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Chapter 9
Development of
Problem-Solving Skills

Thought and Knowledge: An Introduction to Critical Thinking (5th Edition)

Diane F. Halpern

What is a Problem?

  • A Problem is defined as a state where there is a gap or barrier between where the problem solver is and where he or she wants to be
  • Problems have different parts, and problem-solving occurs in stages
  • There are many ways to increase the success of problem-solving that involve increasing alternatives generated to solve the problem and using multiple ways to understand the problem

  • A visual representation of a “problem”

Figure 9.1 A visual representation of a “problem.” To solve the problem, you need to find a way.

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Anatomy of a Problem

  • Newell & Simon (1972) have conceptualized all problems as consisting of parts or components:
  • An initial state, a goal state, and solution paths that link the initial state to the goal state
  • A problem is solved when the problem solver can find “paths” from the initial state to the goal state

Anatomy of a Problem

  • The Initial State is the beginning place in a problem
  • The Goal State is the desired end state in a problem (the problem is solved with paths ending up in the goal state)
  • Solution Paths are methods or means for solving problems; paths from initial to goal states
  • All possible paths from the initial to the goal state in a problem make up the Problem Space
  • Givens are information or rules that place constraints on the problem

Situation Awareness

  • Most real-life problems occur in a context, where a single occurrence or sequence of events creates a problem, sometimes called a situation
  • Situations are different from problems in a math or physics class
  • Contemporary theories of problem solving consider how people gather and use information to handle a situation and implement actions to solve the problems they raise

Situation Awareness (Wiig, 2003)

  • You are a customer service manager and an important customer returns a high-technology piece of equipment because it is not working correctly
  • You recognize that there is a problem with the technology and then have to decide what to do about it.
  • A guiding principle in your decision is that your company is committed to being a leader in your industry; you offer high quality products and service (“givens” in this problem)
  • You decide that the technical problems are serious, and it follows that your company needs to redesign the product

Situation Awareness (Wiig, 2003)

A model of problem solving in which the “problem” occurs in a context in which the problem solver needs to recognize a “situation”

Figure 9.3 A model of problem solving in which the “problem” occurs in a context where the problem solver needs to recognize a “situation” that needs action, and bases that action on a series of steps in which she makes sense out of the situation, decides what action to take based on guiding principles, and then implements the action. Figure from Wiig, K. M. (2003). Journal of Knowledge Management, Vol 7, No 5 (2003), pp 6-24, Knowledge Research Institute. www.krii.com.

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Stages in Problem-Solving

  • 1. Preparation or Familiarization Stage
  • This stage includes the time spent in understanding the nature of the problem, the desired goal, and the givens
  • 2. Production Stage
  • During this stage, the problem solver produces solution paths that define the problem space

Stages in Problem-Solving

  • 3. Judgment or Evaluation Stage
  • The problem solver evaluates solution paths to select the best one
  • Incubation, a period in problem-solving when the problem solver is not actively working on the problem
  • Sometimes people report that a solution comes to them during this “time-out” period
  • Insight is the sudden knowledge of a solution to a problem (the “Aha!” experience)

Persistence

  • Persistence is probably the most important variable in determining success in problem solving
  • An individual who persists at a problem is much more likely to solve it than an individual who gives up
  • Persistence is close to Levine's (1994) idea of "intimate engagement" (p. 3), the willingness to work on a problem in an involved and concentrated way.

Persistence

  • Grit, defined as “perseverance and passion for long-term goals” (Duckworth et al., 2007) is critical for success in many fields
  • Gritty people stay on course despite the inevitable failures, disappointments, and boredom
  • It is a trait found in high achieving individuals, ranging from world class athletes, spelling bee champions, investment bankers, and successful students

Well-Defined and Ill-Defined
Problems

  • Well-Defined Problems are problems with a single correct answer
  • Ill-Defined Problems are problems with many possible correct answers
  • The difficulty with ill-defined problems lies in evaluating possible solutions to decide which one is best
  • Often the goal in these problems is vague or incomplete

Problem Planning and Representation

  • Stating a goal in many different ways results in multiple goal states
  • This strategy is Transcontextual, meaning it is a useful tool in any context with any sort of problem, including well-defined problems
  • The generation of multiple goal states increases the size of the problem space (more solution paths) and provides more opportunities for finding a good solution

Problem Planning and Representation

  • Fear of crime is an important and ill-defined problem
  • Restate the goal in multiple ways to generate different solutions:
  • Goal #1: Reduce crime
  • Possible solutions:
  • Make capital punishment a national law
  • Incarcerate criminals for life if they are convicted of three major crimes

