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Problem Solving

Definition of a problem. A problem exists when you want to get from

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Problem Solving

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    1. Problem Solving Directed thought

    2. Definition of a problem A problem exists when you want to get from “here” (a knowledge state) to “there” (another knowledge state) and the path is not immediately obvious.

    3. A Bit of Problem Typology Well-defined vs. ill-defined problems: Problems where the goal or solution is recognizable--where there is a right answer (ex. a math or physics problem) vs. problems where there is no "right" answer but a range of more or less acceptable answers. Knowledge rich vs. knowledge lean problems: problems whose solution depends on specialized knowledge. Insight vs. non-insight problems--those solved "all of a sudden" vs. those solved more incrementally--in a step by step fashion.

    4. Some Problem Examples Tower of Hanoi Weighing problem Traveling salesman (100 cities = 100! or 10200 or each electron, 109 operations per sec. would take 1011 years!!) but 100,000 cities within 1% in 2 days via heuristic breakup (reduce search!) Missionaries & Cannibals Necklace (4 x 3 links, make 12 link necklace) 2c to break, 3c to join, make for < or = 15c. Flashlight: 1, 2, 5, 10 min. walkers to cross bridge 21 link gold necklace/21 day stay Subway Problem Vases (or 3-door)

    5. Older Work on Problem Solving Kohler’s Apes Bird-brained Crow Functional fixedness (the string & matchbox problem). Problem-solving set (Luchins' water jar problem). Memory for interrupted problems (Zeigarnik effect).

    13. -olk oak

    14. Folk

    15. Polk

    16. Yolk????

    19. Luchins Results Problems 1 thru 5 all solved via b - 2c - a. Probs 6 & 10: b - 2c - a or a - c (83% harder way 6, 7) Probs 7 & 9: b - 2c - a or a + c (79% harder way 9,10) Problem 8: a - c but not b - 2c - a (64% failure) If just given last five problems, <1% use harder method and only 5% fail prob. 8. If given all ten probs. But told “Don’t be blind” after prob. 5 50% overcame set for b - 2c - a

    20. The Cognitive Revolution/Counter-revolution: Newell and Simon Problem-solving as search. Verbal protocols and computer simulation. Algorithms and heuristics,

    21. Problem Spaces Start Goal Move operators Size of space one determinant of difficulty

    27. Some Common Heuristics Hill-climbing Fractionation Working backwards Means-ends-analysis These examples illustrate “weak methods” in contrast to algorithms.

    30. The Centrality of Representation Problem space and representation Problem difficulty and representation The interaction of representation and processing limitations (problem isomorphs)

    31. Representation (cont’) Number scrabble 1 2 3 4 5 6 7 8 9

    55. Expertise Hayes on ten year rule

    58. Expertise: What’s being Learned in the Ten Years? DeGroot and Chase & Simon’s work on chunking and chess Estimates of knowledge base size

    59. Practice Makes Perfect! Power law of practice: Ta = cPb + d

    66. Expertise Expertise in physics problem solving-- chunks strategies problem categorization

    67. Strategy Acquisition in Game-playing Fox and Hounds

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