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

Cognition – 2/eCognition – 2/eDr. Daniel B. WillinghamDr. Daniel B. Willingham

Chapter 11:Chapter 11:

Problem SolvingProblem SolvingPowerPoint by Glenn E. Meyer, Trinity UniversityPowerPoint by Glenn E. Meyer, Trinity University

© 2004 Prentice Hall© 2004 Prentice Hall

© 2004 Prentice Hall© 2004 Prentice Hall 22

How Do People Solve Novel How Do People Solve Novel Problems?Problems?

• Problem Problem In the study of problem solving, a problem In the study of problem solving, a problem is any situation in which a person has a goal is any situation in which a person has a goal and that goal is not yet accomplishedand that goal is not yet accomplished

• Problem SpacesProblem Spaces

• Selecting OperationsSelecting Operations


Problem SpacesProblem Spaces• Problem space Problem space

All possible configurations that a problem can All possible configurations that a problem can take (as seen in Fig. 11.3)take (as seen in Fig. 11.3) Example: Tower of HanoiExample: Tower of Hanoi

Goal: Move all three rings from Goal: Move all three rings from the left peg to the right peg:the left peg to the right peg:

Rules:Rules:

1.1. You can only move one You can only move one right at a timeright at a time

2.2. You can move only the You can move only the top ring on a pegtop ring on a peg

3.3. You cannot put a larger You cannot put a larger ring on a smaller ringring on a smaller ring

• Each position of the problem Each position of the problem state is called the state is called the problemproblem statestate: : A particular configuration of the A particular configuration of the elements of the problem elements of the problem

• Links between different states as Links between different states as seen in Fig. 11.3 indicate paths seen in Fig. 11.3 indicate paths through the problem statethrough the problem state

• Links represent Links represent operatorsoperators: A : A process one can apply to a process one can apply to a problem to change to a different problem to change to a different state in the problem space state in the problem space


Selecting OperationsSelecting Operations• Key to Problem Solving is Selecting OperatorsKey to Problem Solving is Selecting Operators• Brute Force Search: Brute force search Brute Force Search: Brute force search

A problem-solving strategy in which all possible A problem-solving strategy in which all possible answers are examined until the correct solution is answers are examined until the correct solution is found.found. Advantage – easy to applyAdvantage – easy to apply Disadvantage – Number of possibilities increases Disadvantage – Number of possibilities increases

rapidly in a rapidly in a combinatorial explosioncombinatorial explosion: The phenomenon : The phenomenon in which the number of states in the problem space in which the number of states in the problem space increases very rapidly, even with modest increases in increases very rapidly, even with modest increases in the number of attributes of the problem that might be the number of attributes of the problem that might be changed. For example, if one tries to look four moves changed. For example, if one tries to look four moves ahead in a chess game instead of two moves ahead, the ahead in a chess game instead of two moves ahead, the number of states in the problem much more than number of states in the problem much more than doublesdoubles


Selecting Operations - ContinuedSelecting 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 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 that appears to be closer to the goal than you are now

o More likely to make more errors if we must move backwards as seen in the Hobbits and Orcs (Thomas, More likely to make more errors if we must move backwards as seen in the Hobbits and Orcs (Thomas, 1974) in Fig. 11.41974) in Fig. 11.4

Working Backward: A problem-solving heuristic in which one begins at the goal state of the problem and tries to 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 statework back to the starting state

o Useful when the goal state is known but the initial state is not as seen in the double- money problem in Table Useful 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)11.1 (Wickelgren, 1974)

Hill Climbing and Working Backward have a limited range of application as most problems require moving Hill Climbing and Working Backward have a limited range of application as most problems require moving backward and forwardbackward and forward

Means-ends Analysis: A problem-solving heuristic that uses a set of rules about when to work forward or 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 subgoalsbackward and when and how to set subgoals

o Principles:Principles: Compare the current state with the goal state. If no difference, problem solvedCompare 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 differenceIf 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 2Select 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 applicationIf 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 4Return to Step 1 with the new goal set in Step 4

o Tests of Means-ends Analysis (Newell and Simon, 1972)Tests of Means-ends Analysis (Newell and Simon, 1972) General Problem Solver: An artificial intelligence program that uses the means–ends analysis heuristic. The General 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 Problem Solver has been successful in solving a variety of problems. Found impressive degree of correspondence with protocols from subjects solving problemsprotocols from subjects solving problems

Problem behavior graphs: A representation of the problem space as the participant solved (or attempted to solve) a 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 protocolsproblem. Problem behavior graphs typically are derived from verbal protocols . 82% agreement from subjects with a means-. 82% agreement from subjects with a means-ends analysisends analysis

Aggregate data from large numbers of subjects – found means-ends analysis usedAggregate data from large numbers of subjects – found means-ends analysis used


How Do People Apply Experience to How Do People Apply Experience to New Problem?New Problem?

