Problem Solving and Learning Physics
David P. Maloney
Problem solving has been an integral part of physics instruction
for many years. The reason for this is the belief that when the students
solve problems they really learn the physics. Certainly that is the
experience most of us had. However, most of the students in our general
physics courses are not physics majors and do not have anything close
to our motivation to learn physics. Physics education research has
made it clear that students can succeed in general physics courses,
even those which require significant problem solving, without showing
any growth in reasoning skills or the development of a coherent conceptual
framework (McDermott, 1993). Does this mean problem solving is not
a productive instructional activity?
Research on Students' Problem Solving Procedures
-One of the early studies of students' problem solving was that
of Reif, et al. (1976), who pointed out students' tendency to grab
an equation and plug in numbers. They tried, with limited success,
to get the students to follow a more systematic problem solving strategy,
although the intervention was of rather brief duration.
Larkin, et al. (1980) compared the problem solving behaviors of experts
(graduate students and physics professors) with novices (students who
had completed one or two semesters of general physics). In their research
two computer models were developed. The knowledge development model,
which worked from the given information to the goal of the problem,
did a good job modeling the behavior of the experts. The means-end
model, which started with the goal of the problem and worked "backwards" to
the given information, better matched the novices' procedures.
In means-ends analysis the problem solver identifies the goal of
the problem, determines where he/she is relative to that goal, identifies
steps he/she can take to reduce the difference between the current
problem state and the goal state and then applies appropriate steps.
For a novice student trying to solve a textbook physics problem, the
goal they will identify is finding a specific numerical value and what
will look like the most reasonable and efficient way to reach that
goal is to find an equation. Consequently, the characteristic "plug
and chug" behavior is understandable and not unreasonable.
Sweller, et al. (1983, 1988, 1990) have proposed that novices' use
of means-ends analysis on standard textbook problems is counter productive
for learning the physics concepts, principles and relations that underlie
problem solving with understanding. When the students focus on the
goal of finding a specific numerical value that focus will direct their
attention to the equations. With this focus, carrying out a qualitative
analysis involving other representations seems to be of little value.
In addition, applying the means-ends heuristic requires a significant
part of the cognitive resources of the solver, so few resources are
available to consider the concepts and principles and how they apply.
Traditional textbook problems helped us learn physics not because
solving such problems is the best way to learn physics, but because
we were motivated to use them to help us learn. Our success does not
mean such problems will be just as useful to less motivated students.
There may actually be better types of problem structures for helping
the majority of our students learn the concepts, principles and relations
that underlie solving physics problems with understanding. Recent work
in physics education research has led to several ideas for alternative
problem structures, or alternative ways to approach traditional problems;
these have been shown to be more productive for focusing students'
attention on the conceptual basis of problem solving.
Alternative Approaches to Problems and Problem Solving
-Two proposals for using traditional textbook problems in different
ways emphasize the qualitative and conceptual aspects of the solution
process. Van Heuvelen's (1991) technique, called "multiple representation
problem solving," is a direct application of Larkin's sequence
of representations. Students encounter a traditional word problem with
a goal of finding a specific numerical value. However, the problem
is at the top of a page which also contains, in sequence, a region
to draw an everyday sketch of the situation, a region to draw a physics
sketch (a free-body diagram, an electric field map, a lens diagram,
etc.), a region to write the relevant equation, a region to work out
the answer, and finally a region to make comments about what they learned
from solving the problem.
An important part of this approach is that students are explicitly
told that they must have all sections of the page filled in order to
get full credit for solving the problem. One way to enforce this is
to grade the problems by starting with the first representation in
the sequence and as soon as anything is missing, the grading process
stops. In other words, a student who simply writes down the answer,
or the correct equation and the answer, would get a zero for the problem,
since the earlier qualitative representations would not have been found.
This may seem like an unfair procedure which would penalize some students
who actually had the correct answer, but the whole idea is to shift
the focus from the answer to the process, and if students are alerted
at the outset there should be no problems.