Problem Planning and Representation

  • Goal #2: Make life safer for honest citizens
  • Possible solutions:
  • Provide better security for honest citizens
  • Organize anticrime groups in every neighborhood
  • Goal #3: Reduce the number of criminals
  • Possible solutions:
  • Send criminals to Siberia
  • Return to gallows and public floggings to send a strong anticrime message to would-be criminals
  • Begin community programs that act as a deterrent to criminality (e.g., improve education and sports programs)

Problem Planning and Representation: Visual Representations

  • The best way to solve a problem is to devise the best representation
  • Good visual representations can help readers comprehend difficult information, especially textual information
  • Visual and verbal information work together to enhance memory
  • Use multiple representations – add diagrams to printed text and use verbal descriptors with spatial information

Problem Planning and Representation: Draw A Graph or Diagram

  • A venerable old monk leaves the monastery at exactly 6am to climb a winding mountain trail to the solitude of the mountain peak. He arrives at exactly 4pm. After spending the night in sleep and prayer, he leaves the mountain peak at exactly 6am and arrives at the monastery at exactly 4pm. There are no constraints on the speed at which he walks. In fact, he stops several times along the way to rest.
  • Is there some point on the mountain trail that he passes at exactly the same time each day?

  • GRAPHS OF THE MONK’S ASCENT AND DESCENT; SUPERIMPOSED GRAPHS

Figure 9.5 Graphs of the monk’s ascent and descent. The graphs can take any shape because the monk can rest as often as he wishes when he climbs the hill and when he descends.

 

Figure 9.6 By superimposing the ascent and descent graphs, it is easy to see that there must be some place at which the graphs intersect. Thus, there must be a place on the mountain trail that the monk crosses at the same time each day.

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Problem Planning and Representation: Draw A Graph or Diagram

  • To save money and their sanity, Melvin, Brock, Marc, and Claire decide to form a baby-sitting cooperative. They agree to baby-sit for each other’s children with the understanding that when one of them stays with another’s kids, the recipient will repay the sitter with an equal number of baby-sitting hours. They tally hours at the end of the month. This month, Melvin sat with Brock’s kids 9 hours, Marc sat with Melvin’s kids 3 hours, and Claire sat with Melvin’s kids 6 hours. Marc babysat 9 hours with Claire’s children, and Brock baby-sat 5 hours with Claire’s kids.
  • Which of these people has 12 hours of babysitting time due him/her?

Problem Planning and Representation: Draw A Graph or Diagram

  • Begin with a simple diagram illustrating one identified relationship

9 hours

Brock Melvin

  • Add additional relationships in steps, as you add each person

ALTERNATIVE FORM REPRESENTATION (BAR GRAPH OF BABYSITTING HOURS)

Figure 9.7 Alternative form representation.

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TABLE FORMS OF BABYSITTING CO-OP PROBLEM

Table 9.1. A Table Format for Representing the Information in the Babysitting Co-op Problem.

 

Table 9.2. An Alternative Table Format That Can Be Used to Represent the Information in the Babysitting Co-op Program.

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Problem Planning and Representation: Draw A Graph or Diagram

  • Hierarchical Trees are branching diagrams that are useful in solving problems
  • Larger categories of information provide the “nodes” of the trees
  • Hierarchies are most frequently used to assess mathematically the probability or likelihood of uncertain outcomes

HIERARCHICAL TREE DIAGRAM

Figure 9.8 A hierarchical tree diagram of one subject’s attempted solutions to Duncker’s Xray problem. (After Duncker, 1945).

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Problem Planning and Representation: Draw A Graph or Diagram

  • A Matrix is a rectangular array of facts or numbers
  • A matrix is a chart that can be used when the givens in a problem can be broken down into categories
  • Use a matrix to solve the next problem

Problem Planning and Representation: Draw A Graph or Diagram

  • Three men – Fred, Ed, and Ted – are married to Joan, Sally, and Vicky, but not necessarily in that order. Joan, who is Ed’s sister, lives in Detroit. Fred dislikes animals. Ed weighs more than the man who is married to Vicky. The man married to Sally breeds Siamese cats. Fred commutes over 200 hours a year from his home in Ann Arbor to his job in Detroit.
  • Match up the men with the women they married

Problem Planning and Representation: Draw A Graph or Diagram

Joan Sally Vicky

Fred

Ed

Ted_________________________________

Enter NO everywhere you can based on given

information

Problem-Solving Strategies

  • A Means-End Analysis is a general problem solving strategy in which operations are used to reduce the distance between the problem solver’s current state and the nearest possible goal or subgoal
  • When difficult is encountered in solving a problem, it can be broken down into several smaller problems or Subproblems
  • Each subproblem has its own goal called a Subgoal

  • TOWER OF HANOI PROBLEM AND SOLUTION

Figure 9.10 Start state and goal for the Tower of Hanoi problem. Use the strategy of meansends analysis to solve this problem.

