by Annie and John Selden
Expert-Novice Studies Have Often Concentrated on Textbook Problems
Solving Nonroutine Problems
Are Mathematicians Expert Problem Solvers?
How Might One Teach Problem Solving?
The Importance of Affect
The answer to this question depends both on how you define problem and on how you define expert. Problems can be routine textbook exercises or they can be difficult mathematical tasks which take weeks, months, or even years to solve. An expert may mean someone who knows the domain thoroughly and can solve problems in a nearly automatic manner, or it can mean someone who can think of things to do even when no clear solution method suggests itself, marshaling strategies, heuristics, analogies, alternative representations, etc. While both types of "experts" often possess an extensive content knowledge base, the latter are more successful at solving nonroutine problems.
To get a handle on expertise, cognitive psychologists, who want to understand it, and knowledge engineers, who want to use it in AI programs, have extensively examined both general problem-solving heuristics and expertise in particular, often narrow, domains. They have designed artificial intelligence programs which duplicate, and sometimes even exceed, human expertise. Two early examples from the 70's are Newell and Simon's General Problem Solver and MYCIN, a rule-based deduction system for diagnosing bacterial infections. [Cf. Newell & Simon, Human Problem Solving, 1972; Winston, Artificial Intelligence, 3rd ed., 1992, pp.130-132.]
Such attempts required consideration of questions like: Of what does expertise consist? What do experts know that novices don't?
Cognitive psychologists who conducted expert-novice studies in the 80's often employed textbook mathematics or physics problems with beginning students as their novices and teachers or graduate students as their experts. They identified productive behaviors of content experts during relatively routine problem solving and suggested implications for instruction.
For example, Chi, et al, found that after a single course in mechanics, students categorized textbook problems as similar, based on kinds of objects -- pulleys, levers, etc. -- whereas advanced graduate students used principles such as conservation of energy [Cognitive Science 5 (1981), 121-152]. Taking their cue from the artificial intelligence work of the day, Reif and Heller delineated the (often tacit) knowledge needed to describe and solve certain mechanics problems, devised a prescriptive solution procedure, and guided introductory physics students through it step-by-step to get good performance [Educational Psychologist 17 (1982), 102-127]. Implications for instruction included making tacit processes explicit, getting students to talk about processes, providing guided practice, ensuring that the component procedures are well learned, and emphasizing qualitative understanding. [Cf. Heller and Hungate, "Implications for mathematics instruction of research in scientific problem solving," in E. A. Silver (Ed.), Teaching and Learning Mathematical Problem Solving: Multiple research perspectives, Erlbaum, 1985.]
The expert-novice literature suggests that: Experts (1) have a better memory for relevant problem details, (2) classify problem types according to their underlying principles, rather than their surface structure, (3) work forward towards a goal, rather than backwards from it, and (4) use well-established procedures or rule automation. The first three of these can be viewed in terms of schemas, which suggest the category to which a problem might belong, as well as appropriate solutions strategies. Both schemas and rule automation reduce memory load, allowing an expert to handle familiar aspects of a problem routinely, while freeing cognitive capacity for novel aspects of a problem. [Cf. Owen and Sweller, JRME 20, 322-328. Also, our brief discussion of schemas in Research Sampler No. 2].
While these are interesting results, most mathematicians are not primarily concerned with teaching students only a narrow range of "set piece" exercises, but rather with encouraging greater flexibility and the kind of nonroutine problem solving discussed by PÃ³lya in his famous books How to Solve It (1945/1957), Mathematics and Plausible Reasoning (1954), and Mathematical Discovery (1962, 1965/1981).
While PÃ³lya's heuristic strategies, such as exploiting analogies, decomposing and recombining, induction, specialization, variation, and working backwards, ring true to mathematicians, they are descriptive, rather than prescriptive. Mathematicians recognize them, but they appear not to be detailed enough to allow those not already familiar with them to often employ them. Indeed, when Alan Schoenfeld came upon them as a young mathematician in 1974, he was initially excited, only to be disappointed when repeatedly informed by mathematics faculty who coached students for the Putnam Exam that they were of little or no use. He set out to discover why PÃ³lya's widely admired strategies didn't work for most students, looking first to cognitive science and artificial intelligence, e.g., Newell and Simon's General Problem Solver. He discovered, as have others, that general heuristics like means-ends analysis or backward chaining, while good for solving general logic problems such as the missionaries-and-cannibals problem, are almost useless for problems in content rich domains like mathematics. He then began a progression of ever deeper observations of student problem solving using video-tapes of paired problem solving and interviews, the results of which are detailed in his book, Mathematical Problem Solving (Academic Press, 1985). [A short summary of this intellectual journey together with a concise introduction to his findings can be found in "Confessions of an Accidental Theorist," For the Learning of Mathematics 8, 1.]
