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Problem Solving:How Do We Do Think Things Out?

EDIT 600

Fall 2010

Instructor: Elise Christopher

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Topics in Problem Solving

1. Problem Solving Cycle2. Problem Solving Strategies 3. Insight4. Obstacles to Problem Solving

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Problem Solving A dealer in antique coins got an offer to

buy a beautiful bronze coin. The coin had an emperor’s head on one side and the date 544 B.C. on the other. The dealer examined the coin, but instead of buying it, he called the police. Why?In 544 B.C. Christ had not been born, so a coin from that time would not be marked "B.C." (before Christ).

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Problem Solving Basics

1. Initial State• Current situation• Define the problem

2. Goal State• Desired objective

3. Obstacles• Choices made about limitations• Strategy choices• Limited resources

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Problem Solving Cycle

Identify Problem

Evaluate success

MonitorSolving

AllocateResources

Organize Info

Select Strategy

Define Problem

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Sample Problem

15% of the people in Topeka have unlisted numbers. You select 200 names at random from the Topeka phone book. How many of these people will have unlisted numbers?• Did you say 30?• The correct answer is zero

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Sample ProblemA  man wanted to enter an exclusive club but did not

know the password that was required. He waited by the door and listened. A club member knocked on the door and the doorman said, "twelve." The member replied, "six " and was let in. A second member came to the door and the doorman said, "six." The member replied, "three" and was let in. The man thought he had heard enough and walked up to the door. The doorman said ,"ten" and the man replied, "five." But he was not let in.

• What should have he said?

– Three. The doorman lets in those who answer with the number of letters in the word the doorman says.

• Example of Mental Set Effect

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Problem Representation

The importance of determining what information is relevant and what information is irrelevant is the process of problem representation• People pay attention to the wrong

information• People need to focus on the right

information

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Strategy Formation

Select a strategy to solve the problem• Analysis

• Breaking into sub goals• Study for exam sub goals

1. Read textbook & class notes2. Identify most relevant topics3. Create study questions & answers on note cards4. Learn all concepts on note cards5. Test self with note cards6. Recycle through learning and testing until mastery is

achieved

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Strategy Formation

1. Divergent thinking• Generate multiple solutions to problem• Creativity helpful for this

2. Convergent thinking• Narrow down to best answer• Mental set may actually help

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Organization of Information

• Organize to aid solution1. Symbols2. Matrices3. Diagrams

  Mango Peach Steak

Alex x 0 x

Jarod x x 0

Henry 0 x x

Let L = Lucy, S = Sean, 2L=3S, S=10

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Problem Solving Cycle

Identify Problem

Evaluate success

MonitorSolving

AllocateResources

Organize Info

Select Strategy

Define Problem

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Two Main Types of Problems

1. Well-structured problems• Clear path to the solution

• Math problems• Anagrams

2. Ill-structured problems• Dimensions of problem are not

specified or easy to infer• Finding an apartment• Writing a book

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Methods for Studying Problem Solving

1. Error analysis or reaction time• Global measures of performance

2. Verbal protocols• Participants speak their thoughts

out loud while solving problems

• Strategies become evident in protocols

3. Computer stimulation• Create models that can recreate human data

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Newell and Simon (1972)

1. Problem space • All possible actions that can be applied to a

problem

2. Consists of states and operators• States represent the problem

• Initial: given information & prior knowledge• Goal: desired outcome

• Operators transform one state to another state• Example: Permitted or selected

moves

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Newell and Simon (1972)• Use verbal protocol and reproduce

using a production rule system to create a similar representation of the problem

• Created a General Problem Solver (GPS) program– Can solve many

problems, using means-end analysis

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Strategies to Solve Problems

1. Algorithms• Systematic procedure guaranteed to

find a solution

2. Heuristics• Useful rule of thumb based on

experience• Efficient but does not guarantee a

correct solution

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Heuristics for Problem Solving

1. Means-ends analysis (e.g., GPS)2. Working forward3. Working backward4. Generate and test

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Means-End Analysis1. Compare your current state with the goal

and choose an action to bring you closer to the goal

2. Break a problem down into smaller sub goals

• Win at Monopoly• You start by buying properties,

continue to buy until you get a set, buy houses, then buy hotels, wait for others to land on spaces, etc.

3. May not work if sub goals cannot be identified  

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Working Forward

Start at initial state and work to goal state• Math problems • (2 + 6)/(4 x 1) = ?• Complete the math inside parenthesis

first, then divide the quantities to get to solution • (Please Excuse My Dear Aunt Sally)• Memory Test: What is this an example of?

