Problem Solving Vudhichai Boonyanaruthee, MD Department of Psychiatry

Problem Solving Vudhichai Boonyanaruthee, MD Department of Psychiatry
Adapted from Cognitive Psychology: Connecting mind, research, and everyday experience, 2nd & 3rd by E. Bruce Goldstein.

content WHAT IS A PROBLEM?
THE GESTALT APPROACH: PROBLEM SOLVING AS REPRESENTATION AND RESTRUCTURING Representing a Problem in the Mind Restructuring and Insight DEMONSTRATION: Two Insight Problems Obstacles to Problem Solving DEMONSTRATION: The Candle Problem MODERN RESEARCH ON PROBLEM SOLVING: THE INFORMATION- PROCESSING APPROACH Newell and Simon’s Approach DEMONSTRATION: Tower of Hanoi Problem The Importance of How a Problem Is Stated DEMONSTRATION: The Mutilated Checkerboard Problem METHOD: Think-Aloud Protocol

content USING ANALOGIES TO SOLVE PROBLEMS Analogical Transfer
Analogical Problem Solving and the Duncker Radiation Problem DEMONSTRATION: Duncker’s Radiation Problem Analogical Encoding Analogy in the Real World METHOD: In Vivo Problem-Solving Research HOW EXPERTS SOLVE PROBLEMS Differences Between How Experts and Novices Solve Problems Expertise Is Only an Advantage in the Expert’s Specialty CREATIVE PROBLEM SOLVING DEMONSTRATION: Creating an Object SOMETHING TO CONSIDER: DOES LARGE WORKING MEMORY CAPACITY RESULT IN BETTER PROBLEM SOLVING? IT DEPENDS

What Is a Problem? A problem occurs when
there is an obstacle between a present state and a goal and it is not immediately obvious how to get around the obstacle (Lovett, 2002)

Problem Solving Problem solving is defined: a goal to accomplish, with an initial state and goal state, with obstacles to overcome which are not obvious how to get around การบรรลุเป้าหมาย ที่จุดเริ่มต้นและเป้าหมายมีอุปสรรคที่ ต้องเอาชนะ โดยที่ยังไม่รู้ชัดเจนว่าจะใช้วิธีใด

Salient differences between puzzle problems and real-world problems
Puzzles Unfamiliar no prior knowledge necessary information is present in the problem statement unambiguous requirements Real-world problems Familiar require prior knowledge necessary information not present what is the goal?

Greeno’s problem types
Types of problems (Greeno, 1978): Inducing structure (e.g., analogies) Discovery of a pattern relating elements of a problem to each other. Transformation (e.g., Towers of Hanoi, water jar problem) Manipulation of objects or symbols while following certain rules. Arrangement (e.g., anagrams, seating guests) All the elements are given, and the task is to re-arrange them. Any problem could be in more than one category.

The Gestalt Approach: Problem Solving as Representation and Restructuring

Gestalt Viewpoint A structure, arrangement, or pattern of physical, biological, or psychological phenomena so integrated as to constitute a functional unit with properties not derivable by summation of its parts โครงสร้าง การจัดเรียง หรือรูปแบบของปรากฏการณ์ทางกายภาพ ชีวภาพ หรือจิตวิทยาเมื่อรวมกันเข้าเป็นหนึ่งหน่วย ที่มีคุณสมบัติต่างไปจาก ผลรวม ของส่วนประกอบเหล่านั้น "The whole is greater than the sum of the parts"

Gestalt

Gestalt Viewpoint Koehler study with primates in 1910s
Problem-solving use of mental representation Problem-solving is both reproductive and productive Reproductive Problem-Solving involves re-use of previous experience (can be beneficial or detrimental) Productive problem-solving is characterized by restructuring and insight – Metcalfe & Wiebe (1987) insight v noninsight problems Problem-solving: trial-and-error or otherwise? Problem-solving cycle

Problem solving, for the Gestalt psychologists,
(1) how people represent a problem in their mind (2) how solving a problem involves a reorganization or restructuring of this representation.

