Problem Solving Notes

Problem Solving

• 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. (Duncker, 1945; Lovett, 2002)
• A problem, as characterized by psychologists, is difficult, and the solution is not immediately obvious.

The Gestalt Approach: Problem Solving as Representation and Restructuring

• Gestalt psychologists were interested not only in perception, but also in learning, problem solving, and even attitudes and beliefs (Koffka, 1935).
• Problem solving, for the Gestalt psychologists, was about:
  • How people represent a problem in their mind.
  • How solving a problem involves a reorganization or restructuring of this representation.

Representing a problem in the mind

• Success in solving a problem is influenced by how it is represented in the person's mind.
• Circle Problem:
  • The key to solving the circle problem is to recognize x as the diagonal of a rectangle within the circle.
  • Once x is recognized as the diagonal of the rectangle, the representation can be reorganized by creating the rectangle's other diagonal.
  • The length of x equals the length of the radius, r.
  • The solution is obtained by first perceiving the object and then representing it in a different way.
• The Gestalt psychologists called the process of changing the problem's representation restructuring.

Restructuring and insight

• Restructuring is associated with insight— the sudden realization of a problem's solution.
• When people solve problems they are experiencing insight because the solutions usually seem to come to them all of a sudden.
• Metcalfe and Wiebe (1987) Experiment:
  • Compared participants' experiences when they were solving insight and non-insight problems.
  • Participants working on an insight problem, in which the answer appears suddenly, should not be very good at predicting how near they are to a solution.
  • Participants working on a non-insight problem, which involves a more methodical step-by- step process, would be more likely to know when they are getting closer to the solution.
  • Participants made "warmth" judgments every 15 seconds as they were working on the problems.
  • Insight problems:
    • Warmth ratings began at 2 and then didn't change much, until all of a sudden, they jumped from 3 to 7 at the end.
    • 15 seconds before the solution, the median rating was a relatively cold 3 for the insight problems, meaning that at this point participants still didn't feel they were close to a solution.
  • Algebra problems:
    • The ratings began at 3 and then gradually increased until the problem was solved.
  • Solutions for insight problems do, in fact, occur suddenly, as measured by people's reports of how close they feel they are to a solution.
• One weakness of the Metcalfe and Wiebe study is that it relies on the assumption that people have sufficient metacognitive ability to monitor their own problem solving process.
• Another critique is that the measures they used did not provide evidence for restructuring, which is the other defining characteristic of insight.
• Ash and Wiley (2008) found that individuals who successfully solved insight problems produced different importance-to-solution ratings for a specific set of problem elements pre- and post- solution.
• Cushen & Wiley (2012) concluded implicit cues would make the problem less of an insight problem, and accordingly, it was expected that individuals who received these cues would show a more linear solution pattern than non-cued individuals.
• Modern studies support the Gestalt psychologists' belief that restructuring plays a key role in solving insight problems (refer also to Danek, Williams, & Wiley, 2020).
• The Gestalt psychologists also influenced the study of problem solving in another way, based on their strategy to devise problems and situations that made it difficult for people to achieve the restructuring that is needed to solve the problem.

Obstacles to problem solving

• One of the major obstacles to problem solving, according to the Gestalt psychologists, is fixation- people's tendency to focus on a specific characteristic of the problem that keeps them from arriving at a solution.
• One type of fixation that can work against solving a problem is focusing solely on familiar functions or uses of an object; this is called functional fixedness (Jansson & Smith, 1991).
  • Duncker's Candle Problem:
    • Participants presented with boxes as containers found the problem more difficult than did the group presented with the empty boxes.
    • Seeing the boxes as containers inhibits the idea that you can also use them as supports is an example of functional fixedness.
  • Maier's two-string problem:
    • Participants' task was to tie together two strings that were hanging from the ceiling.
    • To solve this problem, participants needed to tie the pliers to one of the strings to create a pendulum, which could then be swung to within the person's reach.
    • Functional fixedness may prevent finding a solution, because people usually think of using pliers as a tool, not as a weight at the end of a pendulum.
    • The solution to the problem occurred once the participants restructured their representation of how to achieve the solution (get the strings to swing from side to side) and their representation of the function of the pliers (they can be used as a weight to create a pendulum).
• Mental set, a preconceived notion about how to approach a problem, which is determined by a person's experience or what has worked in the past.
  • Children (with less "knowledge" and experience) are sometimes better in solving these kind of problems.
  • German and Defeyter (2000) showed that functional fixedness was present in older children but not in 5-year-olds.
• Luchins' classic water jug problem
  • The three jugs had the followin capacities: A = 21 quarts, B = 127 quarts, C = 3 quarts, and the desired volume was 100 quarts.
  • The solution for Problem 1 can be stated as Desired amount = B - A - 2C.
  • Problems 7 and 8 could also be solved in a much simpler way, namely as follows:
    • Problem 7: Desired quantity = A + C (Fill A and C and pour into B.)
    • Problem 8: Desired quantity = A - C (Fill A and pour into C.)
  • The result of this experiment was that only 23 per cent of the participants in the mental s group used the simpler solutions for Problems 7 and 8, whereas all of the participants in the mental set group used the simpler solutions.
• Mental set can influence problem solving both because of preconceptions about the functions of an object (candle and two-string problem) and because of preconceptions about the way to solve a problem (water jug problem).
• Acquired knowledge can often benefit problem solving in common situations it can also work counterproductively if a novel situation presents itself in which one needs t be more cognitively flexible (Blech, Gaschler, & Bilalic, 2020).
• One such training programme is mindfulness, in which participants are taught to purposefully pay attention to the present moment in a non-judgmental way.
  • Greenberg, Reiner and Meiran (2012), typified this as adopting a "beginner's mind,"
  • Experienced mindfulness meditators were better able than a waiting-list group to identify the simple novel solution in the last problems in a series (in their case more than two against their mental set.
  • The positive the between mindfulness and more flexible problem solving has also been confirmed by a study by Ostafin and Kassman (2012), who found that individuals with greater mindful awareness were better able to solve insight problems for which it was needed to overcome habitual responses.

