 |
 |
||||
|
 |
||||
|
|
|
||||
|
|
||||
|
A Contemporary Perspective on the Psychology of Productive Thinking
By Michael Wertheimer, Ph.D.
University of Colorado at Boulder
- Dr. Wertheimer recently retired from the University of Colorado after a teaching career that spanned more than four decades. Dr. Wertheimer has spent the last few years on a biography of his father, the Gestalt psychologist, Max Wertheimer.
The book Productive Thinking by Gestalt psychologist Max Wertheimer, published just over a half century ago (two years after the author's death), has been translated into several foreign languages and has been reissued repeatedly in English; it continues to be cited repeatedly in the Social Science Citation Index. Why are people still reading it, still citing it?
Wertheimer introduced his book by asking, "What occurs when, now and then, thinking really works productively? What happens when, now and then, thinking forges ahead? What is really going on in such a process?"
His answer was that what characterizes productive thought is its fit with the situation to which it is applied. Productive thinking involves going from a state of confusion about some issue that is blind to the core structural features and properties of that issue, to a new state in which everything about the issue is clear, makes sense, and fits together. At the core of the process is a kind of reorganization or restructuring, going from a state that makes no sense to one that does make sense, displays insight, is crystal clear.
In his lectures on thinking, and in his book, Wertheimer used numerous concrete examples to illustrate his principles. They may help clarify his approach. Consider first a perceptual illustration of what he meant by "reorganization," "restructuring," "insight," "understanding."
What does the following mean?: Pas de l'y a Rhône que nous. This example comes from William James (1890). The French might be translated roughly as "Not of there is Rhône (a river) than we," which makes no sense at all. Try saying it out loud. Does that help? Try reading it with an American accent: "Pah de'l ya rown ke-new" -- or "Paddle your own canoe." The reorganization achieves a transition from meaninglessness to a new structure, in which the sequence of sounds symbolized by the letters now makes some sense.
A rebus almost cries out for reorganization.
..........................stood
What does this well mean?
.........................viewEDITOR'S NOTE: Please be aware the words "stood", "well", and "view" should be postitioned one above the other. Depending on the web browser you are using these words might not be positioned as such.
"Well" is under "stood," and both are over "view" -- which, slightly reorganized, becomes "well" under "stood" over "view," or well-understood overview. That's what you need to generate about any problem, to think about it productively.
Such "catching on" characterizes productive thinking and problem solving as well, whether in physics, geometry, or any other field, and Wertheimer analyzed dozens of concrete examples. Here's one instance: why is any sequence of three repeated digits (abc,abc or, efg,efg; 276,276 or 341,341, etc.) divisible without remainder by 13? The solution requires realizing that the factors of the number abc,abc are abc -- and 1001 (1001 times abc equals abc,abc), and that 1001 is divisible by 13 without remainder. Think it through!
A striking example of reorganization is an extension of a popular puzzle. A hunter sees a bear one mile due south of him. He aims his gun, shoots, and misses. The hunter next walks the one mile due south to where the bear was when he fired the shot, then walks one mile due east, then one mile due north -- and finds himself standing at exactly the same place he had been standing when he shot his gun. The usual version next asks, "What color was the bear?" For someone who has never heard this story, the question is astonishing. How could the information provided have anything to do with the color of the bear? To solve the problem, the query has to be reformulated into, "Where on the surface of the earth might it be possible to go successively one mile due south, then one mile due east, then one mile due north, and end up standing at the same place one started from?" Most readers will already know that the spot is the north pole. From there, you go one mile due south, then turn left 90 degrees and walk exactly one mile due east, then turn left again and go exactly one mile due north -- and end up standing on the north pole again. The spherical triangle the hunter traversed looks a bit different from a plane triangle (all three sides are curved and the sum of its interior angles is 270 rather than 180 degrees), but the north pole clearly satisfies the specified constraints. What can you conclude about the color of the bear? Of course: any bear in the arctic is apt to be a polar bear, so the color of the bear must be white.
