Problem solving


Problem solving forms part of thinking. Considered the most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills (Goldstein & Levin, 1987). It occurs if an organism or an artificial intelligence system does not know how to proceed from a given state to a desired goal state. It is part of the larger problem process that includes problem finding and problem shaping.

Contents

Overview

The nature of human problem solving has been studied by psychologists over the past hundred years. There are several methods of studying problem solving, including; introspection, behaviorism, simulation and computer modeling, and experiment.

Beginning with the early experimental work of the Gestaltists in Germany (e.g. Duncker, 1935), and continuing through the 1960s and early 1970s, research on problem solving typically conducted relatively simple, laboratory tasks (e.g. Duncker's "X-ray" problem; Ewert & Lambert's 1932 "disk" problem, later known as Tower of Hanoi) that appeared novel to participants (e.g. Mayer, 1992). Various reasons account for the choice of simple novel tasks: they had clearly defined optimal solutions, they were solvable within a relatively short time frame, researchers could trace participants' problem-solving steps, and so on. The researchers made the underlying assumption, of course, that simple tasks such as the Tower of Hanoi captured the main properties of "real world" problems, and that the cognitive processes underlying participants' attempts to solve simple problems were representative of the processes engaged in when solving "real world" problems. Thus researchers used simple problems for reasons of convenience, and thought generalizations to more complex problems would become possible. Perhaps the best-known and most impressive example of this line of research remains the work by Newell and Simon (1972).

History

However, beginning in the 1970s, researchers became increasingly convinced that empirical findings and theoretical concepts derived from simple laboratory tasks did not necessarily generalize to more complex, real-life problems. Even worse, it appeared that the processes underlying creative problem solving in different domains differed from each other (Sternberg, 1995). These realizations have led to rather different responses in North America and in Europe.

USA and Canada

In North America, initiated by the work of Herbert Simon on learning by doing in semantically rich domains (e.g. Anzai & Simon, 1979; Bhaskar & Simon, 1977), researchers began to investigate problem solving separately in different natural knowledge domains - such as physics, writing, or chess playing - thus relinquishing their attempts to extract a global theory of problem solving (e.g. Sternberg & Frensch, 1991). Instead, these researchers have frequently focused on the development of problem solving within a certain domain, that is on the development of expertise (e.g. Anderson, Boyle & Reiser, 1985; Chase & Simon, 1973; Chi, Feltovich & Glaser, 1981).

Areas that have attracted rather intensive attention in North America include such diverse fields as:

  • Reading (Stanovich & Cunningham, 1991)
  • Writing (Bryson, Bereiter, Scardamalia & Joram, 1991)
  • Calculation (Sokol & McCloskey, 1991)
  • Political decision making (Voss, Wolfe, Lawrence & Engle, 1991)
  • Managerial problem solving (Wagner, 1991)
  • Lawyers' reasoning (Amsel, Langer & Loutzenhiser, 1991)
  • Mechanical problem solving (Hegarty, 1991)
  • Problem solving in electronics (Lesgold & Lajoie, 1991)
  • Computer skills (Kay, 1991)
  • Game playing (Frensch & Sternberg, 1991)
  • Personal problem solving (Heppner & Krauskopf, 1987)
  • Mathematical problem solving (Polya, 1945; Schoenfeld, 1985)
  • Social problem solving (D'Zurilla & Goldfreid, 1971; D'Zurilla & Nezu, 1982)

Europe

In Europe, two main approaches have surfaced, one initiated by Donald Broadbent (1977; see Berry & Broadbent, 1995) in the United Kingdom and the other one by Dietrich Dörner (1975, 1985; see Dörner & Wearing, 1995) in Germany. The two approaches have in common an emphasis on relatively complex, semantically rich, computerized laboratory tasks, constructed to resemble real-life problems. The approaches differ somewhat in their theoretical goals and methodology, however. The tradition initiated by Broadbent emphasizes the distinction between cognitive problem-solving processes that operate under awareness versus outside of awareness, and typically employs mathematically well-defined computerized systems. The tradition initiated by Dörner, on the other hand, has an interest in the interplay of the cognitive, motivational, and social components of problem solving, and utilizes very complex computerized scenarios that contain up to 2,000 highly interconnected variables (e.g., Dörner, Kreuzig, Reither & Stäudel's 1983 LOHHAUSEN project; Ringelband, Misiak & Kluwe, 1990). Buchner (1995) describes the two traditions in detail.