The Information- Processing Approach: Problem Solving as a Search Process

• Newell and Simon developed a "logic theorist" computer program that was designed to simulate human problem solving.
  • This marked the beginning of a research line that viewed problem solving as a process that involves search.

Newell and Simon's approach

• Newell and Simon (1972) saw problems in terms of an initial state— conditions at the beginning of the problem-and a goal state-the solution of the problem.
• Operators- actions that take the problem from one state to another.
• A problem starts with an initial state, continues through a number of intermediate states, and finally reaches the goal state.
• The initial state, goal state and all the possible intermediate states for a particular problem, make up the problem space.
• Means-end analysis:
  • The primary goal of means-end analysis is to reduce the difference between the initial and goal states.
  • This is achieved by creating subgoals — intermediate states that are closer to the goal.
• One of the main contributions of Newell and Simon's (1972) approach to problem solving is that it provided a way to specify the possible pathways from the initial to the goal state.
• The characteristics of the problem space (e.g., length and breadth) provided thereby an objective way of specifying problem difficulty.

The importance of how a problem is stated

• How a problem is stated can affect its difficulty.
• The key to solving the mutilated checkerboard problem is understanding the principle that each domino covers two squares and that these squares must be of different colours, so removing the two corner squares with the same colour makes it impossible to cover the board with 31 dominos.
• Kaplan and Simon (1990) hypothesized that versions of the mutilated checkerboard problem that were more likely to lead participants to become aware of this principle would be easier to solve.
• All four versions of the checkerboard problem have the same board layout and the same solution and hence the same problem space.
• Participants in the bread-and-butter group were found to solve the problem twice as fast as those in the blank group and also managed to finish the task with relatively little help from the experimenter (one hint on average).
• Solving a problem becomes easier when information is provided that helps point people toward the "correct" representation of the problem.
• Think-aloud protocol:
  • By recording people's thought processes as they are ving a problem, the think-aloud protocol reveals a shift in how a person perceives elements of the problem.
  • This is very similar to the Gestalt psychologists' idea of restructuring and the portance-to-solution ratings we discussed earlier (Ash & Wiley, 2008; Cushen & Wiley, 2012).
• The Russian Marriage Problem and the Mutilated Checkerboard Problem:
  • People who read this story are usually able to solve the mutilated checkerboard problem if they realize the connection between the couples in the story and the alternating squares on the checkerboard.
  • This process of noticing connections between similar problems and applying the solution for one problem to other problems is called the method of analogy.

Using Analogies to Solve Problems

• This technique of using an analogy -that is, using the solution to a similar problem to guide solution of a new problem-is called analogical problem solving.
• Research on analogical problem solving aims to specify the circumstances under which the analogy-based approach would be effective and how it can be promoted.

Analogical Transfer

• This transfer from one problem to another is called analogical transfer.
  • Teachers use examples and specific cases in the hope that students will use them as models or analogies for future situations (Gentner, Loewenstein, & Thompson, 2003; Vendetti, Matlen, Richland, & Bunge, 2015).
  • Analogical transfer does not happen automatically.
• Two key terms that are used in research on analogical transfer are target problem, which is the problem the participant is trying to solve, and source problem, which is a similar type of problem in the recent past (or the one that has been shown by the teacher) that illustrates a way to also solve the target problem.
• For the mutilated checkerboard problem described above, the checkerboard problem was the target problem, and the Russian marriage problem was the source problem.
  • Evidence that analogical transfer has occurred in this case was provided by the fact that presentation of the Russian marriage problem enhanced the ability to solve the mutilated checkerboard problem.