Very briefly summarized, Schoenfeld analyzed problem-solving in terms of cognitive resources (one's basic knowledge of mathematical facts and procedures), heuristics (strategies and techniques one has), control or metacognition (how one uses what one knows), and beliefs or weltanschauung (e.g., all math problems can be solved in ten minutes or less if one understands the material). In contrast to successful nonroutine problem solvers, he found college students rush to an answer, use known procedures uncritically, believe there must be a formula for every problem, and go on mathematical "wild goose chases."
While no one doubts the usefulness of an extensive knowledge base of mathematical facts and procedures -- a good mathematical "tool kit" -- in addition, good problem solvers actively monitor their progress, deciding which solution paths to explore or not explore, whether to abandon, pursue, or change approaches/strategies.
Using Schoenfeld's framework, Thomas DeFranco decided to study the problem-solving behavior of mathematicians. An initial pilot study with six male and two female Ph.D. mathematicians proved disappointing when many neither solved the problems posed nor exhibited the kinds of metacognitive behavior Schoenfeld had attributed to problem-solving experts. However, at Schoenfeld's suggestion DeFranco repeated the study -- this time with two groups of male mathematicians (to control for possible gender differences). Group A consisted of eight mathematicians who had achieved national or international recognition in the mathematics community -- they had published 836 articles, received twelve honorary degrees, as well as numerous prizes and medals , and included presidents and vice-presidents of AMS and MAA. Those in Group B were no "slouches" -- eight Ph.D. professional mathematicians, who had not been accorded such honors or held such elected professional positions, but had published a total of 132 articles.
The mathematicians participating in the study were observed and audio-taped as they solved four ill-structured problems, taking as much time as needed. One of the problems was: Prove the following proposition: If a side of a triangle is less than the average of the other two sides, then the opposite angle is less than the average of the other two angles. Of the 32 problem attempts by each group, Group A solved 29, whereas Group B solved just 7. In some cases, mathematicians in both groups, did not recall the necessary mathematical facts, such as the law of cosines, needed to solve a problem in a particular way -- this hindered the performance of Group B mathematicians, but not those of Group A. In 22 (of 32) attempts, Group A mathematicians made better control decisions, sometimes navigating the solution space in meandering but meaningful ways, to arrive at a solution, whereas in 17 (of 32) attempts, Group B mathematicians, while able to avoid the kind of "wild goose chases" students often exhibit, did not exploit their resources well. In general, Group A's control decisions exerted a positive influence on problem solution, whereas Group B's were considered netural. Although both groups of mathematicians were content experts, DeFranco concludes that only those in Group A were problem-solving experts. He asserts, "It is apparent that university mathematics departments train students in subject matter but not in problem solving skills. To the extent that solving problems is important . . . the mathematics community needs to rethink the culture in which students are trained to be mathematicians." [Research In Collegiate Mathematics Education, II, 1996, p. 209].
While there are probably a number of ways, very few have been documented in the literature. Even Alan Schoenfeld who has been teaching a problem-solving course for years, has only recently had his classes video-taped for analysis, with the first papers about to appear in the CBMS volume, Research in Collegiate Mathematics Education, III. One of these deals with the first two days of class from four different perspectives. Because students are used to listening, taking notes, and learning procedures to solve standard problems, it is crucial that the teacher renegotiate the "didactic contract" in order to set up a "mathematical community" in which students propose and evaluate conjectures for themselves using solid mathematical reasons. For Schoenfeld, this involves a variety of traditional and non-traditional techniques with the teacher firmly in control -- he sets the top-level goals, selects the initial problems, directs students' work, and models desirable mathematical actions and dispositions. He selects and structures his problems carefully and knows which solution strategies students tend to proffer and where these will lead. He is thus able to pursue the students' suggestions while furthering his own goal of teaching heuristics.
