Chap 10 Problem solving1. Chapter 10 Problem Solving chap 10 Problem solving2.

chap 10 Problem solving 1

Chapter 10

Problem Solving


Problem SolvingUsing what we know to apply to new situations.

Utilizes Procedural Knowledge - How to do things.

Can draw on Semantic and Episodic Knowledge but the result is a representation of “How” to accomplish some goal.


What is a Problem? Initial state: where you currently are.Goal state: where you want to be.

• goal-directed or purposeful

Obstacles

Operators: Allowable actions for moving from the initial to the goal state.

How many problems have you solved today?


Psychological Studies of Problem Solving

Well Defined Problems.- have clear Initial and Goal states- have clear set of operators for obtaining the goal state.

Tower of Hanoi Problemchap 10 Problem solving 5

Steps to Problem Solving

1. Recognizing that there is a problem.


2. Representing the problem - what are the initial and goal states and possible operators.

Correctly identifying the obstacles.Form a mental model of the problem space.- mental representation of the Initial and Goal states and the allowable operations. For well defined problems it is possible to diagram as tree diagrams or (search trees) that represent all possible set of operations.


This diagram corresponds to a partial search tree in the middle of a Tic-Tac-Toe game.


3. Planning the solution4. Carrying out the plan.


5. Evaluating the solution: Did it obtain the goal?

6. Consolidating Gains: Learning from the experience


Research TechniquesThink Aloud Protocols

Problems with this method?

Response Times

Systematic Errors (misdirection)Problems set-up in such a way that errors will occur that tell us about the cognitive

process that people used.


http://math.ucsd.edu/~crypto/Monty/monty.html

Monty Hall Problem


The inner wheel represents the number of the door that the car is behind, the middle wheel represents the door that is selected by the contestant, and the outer wheel represents the door Monty Hall can show.

The red means that in order to win the contestant needs to switch doors, and the blue means that the contestant should not switch. Notice that there are twice as many red sections as blue. In other words, you are twice as likely to win if you switch than if you don't switch!


Why is the Monty Hall Problem So Difficult? 88% Choose to Stay with original choice.

• Uniformity Fallacy: Heuristic that assumes that all the available options are equally likely whether they are or not.

• Cognitive Load – dual task decreases the number who solve this (De Neys & Verschueren, 2006).

• Monty’s actions are seen as Random – they are not. Misunderstand the effects of Monty’s knowledge on the probabilities.


Knowledge Rich and Knowledge Lean Problems

Knowledge Rich – require knowledge outside of that contained in the problem statement.

- scientific and expert problem solving

Knowledge Lean – require knowledge outside of that contained in the problem statement.


Ill-defined ProblemsTend to offer incomplete or uncertain means of solution and often may be effectively solved with a multitude of potential solutions

Example: from Wertheimer (1945/82) Two boys of different age are playing badminton. The older one is a more skilled player, and therefore it is predictable for the outcome of usual matches who will be the winner. After some time and several defeats the younger boy finally loses interest in playing, and the older boy faces a problem, namely that he has no one to play with anymore.


The Usefulness of Past Experience.

Transfer of Learning

Information or skills related to one topic can sometimes either help or hinder the acquisition of information or skills related to another topic. When learning from one situation assists learning in another, this is referred to as positive transfer.


Negative Transfer

e.g., Functional fixedness Two strings are hanging from the ceiling such that the subject cannot grasp both strings at once. The task is to tie them together using the items in the room. Only 39% of subjects solve this problem in under 10 minutes. This is termed "functional fixedness," since it is hard to see an object as having a use beyond its generally accepted one.


Functional Fixedness

Failure to solve a problem because we assume from past experience that a given object (e.g., pliers) has only a limited number of uses.

Duncker Candle

Problem (1945)


Is it really “fixedness”?Students at Miami University were presented with collections of objects, such that one-third had only the use (e.g., packable-with), another third had two uses 2 (e.g., play catch -with) and the remaining objects had both uses. Neither use was the one for which the objects had been designed (Ye, et al., 2009) .


Tasks 1 and 2 required participants to judge which objects had uses 1 and 2, respectively. The results showed the perception of first use decreased the likelihood of identifying the second use for objects with both affordances (possible uses).


More on FixationChrysikou & Weisberg, (2005)

Participants were instructed to “think aloud” and were assigned to 1 of 3 conditions:

(a)control (standard instructions),

(b) fixation (inclusion of a problematic example, describing its problematic elements),

or

(c) defixation (inclusion of a problematic example, with instructions to avoid using problematic elements).