REPRESENTING A PROBLEM IN THE MIND

A crossword puzzle ( Figure 12.1) This type of problem is represented on the page by a diagram and clues about how to fill in the open squares. How this problem is represented in the mind is probably different for different people, but it is likely to differ from how it is represented on the page.

A crossword puzzle: They may use different ways of representing it in the mind Choose to represent only a small part of the puzzle at a time. Focus on filling in horizontal words and then use these words to help determine the vertical words. Pick one corner of the puzzle and search in their mind for both verticals and horizontals that fit together.

Koehler’s primate studies

Figure 11.4: Results of Metcalfe and Wiebe’s (1987) experiment showing how participants judged how close they were to solving insight problems and algebra problems for the minute just before solving the problems. Figure 11.4: Results of Metcalfe and Wiebe’s (1987) experiment showing how participants judged how close they were to solving insight problems and algebra problems for the minute just before solving the problems.

Problem Solving Cycle

Problem Examples Circle problem Triangle Cheap necklace
Runcker’s Candle-wall problem Maier’s Two-string problem Luchin’s Water jug problem Tower of Hanoi Hobbits-and-Orcs Acrobat & reverse acrobat problem Mutilated checkerboard Monk and the mountain Many others

Figure 11.2

Figure 11.3 (a) Triangle problem and (b) chain problem for “two insight problems” demonstration.

Insight and Non-insight Problems
Insight: problem-solvers poor at predicting success; can’t monitor closeness to solution sudden understanding of what is needed for the solution. combining information in new ways Non-insight or Routine: problem-solvers good at predicting their success; monitor accurately how close they are to solution

Insight and Non-insight Problems
Metcalfe studies Made a distinction between insight and non-insight problems. Non-insight problems solvers are accurate to predict success to a solution Insight problem poor prediction of success to a solution. Used “warmth rating” every 15 seconds while solving a problem Non-sight problems when solvers were closer to the solution the “warmth rating” went up Insight problems not This was support for the Gestalt view of problem solving. Cognitive Psychology: Mind, Research, and Everyday Experience, 2nd Ed. by Bruce Goldstein. Copyright © 2008 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.

Figure Results of Metcalfe and Wiebe’s (1987) experiment showing how participants judged how close they were to solving algebra problems (left column) and insight problems (right column). For the algebra problem, warmth ratings move slowly towards the “hot” end of the scale during the minute before the problem is solved. For the insight problems, the solution is sudden.

Figure 11.4: Results of Metcalfe and Wiebe’s (1987) experiment showing how participants judged how close they were to solving insight problems and algebra problems for the minute just before solving the problems. Figure 11.4: Results of Metcalfe and Wiebe’s (1987) experiment showing how participants judged how close they were to solving insight problems and algebra problems for the minute just before solving the problems.

Figure 11.23 Solution to the circle problem.

Figure 11.24 Solution to the triangle problem.

Figure 11. 25 Solution to the chain problem
Figure Solution to the chain problem. All the links in one chain are cut and separated (3 2 cents = 6 cents). Then each separated link is used to connect the other three pieces and are then closed (3 3 cents = 9 cents). Total = 15 cents.

Obstacles to Problem Solving
Duncker (1945) candle-wall problem A table A candle A box of matches A box of tacks. Functional fixedness – fix a candle to the wall… Participants are better at this task if the matches and the tacks are on the table, not in the box. Adamson (1952) investigate effects of problem presentation and found if the box was empty, problem solving was more successful It is easier to think of empty boxes as something other than containers – to overcome functional fixedness.

Figure 11.5 Objects for Runcker’s (1945) candle problem.

Figure 11.6 Results of Adamson’s (1952) replication of Duncker’s candle problem.

Figure 11.26 Solution to the candle problem.