To sum up, researchers' realization that problem-solving processes differ across knowledge domains and across levels of expertise (e.g. Sternberg, 1995) and that, consequently, findings obtained in the laboratory cannot necessarily generalize to problem-solving situations outside the laboratory, has during the past two decades led to an emphasis on real-world problem solving. This emphasis has been expressed quite differently in North America and Europe, however. Whereas North American research has typically concentrated on studying problem solving in separate, natural knowledge domains, much of the European research has focused on novel, complex problems, and has been performed with computerized scenarios (see Funke, 1991, for an overview).

Characteristics of difficult problems

As elucidated by Dietrich Dörner and later expanded upon by Joachim Funke, difficult problems have some typical characteristics that can be summarized as follows:

  • Intransparency (lack of clarity of the situation)
    • commencement opacity
    • continuation opacity
  • Polytely (multiple goals)
    • inexpressiveness
    • opposition
    • transience
  • Complexity (large numbers of items, interrelations, and decisions)
    • enumerability
    • connectivity (hierarchy relation, communication relation, allocation relation)
    • heterogeneity
  • Dynamics (time considerations)
    • temporal constraints
    • temporal sensitivity
    • phase effects
    • dynamic unpredictability

The resolution of difficult problems requires a direct attack on each of these characteristics that are encountered.

Mathematics standards

Some problem-solving strategies identified by the NCTM are: [1]

  • "Using manipulative materials"
  • "Trial and error"
  • "Making a list"
  • "Drawing a diagram"
  • "Looking for a pattern"
  • "Acting out a problem"
  • "Guess and check"

In standards-based mathematics, less emphasis is placed on specific content-based skills and instruction of traditional methods.

Some problem-solving techniques

  1. Hill-climbing strategy, (or - rephrased - gradient descent/ascent, difference reduction) - attempting at every step to move closer to the goal situation. The problem with this approach is that many challenges require that you seem to move away from the goal state in order to clearly see the solution.
  2. Means-end analysis, more effective than hill-climbing, requires the setting of subgoals based on the process of getting from the initial state to the goal state when solving a problem.
  3. Working backwards
  4. Trial-and-error
  5. Brainstorming
  6. Morphological box
  7. Method of focal objects
  8. Lateral thinking
  9. George Pólya's techniques in How to Solve It
  10. Research: study what others have written about the problem (and related problems). Maybe there's already a solution?
  11. Assumption reversal (write down your assumptions about the problem, and then reverse them all)
  12. Analogy: has a similar problem (possibly in a different field) been solved before?
  13. Hypothesis testing: assuming a possible explanation to the problem and trying to prove the assumption.
  14. Constraint examination: are you assuming a constraint which doesn't really exist?
  15. Take more time: time pressure can cause one to think in circles (the brain, unhelpfully, tends to be "pulled" towards a particular solution, or aspect of the problem)
  16. Incubation: input the details of a problem into your mind, then stop focusing on it. The subconscious mind will continue to work on the problem, and the solution might just "pop up" while you are doing something else
  17. Build (or write) one or more abstract models of the problem
  18. Try to prove that the problem cannot be solved. Where the proof breaks down can be your starting point for resolving it
  19. Get help from friends or online problem solving community (e.g. 3form)
  20. Root Cause Analysis
  21. Wind Tunnel: based on Socratic Method whereby you outrun your logical constraints to reach for new insights to a problem. Developed by Win Wenger.
  22. Rory O'Connor's Inner Vision Deck that combines Socratic Method with methaphorical thinking and assumption breaking.

These are also known as creativity techniques. Also, please see the thinking article.

See also

  • Inquiry
  • Kepner tregoe
  • Lesgold, Alan
  • Newell, Allen
  • Simon, Herbert
  • Soar (cognitive architecture)
  • Transdisciplinary Studies
  • TRIZ
  • Troubleshooting
  • Wicked problem

Notes

  1. [1] Quirk

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