The Disposable Spill-Proof Coffee Cup Problem (Adapted FromJansson & Smith, 1991)Suppose you are asked to construct an inexpensive, disposable, spill-proof coffee cup. You should construct as many designs as possible, write comments with each design, and number each individual design. There are no constraints in the materials you may want to use. The problems to be addressed are:

1. Leaking of the cup if it tips over2. Leaking of the cup when squeezed3. Hot liquid burning the user’s mouth


This is an example of a present day disposable, spill-proof coffee cup. It is a Styrofoam cup, with a mouthpiece and a straw. Theproblems in this case are that the straw will leak if the cup tips over and if it is rotated 90° from the angle shown in the diagram; the cup will also leak if it is squeezed, another negative characteristic; finally, the hot liquid emerging uncooled from the straw shown in the example would burn one’s mouth.

In your designs TRY TO AVOID:1. Using straws2. Using mouthpieces3. Using an overflow device


Negative transfer due to examples was found in conditions b and c.

Examples can constrain creativity.

Junior writing exam example. – Say no to Piaget!


Mental SetThe tendency to approach situations in a certain way because that method worked in the past.


Luchins’ water-jug experiment (Lurchin 1942, 1959) - Mental SetThe subject is given a set of jugs of various stated capacities, and is asked to measure out a desired quantity of water.

Problem

Capacity of Jug A

Capacity of Jug B

Capacity of Jug C

Desired quantity

1 21 127 3 100

2 14 163 25 99

3 18 43 10 5

4 9 42 6 21

5 20 59 4 31

6 23 49 3 20

7 15 39 3 18

8 28 76 3 25

9 18 48 4 22

10 14 36 8 6chap 10 Problem solving 27

All problems except 8 can be solved by B - 2C - A.

For problems 1 through 5 this solution is simplest.

For problem 7 and 9 the simpler solution is A + C.

Problem 8 cannot be solved by B - 2C - A, but can be solved by A - C.

Problems 6 and 10 can be solved more simply as A - C.


Subjects who worked through all problems in order: •83% used B- 2C - A on problems 6 and 7. •64% failed to solve problem 8. •79% used B - 2C - A on problems 9 and 10.

Subjects who saw only last 5 problems. •Fewer than 1% used B - 2C - A. •Only 5% failed to solve problem 8.

Past solutions can blind us to better solutions.


Levine (1971) proposed that

(a)subjects will test hypotheses from the domain that contained the solution to previous problems and

(b) once subjects start testing hypotheses from a domain, they will not switch domains until they have tested all the hypotheses in the domain.

This implies that if subjects are testing hypotheses from an infinitely large domain that does not contain the solution, they will never find the solution.


Levine showed that subjects induced to test hypotheses from the domain of position sequences (e.g., left, left, right, left, left, right, etc.) will overlook a simple solution to a problem (i.e., always choose the stimulus with the "A" on it). 80% fail to solve the problem even after hundreds of trials.

A

B


Module 27 - Thinking 32

Mental set can cause us to put non-existent conditions on a problem. Video

You Tube Escalator

Insight ProblemsAnswer occurs suddenly “aha” or not at all.

Metcalfe and Wiebe (1987) Study – rated “Warmth” (how close to a solution) every 30 secs.

Non-Insight problems -knew when they were close.Insight problems - did not know they were close.


Mutilated Chessboard ProblemSuppose a standard 8x8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2x1 so as to cover all of these squares?

Very difficult problem for computers to solve.


The correct answer is that it is not possible to cover the board in dominos. Consider the board with pink and black squares.

Each domino must cover both a black and a pink square. On the mutilated board there are 32 black squares but only 30 pink squares. Therefore, it is not possible to cover the board exactly.


A person who simply tries to visualize placing dominos on the board is likely to run into memory difficulties (758,148 permutations).

Very few people solve the problem without assistance.


Kaplan and Simon (1990)four versions of this problem:1.A board filled with blank squares.2.A board with pink and black squares 3.A board in which the squares were labeled with the words "pink" and "black", instead of the colors themselves.4.A board in which the squares were labeled with the words "bread" and "butter".They found that performance increased from problems 1 - 4.


A better way to represent the problem is to include the exact numbers of black and pink squares, plus the information that a domino will cover one of each. This representation becomes increasingly likely with versions 1 - 4 above. Shifting from a naive representation of the problem to the more sophisticated one can be considered an example of restructuring (insight).