Functional fixedness – fixed on the use of certain objects Another example is Maier’s (1931) two-string problem – F 11.7 Two strings A chair A pair of scissors or pliers Participants did not do well in solving the problem, 60% fail to solve. If the participants have not solve the problem after 10 minutes, the strings are set in motion. It has been found that 23/37 participants who have not solved the problem solve it within 10 second! Other examples: panty-hose can be used to make a fan belt… water in a car radiator can be drunk…

Figure 11. 7 Maier’s (1931) two-string problem
Figure Maier’s (1931) two-string problem. As hard as Sebastian tries, he can’t grab the second string. How can he tie the two strings together?

Mental set is the obstacle to problem solving. Set is a preference for certain operators (things you can do, actions you can take to solve a problem). The Gestaltist called these obstacles Einstellung (mechanization of thought) – people kept using a strategy that worked even when a better one was available.

Figure 11. 8 Luchins’ (1942) water-jug problem
Figure Luchins’ (1942) water-jug problem. Each problem specifies the capacities of jugs A, B, and C, and a final desired quantity. The task is to use the jugs to measure out the final quantity. The solution to problem 1 is shown. All of the other problems can be solved using the same pattern of pouring, but there are more efficient solutions to problems 7 and 8.

Cognitive Psychology: Mind, Research, and Everyday Experience, 2nd Ed
Cognitive Psychology: Mind, Research, and Everyday Experience, 2nd Ed. by Bruce Goldstein. Copyright © 2008 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.

Luchins’ (1942) water jar Note that the strategy for solution changes after problem 6. There are more efficient solutions. Participants who began the water jar problem with problem 7 used the shorter more efficient Possible

Figure All of the participants who began the Luchins water-jug problem with problem 7 used the shorter solution (right bar), but less than a quarter who had established a mental set by beginning with problem 1 used the shorter solution to solve problem 7 (left bar).

Slide 27 Solution to problem 7 of the Luchins water-jug problem
Slide 27 Solution to problem 7 of the Luchins water-jug problem. Problem 8 can also be solved using just jugs A and C.

Modern Approaches to Problem Solving
Problem-Space Theory Problem space: A mental representation of a problem that contains knowledge of the initial state and the goal state of the problem possible intermediate states that must be searched in order to link up the beginning and the end of the task. McGraw-Hill Science & Technology Dictionary:

Problem-Space Theory – Newell and Simon – view problem solving as a search process – GPS computer program Solving a problem involves Negotiating alternative paths to a solution Initial state is linked to goal state Problem-space refers to the abstract structure of a problem Operators are specific knowledge structures that transform data Knowledge states are produced by the application of mental operators

Algorithms vs. Heuristics Algorithms a step-by-step procedure that guarantees a solution Heuristics a general approach (rule of thumb) to problem solving not guarantee a solution.

Heuristics from the Greek "Εὑρίσκω" for "find" or "discover" a general approach (rule of thumb) to problem solving not guarantee a solution. a rule of thumb, from trial-and-error experience, used to guide decisions when algorithms are unavailable. a commonsense rule (or set of rules) intended to increase the probability of solving some problem experience-based techniques that help in problem solving, learning and discovery

Heuristics: means-end analysis: calculate difference between current state and goal; create a subgoal to reduce that difference; select an operator that will solve this subgoal initial and goal states and operators Limited processing resources Move problems: – Towers of Hanoi, Hobbits-and-Orcs Other Heuristics: working forward, working backwards, generate and test, hill climbing strategy

Figure (a) Initial and goal states for the Tower of Hanoi problem. (b) Operators that govern the Tower of Hanoi problem.

Figure 11. 11: Problem space for the Tower of Hanoi problem
Figure 11.11: Problem space for the Tower of Hanoi problem. The arrow on the right indicates that the most efficient solution involves reaching the goal state (8) by traversing intermediate states 2–7 (Dunbar, 1998). Figure 11.11: Problem space for the Tower of Hanoi problem. The arrow on the right indicates that the most efficient solution involves reaching the goal state (8) by traversing intermediate states 2–7 (Dunbar, 1998).

Means-Ends analysis

Figure Initial steps in solving the Tower of Hanoi problem, showing how the problem can be broken down into subgoals.

Initial and goal states for the hobbits – and - orcs problem
Initial and goal states for the hobbits – and - orcs problem. Conditions: at least one in the boat and at no time can there be more orcs than hobbits on a river bank.