 

Figure 9.11 Solution to the Tower of Hanoi problem. Notice how subgoal are planned and obtained in reaching the goal.

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Problem-Solving Strategies

  • Working Backwards is a problem-solving strategy in which operations are planned that move from the goal to the present or initial state
  • A Forward-Looking Strategy is a problem-solving strategy in which all of the planning is done by considering the operations that move the problem solver closer to subgoals and the goal

  • WORKING BACKGROUND WITH THE GARDEN HOSE

Figure 9.12 Working backward is a good strategy when there are fewer paths from the goal than from the start.

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Problem-Solving Strategies

  • Generalization is a problem-solving strategy in which the problem is considered as an example of a larger class of problems
  • Specialization involves considering the problem as a special case drawn from a larger set of problems

  • TREE DIAGRAM FOR DESIGNING A CHAIR FOR READING

Figure 9.13 Possible tree diagram for the problem of designing a chair for reading.

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Problem-Solving Strategies

  • Random Search is a problem-solving strategy in which all possible solution paths from initial to goal states are considered in an unsystematic (random) manner
  • Trial-and-Error is a problem-solving strategy in which all solution paths from initial to goal states are searched systematically
  • Hints are additional information that is given after an individual has begun to work on a problem

  • THE TWO-STRING PROBLEM

Figure 9.14 The two-string problem. How could you reach both strings at the same time? (After Maier, 1931).

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Problem-Solving Strategies

  • The Split-Half Method is a problem-solving strategy that is useful when there is no a priori reason for selecting among a sequentially organized set of possible solution paths
  • The method involves continually selecting a point that is halfway between the present state and the goal as a systematic means for “guessing” at the solution

Problem-Solving Strategies

  • Brainstorming is a group or individual method for generating solutions paths to problems
  • Problem solvers are encouraged to think up imaginative solutions and defer judgment on solutions until a later time
  • The goal is to produce a large number of possible solutions

Problem-Solving Strategies

  • Sometimes problems involve contradictions
  • The pizza box needs to keep the pizza hot, but allow air to circulate so the pizza doesn’t get soggy
  • A desk needs to have lots of surface area, but can’t take up too much space or be to heavy
  • Thinking of contradictions increases the number of possible solutions we generate for a problem, increasing the likelihood of a good solution

Problem-Solving Strategies

  • The use of analogies is a problem-solving strategy in which similarities are noted between two or more situations, while simultaneously discerning that there are also differences
  • E.g., by noting similarities between two different problems, the problem solver may discover that similar solutions are applicable

Problem-Solving Strategies

  • Four types of analogies useful in solving problems (Gordon, 1961):
  • A Personal Analogy is a problem-solving strategy in which you think of yourself as a participant in the phenomenon that you want to understand
  • A Direct Analogy is a problem-solving strategy in which you note similarities between your problem and related problems in other domains

Problem-Solving Strategies

  • A Symbolic Analogy is a problem-solving strategy involving the deliberate use of visual imagery or other symbolic representations
  • A Fantasy Analogy is a problem-solving strategy in which problem solvers utilize their imagination to conceptualize ideal solutions

Problem-Solving Strategies

  • Experts are often helpful in solving problems, especially if the problem is very complex and involves expertise and skills that we do not have
  • This is the idea behind crowdsourcing, which is posing a job or a problem to a group of people, usually by posting it to the internet
  • The advantage of crowdsourcing is that it attracts people with a broad range of information, experience, and knowledge
  • Groups have a collective intelligence, which depends much less on intelligence of group members than it does on how they relate to each other
  • Research indicates that social sensitivity is important (Wooley et al., 2010)

Problem-Solving Problems

  • Mental Sets are predispositions to respond to the world in certain ways
  • Functional Fixedness is a type of mental set in which individuals only consider the usual use (function) of objects
  • Worldview Constraints involve limitations on the way we approach problems placed on us by our social class, nationality, or political views