Cognition – 2/e Dr. Daniel B. Willingham Chapter 11: Problem Solving PowerPoint by Glenn E. Meyer, Trinity University © 2004 Prentice Hall © 2004 Prentice Hall

© 2004 Prentice Hall2 How Do People Solve Novel Problems? Problem In the study of problem solving, a problem is any situation in which a person has a goal and that goal is not yet accomplishedProblem In the study of problem solving, a problem is any situation in which a person has a goal and that goal is not yet accomplished Problem SpacesProblem Spaces Selecting OperationsSelecting Operations

© 2004 Prentice Hall3 Problem Spaces Problem Spaces Problem space All possible configurations that a problem can take (as seen in Fig. 11.3)Problem space All possible configurations that a problem can take (as seen in Fig. 11.3)  Example: Tower of Hanoi Goal: Move all three rings from the left peg to the right peg: Rules: 1.You can only move one right at a time 2.You can move only the top ring on a peg 3.You cannot put a larger ring on a smaller ring Each position of the problem state is called the problem state: A particular configuration of the elements of the problemEach position of the problem state is called the problem state: A particular configuration of the elements of the problem Links between different states as seen in Fig indicate paths through the problem stateLinks between different states as seen in Fig indicate paths through the problem state Links represent operators: A process one can apply to a problem to change to a different state in the problem spaceLinks represent operators: A process one can apply to a problem to change to a different state in the problem space

© 2004 Prentice Hall4 Selecting Operations Selecting Operations Key to Problem Solving is Selecting OperatorsKey to Problem Solving is Selecting Operators Brute Force Search: Brute force search A problem-solving strategy in which all possible answers are examined until the correct solution is found.Brute Force Search: Brute force search A problem-solving strategy in which all possible answers are examined until the correct solution is found.  Advantage – easy to apply  Disadvantage – Number of possibilities increases rapidly in a combinatorial explosion: The phenomenon in which the number of states in the problem space increases very rapidly, even with modest increases in the number of attributes of the problem that might be changed. For example, if one tries to look four moves ahead in a chess game instead of two moves ahead, the number of states in the problem much more than doubles

© 2004 Prentice Hall5 Selecting Operations - Continued Selecting Operations - Continued Problem Solving Heuristics:Problem Solving Heuristics:  Hill Climbing: A heuristic in which one searches for an operator that will take you to a state in the problem space that appears to be closer to the goal than you are now oMore likely to make more errors if we must move backwards as seen in the Hobbits and Orcs (Thomas, 1974) in Fig  Working Backward: A problem-solving heuristic in which one begins at the goal state of the problem and tries to work back to the starting state oUseful when the goal state is known but the initial state is not as seen in the double- money problem in Table 11.1 (Wickelgren, 1974)  Hill Climbing and Working Backward have a limited range of application as most problems require moving backward and forward  Means-ends Analysis: A problem-solving heuristic that uses a set of rules about when to work forward or backward and when and how to set subgoals oPrinciples:  Compare the current state with the goal state. If no difference, problem solved  If there is a difference, set a goal to solve that difference. With more than one difference, solve the largest difference  Select an operator that will solve the difference identified in Step 2  If the operator can be applied, do it. If not, set a new goal to reach a state to allow application  Return to Step 1 with the new goal set in Step 4 oTests of Means-ends Analysis (Newell and Simon, 1972)  General Problem Solver: An artificial intelligence program that uses the means–ends analysis heuristic. The General Problem Solver has been successful in solving a variety of problems. Found impressive degree of correspondence with protocols from subjects solving problems  Problem behavior graphs: A representation of the problem space as the participant solved (or attempted to solve) a problem. Problem behavior graphs typically are derived from verbal protocols. 82% agreement from subjects with a means-ends analysis  Aggregate data from large numbers of subjects – found means-ends analysis used

© 2004 Prentice Hall6 How Do People Apply Experience to New Problem? Background KnowledgeBackground Knowledge AnalogyAnalogy Functional FixednessFunctional Fixedness