But that is where the extension of this problem starts. Where else on the earth's surface, other than at the north pole, can one go one mile due south, then one mile due east, then one mile due north, and end up standing at the same place one started from? Perhaps you should stop here and ponder the puzzle for a while.
Such examples may help convey what Max Wertheimer meant by "reorganizing" or "restructuring." He argued that productive thinking requires an insightful revision of one's representation of the problem domain, to use more modern terminology. In summary, he proposed three broad generalizations about productive problem solving, all of which can be viewed as challenges to modern cognitive psychology, and all of which have been addressed by contemporary writers. First, productive thought involves transforming the representation of a problem from a vague, fuzzy, incomplete and confused one that is blind to essential structural features of the problem to one that is clear, has no gaps in it, makes sense, and views each part of the problem in terms of its place, role, and function within the problem as a whole.
Second, such transformations are (a) hampered by blind search, "functional fixedness," empty associations, "and-sums," conditioning, school drill, bias, and so on, and are (b) aided by open-minded exploration of the problem, searching for its essential, crucial features, and its "rho relations." By "functional fixedness" Wertheimer meant that if an object is seen as fulfilling a particularly useful function in one context, it makes is less likely that one will see that it could perform a different function as well in another context. An "and-sum" is a mere conglomeration of items that are arbitrarily connected, without regard to the attributes of those items or their meaningful relations to one another. The term "rho relation" was used by Wertheimer to indicate a feature that is crucial to the essence of a problem. For example if you are to build a toy bridge of wooden blocks, there is a rho relation between the distance separating the two uprights and the length of the horizontal member (it can't be shorter than the distance between the two uprights), as well as a rho relation between the heights of the two vertical blocks -- they must be at least roughly comparable if the bridge is to stand. But the color of the blocks bears no rho relation to whether the bridge will stand or not.
Third, this perspective generates several potentially productive areas for research: (a) laws governing segregation, grouping, centering, and structural transposability, (b) how relations between parts and their wholes govern the possible operations on parts that take into account the part's place, role and function within the whole of which it is a part, and (c) the nature of "outstanding wholes," "good Gestalten," indeed of "rho relations" themselves.
Wertheimer illustrated these observations with numerous examples, ranging from finding the area of a parallelogram to how Albert Einstein formulated the theory of relativity. To paraphrase Ericsson and his colleagues in their preface to the 1982 edition of Productive Thinking, the examples set a challenge for the modern cognitive psychologist -- indeed for any thoughtful human being. They contrast pure memory, or reproductive thinking, which can be accounted for reasonably well by the associationist paradigm that prevailed half a century ago (and by its modern counterpart, the connectionist approach to computer modeling) with productive thinking, or insight-based reasoning, which is not so easily handled by an associationist or connectionist strategy. Examples of productive problem solving and thinking compel consideration of complex mental structures and processes, typically ones that are idiosyncratic to a particular problem and do not generalize from one problem domain to another.
The advent of the computer a few decades ago generated what is now called the "cognitive revolution." The computer became the model for the human mind. Newell, Shaw and Simon (1958, 1962), Newell and Simon (1972), and Simon (1978) formalized what has become the prototype of the kinds of paradigms that have been taken for granted by cognitive psychologists ever since. Problem solving is conceived as goal-directed search among possible perceived solutions within a specified domain called the "problem space." Such a conception works well in simulations of the problem-solving efforts of novices who have little experience with attempting to solve novel problems, but cannot readily account for how experts like chess masters, physicists or designers, who have a thorough knowledge and an organized understanding of a domain, go about solving difficult problems in the area of their expertise. One consequence of this failure was the postulation by Kintsch (e.g. Kintsch and van Dijk, 1978) and others of complex abstract knowledge structures such as schemas, scripts, or frames to account for text comprehension and other complex cognitive processes. From this perspective, as Greeno (1977) put it, "insight" involves the discovery of the applicability of an existing general schema to a novel situation. But what processes generate genuinely productive thinking, that is, yield representations that can in fact be used successfully to solve a novel problem, remained -- and remains -- elusive. Blind schema-generalization cannot work; the restructuring and insight emphasized by Wertheimer are missing in computer models of cognitive processes. Ericsson and his co-authors in 1982 (pp. xv-xvi) concluded that while modern cognitive science has made some modest progress on several issues raised in the book Productive Thinking, "it has by no means solved all of them. All of Wertheimer's examples raise serious problems for an associationistic paradigm of mental processes. Today, the information-processing psychologist considers the solution of the issues raised by Wertheimer central to progress in [the] understanding of problem solving and productive thinking. Many of the examples so lucidly discussed by Wertheimer remain only partially understood and continue to represent significant challenges to cognitive scientists."