Analogical problem solving and the Duncker radiation problem

• This problem is described in the following demonstration. Please try to solve this problem first before reading on.
• When Duncker (1945) originally posed this problem, most of his participants could not solve it, and Gick and Holyoak (1980, 1983) found that only 10 per cent of their participants arrived at the correct solution.
• The solution is to bombard the tumour with a number of low-intensity rays from different directions, which destroys the tumour without damaging the tissue that the rays are passing through.

Effect of making surface features more similar

• The lightbulb problem is a problem with surface features similar to the radiation problem.
• Participants in one group were taught about the radiation problem and its solution in an introductory psychology class, just prior to being given the lightbulb problem.
• Participants in the control group did not know about the radiation problem.
• 81 per cent of participants who knew about the radiation problem solved the lightbulb problem, compared to only 10 per cent of the participants in the control group.

Effect of varying the structural features

• Structural features are the underlying principle that governs the solution.
• The Fragile-Glass Version structural features are similar to the Radiation Problems.

Analogical Encoding

• Analogical encoding is the process by which two problems are compared and similarities between them are determined.
• When learners actively compare two examples, they come to understand the underlying structure common to both.
• "Learners who compare cases will develop a more general problem-solving schema that primarily captures the common structure of the cases rather than the surface elements" (Gentner et al., 2003, p. 394).
• Analogical encoding is believed to promote both recall and transfer (note that analogical encoding can also be seen as a special case of deep level processing, as discussed in Chapter 7).
• The underlying principle of analogical encoding is demonstrated in several experiments by Gentner et al. which involved several negotiation problems.
• The compromise strategy is typically adopted by novice negotiators.
• Having people compare source stories is an effective way to achieve analogical encoding because it forces them to pay attention to structural features that enhance their ability to solve other problems.
• Making comparisons promotes schema abstraction and transfer" (Gentner et al., 2003, p. 400).

Analogy in the Real World

• Dunbar (2001) has called the analogical paradox: While it is difficult to apply analogies in laboratory research, people routinely use analogies in real-world settings.
• Dunbar studied the use of analogies in real-world settings using a technique called in vivo problem-solving research.

How Experts Solve Problems

• Experts are people who are socially recognized as being extremely knowledgeable or skilled in a particular field, like chess, music, physics, etc.
• Based on this example of the mythical 10,000 hour rule, can it be assumed that practice and experience are necessary and sufficient to achieve expertise level in all cases?

Differences between how experts and novices solve problems

• Experts usually solve problems faster and with a higher success rate than do novices, people who are beginners or who have not had the extensive training of experts (Chi, Glaser, & Rees, 1982; Larkin, McDermott, Simon, & Simon, 1980).

Experts possess more and better-structured knowledge about their fields

• Chase and Simon (1973a, 1973b) compared how well a chess master with more than 10,000 hours of experience, and a beginner with fewer than 100 hours of experience, were able to reproduce the positions of pieces on a chessboard after looking at an arrangement for five seconds.
  • Experts excelled at this memory task when the chess pieces were arranged in actual game positions but were no better than the beginners when the pieces were arranged randomly.
  • The chess masters in this situation could make use of the patterns they had stored in their long-term memory based on the many games they played.
• Novices have schemas that involve relatively few, disconnected knowledge elements that are organized according to superficial similarities.
• Experts possess schemas that involve large, highly inter-connected units of knowledge, organized according to underlying structural similarities (Bryson, Bereiter, Scarmadalia, & Joram, 1991).
• Chi et al. (1982) presented 24 physics problems to a group of experts (physics professors) and a group of novices (students with one semester of physics) and asked them to sort the problems into groups based on their similarities.
  • The novice sorted the problems based on surface characteristics, such as how similar the objects in the problem were.
  • The expert, in contrast, sorted problems based on structural features, such as general principles of physics.

Experts spend more time analyzing problems

• Experts often get off to what appears to be a slow start on a problem, because they spend time trying to understand the problem rather than immediately trying to solve it (Brand-Gruwel, Wopereis, & Vermetten, 2005).
• Chi et al. (1982) suggest that experts therefore seem to be able to work forward, going from "What do I know?" to "What do I need to find out?"
• Novices often seem to work the other way around, meaning means end analysis (refer to earlier in this chapter under Newell and Simon's approach), which takes up a lot of time.

Expertise is only an advantage in the expert's speciality

• Experts are experts only within their own field and perform like anyone else outside of their field (Bedard & Chi, 1992).
• One disadvantage is that knowing about the established facts and theories in a field may make experts less open to new ways of looking at problems (refer to functional fixedness, in the Obstacles to problem solving section, earlier in this chapter).
• Being an expert may actually be a disadvantage when being confronted with a problem that requires flexible thinking—a problem whose solution may involve rejecting the usual procedures in favour of other procedures that might not normally be used (Frensch & Sternberg, 1989).