• Background KnowledgeBackground Knowledge• AnalogyAnalogy• Functional FixednessFunctional Fixedness


Background KnowledgeBackground 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 You to be better able to classify the problem and understand its

critical components. critical components. o Ex: Chunking in the perception of chess boardEx: Chunking in the perception of chess board

Automatizing some of the problem-solving steps so that do not Automatizing some of the problem-solving steps so that do not demand attentiondemand attention

• Background knowledge and the frontal lobe:Background knowledge and the frontal lobe: Shallice and Evans (1978) asked frontal lobe patients (as seen in Shallice and Evans (1978) asked frontal lobe patients (as seen in

Fig. B11.3) questions whose answers needed a problem solving Fig. B11.3) questions whose answers needed a problem solving strategy to answer. Most of their answers were divergent or strategy to answer. Most of their answers were divergent or bizarre as compared to normalsbizarre as compared to normals

Similarly, on the Wisconsin Card Sorting task, patients had much Similarly, on the Wisconsin Card Sorting task, patients had much difficulty in changing sorting rules when needed (perseveration)difficulty in changing sorting rules when needed (perseveration)


AnalogyAnalogy• 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 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 problemproblem did not show much transfer to the Duncker (1945) X ray problem

Predictor of using analogy is Predictor of using analogy is surface similaritysurface similarity: Whether two problems share : Whether two problems share similar elements (e.g., if both problems entail inclined planes, the problems have 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. surface similarity even if very different strategies are necessary to solve them. Ex. Keane (1987))Keane (1987))

Structural Similarity - (Refers to whether two problems share content that allows 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 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 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)body and the other an inclined plane) – has some effect (Catarambone, 2002)

Studies indicate that surface similarity is more powerful than structural similarityStudies indicate that surface similarity is more powerful than structural similarity Two processes important are needed to make effective use of analogy:Two processes important are needed to make effective use of analogy:

o It must occur to the person that an analogous problem would be helpfulIt must occur to the person that an analogous problem would be helpfulo A mapping or correspondence between elements of the two problems must be drawnA 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 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 to handle other problems of that type (Novick and Holyoak, 1991; Donnelly and McDaniel, 1993; Robins and Mayer, 1993)McDaniel, 1993; Robins and Mayer, 1993)


Functional FixednessFunctional Fixedness• Functional Fixedness:Functional Fixedness:

In problem solving, one is fixated on an object serving its typical 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, function, and one fails to think of an alternative use of the object, even though it would be quite useful in the problem 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 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 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 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 effective. How can you get the lit candle to be 5 feet off the ground without having to hold it there.having to hold it there.

Answer: Use the box and tack it to the wall.Answer: Use the box and tack it to the wall.• Dunckers’ is an example of an Dunckers’ is an example of an insight probleminsight problem: A problem in which : A problem in which

the solver feels that the answer comes all at once, in an “Aha!” the solver feels that the answer comes all at once, in an “Aha!” moment of illumination. moment of illumination.

Several are seen in Table 11.3 from Metcalfe and Wiebe (1987). The study Several are seen in Table 11.3 from Metcalfe and Wiebe (1987). The study demonstrates that for most subjects insight is a sudden solution.demonstrates that for most subjects insight is a sudden solution.


Functional Fixedness - ContinuedFunctional 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 If the key object is presented in an atypical manner. this helps break functional fixedness (Adamson, 1952)functional fixedness (Adamson, 1952)

Kids are less likely to show functional fixedness (German & Defeyter, 2000)Kids are less likely to show functional fixedness (German & Defeyter, 2000) Impasses can be broken by restructuring (a Gestalt concept): A process Impasses can be broken by restructuring (a Gestalt concept): A process

emphasized by Gestalt psychologists, applied to problem whereby one emphasized by Gestalt psychologists, applied to problem whereby one perceives a whole that had not been seen beforeperceives a whole that had not been seen before

o Example of restructuring is the Necker Cube to the leftExample of restructuring is the Necker Cube to the lefto Restructuring was thought to be unconsciousRestructuring was thought to be unconsciouso Bowers , et al. (1990) demonstrated that while the feeling of insight might feel Bowers , et al. (1990) demonstrated that while the feeling of insight might feel

sudden it is preceded by a more gradual cognitive process. Subjects had sudden it is preceded by a more gradual cognitive process. Subjects had meaningful intuitions before they solved problemsmeaningful intuitions before they solved problems

o Knoblich, et al. (1999) suggest impasses are caused by a constraint in which the Knoblich, et al. (1999) suggest impasses are caused by a constraint in which the problem solver uses a concept, To solve a problem:problem solver uses a concept, To solve a problem:

The constraint must be relaxed. For example, in the Duncker problem to think of the 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 holdertack 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 The initial chunking of the problem situation might have to be broken and the elements rechunked into a new configurationrechunked into a new configuration

An example (as seen in Figure 11.10) – move one matchstick to make the arithmetic An example (as seen in Figure 11.10) – move one matchstick to make the arithmetic expression true expression true


What Makes People Good at Solving What Makes People Good at Solving Problems?Problems?