Problem space – a version of hobbits-and-orcs: missionaries and cannibals

Solution to the hobbits-and-orcs problem
Solution to the hobbits-and-orcs problem. Each trip indication on the left (trip 1, trip 2, etc.) indicates the number of hobbits and orcs that remain after the trip. The hobbits and orcs in the next row down indicate how many hobbits and orcs there are each time the boat comes back.

Representations of the Problem
Some problems are more easily understood and solved if they are represented in concrete terms (e.g. a mental image), others are more easily solved in abstract terms. Acrobat & Reverse Acrobat Problem One small change in wording of problem Not just analyzing structure of problem space How a problem is stated can affect its difficultly Mutilated checkerboard problem

Figure Mutilated-checkerboard problem. See demonstration for instructions.

Finding the right representation of a problem can be crucial for finding the solution. Kaplan and Simon (1990) used four version of the mutilated checkerboard The key to problem solving was to arrive a parity representation – take into account that each square has a neighbor. Most Ps started by thinking about the number of squares: 64-2=62, there is room for 31 dominos. When the various board emphasized the “difference” between adjoining squares the problem was found to be easier. Bread and Butter – average on 1 hint Bland board – average of 3.18 hints The color and black/pink boards somewhere between. There are other ways to arrive at parity representation Monk and Mountain problem Use of image representation to solve problem

Figure 11.15 Conditions in Kaplan and Simon’s (1990) study of the mutilated-checkerboard problem.

Monk and Mountain Problem

Monk and the mountain Progress of Charlie, who starts climbing up the mountain in the morning, and Susan, who starts climbing down at the same time, showing that at some point in time they are both at the same place on the mountain, just as the monk, in the “monk and the mountain” problem is, during his ascent and descent.

Heuristics for problem solving
Other Heuristics Working forward Start at the beginning and try to solve the problem from the start to the finish Working backward The problem-solver start at the end and tries to work backward from there Generate and test The problem-solver generates a list of alternative ways of action, not necessarily in systematic way, and then notices in turn whether each course of action will work Hill climbing strategy For any particular state, carry out the operation that moves you closest to the final goal state. (often not a good strategy)

Working backwards Heuristic: Example

One (painful) way to solve the water lilies problem
Initial number of water lilies = 1 double the initial value 90 times Record each of these values Find the value that is 1/2 of the 90th day value. This problem might be solved by considering the initial number of water lilies..figuring out what number would be there after each double, and figuring out which doubling equaled half of the number present after 90… 618,970,019,642,690,000,000,000,000 lilies on the 90th day….about 6.2 times 10 to the 26th…. So what’s half of that? Working backwards: or you could think about the fact that just before 90…there must have been half as many, since each time the number doubled….thinking of transformations of the end state makes the problem trivially easy.

The way to solve the water lilies problem
Working backwards: - value doubling every day is equivalent to say that the value is halved each preceding day - the field was full Day 90th - the field was half full on day 89th This problem might be solved by considering the initial number of water lilies..figuring out what number would be there after each double, and figuring out which doubling equaled half of the number present after 90… 618,970,019,642,690,000,000,000,000 lilies on the 90th day….about 6.2 times 10 to the 26th…. So what’s half of that? Working backwards: or you could think about the fact that just before 90…there must have been half as many, since each time the number doubled….thinking of transformations of the end state makes the problem trivially easy.

Possible or Impossible?
Think about this one for a bit… the problem is impossible. Turns out the easiest way to figure this out is to think abstractly rather than visually…first, not how many black and white squares there are (13 black and 12 white). Now consider the fact that the rules of movement require that the pencil must alternate between black and white with each move…now it is easy to calculate that if you start on white, you’ll always run out of white squares before you can get to the last black one... Starting in the square marked by the circle, draw a line through all the squares without picking up your pencil, without passing through a square more than once, without diagonal lines and without leaving the checkerboard.

