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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. [Site] FPSPI
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Overview
The nature of human problem solving methods 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.[News] Plans call for solving dropout problem
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Beginning with the early experimental work of the Gestaltist 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). [Image] 
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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).[Video] Watch iCarly Episodes For Free
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)
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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.[Post] Problem Solving :: RE: VIC
beater wrote:. OA - C. Could someone show me how to solve this problem by plugging in values THanks! Let x=20, y=10, total distance = 40 miles (long way to school, right?) therefore, t=3 (20/20 + 20/10) and the answer we get when we ... 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). [Book] A Problem Solving Approach to Mathematics for Elementary School Teachers (9th Edition) Addison Wesley
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:[Site] Problem Solving Techniques - from MindTools.com
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- 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
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The resolution of difficult problems requires a direct attack on each of these characteristics that are encountered. [Image] 
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In reform mathematics, greater emphasis is placed on problem solving relative to basic skills, where basic operations can be done with calculators. However some "problems" may actually have standard solutions taught in higher grades. For example, kindergarteners could be asked how many fingers are there on all the gloves of 3 children, which can be solved with multiplication. [2007 Draft, Washington State Revised Mathematics Standard] [Video] Fairvalue #145 Fair Value Accounting Standards
Some problem-solving techniques
There are many approaches to problem solving, depending on the nature of the problem and the people involved in the problem. The more traditional, rational approach is typically used and involves, eg, clarifying description of the problem, analyzing causes, identifying alternatives, assessing each alternative, choosing one, implementing it, and evaluating whether the problem was solved or not.[Auction] CLINTON ANDERSON 4 DVD PROBLEM SOLVING 
Only $68.99 Another, more state-of-the-art approach is appreciative inquiry. That approach asserts that "problems" are often the result of our own perspectives on a phenomena, eg, if we look at it as a "problem," then it will become one and we'll probably get very stuck on the "problem." Appreciative inquiry includes identification of our best times about the situation in the past, wishing and thinking about what worked best then, visioning what we want in the future, and building from our strengths to work toward our vision. [ Problem Solving] [Post] ICAWC 2008 Presentation: Population Dynamics (Problem Solving by ...
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- divide and conquer: break down a large, complex problem into smaller, solvable problems.
- 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.
- 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.
- Working backwards
- Trial-and-error (also called guess and check)
- Brainstorming
- Morphological analysis
- Method of focal objects
- Lateral thinking
- George Pólya's techniques in How to Solve It
- Research: study what others have written about the problem (and related problems). Maybe there's already a solution?
- Assumption reversal (write down your assumptions about the problem, and then reverse them all)
- Analogy: has a similar problem (possibly in a different field) been solved before?
- Hypothesis testing: assuming a possible explanation to the problem and trying to prove the assumption.
- Constraint examination: are you assuming a constraint which doesn't really exist?
- 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
- Build (or write) one or more abstract models of the problem
- Try to prove that the problem cannot be solved. Where the proof breaks down can be your starting point for resolving it
- Get help from friends or online problem solving community (e.g. 3form, InnoCentive)
- delegation: delegating the problem to others.
- Root Cause Analysis
- Working Backwards (Halpern,2002)
- Forward-Looking Strategy (Halpern, 2002)
- Simplification (Halpern, 2002)
- Generalization (Halpern, 2002)
- Specialization (Halpern, 2002)
- Random Search (Halpern, 2002)
- Trial-and-error (Halpern, 2002)
- Split-Half Method (Halpern,2002)
- Analogies (Halpern, 2002)<
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