A second way to use traditional problems, described by Leonard et
al. (1996), requires students to provide a qualitative strategy for
solving a problem. The strategies contain three components: (1) identification
of the appropriate concepts, principles, and relations that apply to
the problem; (2) a reasonable and appropriate explanation of why
they apply; and (3) a description of how they apply. Students
in the section where the strategy writing was employed were found to
be better at problem classification tasks and at recall of major ideas
from the course when tested several months after the end of the course.
Other approaches proposed for problem solving employ different problem
structures from the traditional items. In one approach D'Alessandris
(1995) has developed problems which do not ask the students to find
any specific numerical value. Instead the students must thoroughly
analyze the given situation, determine exactly what is happening and
essentially find all major values associated with the situation. With
the focus of finding a specific numerical value removed, the students
cannot simply look for an equation into which to plug numbers. Before
deciding what values to find, and what equations to use, the students
must figure out what is happening in the situation and what physically
important quantities are relevant.
D'Alessandris has developed this alternative format as a part of
an entirely different way to run the introductory course. The problem
format is an integral part of his "Spiral Physics" approach.
However, the idea of modifying the format of the problems to make students
thoroughly analyze physical situations is certainly one that can be
adopted by other instructors. There are actually several ways to produce
problems of this type. One way is to present a traditional problem
without the identification of a specific value to find and instead
ask the students, "What can you assert about this situation?" They
then have to determine, and calculate, all of the major quantities
they can from the given values.
-Research in physics education has shown that having students solve
traditional textbook problems is of limited usefulness in helping them
learn the concepts, principles and relations. One possible explanation
for why these problems are less productive than expected is that the
students' use of means-ends analysis in trying to solve the problems
leads them to plug and chug procedures which ignore the qualitative
analysis that is involved in problem solving with understanding. In
employing plug and chug approaches the students do not work with the
other representations, such as physics diagrams, which is where the
links are to the appropriate conceptual knowledge. Modifying how traditional
problems are done, or modifying the problem format has been shown,
in certain cases, can produce better outcomes for conceptual understanding.
A fuller review of the physics education research relating to problem
solving can be found in Maloney (1994).
D'Alessandris, P., "Assessment of a Research-Based Introductory
Physics Curriculum" AAPT Announcer 25 (4), 77 (1995)
Larkin, J.H., McDermott, J., Simon, D.P., and Simon, H.A. "Expert
and Novice Performance in Solving Physics Problems" Science 208,
1335-1342 (1980); also, "Models of Competence in Solving Physics
Problems," Cognitive Science 4, 317-345 (1980).
Leonard, W.J., Dufresne, R.J. and Mestre, J.P. "Using Qualitative
Problem-solving Strategies to Highlight the Role of Conceptual Knowledge
in Solving Problems," Am. J. Phys. 64, 1495-1503 (1996)
Maloney, D.P., "Research on Problem Solving: Physics" in
Handbook of Research on Science Teaching and Learning D. Gabel (Ed.),
MacMillan Publishing Co., New York (1994).
McDermott, L. C., "How We Teach and How students Learn-A Mismatch?" Am.
J. Phys. 61, 295-298 (1993).
Reif, F., Larkin, J.H. and G.C. Brackett "Teaching General Learning
and Problem-Solving Skills" Am. J. Phys. 44, 212-217 (1976)
Sweller, J., Mawer, R.F. and M.R. Ward "Development of Expertise
in Mathematical Problem Solving,"J.Exp. Psych.: General 112, 639-661
Sweller, J. "Cognitive Load During Problem Solving: Effects
on Learning" Cognitive Science 12, 251-285 (1988)
Van Heuvelen, A. "Overview, Case Study Physics" American
J. Phys., 59, 898-906 (1991)
Ward, M. and J. Sweller "Structuring Effective Worked Examples" Cognition
and Instruction 7, 1-39 (1990)
David Maloney is professor of physics at Indiana University-Purdue
University, Fort Wayne.