Module 27 - Thinking 39

Kounios & Beeman (2009) have identified where Insight flashed come from. In the seconds before the insight appears, a brain area in the right temporal lobe (shown here) shows a spike in activity. This region of the ‘right brain’ in particular excels in drawing together distantly related information – exactly what is needed when working on a hard creative problem.

There are distinct neural processes (right hemisphere) associated with solving problems with sudden insight.Right Hemisphere integrations of distant and lose associations.Left Hemisphere closely connected associations.


Insight seem effortless and automatic.

Analytic problem solving is effected by dual task procedure - but not insight problems (Lavric et al., 2000)

Individual differences in WM correlate with analytic problem solving – but not insight problems (Fleck, 2008)

People with less focused attention sometimes perform better on tests of insight and creative problem solving (Bowden et al. 2005)


EEG evidence shows neural processing is occurring - but the stages are not open to conscious awareness (Sheth et al., 2009)


Representational Change TheoryOhlsson (1992)

Three ways to change representations

1.Constraint relaxation (Knoblich et al., 1999).

2.Re-encoding

3.Elaboration


VI = VI I + IConstraint Relaxation

Move one toothpick to make equation true.

VIII VII - IӦllinger et al. (2008) – facilitation of same kind of insight problems but interference when an insight problem required a different kind of insight from the previous problems.

chap 10 Problem solving44

Can brain trauma cause cognitive enhancement?

Reverberi et al. (2008) Patients with lateral Frontal Cortex damage solved 82% of "insight"-based task problems (math problems arranged in toothpicks) compared to 43% for healthy controls.

Lateral Frontal lobe plays role in imposing constraints.



On December 10, 1996, Jill Bolte Taylor, a thirty-seven- year-old Harvard-trained brain scientist experienced a massive stroke in the left hemisphere of her brain. As she observed her mind deteriorate to the point that she could not walk, talk, read, write, or recall any of her life-all within four hours-Taylor alternated between the euphoria of the intuitive and kinesthetic right brain, in which she felt a sense of complete well-being and peace, and the logical, sequential left brain, which recognized she was having a stroke and enabled her to seek help before she was completely lost. It would take her eight years to fully recover.

For Taylor, her stroke was a blessing and a revelation. It taught her that by "stepping to the right" of our left brains, we can uncover feelings of well-being that are often sidelined by "brain chatter."

Link to TED Talk

Can Insight be improved?

•Training – but little transfer to real life problems.•Positive mood•Context cues

Slepian et al. (2010) 73 college students watched as words were flashed across a computer screen. They viewed 10 words associated with insight — such as create, conceive, and envision —10 other words and 20 non-words. They were then asked to respond as quickly and as accurately as possible if what they were shown was a word or non-word.


The students had either a bare, unshaded incandescent 25-Watt light bulb or an overhead fluorescent light turned on in the room. Volunteers exposed to the light bulb responded quicker to words linked to insight than other words, supporting the notion that light bulbs were indeed connected to insight in their minds.


To see if light bulbs could actually promote insights, Slepian et al. next gave college students spatial, math and verbal problems to solve and had either a bare light bulb or an overhead fluorescent light turned on in the room partway into the problem. The volunteers either solved the problems faster or more often with the light bulb than with the fluorescent light.


Incubation – taking a breakSio & Ormerod (2009). Meta-analysis•There is an small but significant incubation effect•Better with divergent thinking tasks benefiting more than linguistic and visual insight problems.• Longer preparation periods gave a greater effect•filling an incubation period with high cognitive demand tasks gave a smaller incubation effect. •Surprisingly, low cognitive demand tasks yielded a stronger incubation effect than did rest during an incubation period.


The cheap-necklace problem experiment (Silveira, 1971)“You are given four separate pieces of chain that are each three links in length. It costs 2¢ to open a link and 3¢ to close a link. All links are closed at the beginning of the problem. Your goal is to join all 12 links of chain into a single circle at a cost of no more than 15¢.”


Incubation effects - taking a breakGroup 1 - no break: 55% solved the problem.Group 2 - 30 min break: 64% solved the problemGroup 3 - 4 hour break: 85% solved the problem

Are people working on it without knowing it?

Talk aloud protocols (Simon, 1966) - did not come back with a solution. - break counteracted mental set effects. - new solutions were tried.chap 10 Problem solving 52

Remote Associates Tests

• Falling Actor Dust

• Coin Quick Spoon

• Cracker Union Rabbit

• Rock Times Steel


Vul & Pashler (2007) - three potential accounts of incubation (subconscious work, spreading activation, and fixation forgetting) •Internet-based participant pool. •Distribution of time was manipulated in several incubation conditions. Incubation benefits were found when participants were given misdirecting clues (probably because time delays facilitated forgetting of these clues).