© 2004 Prentice Hall7 Background Knowledge Background knowledge of the domain allows:Background knowledge of the domain allows:  You to be better able to classify the problem and understand its critical components. oEx: Chunking in the perception of chess board  Automatizing some of the problem-solving steps so that do not demand attention Background knowledge and the frontal lobe:Background knowledge and the frontal lobe:  Shallice and Evans (1978) asked frontal lobe patients (as seen in Fig. B11.3) questions whose answers needed a problem solving strategy to answer. Most of their answers were divergent or bizarre as compared to normals  Similarly, on the Wisconsin Card Sorting task, patients had much difficulty in changing sorting rules when needed (perseveration)

© 2004 Prentice Hall8 Analogy People don’t use analogies well in solving problemsPeople don’t use analogies well in solving problems  Gick and Holyoak (1980, 1983) – subjects given the solution to a Fortress approach problem did not show much transfer to the Duncker (1945) X ray problem  Predictor of using analogy is surface similarity: Whether two problems share similar elements (e.g., if both problems entail inclined planes, the problems have surface similarity even if very different strategies are necessary to solve them. Ex. Keane (1987))  Structural Similarity - (Refers to whether two problems share content that allows them to be solved by the same strategy (e.g., if problems can both be solved by Newton’s second law, they share structural similarity, even if one involves a falling body and the other an inclined plane) – has some effect (Catarambone, 2002)  Studies indicate that surface similarity is more powerful than structural similarity  Two processes important are needed to make effective use of analogy: oIt must occur to the person that an analogous problem would be helpful oA mapping or correspondence between elements of the two problems must be drawn  Practice with a class of problems aids in developing a schema that is general enough to handle other problems of that type (Novick and Holyoak, 1991; Donnelly and McDaniel, 1993; Robins and Mayer, 1993)

© 2004 Prentice Hall9 Functional Fixedness Functional Fixedness: In problem solving, one is fixated on an object serving its typical function, and one fails to think of an alternative use of the object, even though it would be quite useful in the problemFunctional Fixedness: In problem solving, one is fixated on an object serving its typical function, and one fails to think of an alternative use of the object, even though it would be quite useful in the problem The Classic Example from Duncker (1945):The Classic Example from Duncker (1945): In an empty room are a candle, some matches, and a box of tacks. The goal is to have the lit candle about 5 feet off the ground. You’ve tried melting some of the wax on the bottom of the candle and sticking it to the wall, but that wasn’t effective. How can you get the lit candle to be 5 feet off the ground without having to hold it there. Answer: Use the box and tack it to the wall. Dunckers’ is an example of an insight problem: A problem in which the solver feels that the answer comes all at once, in an “Aha!” moment of illumination.Dunckers’ is an example of an insight problem: A problem in which the solver feels that the answer comes all at once, in an “Aha!” moment of illumination.  Several are seen in Table 11.3 from Metcalfe and Wiebe (1987). The study demonstrates that for most subjects insight is a sudden solution.

© 2004 Prentice Hall10 Functional Fixedness - Continued Impasses in Insight Problems are attributed to functional fixednessImpasses in Insight Problems are attributed to functional fixedness  If the key object is presented in an atypical manner. this helps break functional fixedness (Adamson, 1952)  Kids are less likely to show functional fixedness (German & Defeyter, 2000)  Impasses can be broken by restructuring (a Gestalt concept): A process emphasized by Gestalt psychologists, applied to problem whereby one perceives a whole that had not been seen before oExample of restructuring is the Necker Cube to the left oRestructuring was thought to be unconscious oBowers, et al. (1990) demonstrated that while the feeling of insight might feel sudden it is preceded by a more gradual cognitive process. Subjects had meaningful intuitions before they solved problems oKnoblich, et al. (1999) suggest impasses are caused by a constraint in which the problem solver uses a concept, To solve a problem:  The constraint must be relaxed. For example, in the Duncker problem to think of the tack box as the candle holder and not the tack holder  The initial chunking of the problem situation might have to be broken and the elements rechunked into a new configuration  An example (as seen in Figure 11.10) – move one matchstick to make the arithmetic expression true

© 2004 Prentice Hall11 What Makes People Good at Solving Problems? How Do Experts Differ From Novices?How Do Experts Differ From Novices? How Do People Become Experts?How Do People Become Experts? What Makes Nonexperts Good at Problem Solving?What Makes Nonexperts Good at Problem Solving?

