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Teach Talk
The first of three Teach Talk columns to focus on the implications of research into learning for actual classroom practice
It's a common refrain that came up again recently during a conversation among several faculty members after a seminar on new educational technologies. The discussion had winded its way around to the intellectual strengths and weaknesses of the students, and the question popped up, as it often does around this subject: Why can't students be better problem solvers? Professor Heidi Nepf from Civil and Environmental Engineering summed up the faculty's sense of frustration particularly well. "I can give my students a set of problems that all follow a certain model, and they'll do fine," she said. "The minute I throw in a novel condition or create a problem that doesn't look like something they've seen before, they're lost." Then she turned to me and asked, "How come?"
I don't think anyone would argue that the problem is a complex one. It is connected to such factors as the kind of high school education our students received, their own proclivities, and their stage of intellectual maturity. But I'd like to suggest that at least part of the answer lies in the fact that too often we don't explicitly teach students the process of problem solving. We expect that as they listen to us in lecture or watch us in recitation they will somehow absorb the skills they need to make the jump from using "plug 'n' chug" to employing more sophisticated problem solving strategies. But as Donald Woods, professor emeritus of chemical engineering at McMaster University and a leading developer of problem-based learning curricula, writes, "In a four-year engineering program, students observed professors working more than 1,000 sample problems on the board, solved more than 3,000 assignments for homework, worked problems on the board themselves, and observed faculty demonstrate the process of creating an acceptable internal representation about fifteen times. Yet despite all this activity, they showed negligible improvement in problem-solving skills . . .." (Donald Woods, "How Might I Teach Problem Solving," in J. E. Stice, ed., Developing Critical Thinking and Problem-Solving Abilities. New Directions for Teaching and Learning, no. 30, 1987, pp.58-59) Yet I don't think instructors should be blamed: My guess is that if a representative sample of MIT faculty were asked to describe how they go about solving problems, they wouldn't be able to. In that regard, they wouldn't be any different from most experts who have so internalized their problem solving abilities that these skills have become transparent to them.
Happily, thanks to the work of cognitive psychologists, educators, and researchers in artificial intelligence, who have been studying problem solving for at least the last 30 years, we do know something about how skilled problem solvers recognize, approach, and ultimately solve problems. Much of this research has revolved around examining what distinguishes expert problem solvers from novices. Educators have then gone a step further to develop methods that can be used both inside and outside of the classroom to strengthen the novice's problem solving skills.
In this Teach Talk I'd like to focus on the expert/novice dichotomy, because I believe it contains an especially rich lode of information regarding the skills our students need to develop. In fact, this column is the first of three Teach Talks that will be devoted to describing recent research in learning in higher education. (The next two columns will deal with the theories of constructivism and situated learning.) Each column is designed to inform readers on how this research can be applied to improving actual classroom practice, for this knowledge has direct implications for structuring the MIT educational experience.
The Components of Problem Solving
The most useful definition I have found for problem solving begins by conceptualizing a continuum that runs from "learning" to "problem solving" to "creativity." In this schema, learning refers to the students' ability to demonstrate they have internalized the material to which they have been exposed by displaying it in a context similar to that in which they were taught. "Transfer of learning" is demonstrated when the situation is somewhat different from the original one. If, however, the transfer situation is substantially different from the original, or if students meet some barrier or difficulty in using the learning, then they are faced with problem solving. (This is the situation to which Professor Nepf referred.) Creativity is at the far end of the continuum where the situation is so vastly different that what has been learned is transferred to a totally new context.
Several scholars, including Donald Woods, have sought to break down the process of problem solving into its component parts. Woods' six-step plan, which he credits as an extension of the plan devised by György Polya in his classic book How to Solve It, directs problem solvers to: read about the situation; define the given situation or problem; define the "real" problem and create a "representation" of it (more on this below); plan; do it; and check, look back, and implement. Woods further decomposes each step into smaller parts. For example, "defining the situation" (step two) is rooted in analysis, which consists of reasoning, classifying, identifying series and/or relationships, creating analogies, and checking for consistency. While there may be disagreement about the exact nature or order of the steps in the problem solving process, the underlying point remains valid: Problem solving can be dissected into a set of skills that students can be exposed to along with course content. One cannot substitute for another. (Interestingly, attempts to teach problem solving as a separate course have not been as successful as when problem-solving skills are interwoven into a "content" course. Giving students problems from the "real world" and using those problems as the basis for teaching problem solving is particularly effective. In fact, Woods maintains that the types of problems students are typically given in science and engineering classes are not appropriate at all for teaching problem-solving skills.)
Finally, while we are likely to think of problem solving as a cognitive capability, a number of researchers have also looked at the role of attitudes, values, beliefs, and emotions in successful problem solving. (Actually, the research of neurologist Antonio Damasio suggests that emotion and cognition should not be viewed as separate activities in the brain at all; rather, they work in concert.) We know, for example, that if students believe they are incapable of solving a certain kind of problem, they are likely to be unable to do it. De Bellis and Goldin have examined the "influence of values, i.e., one's psychological sense of what is right or justified, on problem solving," report Annie and John Selden in "What Does It Take to Be an Expert Problem Solver?" The Seldens go on to write, "For example, some students may feel they 'should' follow established procedures, whereas others may value originality and self-assertiveness." (MAA Online, 8/30/97, p. 4) Other students who feel they should know the answer to a problem may become easily frustrated, which can "lead them to guess or use plausible, but inappropriate, procedures," the Seldens write. (MAA Online, 8/30/97, p. 4)
Good problem solvers are more often than not intrinsically motivated by curiosity, challenge, and fantasy. (Joanne Gainen Kurfiss, "Critical Thinking: Theory, Research, Practice, and Possibilities," ASHE-ERIC Higher Education Report No. 2, 1988, p. 47) Good problem solvers are not daunted by the unknown, but are challenged by it. They may experience frustration in their work, but it doesn't defeat them; instead, it spurs them on. What else differentiates the experts from the novices?