Summary•Experimental effects of Incubation are mixed

•Likely depends on type of problem to be solved and the reason for it not being solved.

•If the problem is mental set – then taking a break is likely to help.

Sleep and Problem Solving?REM is necessary for cognitive tasks.Newly learned procedures can be practiced duringREM sleep!!!


General Problem Solver (Newell & Simon, 1972)was a theory of human problem solving stated in the form of a simulation program

- • limited Working Memory capacity• Serial processing

While GPS solved simple problems such as the Towers of Hanoi that could be sufficiently formalized, it could not solve any real-world problems because search was easily lost in the combinatorial explosion of intermediate states


Algorithms vs. Heuristics Algorithm (computers use these) - generate and evaluate all possible solution paths

– a procedure that is guaranteed to produce a solution to the problem

– Examples: • solving the anagram "xbo“ by enumerating all possible

combinations: xbo, xob, oxb, obx, bxo, box

• What about "ntraoc"? There are 6! (or 72) possible combinations.


Heuristics in Problem Solving• Heuristic = a rule of thumb, "mental shortcut“,

“educated guess” for solving a problem • not guaranteed to give the right answer• usually much more efficient than an algorithm• Heuristics for solving anagrams:

– “xbo” vowel in the middle assume “x” is not word-initial

– “ntraoc” start with likely groupings of letters: "ant, car, tan, tar, ton"


Means-ends Analysis - tries to eliminate difference between current and goal state

setting up sub-goals that are not the solution, however are a “means”to the end.- If a operator is blocked, an operator sub-goal is set up to enable blocked operators.- The Means (blocked operator) temporarily becomes the end state.


E.g., Writing a term paperSub-goals and actions for- choosing a topic.- gathering literature.- organizing the material.- writing a draft.- editing the draft.


Difference Reduction Method (Hill Climbing)Changing the present state to one closer to the solution.

The operation that appears to reduce this difference the most will generally be selected.

Focus is on short term goals rather than the larger goal.


Can Planning Help? Delaney, 2004Instructions to completely plan a task that normally produces little planning before beginning.•Control group show little evidence of planning•Ability to plan •Planning group used fewer moves

•experience with planning led to better solutions even after planning was no longer required, whereas control participants showed little improvement. chap 10 Problem solving 62

Back-up Avoidance - reluctance to return to anearlier state in the problem.

- generally this is good because it keeps you frommoving in circles, but it is not always effective.

Mr. Big (200 lbs) , Mr. Medium (120 lbs) and Mr. Small (80 lbs) want to cross a river. They have a boat that can hold only 200 lbs. How do they get across the river.


Working Backwards

Try this problemWater lilies are growing on Blue Lake. The water lilies grow rapidly, so that the amount of water surface covered by lilies doubles every 24 hours.On the first day of the summer, there was just one water lily. On the 90th day of the summer, the lake was entirely covered. On what day was the lake half covered?


Using analogies – similarities between past and current problems.


The tumor problem (Duncker, 1945): A doctor wants to destroy a tumor inside a patient's body without damaging the surrounding healthy tissue. There is a device for delivering rays that can destroy the tumor, but at the intensity needed these rays also destroy the surrounding healthy tissue. What should the doctor do?


Reasoning By Analogy Gick & Holyoak’s study on the Tumor and the Fortress problems.

The fortress story (Gick & Holyoak, 1980) helps subjects solve the tumor problem. • Without hearing the fortress story, about 10% of college students solve the tumor problem. • After hearing the fortress story, about 75% of college students solve the tumor problem. • Similarity between the target and the source problem helps people notice and use the source problem as an analogy for the

target problem. chap 10 Problem solving 67

Superficial vs. Structural SimilaritiesStudents given a set of math word problems and helped to work through the solutions.New problems given (child is told that earlier examples will help). Children often tried to use procedures that applied to problems with similar context (problems about apples) rather than problems with similar form(addition or subtraction).


Hypothesis Testing and Science

Confirmation Bias

Falsification

Cowley and Byrne (2005)My hypothesis – confirmation biasOther person’s hypothesis – falsification.


Strong and Weak MethodsWeak Methods

Unusualness heuristic – focusing on unusual or surpising findings!

Leading scientists’ heurisitc

-Challenge conventional thinking

-Adopt a step-by-step approach

-Carry out lots of experiments of trial and error basis/



General Principles(Theory)

Induction Deduction

Observations(data)

Advantage: Self correcting cycle.

