PSY 323 – Cognition

Chapter 12: Problem Solving

Problem SolvingMental processes that occur when people work toward determining the solution to a problemProblems

Math, chemistry, physics problems; Writing a term paper Selecting the movie you want to watch Finding a roommate Finding a solution for a chess problem

Representing a problem in the mindSolving a problem involves a reorganization or restructuring of problem representation

How people represent a problem matters a lot. Many different ways to solve a crossword puzzle

Insight problemsThose that require Aha! Solutions for insight problems often require gestalt process (re-organizing representation)

Metclafe & Wiebe (1987)Attempted to distinguish between insight and noninsight problemsResearchers felt that participants working on insight problems would not be good at predicting how close they were to a solution; noninsight problem solvers would be more methodical and thus better at determining how close they were See next slide

Show how you can move 3 of the circles to get the triangle to point to the bottom of the page

Metcalfe & Wiebe (1987)

A woman has 4 pieces of chain. She wants to join the pieces into a single closed loop of chain. To open a link costs 2 cents and to close a link costs 3 cents. She only has 15 cents. How can she do it?

Metcalfe & Wiebe (1987)

Solve for x: (1/5)x + 10 = 25

Factor 16y^2 - 40yz + 25z^2

Metcalfe & Wiebe (1987)

Two classes of problems: Insight and Non-insightSubjects gave “warmth” ratings every 15 seconds

Metcalfe & Wiebe (1987)

Warmth ratings during the final minute

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InterpretationInsight problems really do have an “Aha!” moment

Instant problem solving

Non-insight problems have specific steps to be taken

No “Aha!” moment Gradual problem solving

What prevents us from finding a solution? We often get fixated, and keep applying the same approach to solve problemsFunctional FixednessRestricting the use of an object to its familiar functions

Duncker (1945)Group 1: Presented with small cardboard boxes containing the materialsGroup 2: Presented with the same materials, but outside the boxes, so the boxes were empty

You are in a room with a corkboard on the wall. Your task is to mount the candle on the board so it will burn without dripping wax on the floor.

Candles

Matches in a matchbox Tacks

Duncker’s (1945)

ResultsThe group that had been presented with the boxes as containers found the problem more difficultInterpretationFunctional fixedness gets in the way of the solving of problems

Adamson (1992)Replicated Duncker’s candle problem experimentGroup 1: Got the matches in the matchboxGroup 2: Got the matches and matchbox separately so matchbox was empty

Results

Maier’s (1931)•Participants’ task was to tie together two strings that were hanging from the ceiling•The strings are separated so that you can’t reach one of them while holding the other. You have a chair and pliers.

Two-string problem

Results60 percent of the participants did not solve the problem When researcher set the string into motion by “accidentally” brushing against it, 23 of 37 participants who hadn’t solved the problem after 10 minutes proceeded to solve it within 60 secondsInterpretationFunctional fixedness for the 60%Restructured representation of how to solve problem occurred

Mental SetA preconceived notion about how to approach a problem determined by a person’s experience or what has worked in the past

Luchins (1942)Task: Participants given three jugs of different volumes; required to use these jugs to measure out specific amount of water for 8 problemsAll problems can be solved by B - A - 2C

A: 21 B: 127.C: 3 Desired: 100

B - A - 2C = Desired Quantity

All eight problems can be solved as B - A - 2C

But problems 7 and 8 can be solved in fewer steps

Procedure Mental set group: Solve problems in orderNo mental set group: Solve 7 & 8 first

Results

Luchins (1942)

InterpretationThe mental set created by solving problems 1 to 6 inhibited them from using the simpler solution for 7 and 8.

Luchins (1942)

Newell & Simon (1972)Artificial Intelligence approachTreat problem solving as a search process

They used the Tower of Hanoi problem to illustrate

Problem Space

Newell & Simon (1972)

Means-end analysis:

Reduce the difference between the initial and goal states by reaching sub-goals (intermediate goals).

Newell & Simon (1972)

Problem solving is more than just finding the path to reach a goalHow problems are stated and presented affects problem solving a great deal

Importance of how a problem is stated…

Initial state Goal state

Restrictions:1. Only one acrobat may jump at a time.

2. Whenever two acrobats are on the same flagpole, one must be standing on the shoulders of the other.

3. An acrobat may not jump when some is standing on his or her shoulders.

4. A bigger acrobat may not stand on the shoulders of a smaller acrobat.

Kotovsky et al. (1985)Acrobat Problem

ResultsParticipants took an average of 5.63 minutes to solve the problemBut when the experimenter changed the problem slightly, the problem became much more difficult

Initial state Goal state

Restrictions:1. Only one acrobat may jump at a time.