What Do the Experts Do?
There are a number of characteristics that differentiate the expert from the novice problem solver. But at the heart of the matter is that experts think about, consider, and examine the problem as a whole before beginning to work on a solution. They classify a problem according to its underlying principles, deciding to what class of problem it belongs. They engage in a planning stage before even attempting a solution. Novices jump right in.
In a classic 1978 study comparing individuals who were expert at solving problems in physics with novices, Simon and Simon found that experts use a "working forward" method, looking at the givens of the problem first and moving from the statement of the problem to a physical representation of it. Only after they do this analysis, identifying likely ways to reach an answer, do they employ equations. Then they call upon successive layers of equations, first using ones that can be solved with the givens in the problem. They also add information that will help them solve the problem from their own reservoir of learning. The experts' use of equations, in other words, is guided "by the planning already done." (D. P. Simon and H.A. Simon, "Individual Differences in Solving Physics Problems," in R. S. Siegler, ed., Children's Thinking: What Develops? 1978, as reported in Larkin, Heller, and Greeno, "Instructional Implications of Research on Problem Solving," New Directions for Teaching and Learning, 2, 1980, pp. 55-57)
Novices, on the other hand, use a "working backward" strategy trying to determine what procedure will get them to an answer. They tend to take more "piecemeal approaches" (Larkin, Heller, and Greeno, p. 59), working by trial and error. They memorize, then try to apply equations independent of context or any relationship to the inherent characteristics of the problem. Especially problematic is that they try to translate the problem directly into a mathematical representation, using a means-ends analysis. Or as one writer characterized it "[they] . . . select a 'first impression solution.'" "In effect," write Larkin, Heller, and Greeno, "experts understand problem situations better than novices." (p. 59)
The good news is that when studies compared successful students with those having difficulty solving problems, the former looked much like the effective problem solvers of the Simon and Simon study. Successful students are able to apply specific pieces of knowledge to help answer the problem. Unsuccessful students can't relate what they have learned to the question if the question is asked in a form that is different from the one they have seen. (Greenfield, p. 15) Successful students work more actively; unsuccessful students more passively. Successful students are careful and systematic. Unsuccessful students leap into a problem with at best a haphazard plan, move without direction, and are unable to focus on any particular starting point. Their knowledge base has no hierarchical organization to it, and they are easily distracted by some difficulty or something irrelevant. On the other hand, like their professional counterparts, successful students begin with a plan, modifying it as needed. They carefully develop and organize their knowledge base, structuring it around fundamental principles and abstractions. (Greenfield, p. 15)
If we accept the premise that good problem solvers are made and not born (allowing, of course, for differences in innate capabilities), and that we have a responsibility to instruct in this area as well as in content, the simple question is, how? In other words, what are the implications of this research for what happens in our classrooms?
Teaching Problem Solving
I'd like to reiterate what I wrote earlier: The process of problem solving has to be taught explicitly if we want to raise the general level of students' problem-solving abilities. Although many students will eventually internalize the habits of good problem solving, this can occur earlier for more students if the necessary skills are described, modeled, and practiced, and if the instructor provides students with feedback on their behavior. As with many skills, learning happens when a discussion of best practices are combined with opportunities for learners to try their hands at the skill, and are told both what they are doing correctly and how to improve.
Greenfield suggests six things instructors can do to teach problem solving. They should:
The author also suggests using the "think aloud" process first developed by Jack Lochhead and Arthur Wimbey in the early 1980s. In this instructional method, two students work together to solve a series of short problems. One student becomes the problem solver, and he/she reports out loud everything that is going on in his/her head as he/she attacks the problem. The other student is the listener whose "primary objective," write Lochhead and Wimbey, "is to understand in detail every step and every diversion or error made by the problem solver." The listener can also use a checklist that the authors have developed to help him/her notice errors in the problem solver's reasoning process. ("Teaching Analytical Reasoning through Thinking Aloud Pair Problem Solving," in Stice, p.75) After the first student solves his/her problem, the two students switch roles and work on another problem. There are obviously a number of benefits to this method: students call direct attention to the process they are using and reflect on it; the process is monitored and can be called into question by another; and students practice working with others as they will be doing in the professional world.
Some educators say that what is needed is a "cognitive apprenticeship" approach to instruction. The elements of such a pedagogical method would consist of modeling, coaching, scaffolding (i.e., providing expert guidance at the beginning of the process and then removing it), articulating, reflecting, and exploring. (Kurfiss, p. 45) This is a very different model from the one in which the instructor does the problem solving for the class, but doesn't reveal the "secrets" of his/her success. If we want students to be better problem solvers, we have to be like magicians who are willing to show our audience how we do our sleight of hand. If we want students to be better problem solvers, we need to be better teachers of the process for solving those problems.
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