Cryptarithmetic DONALD + GERALD = ROBERT SEND + MORE = MONEY
EAT + THAT = APPLE CROSS + ROADS = DANGER COUNT – COIN = SNUB With each cryptarithmetic problems find a number to substitute for a letter. Only one number can be used for any given letter and once a number is use for a letter it cannot be used for another letter. If the letter appears more than once in a problem the same number can be used again. Note: Newell & Simon’s GPS (General Problem Solver) was able to solve these problems

Problem Solving by Analogy
Use of analogy, a structural similarity between situations or events. An atom is a miniature solar system Use to explain many abstract concepts Restructuring, used in the Gestalt approach, could be aided by retrieving analogous instances. Duncker’s (1945) Radiation Problem

Duncker’s (1945) Radiation Problem
Suppose you are a doctor faced with a patient who has a malignant tumor in his or her stomach. It is impossible to operate on the patient, but unless the tumor is destroyed the patient will die. There is a special type of ray that can be used to destroy the tumor, as long as the rays reach the tumor with sufficient intensity. However, at the necessary intensity, the healthy tissue that the rays pass through will also be destroyed and the patient will die. At lower intensities, the rays are harmless but they will not affect the tumor either.

Duncker’s (1945) Radiation Problem
What procedure might the doctor employ to destroy the tumor with the rays, at the same time avoiding destroying any healthy tissue?

Analogy in Problem Solving
Duncker found that most of the Ps could not solve this problem without hints. He used an analogy to the problem to assist the problem solvers, “The General” , summarized as follows: A general is attacking a fortress. He can’t send all his men in together as the roads are mined to explode if large numbers of men cross them. He therefore splits his men into small groups and sends them in on different roads. 70 % were unable to solve problem given the surface analogy Gick and Holyoak (1983) used four versions of the “military problem” and found that 75% of Ps solved the radiation problem when given the analogous “military problem”. The solution to the radiation problem is how modern radiosurgery is done.

Figure 11. 16a (b) Solution to the radiation problem
Figure 11.16a (b) Solution to the radiation problem. Bombarding the tumor, shown in the center, with a number of low-intensity rays from different directions destroys the tumor without damaging the tissue it passes through.

Figure 11.16b (c) Radiosurgery, a modern medical technique for irradiating brain tumors with a number of beams of gamma rays, uses the same principle. The actual technique uses 201 gamma-ray beams.

Figure 11.16c (a) The general’s plan from the fortress problem.

According to Gick and Holyoak (1980), the process of analogical problem solving involves these three steps Noticing parallels – the most difficult step, easier with greater similar between story and problem – positive transfer effects. 2. Mapping corresponding elements between source story and target problem 3. Appling the mapping to generate a solution

Using Analogies to Solve Problem
Analogical encoding: comparing two cases that illustrate a principle – Gentner et al. (2003) Effective way to get participants to pay attention to structure features that aide problem solving Used trade-off and contingency negotiation strategies Analogical paradox – Dunbar et al. (2001) Participants in experiments focus on surface features People in the real-world use structural features In-vivo problem solving research – Christensen & Schunn (2007) People are observed to determine how they solve problems in the real world Advantage: naturalistic setting Disadvantages: time-consuming, cannot isolate and control variables

Expertise A large organized body of domain knowledge is a prerequisite to expertise; this knowledge influences the perceptual processes & strategies of problem solving. One’s knowledge can be measured in terms of quantity. For example, Chase & Simon (1973) found, in a memory study, that chess experts recalled more chess pieces because they had a greater number of patterns recognize (about 50,000) than good players (1,000). It seems that structure (organization) of knowledge is more important because it makes that knowledge more accessible, functional, and efficient.

Expertise To examine this we can look at what kind of information do experts and novices use to make categorical decisions Chi, Feltovich, and Glaser (1981) used a card sorting task to classify physics problems; novices sorted based on surface features (e.g., type of object involved) while experts sorted based on physical principles, Figure , Experts use different solution strategies like using backwards reasoning with mathematics problems. Experts spend more time analyzing the problem and less time searching for a solution, see problem in Slide 17 Expertise seems to be task specific; there seems to be little transfer from one domain to another – Voss et al. (1983) study.