2. Whenever two acrobats are on the same flagpole, one must be standing on the shoulder of the other.

3. An acrobat may not jump when some is standing on his or her shoulders.

4. A smaller acrobat may not stand on the shoulders of a bigger acrobat.

Reverse Acrobat Problem Kotovsky et al. (1985))

InterpretationThe second problem became much harder because the idea that a smaller acrobat cannot stand on the shoulders of a bigger acrobat is inconsistent with what we know about the worldProblem solving is much more than just analyzing problem space

Reverse Acrobat Problem

Kotovsky et al. (1985)

The Mutilated Checkerboard Problem

Task: A checkerboard consists of 64 squares. These 64 squares can be completely covered by placing 32 dominos on the board so each domino covers two squares. If we eliminate two corners of the checkboard, can we cover the remaining squares with 31 dominos?See whether you can solve this problem. A solution would be either a “yes” or “no” answer plus a statement of the rationale behind your answer.

Kaplan & Simon (1990)

Kaplan & Simon (1990) tested 4 groups of subjects.

Each group received different boards.

Results:

The bread and butter board group solved the problem fastest and with fewest hints.

The blank board group took much longer and needed many more hints.

Kaplan & Simon (1990)

InterpretationParticipants who were presented boards that emphasized the difference between adjoining squares, found the problem to be easier to solveHow a problem is stated is a major determinant related to successKaplan & Simon (1990)

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Kaplan & Simon (1990)Used think-aloud protocol to gain a better understanding of what was actually taking place during this experimentHelped them to conclude that the Gestalt idea of restructuring a problem leading to insight took place

AnalogyThe process of noticing connections between similar problems and applying the solution for one problem to other problemsAnalogical TransferTransfer of one’s experiences from solving one problem to solving another similar problem

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The Russian Marriage ProblemIn a small Russian village, there were 32 bachelors and 32 unmarried women. The matchmaker succeeded in arranging 32 satisfactory marriages. Then one drunken night, two bachelors, in a test of strength, killed each other. Can the matchmaker come up with 31 heterosexual marriages among the 62 survivors?

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ExpertsPeople who, through intensive study, have become acknowledged as being knowledgeable about their fieldThey solve problems in their field better and faster than novices

• Chase & Simon (1973)• Chess master vs. beginners•Memorize chess pieces positioned for a real chess game for 5 seconds•Reproduce the arrangement shortly after

Actual Game Random Game

Chase & Simon (1973)

(a) The chess master is better at reproducing actual game positions

(b) Master’s performance drops to level of beginner when pieces are arranged randomly

Chase & Simon (1973)

Interpretation•Chess master did not have a superior STM (as some had suggested); rather he had stored many of the patterns that occur in real chess games in LTM•The chess master’s advantage vanished when the board was arranged randomly – familiar patterns were destroyedChase & Simon (1973)

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Chi et al. (1981) Give 24 physics problems to experts (professors) and novices (students with one semester of physics)Ask each group to organize them

See next slide

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Lesgold (1988)An expert will try to understand the problem and underlying concepts before diving inExample: Drawing a picture before writing equations

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Voss et al. (1983) Gave a problem involving Soviet agriculture to three groups

Expert political scientists Novice political scientists Expert chemists

Only the expert political scientists solved the problem well

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Kuhn (1970)Younger scientists are often responsible for revolutionary discoveries Frensch & Sternberg (1989)Experts are worse than novices at situations that require flexible thinking

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Creativity tests employ divergent thinkingOpen-ended thinking; no “correct” answerAsking participants to determine as many uses as possible for familiar objects like bricks

Finke (1990) Attempted to

train people to think creatively

Asked participants to randomly select three object parts

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Finke (1990) Preinventive forms

were the ideas (“inventions”) that preceded the creation of the finished creative product that participants came up with ▪ Example

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A panel of judges rated 360 created objects 120 were rated as “practical inventions” 65 were rated as “creative inventions”

Anyone can be creative - you don’t need training or even practice

Some of the slides in this presentation prepared with the assistance of the following web sites:www.tamu.edu/faculty/.../Ch%2012%20Problem%20solving.p...archlab.gmu.edu/people/jthompsz/11-ProblemSolving_1.pptarchlab.gmu.edu/people/jthompsz/11-ProblemSolving_2.ppt