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Working Backward

1. Figure out the last step needed to reach your goal, then the next-to-the-last step, and so on

• You have lost your keys• Try to remember the last time you

used them and work backwards

2. Work backwards from goal state 3. Often works well for BIG problems

• Long-term goals, for example

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Generate and Test

1. Trial and error strategy2. Create possibilities, test them and

discard the ones that are incorrect • Your car will not start• Wait a moment and try again, may be

flooded• Check to see if there is gas, if no success• Check to see if the battery is charged… etc.

3. This may not be the most efficient strategy

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Transformation Problem

1. Hobbits & Orcs • Three hobbits and three Orcs come to

a river and find a boat that holds two. If the Orcs ever outnumber the Hobbits on either bank, the Hobbits will be eaten.

2. How do you get them all to the other side?

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Recognizing the Isomorphic1. Reed (1987) found that participants have

difficulty recognizing that a past problem’s solution will help them to solve the current problem

• Difficulty in recognizing crucial commonalities• Surface features of the problem distract

2. Current research focuses on factors that help the transfer of solutions

3. How can this be used in teaching?

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Insight and Problem Solving1. Insight is the apparent sudden

solution to a problem some time after the problem has been presented

2. Metcalfe & Weibe (1987)• Participants were given either

insight or algebra problems to solve• Insight: A prisoner was attempting escape

from a tower. He found in his cell a rope which was half long enough to permit him to reach the ground safely. He divided the rope in half and tied the two parts together and escaped. How could this be?

• Algebra: (3x2 + 2x = 10)(3x) = ?

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Insight Results: Metcalf & Wiebe (1987)

1. Participants indicated how close they were to solution every 15 seconds

• 1 being very cold to 7 being very warm

2. For insight problems• Sudden shift in warmth rating, “cold” to

“hot!”

3. For algebra problems• A gradual “getting warmer” pattern

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Insight and Brain Activity

1. Neural activity associated with insight

2. fMRI studies found • Right hippocampus is active during

problem solving• Luo & Niki (2003)

• Another found spike EEG from temporal lobe just before insight• Jung-Beeman et al (2004)

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Insight Improved by Sleep?• Wagner, Gais, Haider, Verleger & Born (2004)• Subjects performed cognitive task requiring learning of

stimulus–response sequences, in which they improved gradually by increasing response speed across task blocks

• Could also improve abruptly after gaining insight into hidden abstract rule underlying all sequences

• Initial training establishing task representation was followed by 8h nocturnal sleep, nocturnal wakefulness, or daytime wakefulness

• At subsequent retesting, >2x subjects gained insight into hidden rule after sleep as after wakefulness

• Sleep did not enhance insight in the absence of initial training

• Researchers concluded: sleep, by restructuring new memory representations, facilitated extraction of explicit knowledge and insightful behavior

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Gestalt View of Insight

1. Wertheimer• Sudden rearrangement

of elements creates “insight”• Productive thinking goes beyond

previously learned associations

2. Kohler• Animal Model of Insight• Sultan stacked boxes to get banana

• Figure 11.10 in your book

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Three-Process View

Davidson & Sternberg (1984)1. Selective-encoding insights

• Sorting relevant from irrelevant

2. Selective-comparison insights• Make connections to previously learned

information

3. Selective-combination insights• Combine elements in a novel way

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Insight

Current Debate• Is insight a special process or just a

normal process in problem solving?

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Insight vs. Logic

1. Schooler, Ohlsson & Brooks (1993)2. Proposed that solving insight problems rely on

different mental structures than solving logical transformation problems

• Logical, transformation problems were solved with verbal systems, but insight problems were solved with nonverbal systems

3. Participants were asked to solve a series of insight and logic problems

• Half the participants were required to verbalize their strategies as they tried to solve the problem

• The control group did not verbalize as they solved the problem

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Schooler, Ohlsson & Brooks (1993) Results

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10

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50

60

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80

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Insight Logic

Verbal

Control

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Obstacles to Problem Solving

1. Mental set 2. Functional fixedness 3. Incorrect or incomplete

representation of the problem 4. Lack of domain knowledge

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Mental Set• Seeing a problem in a particular

way instead of other plausible ways due to experience or context

1. May cause you to adopt an ineffective strategy and prevents problem solving

2. May make assumptions without realizing it

3. May find it hard to approach the problem in a new way

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Water Jar Problem Luchins (1942)

How would you use 3 jars with the indicated capacities to measure out the desired amount of water?

Problem Jar A Jar B Jar C Desired

1 29 3 2 20

2 21 127 3 100

3 14 163 25 99

4 18 43 10 5

5 9 42 6 21

6 20 59 4 31

7 23 49 3 20

8 15 39 3 18

9 28 76 3 25

B-A-2C

A-C

A+C

2B-4A-5C

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Bar Problem

1. A man walked into a bar and asked for a drink. The man behind the bar pulled out a gun and shot the man. Why should that be so?