I have proposed (1985) that the inherently blind connections that make up a computer and a computer program can never achieve insight: understanding and meaning are in principle outside the capacity of any computer or computer program; to the extent that a program might be able to mimic or simulate productive thinking, the insight or understanding is not in the program or computer itself, but in the programmer.
Library research and suggestions of several colleagues yielded many recent publications that are clearly relevant to the issues raised in Wertheimer's book. People are still thinking about, writing about, and doing empirical work on these matters. Consider a brief sample of these publications. The question about all these items, I believe, should be whether recent developments demonstrate real progress on the central problem that Max Wertheimer addressed in his analyses of productive thinking: the crucial role of reorganization, of restructuring, of insight. An old friend, Ward Edwards of southern California, a long-time systems analyst, wrote me in another context that he believes that one should let computers do what they do well, the "intellectual" processes of evaluation, inference, and decision -- and let people do what they are good at, which are the tasks required to structure the problem in the first place and to provide inputs to those three processes. To repeat, computers and an information-processing model are, because they are inherently blind, excruciatingly literal, and incapable of processing meaning, in principle unable to simulate the most critical property of productive thinking, restructuring.
Holyoak and Spellman's chapter on thinking for the 1993 Annual Review of Psychology contrasts what they call the production-systems approach of Simon and his colleagues, which handles "well-defined" problems that have clear goals, a clear starting state, and obvious operators reasonably well, with the approach to less well-defined problems on which Gestalt psychologists like Max Wertheimer worked, that typically require "restructuring" of the problem representation if a solution is to be achieved. They write (p. 269) that "It is unlikely... that connectionism will undermine the traditional view that human thinking requires a symbol system," and (p. 273) give credit to Tweney (1990) for indicating that the complex interrelatedness of hypotheses provides a major challenge for computational theories of scientific reasoning.
Holyoak and Spellman point out that "A crucial question for theories of thinking concerns relevance," or what Wertheimer meant by rho relations. Yet another issue (p. 297) is transfer, the transfer of knowledge learned in one context to other related situations: "Essentially by definition, transfer is based on the perception that prior knowledge is relevant to the current context." How can a computer be programmed to make such metaphorical and analogical jumps? Another aspect of the relevance issue is stated as follows by Holyoak and Spellman (p. 302): "A crucial aspect of the general characterization of a representational system is that it involves specifying which aspects of the represented world are relevant." Once again: how do you program a computer so it will be able to recognize the difference between rho relations and trivial, superficial attributes of a problem?
Sternberg and Davidson's 1995 book, The Nature of Insight, is full of references to Productive Thinking, and Murray published a book in 1995 entitled Gestalt Psychology and the Cognitive Revolution. Murray claims, and documents in detail, that "the Gestalt psychologists... foreshadowed the cognitive revolution" (p. xi); he "emphasizes the value of the insights of Gestalt psychology for our understanding of cognitive psychology, and argues that we need to re-evaluate many of Gestalt psychology's ignored insights" (back cover).
