Student Problem Solving
Alan Van Heuvelen
A recent survey (Blake, 1995) by the American Institute of Physics
(AIP) asked former physics majors who are now in the workplace to identify
the most important skills needed for their work. Solving problems was
rated first, followed closely by the interpersonal skills needed to
work effectively in groups, and by technical writing. The 864 respondents
judged physics knowledge as low in importance in the workplace.
This outcome-the lack of importance of physics knowledge and the
considerable importance of problem solving-may be a blessing in disguise
for our physics education. We can select a reduced content of the most
important principles-a less is more philosophy endorsed by the Introductory
University Physics Project. Students can be given more responsibility
for acquiring that knowledge-a strategy that enhances learning. Time
made available by this reduced content can be used to help students
develop the skills needed to address more complex problems.
The ability to work effectively in groups was also judged very important
in the AIP survey. Eighty percent of our former physics majors work
in a group or supervise a group. Fortunately, promoting group work
in education is a win-win situation. Johnson, et al. (1981) analyzed
student achievement in 51 high-quality studies comparing cooperative
learning to so-called competitive lecture-based learning. They found
that the cooperative groups on average scored 0.81 standard deviations
(almost one grade point) higher than the lecture-based groups.
What about the recent emphasis on conceptual understanding in our
college courses? This is yet another win-win situation. Research indicates
that good problem solving starts with a strong conceptual foundation.
Hake (1997) reports a strong correlation between student scores on
a conceptual test and on a problem-solving test. Researchers at the
University of Massachusetts have found that problem-solving performance
improves when students use a hierarchically structured conceptual analysis
strategy (Dufresne, et al., 1992) and when they integrate qualitative
strategies into their problem solving (Leonard, et al., 1996). Ron
Gautreau at New Jersey Institute of Technology and I have found that
student problem-solving scores in university physics courses improve
when concepts are introduced and used qualitatively before their use
in mathematical form.
There are important reasons why a strong conceptual foundation is
correlated with the ability to use with understanding the principles
of physics in their mathematical form. Cognitive research indicates
that the mind is essentially a symbol-processing device. The symbols
in our minds are not mathematical symbols but are some special brain
descriptions in a sort of internal brain language. A person makes sense
of spoken language, written language, and the math symbols in an equation
by a dynamic interplay between internal imagery and these external
representations. If the external representations have no links to a
person's internal imagery, then the person cannot construct meaning
for the external representations. The symbolic language of physics
is very abstract. For the symbols to make sense, they must elicit internal
mental images that give meaning to the symbols.
To address this difficulty, we can integrate the mathematical descriptions
of physical processes with qualitative descriptions that students learn
while building their qualitative foundation-a multiple representation
strategy. These representations provide links between the abstract
math and the more qualitative picture-like and diagrammatic descriptions.
Technology that includes intuitive diagrammatic and picture-like representations
show promise in helping students visualize the quantities and concepts
of physics. Familiar context also helps relate the physics to imagery
in students' minds.
As students develop better understanding, they can be asked to "read" an
equation and then invent a process that is consistent with the equation-I
call these "Jeopardy" problems. Their description can involve
words, pictures, or some other more intuitive representation. In the
example below, students are to invent a process represented by the
(100 kg)(9.8 m/s2)(50 m sin 37o) = («)k(50 m)2
Finally, having developed a better qualitative understanding and
facility with the mathematical language, the student is ready for more
complex multipart problems. To solve these latter problems, students
learn to add definition to poorly-defined problems, divide complex
problems in parts, access the appropriate knowledge to solve each part,
choose quantities whose values must be determined in order to solve
the problem, make rough estimates in order to supply missing information,
interpret data tables and their graphs, and justify approximations.
The problems can involve experimental apparatus, so-called "experiment
problems" (Van Heuvelen, 1995), and context-rich problems (Heller,
et al. 1992).
Does such a system improve learning? There is considerable evidence
that strategies such as those described here enhance students' abilities
to reason effectively about physical processes without using mathematics
and to apply the symbolic language of physics with better understanding.
There is also growing evidence that the strategies also enhance students'
abilities to analyze and solve more complex problems.
Blake, G. (1995), "Skills used in the workplace: What every
physics student (and professor) should know," American Institute
of Physics, College Park, MD.
Dufresne, R., W. J. Gerace, P.T.Hardiman, and J.P.Mestre (1992),
J. Learning Sciences 2, 307-331.
Heller, P., R. Keith, and S. Anderson (1992), Am. J. Phys. 60, 627-636.
Johnson, D. W., G. Maruyama, R. T. Johnson, D. Nelson, and L. Skon
(1981), Psychological Bull. 89, 429-445.
Leonard, W. J., R.J.Dufresne, and J. P. Mestre (1996), Am. J. Phys.64,
Van Heuvelen, A. (1995), Phys. Teach. 33, 176
Instructions for about 30 experiment problems can be accessed from http://physics.www.ohio-state.edu/~physedu/expros/index.cfm].
Alan Van Heuvelen is professor of physics at Ohio State University,
where he does research on physics education. He conducts workshops
on physics problem solving.