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Characteristics of Problem Solvers
Driscoll (1982) gives us a list of attributes that are characteristics of good problem solvers versus poor problem solvers. His abbreviated table (p.71) follows:
Successful Problem Solvers
Unsuccessful Problem Solvers
Do more re—reading,
rechecking, and
reviewing.1. Proceed on the basis
rechecking, and of a few clues.
reviewing.2. Are able to pull key
ideas from a problem,
to distinguish between
relevant and irrele—
vant information to
bear on a problem.2. Often know what is
ideas from a problem, relevant, but even
to distinguish between when they do, they do
relevant and irrele— not bring the infor-
vant information to mation to bear on
bear on a problem. solving problems.3. Exhibit goal-oriented planning--that is, they as well.
identify a solution and
a plan of attack.3. Do not, as often or as well. 4. Perceive the mathematical structure of problems.
4. Tend to focus on question form or content. 5. Can generalize across problems, seeing mathematical trends. 5. Cannot as well.
With the work of Dodson (1970), we can also add the following characteristics of “insightful” problem solvers. They scored higher on tests of verbal and general reasoning #40 than the less successful problem solvers. They were good in determining spatial relationships. They were also able to resist distraction and disregard irrelevant elements in a problem statement. They were, in general, unconcerned about “messiness” or “neatness” in their work. Lastly, they had a teacher whose academic work was beyond a bachelor’s degree, frequently with graduate degrees.
Successful problem solvers presumably use information, procedures, and more general notions that have been acquired via past experiences and training. Krutetskii (1976) suggests that good problem solvers exhibit qualitative differences in their recall of information from previously encountered problems. Also, good problem solvers apparently tend to recall the structural features of a problem, where as poor problem solvers tend to recall, if anything, the specific details of a problem statement.
Silver (1981) showed that poor problem solvers tended to remember the questions asked in a problem and also the context of the problem statement. In recalling structural information in a problem that they failed to solve, poor problem solvers did not notice the structure before solving the problem, and their unsuccessful attempts at a solution probably revealed very little about the problem structure. In essence, the results of the study suggest that good and poor problem solvers also differ with respect to the degree to which they can remember and utilize structural information obtained from a presented problem solution, suggesting that there exist important individual differences in recall behavior which are intimately related to problem solving performance.
Kulm and Days (1979) claim, in reference to algebraic problems, that the solution set-up is the most difficult of the problem solving stages for both good and poor problem solvers. Also, students learned more by solving a more general problem or a similar problem than by solving a simpler case or an equivalent problem. Muth (1986) further addressed the issue of students scoring lower on problems containing extraneous information. Their results suggest that math anxiety is often experienced when students are given problems with extraneous information. The implication is that teachers should systematically include extraneous information in word problems in an effort to help students overcome this handicap.
Obviously, many of the above characteristics are ones generally considered teachable. But is problem solving itself a teachable skill? Schoenfield (1982) proposed that, in addition to a knowledge of basic facts, a successful problem solver must possess a mastery of basic problem solving techniques and a managerial strategy. These help the individual select appropriate solutions and terminate the fruitless ones. The results of this study support the ideas that students can learn to use a general problem solving strategy when presented with it and that the learning of problem solving heuristics and managerial strategies can help students improve their problem solving skills. Schoenfield also found that when students do learn a problem solving heuristic, there is a transfer to problems unrelated to those used in the instruction and the student’s subjective assessment of their problem solving performance is improved. Lee (1978), who instructed students on a Polya-like problem solving method, concluded that the students became better problem solvers as a result of being taught the heuristic.
The role of the teacher in modeling problem solving skills can be supported by the findings of several experiments. First, Lester (1980)demonstrated that problem solving instruction is most effective when: (1) the teacher regards problem solving as an important event, (2) the teacher actually engages in problem solving as a part of mathematics instruction, and (3) the teacher values each student as a problem solver. Schoen (1980) showed that the teacher’s attitude toward problem solving was related positively to the problem solving ability of that teacher’s students. In other words, students need to see the teacher modeling appropriate behaviors and to sense in their teachers appropriate attitudes about problem solving. Closely associated with a teacher’s modeling of desired behaviors is the student’s acquisition of attitudes towards problem solving, attitudes which may or may not be reinforced by the teacher.
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