A paper by Newell (1980) extolled the virtues of the concept of a problem space, arguing that people construct and improve such spaces as they gain experience in a problem domain, and that the problem-space idea (p. 715) "has strong implications for the transfer of skill.... If a [person] maps a task into an existing problem space, then the transfer of this knowledge to the new task is implied." But Newell does not address the critical issue of rho relations: how does one know into which (already-familiar) problem space to transfer a new problem? Studies by Metcalfe (e.g., 1986) and her colleagues provide an empirical, functional distinction between the processing of memory tasks and of problem-solving tasks, the same distinction that Wertheimer made between reproductive and productive thinking. While people are generally able to predict their future performance on reproductive memory tasks (p. 292), they cannot predict future performance on productive problems that require transformation of the problem representation for their solution. Kounios and Smith (1995) provide comparable findings, using a sophisticated method to study the time-course of partial information accumulation during the processing of anagram tasks, which require some degree of reorganization or insight. Both of these lines of research imply that it may be inherently impossible for the current continuity models of information processing to account for the all-or-none or discontinuity features of problem solving that requires a changed representation.
Winston, Chaffin and Herrmann (1987), in their taxonomy of part-whole relations, recognize that such relations are not limited to logical inclusion or class membership -- indeed there are many kinds of part-whole relationships, some relatively empty and some relatively rich and pregnant. Rho relations again? The authors do not mention them, nor do they refer to an appendix in the later editions of Productive Thinking in which Wertheimer distinguished at length between "arbitrary components" and "necessary parts."
Kaplan and Simon's 1990 paper, "In search of insight," refers extensively to the Gestalt literature and then reports empirical work on a classic mathematical puzzle, the mutilated-checkerboard problem. Attaining "insight," they write, requires discovering an effective problem representation, and the likelihood that such a representation will be discovered is related to the search for invariants, what they call the "notice invariants heuristic." Yet it is unlikely that such a heuristic could be generalized to other problems since it is specific to this particular problem -- and it also remains unclear how one should go about generating a good problem representation in the first place. What commands could one give a computer that would have this desired effect? Nobody knows yet how to program a computer so that it can be sensitive to rho relations.
Most of the chapters in Sternberg and Davidson's 1995 book on insight are directly relevant. Mayer's opening chapter, for instance, is on "The Search for Insight: Grappling with Gestalt Psychology's Unanswered Questions." Dominowski and Dallob, in "Insight and Problem Solving," deal with characteristics of problem solving, the difference between reproductive and productive thinking, the nature of insight, understanding, functional fixedness, and restructuring. Schooler et al.'s "epilogue," entitled "Insight in Perspective," touches on the definition of insight, the causes of impasses during the process of solving a problem, how impasses are overcome, coherence, and other crucial issues.
Two things remain. First, will the modern computer-based information-processing paradigm that is dominating cognitive psychology be able to deal adequately with the central issue of productive thinking? I won't belabor my answer to that question. In any event, people today are still reading and pondering Max Wertheimer's book, Productive Thinking. Its striking descriptions and analyses of insights are as fresh today as they were a half century ago, and pose a serious challenge to any blind or mechanical models of human thinking. No theory of cognition can afford to ignore that productive thought is often insightful, indeed sometimes exhilarating.
The other item concerns spherical triangles. Where on the surface of the earth, other than at the north pole, can you go one mile south, then one mile east, then one mile north, and end up at the spot you started from? The solution requires a major reorganization of the concept of a triangle. Start anywhere one mile north of a circle just north of and surrounding the south pole, that is exactly one mile in circumference. You go a mile south to that circle, go east around the circle, and then head north for one mile, exactly retracing, in the opposite direction, the route you had taken south. This spherical triangle doesn't look anything like a plane triangle. Further: the locus of points from which you could start is a circle, but that circle is not the only solution. The critical circle on which you go east need not be exactly one mile in circumference; any perfect fraction of a mile would work just as well. If it were, say, one third of a mile in circumference, you could walk one mile due south to the circle, then go east around it three times, then head back north for one mile, exactly retracing the route you had gone south. It's a fascinating problem; you might enjoy thinking about it some more.
References: Available upon request to Psychology Teacher Network.
The above article was originally published in the January/February 1997 issue of The Psychology Teacher Network. The article is reprinted here with the permission of the Education Directorate of the APA. Further publication of the article is not permitted without the express written consent of the Education Directorate.
For information on subscribing to The Psychology Teacher Network, write to:
- Psychology Teacher Network
Education Directorate
APA
750 First Street, NE,
Washington, DC 20002-4242.