What if thinking!

Expertise EXPERT : “one, who has acquired special skill in or knowledge about a particular subject through professional training and practical experience” (Webster’s, 1976, p. 800).


Knowledge Rich Problem Solving

What are you an expert At???


Skill at chess, the "Drosophila (Fruit Fly) of Expertise Studies.

• Can be measured • Broken into components (strategies, memory load etc.)• Subjected to laboratory experiments • Readily observed in its natural environment.

Adriaan de Groot Dutch chess master and psychologist


Experts vs. Novices

How do experts differ from novices?

Practice! (10,000 hours to be grand master)

Templates – e.g., King's GambitShould give them better memory for chess positions.Yes and no!


Memory and Chess Playingde Groot (1965) compared average and strong players with the world's leading grandmasters.

•Asked players to describe their thoughts as they examined a position taken from a tournament game.

•Although experts--the class just below master--did analyze considerably more possibilities than the very weak players, there was little further increase in analysis as playing strength rose to the master and grandmaster levels.


The better players did not examine more possibilities, only better ones!

de Groot had subjects examine a position for a limited period and then try to reconstruct it from memory. Beginners could not recall more than a very few details of the position, even after having examined it for 30 seconds, whereas grand-masters could usually get it perfectly, even if they had perused it for only a few seconds.


Chase & Simon (1973)

Players of various strengths reconstructed chess positions when the pieces had been placed randomly on the board.

Experts were no better than novices!

Burns (2004) – experts are better under pressure.General intelligence is correlated with ability BUTPractice is the best predictor.


To a Novice, a position with 20 chessmen on the board may contain far more than 20 chunks of information, because the pieces can be placed in so many configurations. A grandmaster, however, may see one part of the position as "fianchettoed bishop in the castled kingside," together with a "blockaded king's-Indian-style pawn chain," and thereby cram the entire position into perhaps five or six chunks. Simon estimated that a typical grandmaster has access to roughly 50,000 to 100,000 chunks of chess information. A grandmaster can retrieve any of these chunks from memory simply by looking at a chess position.


Frames can help problem solving by making the problem more manageable. For example, say a chess problem requires nine moves to reach its solution. A novice would treat this as a problem with nine distinct steps, while an expert might see that the first five moves flow naturally from one another, and the next four are a well-known sequence. Thus, the expert can treat the problem as if it has only two steps. Reducing the number of operators needed to reach the goal reduces the load on working memory, so that its limited resources can be used more efficiently.


Deep Blue was then heavily upgraded (unofficially nicknamed "Deeper Blue") and played Kasparov again in May 1997, winning the six-game rematch 3½–2½, ending on May 11, finally ending in game six, and becoming the first computer system to defeat a reigning world champion in a match under standard chess tournament time controls.The system derived its playing strength mainly out of brute force computing power. It was capable of evaluating 200 million positions per second, twice as fast as the 1996 version. Deep Blue's evaluation function was initially written in a generalized form, with many to-be-determined parameters (e.g. how important is a safe king position compared to a space advantage in the center, etc.). The optimal values for these parameters were then determined by the system itself, by analyzing thousands of master games. The evaluation function had been split into 8,000 parts, many of them designed for special positions. In the opening book there were over 4,000 positions and 700,000 grandmaster games. The endgame database contained many six piece endgames and five or fewer piece positions.

Deep Blue versus KasparovOn February 10, 1996, Deep Blue became the first machine to win a chess game against a reigning world champion (Garry Kasparov) under regular time controls. However, Kasparov won three games and drew two of the following games, beating Deep Blue by a score of 4–2. The match concluded on February 17, 1996.


After the loss, Kasparov said that he sometimes saw deep intelligence and creativity in the machine's moves, suggesting that during the second game, human chess players, in violation of the rules, intervened.

Kasparov accused IBM of cheating and demanded a rematch, but IBM declined and dismantled Deep Blue. Kasparov beat a

previous version of Deep Blue in 1996.

From Wikipedia, the free encyclopedia


Medical ExpertiseImplicit vs. Explicit Knowledge

Automated Skills (practice makes perfect)

- involves procedural knowledge.

Krupinsky et al. (2006) eye-movement studies-Experts: more info extracted on first fixation (global) -Novices: serial search


Deliberate Practice

• Appropriate level.

• Informative Feedback

• Repetition

• Error correction

“Practice does not make perfect – perfect practice makes perfect!”


Chap 10 Problem solving1. Chapter 10 Problem Solving chap 10 Problem solving2.

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