• How Do Experts Differ From How Do Experts Differ From Novices?Novices?

• How Do People Become Experts?How Do People Become Experts?• What Makes Nonexperts Good at What Makes Nonexperts Good at

Problem Solving?Problem Solving?


How Do Experts Differ From Novices?How Do Experts Differ From Novices?

• Two expected differences between experts and Two expected differences between experts and novices:novices: Experts have more knowledge about the domainExperts have more knowledge about the domain

o Demonstrated in Chess (Chase and Simon, 1973; DeGroot, Demonstrated in Chess (Chase and Simon, 1973; DeGroot, 1946/1978) with memory tasks1946/1978) with memory tasks

o Similarly demonstrated in other domains such as bridge, Similarly demonstrated in other domains such as bridge, circuit design and computer programmingcircuit design and computer programming

o Information is organized differently (Chi, et al., 1981) – In Information is organized differently (Chi, et al., 1981) – In physics problems, novices organized by surface features, physics problems, novices organized by surface features, experts by underlying physics principlesexperts by underlying physics principles

Experts might be better at selecting operators:Experts might be better at selecting operators:o Originally thought to be the caseOriginally thought to be the caseo Evidence does not support it now – many examples in the chess Evidence does not support it now – many examples in the chess

expertise literatureexpertise literature


How Do People Become Experts?How Do People Become Experts?• Two Important factors:Two Important factors:

PracticePractice Inherent TalentInherent Talent

• Practice:Practice: Ericsson’s definitionEricsson’s definition

o Subject must be motivatedSubject must be motivatedo Task must be at the appropriate levelTask must be at the appropriate levelo There must be immediate corrective feedbackThere must be immediate corrective feedbacko Repetition of the same or similar tasksRepetition of the same or similar tasks

Ten-Year Rule: The phenomenon that experts in almost all fields are seldom able to compete at the 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. very highest levels with less than a decade of intense practice.

o Found in chess by Simon and Chase (1973)Found in chess by Simon and Chase (1973)o Other domains: musical composition and performance, mathematics, tennis, long distance running, livestock Other domains: musical composition and performance, mathematics, tennis, long distance running, livestock

evaluation, medical diagnosisevaluation, medical diagnosis Best experts practice more as seen in Fig. 11.12. One might argue they need less practice because of Best experts practice more as seen in Fig. 11.12. One might argue they need less practice because of

inherent talent but this seems not to be the case.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)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)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 eminence1.1. Child becomes exposed to domain under playful conditions & shows promise Child becomes exposed to domain under playful conditions & shows promise 2.2. Parents arrange for instruction from expert who works well with children. Practice emphasizedParents 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 expertise3.3. Decision made to commit to activity full time and get the best instructionDecision made to commit to activity full time and get the best instruction


What Makes Nonexperts Good at Problem What Makes Nonexperts Good at Problem Solving?Solving?

• Working Memory CapacityWorking Memory Capacity Several things might be needed working-memory for means-ends analysisSeveral things might be needed working-memory for means-ends analysis Needed for shuttling info between working and secondary memory Needed for shuttling info between working and secondary memory Example: Kotovsky, et al. (1985) finding that a verbal isomorph (an altered version that 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 maintains the same problem space) of the Tower of Hanoi is extremely difficult because of working memory limitsworking memory limits

Christoff, et al. (2001) find activation of left rostrolateral prefrontal cortex with 2-relational Christoff, et al. (2001) find activation of left rostrolateral prefrontal cortex with 2-relational problems as compared to 1-relational problems. Area supports relational integrationproblems as compared to 1-relational problems. Area supports relational integration

Instructions which reduce working memory load aid problem solving and vice versa (Gilhooly, Instructions which reduce working memory load aid problem solving and vice versa (Gilhooly, et al., 1993; Barrouillet, 1996)et al., 1993; Barrouillet, 1996)

Dual task paradigms indicate that working memory needs of the simultaneous task interferes Dual task paradigms indicate that working memory needs of the simultaneous task interferes with problem solving (Phillips, et al. 1999 as seen in Fig. 11.13)with problem solving (Phillips, et al. 1999 as seen in Fig. 11.13)

High correlations found between working memory capacity and problem solving (Kyllonen and High correlations found between working memory capacity and problem solving (Kyllonen and Christal, 1990) as seen in Table 11.5Christal, 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 Teaching people to use subgoals would seem to break people’s attempt to use memorized

solutions stepssolutions steps Attempts to teach to do this has not been very effectiveAttempts 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 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 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.tend not to transfer unless they see an explicit reason to do so.

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