2. Solution: The man behind the bar wasn’t a bartender. He was a robber.

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Functional Fixedness

An inability to assign new functions and roles to elements of a problem • Two string problem• Duncker’s candle problem

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Transfer

1. Negative Transfer• Solving prior problem makes it more

difficult to solve later problem

2. Positive Transfer• Solving earlier problem helps to solve

later problem• Gick & Holyoak examine factors

contributing to positive transfer

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Gick & Holyoak (1980)

1. Give participants one problem to read with a solution

2. Give same participants a second problem which can be solved using a similar solution

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Gick & Holyoak (1980)• Analogous General/Fortress problem

A dictator ruled a small country from a fortress. The fortress was situated in the middle of the country and many roads radiated outward from it, like spokes on a wheel. A great general vowed to capture the fortress and free the country from the dictator. The general knew that if his entire army could attack the fortress at once it could be captured. But a spy reported that the dictator had planted mines on each of the roads. The mines were set so that small bodies of men could pass over them safely, since the dictator needed to be able to move troops and workers about, however, any large force would detonate the mines. Not only would this blow up the road, but the dictator would destroy many villages in retaliation. A full-scale direct attack on the fortress therefore seemed impossible.

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Gick & Holyoak (1980)

• Solution to general problemThe general, however, was undaunted. He divided his army up into small groups and dispatched each group to the head of a different road. When all was ready he gave the signal, and each group charged down a different road. All of the small groups passed safely over the mines, and the army then attacked the fortress in full strength. In this way the general was able to capture the fortress.

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Ask Participants to Solve this Problem

• Radiation problemGiven a human being with an inoperable stomach tumor, and rays that destroy organic tissue at sufficient intensity, by what procedure can one free him of the tumor by these rays and at the same time avoid destroying the healthy tissue that surrounds it?

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Gick & Holyoak (1980)

3 groups of participants 1. Control group that only tried to solve

the radiation problem 2. A group previously given the

analogous General/Fortress problem & solution

3. A group given the General/Fortress problem and told that its solution would help in solving the radiation problem

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Gick & Holyoak (1980) Results

01020

3040506070

8090

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Control Analogy Analogy & Hint

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Factors Affecting Use of Analogies

1. Similarity2. Number of examples exposed to

• Gick and Holyoak conducted a study in which the dictator story was just one of three other stories participants heard before radiation problem

• Only 20% got the problem correct3. Whether schema for problem is activated

• If the two problems are separated by a delay or if they are presented in different contexts, almost none of the participants use the analogy

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Incubation

Time away from a problem provides new insights or otherwise facilitates the problem solving process

1. Release from a problem solving set, or functional fixedness

2. Retrieval of new information by changing context

3. Recovery from fatigue

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Neuropsychology of Planning1. Frontal lobe active in problem solving2. Prefrontal cortex active

in planning3. Planning requires

many brain regions1. Orbitofrontal cortex helps

weigh immediate payoff against later rewards

2. Amygdala marks what is emotionally important

3. Anterior cingulate responds when we make errors

4. Hippocampus coordinates memories of past events. 5. Prefrontal cortex helps store items in working memory

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Expertise

1. Not a general ability2. Experts have extensive knowledge

that is used to organize, represent, and interpret information

3. Thus affecting their abilities to remember, reason, and solve problems

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Expertise

1. Chase & Simon (1973) & DeGroot (1965)2. Participants were chess masters and

beginning chess players3. Studied a chess board that had the

pieces randomly displayed or a chess board with pieces in the middle of a game.

4. Beginners and experts had to recall as many pieces as they could

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Results

1. Master chess players and beginning players recalled a similar number of pieces from the random board

2. Master chess players remember significantly more chess pieces from the game board in play than did the beginning chess players

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Beer StudyValentin, Chollet, Beal & Patris

(2007)1. Beer experts

• Two year beer training program in France

2. Beer Novices• No prior training

3. Tasted a series of 8 different beers

4. Assessed memory of beers between experts and novices• Experts remembered more

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Experts Differ From Novices

1. Better schemas2. Well organized knowledge in

specific domain3. Less time to set up problem4. Select more appropriate

strategies5. Faster at solving problems6. Are more accurate

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Summary: Problem Solving1. Algorithms vs. Heuristics2. Heuristic Strategies of Problem Solving

• Mean-ends analysis• Working forward• Working backward• Generate and test

3. Insight is that “Aha!” moment4. Obstacles to Problem Solving

• Mental set • Functional fixedness • Incorrect or incomplete representation of the

problem • Lack of domain knowledge

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Recommended Book: Problem-Solving 101

• Watanabe, Ken. (2008) Problem Solving 101: A Simple Book for Smart People.

• Fun to read!• For kids AND

adults!