropriate kinematics equation or equations
to relate and solve for the problems' unknown. Evidently some
students have been unable to figure out by observation the relatively
sophisticated black box mental process the instructor goes through
to select the appropriate kinematics equation.
What was not self-evident to the physics instructors is that
students would appear, in some cases, not have a good understanding
of the equation-selecting process that goes in quickly in instructors'
minds. Though instructors argue that students appear to learn
from example, one of the most important examples that is lacking
is that which illustrates the thinking process that the course
instructor goes through to select the appropriate equation among
those available in kinematics. In one case a NPS had a clearer
view of this than, perhaps, an EPS. This same EPS noted that he
didn't think there was a general problem-solving process that
students could comprehend. Perhaps this is so because that EPS
never established a clear procedure for himself as is evidenced
by the rigid, lock-step procedure of attempting to solve the kinematics
problems by starting with equation 7 each time.
It is clear from subsequent discussions with each of the faculty
members participating in this project that they may well generally
lack a clear understanding of students' problem-solving difficulties.
They tend to see a host of student problem-solving difficulties
such as: (a) failing to use a systematic process to solve problems,
(b) failure to identify variables with known quantities, (c) adding
dissimilar knows together such as velocity and acceleration, (d)
trying to solve equations without writing them down, (e) using
calculators to solve the problems rather than the equation for
the unknown, (f) randomly selecting equations to be solved for
the unknown variable, (g) making algebraic errors, (h) confusing
with ., (i) failing to recognize simplifying conditions (
at top of flight path for a projectile, for instance), and that
(j) novices are much less systematic than experts in both thinking
and writing down their work. The instructors studied do not seem
to be aware, however, of the difficulties students face when attempting
to figure out what is going on in the black box of establishing
relationships between variables. How widespread this evident unawareness
on behalf of instructors is not known.
Because the faculty members interviewed possibly have never taken
the time to analyze student problem-solving difficulties, and
then triangulated those observations to lend credibility to their
findings, they seem not to be aware of the central issue of problem
solving by NPSs. Additionally, if the instructors studied were
to more closely examine the nature of the questions that so many
students ask during class, they might be more aware of the need
for students to have a metacognitive understanding of the problem-solving
process being used, and particularly those occurring in the dark
recesses of the black box known as "establish relationship."
Two questions that arose in the mind of the interviewer as he
talked with students and faculty members alike were, "Why
don't faculty members take the time to take a metacognitive approach
to problem solving?" and "Why don't faculty members
talk about the entire problem-solving rather than expecting students
merely to learn by example?" If instructors were to clarify
for themselves the most efficient approaches for solving problems,
this might enhance their teaching and student problem solving
as well. A more systematic analysis of problem-solving difficulties
in all areas of physics teaching promises to pay dividends.
Guba, E. G., & Lincoln, Y. S. (1981) Effective evaluation:
Improving the usefulness of evaluation results through responsive
and naturalistic approaches. San Francisco, Jossey-Bass.
Heller, R., Keith, R., & Anderson, S. (1992). Teaching problem
solving through cooperative grouping. Part 1: Group versus individual
problem solving. American Journal of Physics, 60(7), 627-636.
Larkin, J. H., McDermott, J., Simon, D. P., & Simon, H. A.
(1980). Models of competence in solving physics problems. Cognitive
Science, 4, 317-345.
Maloney, D. (1994). Research on alternative conceptions in science,
In Handbook of research on science teaching and learning (Dorothy
L. Gabel, Ed.) Washington, DC: National Science Teachers Association.
Merriam, S. B. (1991). Case study research in education: A quantitative
approach. San Francisco: Jossey-Bass.
Simon, D. P., & Simon, H. A. (1978). Individual differences
in solving physics problems. In Children's thinking: What develops?
(Robert S. Siegler, Ed.) Hillsdale, NJ: Lawrence Erlbaum.
Yin, R. K. (1994). Case study research: Design and methods. Newbury
Park, CA: Sage.
