1. PSY 323 – Cognition
Chapter 12:
Problem Solving

2. What is a problem? Problem Solving
Mental processes that occur when people work toward determining the solution to a problem
Problems
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

3. Gestalt Approach Representing a problem in the mind
Solving 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 

4. Restructuring & Insight
Insight problems
Those that require Aha!
Solutions for insight problems often require gestalt process (re-organizing representation)

5. Restructuring & Insight
Metclafe & Wiebe (1987)
Attempted to distinguish between insight and noninsight problems
Researchers 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 

6. Example Insight Problems: The Triangle Problem
Show how you can move 3 of the circles to get the triangle to point to the bottom of the page
Metcalfe & Wiebe (1987)

7. Example Insight Problems: The Chain Problem
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)

8. Example Noninsight Problems: Algebra!
Solve for x: (1/5)x + 10 = 25
Factor 16y^2 - 40yz + 25z^2
Metcalfe & Wiebe (1987)

9. Studying Insight Two classes of problems: Insight and Non-insight
Subjects gave “warmth” ratings every 15 seconds
Warmth ratings during the final minute 
Metcalfe & Wiebe (1987)

10. What does this mean? Interpretation
Insight 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

11. Obstacles to Problem Solving
What prevents us from finding a solution? We often get fixated, and keep applying the same approach to solve problems
Functional Fixedness
Restricting the use of an object to its familiar functions
Duncker (1945)
Group 1: Presented with small cardboard boxes containing the materials
Group 2: Presented with the same materials, but outside the boxes, so the boxes were empty

12. Functional Fixedness: The Candle Problem
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.
Matches in a matchbox
Tacks
Candles
Duncker’s (1945)

13. Obstacles to Problem Solving
Results
The group that had been presented with the boxes as containers found the problem more difficult
Interpretation
Functional fixedness gets in the way of the solving of problems

14. Obstacles to problem solution…
Adamson (1992)
Replicated Duncker’s candle problem experiment
Group 1: Got the matches in the matchbox
Group 2: Got the matches and matchbox separately so matchbox was empty
Results 

15. Obstacles to problem solution…
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 

16. Obstacles to problem solution…
Results
60 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 seconds
Interpretation
Functional fixedness for the 60%
Restructured representation of how to solve problem occurred

17. Obstacles to problem solution…
Mental Set
A preconceived notion about how to approach a problem determined by a person’s experience or what has worked in the past

18. Mental Set: Luchins’ Water Jug Problem
Task: Participants given three jugs of different volumes; required to use these jugs to measure out specific amount of water for 8 problems
All problems can be solved by B - A - 2C

19. Solution for Problem 1 A: 21 B: 127. C: 3 Desired: 100
B - A - 2C = Desired Quantity

20. Problems 7 and 8 All eight problems can be solved as B - A - 2C
But problems 7 and 8 can be solved in fewer steps

21. Mental Set Experiment Procedure
Mental set group: Solve problems in order
No mental set group: Solve 7 & 8 first
Results 
Luchins (1942)

22. Luchins’ Methodology Interpretation
The mental set created by solving problems 1 to 6 inhibited them from using the simpler solution for 7 and 8.
Luchins (1942)

23. Problem solving: the information processing approach
Newell & Simon (1972)
Artificial Intelligence approach
Treat problem solving as a search process
They used the Tower of Hanoi problem to illustrate 

24. Newell & Simon (1972)
Problem Space

25. Newell & Simon (1972)
Means-end analysis:
Reduce the difference between the initial and goal states by reaching sub-goals (intermediate goals).

26. Importance of how a problem is stated…
Problem solving is more than just finding the path to reach a goal
How problems are stated and presented affects problem solving a great deal

27. Restrictions: Goal state Initial state Acrobat Problem
Only one acrobat may jump at a time.
Whenever two acrobats are on the same flagpole, one must be standing on the shoulders of the other.
An acrobat may not jump when some is standing on his or her shoulders.
A bigger acrobat may not stand on the shoulders of a smaller acrobat.
Goal state
Initial state
Acrobat Problem
Kotovsky et al. (1985)

28. Importance of how a problem is stated…
Results
Participants took an average of 5.63 minutes to solve the problem
But when the experimenter changed the problem slightly, the problem became much more difficult

29. Reverse Acrobat Problem Kotovsky et al. (1985))
Restrictions:
Only one acrobat may jump at a time.
Whenever two acrobats are on the same flagpole, one must be standing on the shoulder of the other.
An acrobat may not jump when some is standing on his or her shoulders.
A smaller acrobat may not stand on the shoulders of a bigger acrobat.
Goal state
Initial state
Reverse Acrobat Problem
Kotovsky et al. (1985))

30. Reverse Acrobat Problem
Interpretation
The 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 world
Problem solving is much more than just analyzing problem space
Kotovsky et al. (1985)

31. 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)

32. Kaplan & Simon (1990) tested 4 groups of subjects.
Each group received different boards.

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

34. Importance of how a problem is stated…
Interpretation
Participants who were presented boards that emphasized the difference between adjoining squares, found the problem to be easier to solve
How a problem is stated is a major determinant related to success
Kaplan & Simon (1990)

35. What does this show? Kaplan & Simon (1990)
Used think-aloud protocol to gain a better understanding of what was actually taking place during this experiment
Helped them to conclude that the Gestalt idea of restructuring a problem leading to insight took place

36. Using analogies to solve problems…
Analogy
The process of noticing connections between similar problems and applying the solution for one problem to other problems
Analogical Transfer
Transfer of one’s experiences from solving one problem to solving another similar problem

37. Telling an analogous story
The Russian Marriage Problem
In 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?

38. Experts are good at solving problems…
People who, through intensive study, have become acknowledged as being knowledgeable about their field
They solve problems in their field better and faster than novices

39. Experts Possess More Knowledge About Their Fields
Chase & Simon (1973)
Chess master vs. beginners
Memorize chess pieces positioned for a real chess game for 5 seconds
Reproduce the arrangement shortly after

40. Actual Game
Random Game
Chase & Simon (1973)

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

42. Experts Possess More Knowledge About Their Fields
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 destroyed
Chase & Simon (1973)

43. Experts’ Knowledge is Organized Differently than novices’
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 

44. Chi et al. (1981): Results Same physics principals Look the same
NOVICES
EXPERTS
Look the same
Same physics principals

45. Experts Spend More Time Analyzing Problems
Lesgold (1988)
An expert will try to understand the problem and underlying concepts before diving in
Example: Drawing a picture before writing equations

46. Experts Are No Better Outside of their Domain
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

47. Experts do not always know best
Kuhn (1970)
Younger scientists are often responsible for revolutionary discoveries
Frensch & Sternberg (1989)
Experts are worse than novices at situations that require flexible thinking

48. Divergent Thinking Creativity tests employ divergent thinking
Open-ended thinking; no “correct” answer
Asking participants to determine as many uses as possible for familiar objects like bricks

49. Creative Cognition Task
Finke (1990)
Attempted to train people to think creatively
Asked participants to randomly select three object parts

50. Creative Cognition Task
Finke (1990)
Preinventive forms were the ideas (“inventions”) that preceded the creation of the finished creative product that participants came up with
Example 

51. Finke (1990): Results 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

52. Credits
Some of the slides in this presentation prepared with the assistance of the following web sites:
archlab.gmu.edu/people/jthompsz/11-ProblemSolving_1.ppt
archlab.gmu.edu/people/jthompsz/11-ProblemSolving_2.ppt