Creative Problem Solving

• If we define creativity as producing original answers, or as being able to come up with multiple solutions to a problem the student was being creative.

What is creativity?

• Creativity is one of those terms, like intelligence, for which many definitions have been proposed.
• Many examples of creativity focus on divergent thinking-thinking that is open-ended, involving a large number of potential "solutions" (although some proposals might work better than others; refer to Guilford, 1956; Ward, Smith, & Vaid, 1997).
• Kaufman proposes that in addition to being original, a creative response to a problem must be useful.
• This approach to creativity is captured in the definition of creativity as "anything made by people that is in some way novel and has potential value or utility" (Smith, Kerne, Koh, & Shah, 2009).

Practical Creativity

• Many examples of how inventions were created involve analogical thinking, in which observing a phenomenon has led to a new, novel and useful solution to a practical problem.

Examples of Creative Analogical Problem Solving

• Velcro:
  • George de Mestral inspected the burrs under a microscope to design a fabric fastener with many small hooks on one side and soft loops on the other side.
• Odón Device:
  • Jorge Odón, designed a device to deal with the life-threatening situation of a baby stuck in the birth canal during delivery.
  • The beginnings of Odón's design can be traced to viewing a YouTube video that demonstrated how to remove a cork that had been pushed inside a wine bottle (refer to Dvorak Uncensored, 2007).

Problem Solving as a Process

• Most creative problem solving includes far more than just getting an idea. It also involves a lengthy period of trial-and-error development to turn the idea into a useful device.
• Creative problem solving as a four-stage process that begins with generation of the problem and ends with implementation of the solution(Basadur, Runco, & Vega, 2000).
• Realizing that there is a problem in the first place, which then leads to ideas, which are evaluated and eventually turned into a product (also refer to Finke, 1990; Mumford, Medeiros, & Partlow, 2012).
• The Wright brothers' example also illustrates that problem solving is not simply about getting an idea in a flash of insight, although that may happen, but about having a base of knowledge that makes the idea possible.
  • The Wright brothers were successful because their knowledge of physics and mechanics, plus their extensive experience with bicycles in their bicycle repair shop, provided a basis for their creative ideas about how to combine a number of components to create an airplane.

Generating Ideas

• The example of the Wright brothers illustrates that ideas depend on having a base of knowledge.
• As important as knowledge is, sometimes too much knowledge can hinder creative problem solving.
• Odón (who obviously is not a doctor, developed a device for birthing. As one of the doctors who had worked with Odón said, "Doctors are very structured in their thinking and Jorge is a free mind, he can think of new things" (Venema, 2013).
• Smith, Ward and Schumacher (1993), who showed that providing examples to people before they solve a problem can influence the nature of their solutions.
• Compared with the life forms created by a control group that had not seen any examples, the designs generated by the example group incorporated many more of the example features.
• The greater use of these features by the example group is related to the idea of functional fixedness, described earlier in the chapter.
• Functional Fixedness leads to preconceptions that can inhibit creativity.
• Brainstorming:
  • Osborn (1953) proposed the technique of group brainstorming. The purpose of this technique is to encourage people to freely express ideas that might be useful in solving a particular problem.
  • Instructions given to participants in brainstorming groups emphasize that they should just say whatever ideas come into their mind, without being critical of their own ideas or of the ideas of others in the group.
  • Research has shown that placing people in groups to share ideas, results in fewer ideas than adding up the ideas generated by the same number of people asked to think of ideas individually (Mullen, Johnson, & Salas, 1991; Stroebe, Nijstad, & Rietzschel, 2010).
• Individual brainstorming to generate ideas, however, can be effective.
• One method of individual idea generation that does seem to work has been proposed by Ronald Finke, who developed a technique called creative cognition to train people to think creatively (Finke, 1990, 1995).
• Finke called these "inventions" pre-inventive forms because they are ideas that precede the creation of a finished creative product.

Effects of Mood and Physical Exercise

• The idea behind individual brainstorming and Finke's creative cognition technique -to keep the mind open— is to avoid fixations that limit creativity.
• One factor that may facilitate creative thinking is being in a positive mood.
  • Baas, De Drue and Nijstad (2008) reviewed 25 years of research on the mood-creativity relationship and concluded that creativity is enhanced by positive and "cheerful" mood states that are both activating and associated with an approach motivation, like happiness.
• Other factors that are believed to facilitate creative thinking are physical activity (e.g., Steinberg et al., 1997) or being in a nature environment (Tyrväinen et al., 2014).
• Even walking on a treadmill has been found to facilitate creative or divergent thinking, as measured by the alternate uses test (Oppezzo & Schwartz, 2014).