For example, his second problem on the first day was: A friend of mine claims that he can inscribe a square in the triangle -- that is, that he can find a construction that results in a square, all of whose corners lie on the sides of the triangle. Is there such a construction -- or might it be impossible? Do you know for certain that there's an inscribed square? Do you know for certain there's a construction that will produce it? Schoenfeld used this problem to introduce two of PÃ³lya's heuristics -- find an easier, related problem, and if there is a special condition, relax it and look for the solution in the resulting family of solutions. After working in groups using the first heuristic, students suggested trying an inscribed rectangle or a circle instead of a square, as well as looking for a counterexample, a suggestion that was put on hold, while Schoenfeld evaluated the other suggestions with the class and pointed out the difficulties of employing this heuristic. He then introduced the "relax a condition" heuristic, asking what would be easier to inscribe than a square. A student quickly suggested a rectangle and it was noted, using a continuity argument, that between inscribed "short and fat" and "tall and skinny" rectangles, there would be a square. While this provided an existence proof, it did not provide a constructive one, the discussion of which was deferred to the next class. While more went on in the class, the above gives an indication of Schoenfeld's approach.
Teaching heuristics is not easy -- one must introduce them quickly so students can appreciate their power, yet slowly enough so students learn to apply them over a wide range of problems. There is the specificity problem, that is, heuristics such as "find an easier problem" are too general to be useful, but presenting specific versions of each strategy would make the list of useful heuristics too long and cumbersome to teach and learn. So generality of strategies and their attendant vagueness must be retained. There is the implementation problem, that is, even if a student selects a potentially productive strategy, this can be undermined by mistakes at any step. There is the resource problem, that is, even if a student selects a workable strategy, failure to recall the necessary mathematical concepts or procedures can cause it to fail. [Cf. Arcavi, Kessel, Meira & Smith, "Teaching mathematical problem solving: An analysis of an emergent classroom community," RCME, III, to appear.]
Whereas Schoenfeld investigated beliefs like "all math problems can be solved in ten minutes or less," others have considered affective factors such as attitudes and emotions. Emotions are regarded as the most intense and least stable, often disappearing when the "frustration of trying to solve a hard problem is followed by the joy of solution." Attitudes are seen as moderately intense, reasonably stable responses that develop through the automatization of repeated emotional reactions or through transfer of pre-existing attitudes to new, but related situations. Beliefs, which may be about mathematics or about oneself in relation to it, are viewed as mainly cognitive and develop comparatively slowly.
Experts and novices exhibit similar kinds of emotional reactions during problem-solving, but experts handle them better. In studying affect, Douglas McLeod has used Mandler's general theory of emotion, which indicates that physiological phenomena, such as an increase in heartbeat or muscle tension, often occur in response to the interruption of planned behavior. Such interruptions can be pleasant surprises, unpleasant irritations, or major catastrophes. McLeod, et al, found that, whatever their attitude towards mathematics generally, nonmajors reported experiencing similar up and down mood swings as they made, or did not make, progress in solving nonroutine problems. The authors suggest emotions are relatively independent of traditional attitude constructs. [McLeod, Craviotto & Ortega, Proceedings of 14th PME, 1(1990), 159-166.]
Recently, DeBellis and Goldin have considered the influence of values, i.e., one's psychological sense of what is right or justified, on problem solving. For example, some students may feel they "should" follow established procedures when tackling problems, whereas others may value originality and self-assertiveness. Good problem solvers exhibit productive responses to insufficient understanding, while others, not wanting to admit deficiencies in their mathematical knowledge, may feel they "should" know and this may lead them to guess or use plausible, but inappropriate, procedures.
DeBellis and Goldin, view beliefs, attitudes, emotions, and values -- and their interplay with cognition -- as fundamental to problem solving; these provide information that can facilitate or hinder monitoring. Emotions are often fleeting, whereas the others are relatively stable, self-regulating structures of an individual. Affective pathways (i.e., established sequences of states of feeling) can be positive or negative. For example, if a positive pathway is invoked at the onset of problem solving, curiosity may motivate the solver to better understand the problem and lead to exploratory heuristics; frustration at a subsequent impasse can cause a revision of strategies. If a negative pathway is invoked , bewilderment can led to a search for "safe" procedures, rather than exploration; when these fail, frustration may lead to anxiety and reliance on authority or avoidance. [Proceedings of 21st PME, 2(1997), 209-216.]
These results on affect are just a beginning. More studies are needed on how cognition and affect interact during problem solving, as well as on how teachers might engender and harness positive affect.