© 2004 Prentice Hall12 How Do Experts Differ From Novices? Two expected differences between experts and novices:Two expected differences between experts and novices:  Experts have more knowledge about the domain oDemonstrated in Chess (Chase and Simon, 1973; DeGroot, 1946/1978) with memory tasks oSimilarly demonstrated in other domains such as bridge, circuit design and computer programming oInformation is organized differently (Chi, et al., 1981) – In physics problems, novices organized by surface features, experts by underlying physics principles  Experts might be better at selecting operators: oOriginally thought to be the case oEvidence does not support it now – many examples in the chess expertise literature

© 2004 Prentice Hall13 How Do People Become Experts? How Do People Become Experts? Two Important factors:Two Important factors:  Practice  Inherent Talent Practice:Practice:  Ericsson’s definition oSubject must be motivated oTask must be at the appropriate level oThere must be immediate corrective feedback oRepetition of the same or similar tasks  Ten-Year Rule: The phenomenon that experts in almost all fields are seldom able to compete at the very highest levels with less than a decade of intense practice. oFound in chess by Simon and Chase (1973) oOther domains: musical composition and performance, mathematics, tennis, long distance running, livestock evaluation, medical diagnosis  Best experts practice more as seen in Fig One might argue they need less practice because of inherent talent but this seems not to be the case. Inherent TalentInherent Talent  Some suggestion that talents like “perfect pitch” can be acquired (Takeuchi & Hulse, 1993)  However, twin studies suggest genetic component is greater (Drayna, et al., 2001) Practice and Talent Interact (Bloom, 1985) – Developmental stages in achieving eminencePractice and Talent Interact (Bloom, 1985) – Developmental stages in achieving eminence 1.Child becomes exposed to domain under playful conditions & shows promise 2.Parents arrange for instruction from expert who works well with children. Practice emphasized Parents show a great deal of enthusiasm provide teachers of increasing expertiseParents show a great deal of enthusiasm provide teachers of increasing expertise 3.Decision made to commit to activity full time and get the best instruction

© 2004 Prentice Hall14 What Makes Nonexperts Good at Problem Solving? Working Memory CapacityWorking Memory Capacity  Several things might be needed working-memory for means-ends analysis  Needed for shuttling info between working and secondary memory  Example: Kotovsky, et al. (1985) finding that a verbal isomorph (an altered version that maintains the same problem space) of the Tower of Hanoi is extremely difficult because of working memory limits  Christoff, et al. (2001) find activation of left rostrolateral prefrontal cortex with 2- relational problems as compared to 1-relational problems. Area supports relational integration  Instructions which reduce working memory load aid problem solving and vice versa (Gilhooly, et al., 1993; Barrouillet, 1996)  Dual task paradigms indicate that working memory needs of the simultaneous task interferes with problem solving (Phillips, et al as seen in Fig )  High correlations found between working memory capacity and problem solving (Kyllonen and Christal, 1990) as seen in Table 11.5 Setting SubgoalsSetting Subgoals  Teaching people to use subgoals would seem to break people’s attempt to use memorized solutions steps  Attempts to teach to do this has not been very effective Comparing ProblemsComparing Problems  Transfer to new problems can be effective if subjects see the deep structure of a problem. Might be done by having subjects compared problems. However, this is difficult to teach as subjects tend not to transfer unless they see an explicit reason to do so.