Figure The kinds of physics problems that were grouped together by novices (left) and experts (right) (Chi et al, 1981).

Experts and Novices from Sternberg (2003) textbook

Summary of the Nature of Expertise
Experts have well-organized knowledge -- not just “problem solving” strategies; their knowledge is organized to support understanding (qualitative before quantitative) and it is “conditionalized” for use. Experts have fluent access to their knowledge. Such knowledge is acquired over time and depends on multiple, contextualized experiences. Implications -- “wisdom” can’t be taught directly and instruction must be directed towards the gradual acquisition of understanding and expertise

Problem Solving and Creativity
We have explored problem solving by restructuring (Gestalt), searching (Newell & Simon), and analogy. What about creativity? Where the problem solver is not required to find a “correct answer” but to be creativity with possible solutions. Design fixation study – Jansson & Smith (1991) – data F 11.19 Fixated on what not to do as demonstrated by sample Fixation can inhibit problem solving

Demo in Figure 11.20, Finke (1990) had Ps choose 3 objects with eyes closed. Then with these 3 objects construct a “new” object. Finke (1990, 1995) asked Ps to pick 3 “preventive forms” - F 11.21, and make an unfamiliar objects. Experimenter provides an object’s category name - Table 11.1, and then Ps were to interpret his/her object accordingly. It was found that 360 objects were created 120 were judged as “practical inventions” 65 were judged as “creative inventions”. The study participants had no special aptitude in creativity. Creative thinking is ubiquitous.

Creativity has been characterized as more divergent thinking – open ended - than convergent thinking focus on specific solutions.

Figure 11. 19: (a) Sample design for coffee cup
Figure 11.19: (a) Sample design for coffee cup. (b) Percentage of designs with straws and mouthpieces for the control group (C), which didn’t see the sample design, and the fixation group (F), which did (Jansson & Smith, 1991). Figure 11.19: (a) Sample design for coffee cup. (b) Percentage of designs with straws and mouthpieces for the control group (C), which didn’t see the sample design, and the fixation group (F), which did (Jansson & Smith, 1991).

Figure 11.20 Objects used by Finke (1990).

Figure How a preinventive form that was constructed from the half-sphere, wire, and handle can be interpreted in terms of each of the eight categories in Table 11.2 (Finke, 1995).

Problem Solving and the Brain
Bowden and Beeman (1998; 2002) When solving insight-like problems (remote associates), subjects show a sudden increase in neural activity in the right temporal lobe immediately before solving a problem (this activation is not shown for analytic problems). Lavric, Forstmeier, and Rippon (2000) In an ERP( event related potential) study, the P300 wave (related to cognitive processing demands), was more frontal for an insight problem than for an analytic problem. Nichelli et al. (1994) found in a study of chess player using PET scans, that different locations of the cortex were more active depending on the task. See Slide 26

Problem Solving and the Brain
Study of brain damage patients have difficulties with problem solving due to perseveration. Patients with damage to the PFC(prefrontal cortex) has Difficult with the rules of a card-sorting task Difficulties with planning Decrease performance on tasks like the Tower of London (a task similar to Towers of Hanoi) and other problems. The PFC is important to connect different parts of stories determine the steps required to solve a problem

Areas of the brain that are activated by playing chess, according to Nichelli et al. (1994).

Sleep Inspires Insight
Sleep facilitates discovering hidden structure needed to solve problems Sleep is an effective study break! Restorative Helps consolidate information

Figure 11.22: Percentage of problems solved by three different groups (see text). Performance is best for participants who “slept on it.” (Based on data in Wagner et al., 2004.) Figure 11.22: Percentage of problems solved by three different groups (see text). Performance is best for participants who “slept on it.” (Based on data in Wagner et al., 2004.)

The End Reference: - Cognitive Psychology: Connecting mind, research, and everyday experience, 2nd & 3rd edition by E. Bruce Goldstein.

Problem Solving Vudhichai Boonyanaruthee, MD Department of Psychiatry

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