Appendix A
Physics Problems
Think Aloud Problems
Please use a "think aloud" protocol as you solve the following problems. Use a separate sheet of paper for each problem. Clearly label each problem with the corresponding numbers below. A calculator is provided. Take the magnitude of the acceleration due to gravity ( ) to be equal to 9.8 m/s2. Below are formulas for your use.
where is the distance traveled by an object during a time , with constant acceleration , initial speed , final speed , and average speed .
1. A bullet is shot from a rifle with a speed of 160 m/s. If the barrel of the gun is 0.8 m in length, what is the average speed of the bullet while in the barrel assuming constant acceleration? For how long is the bullet in the barrel?
2. A "dragster" accelerates uniformly from rest to 100 m/s in 10 s. How far does it go during this interval?
3. A toy rocket is shot straight upward from ground level with an initial speed of 49 m/s. How long does it take the rocket to return to earth? Assume the absence of wind resistance.
4. A landing commercial airliner, upon "reversing" its engines, uniformly slows from 150 m/s to 30 m/s using 1,800 m of runway. What is the acceleration of the plane during this procedure?
5. A little girl glides down a long slide with a constant acceleration of 1 m/s2. If the girl gives herself an initial speed of 0.5 m/s and the slide is 3 m long, what is her speed upon reaching the bottom of the slide? How long does it take her to reach the bottom of the slide?
Appendix B
Interview Questions
For novices and experts:
1. What is the first thing you search for in a problem statement?
2. What is the first thing you do after determining what you are
to find?
3. Do you follow any particular pattern or procedures when you
solve physics problems? If so, please explain.
For novices only:
4. When you have difficulties solving a physics homework problem,
what do you do?
5. What use do you make of examples when attempting to solve problems
with which you are having problems?
6. How did you learn to solve physics problems?
For experts only:
7. How do you teach your introductory physics students how to
solve physics problems?
8. Do you ever talk about the problem-solving process? If so,
what do you say?
Appendix C
Coding Plan for Observations of Physics Problem Solving
1. Visualize the problem.
o draw a sketch
o identify the known variables and constraints
o restate the question
o identify the general approach to the problem
2. Describe the problem in physics terms.
o use identified principles to construct idealized diagram
o symbolically specify the relevant known variables
o symbolically specify the target variable
3. Plan a solution.
o start with the identified physics concepts and principles in
equation form
o apply the principles systematically to each type of object or
interaction
o add equations of restraint that specify any special conditions
o work backward from the target variable until you have determined
that there is enough information to solve the problems
o specify the mathematical steps to solve the problem
4. Execute the plan.
o use the rules of algebra to obtain an expression for the desired
unknown variable
o instantiate the equation with specific values to obtain a solution
o solve the equation for the desired unknown
5. Check and evaluate.
o check - is the solution complete?
o check - is the sign of the solution correct?
o check - does the solution have the correct units?
o check - is the magnitude of the answer reasonable?
6. Makes an Error.
o makes error in solution of algebraic equation
o makes error in statement of fact
7. Expresses Confusion.
o admits confusion
o expresses doubt
o expresses anger
o admits inability / gives up
Table 1.
Logical approaches used by expert and novice problem solvers to solve problem one.
Model EPS #1 EPS #2 EPS #3 NPS #1 NPS #2 NPS #3
1 1 2 2 1 2 1
2 2 3 1 2 1 3
3 3 1 3 3 3 4
4 4 4 4 4 4 6
5 5 5 5 5 5
7
5
3
7
3
4
5
Table 2.
Mathematical approaches used by expert and novice problem solvers.
# EPS #1 EPS #2 EPS #3 NPS #1 NPS #2 NPS #3
1
2
3
4
5 *
* Did not solve for .
Figure 1.
Problem-solving flow-chart.
Caption: The general problem-solving procedure appears to consist
of identifying the known and unknown variables, finding a mathematical
relationship between the variables, and then solving for the unknown.
Unfortunately, some students do not appear to have a clear understanding
of the thought processes that take place in the black box entitled
"Establish Relationship."