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The Fourth Way of Learning
LEARNING TO SOLVE PROBLEMS AND MAKE DECISIONS
Using Mental Models for Learning

HOW DO YOU REACT when someone gives you a problem? Try this one (Fixx, 1978)¹:

If you have black socks and brown socks in your drawer—mixed in a ratio of four to five, how many socks will you have to take out to make sure of having a pair of the same color?

Are you eager for the challenge posed by a problem? Do you know how to approach problems? Does a chapter that begins like this make you want to turn to the next chapter?

Problems—big and small—are everywhere. In the workplace they arise from things that go wrong, but they are also embedded in opportunities. A problem is a question proposed for solution or discussion—usually a matter involving doubt, uncertainty, or difficulty. Most problem solving and decision making gets complicated. Usually several considerations enter the picture at once, and it becomes difficult to keep them straight. You can feel like a juggler trying to keep all the tenpins in the air, or a circus performer trying to spin a dozen plates at once. Such an act “boggles the mind,” we say, and to boggle, the dictionary tells us, is to alarm, astound, shock, or stagger. So the mind must find some alternative to getting boggled. Psychologists call this boggling cognitive overload. The mind needs some system for dealing with the complexity posed by problem solving and decision making. This is why we turn to mental models.

The first thing you can do, when the agenda calls for problem solving, is to maintain your composure and avoid panic. Frustrated by years of struggling with homework problems, and because there was no school subject called “problems in general,” many people have come to hate problem solving. Without workable problem-solving strategies, we experience high levels of frustration. What we learn from this frustration is to avoid problems, what Brandsford and Stein (1993, 8–9)² call the “let me out of here approach” to problem solving. The first step is to persist.

AVOIDING A BOGGLED MIND
Understanding Mental Models

We use mental models all the time in our daily lives to develop simplified pictures of how things work—such as the human digestive process, the clutch on a car, or the orbiting of the planets around the sun. “The mental pictures we form of the component parts of these systems and how these parts interact are called mental models” (Ward, et. al., 1995, 53).³ When we are asked, for example, which dogs have ears that stick up above the head, we don’t go through a memorized list, such as shepherd-up and beagle-down; we recall a mental image of what the dog looks like (Glucksberg, 1988).⁴

Most expert problem solvers use mental models to proceed through the various steps involved in solving a problem. Decision makers use mental models to weigh the options in a decision and to predict likely outcomes.

The mental models theory has a fascinating history. Some of the early research on apes done by Wolfgang Koehler suggests that even lower animals have problem-solving abilities. When given boxes and sticks to arrange in order to be able to go after a banana at the top of their cage, apes appear to have moments of insight (Dworetzky, 1985, 237–38).⁵ Were they using mental models? Modern research on problem solving proceeds from the work of Karl Dunker in 1945, who asked his subjects to think aloud as they attempted to solve problems so that he could track their techniques (Dellarosa, 1988).⁶ The classic work on problem-solving theory is Allan Newell and Herbert Simon’s Human Problem Solving (1972).⁷ Decision-making theory has its roots in the philosophy of Blaise Pascal (1623–1662), who developed what has come to be regarded as the first decision analysis technique (Baron, 1994, 315).⁸

As with critical, creative, and dialogical thinking there is a growing awareness that the skills needed for problem solving and decision making can be learned. There are guidelines to follow and techniques to use. Practice in applying those guidelines and techniques usually comes through cases or projects that embody real world problems to be solved or decisions to be made. There are things you can do to maximize your learning in these settings.

ESTABLISHING A FRAMEWORK
A Basic Problem-Solving Model

Newell and Simon (1972, 53–63, 787–791)⁹ provide the classic problem-solving model now found in most texts on the subject, the components of which are reviewed briefly here.

• Goal state. A problem calls for a solution. It is important to have some idea of the goal, some picture of what things will be like when the problem is solved. What are the criteria against which a solution will be judged? What is the problem statement calling on you to do or to determine?

• Initial state. The conditions and information provided up front can be considered the “initial state.” What information do you have to work with when you start?

• Problem space. The gap between the goal state and initial state is the problem space, the bounded area where the problem can be worked out.

• Solution paths. These are the options generated as potential solutions to the problem. Solution paths are the ideas people generate to try to solve the problem.

• Operations. Certain mental activities often need to be performed to move from initial state to goal state. Mental models are often useful here because the operations can involve a flood of information and ideas.

• Barriers. The problem space presents certain barriers. It is not easy to move from initial state to goal state; if it were easy, there would not be a problem.

Newell and Simon’s general framework for thinking about problems is in itself a useful mental model into which more specific models fit. Using this framework is an essential first step.

GENERATING SOLUTIONS
Using Mental Models

What happens in the problem space is crucial, of course, and that is where mental models come into play directly. These are some of the mental models that can be used to attack problems directly.

Random search. This is sometimes called trial and error. Random search works when the number of options is small and every possible option can be examined directly. For example, in working with the anagram THA, there are only six possible solution paths for arranging the letters to make a word: THA, TAH, ATH, AHT, HTA, and HAT. The problem solver will arrive at HAT, eventually and inevitably, even if it is last, through trial and error. This model works in some situations, but if there are many potential solution paths and the paths themselves are complex, a better model is needed. Some would say that random search is not an intelligent approach to problem solving, and for that reason is not really a mental model (Halpern, 1984, 189).¹⁰

Hill climbing. Picture yourself blindfolded on a hill. Your goal is to get to the top. You start out in some direction and you get some feedback from your legs that this is downhill. You try again and start moving up. Your steps are small but you can see that you are getting closer to the goal. Physicians sometimes use the hill-climbing model to arrive at the right dose of medicine (Baron, 1994, 68).¹¹ With this model you say, “Let’s try this and get some feedback; then let’s try this.”

Means-ends analysis. If the goal is the end, and the means of getting there is not clear, it is sometimes useful to find subgoals and then devise the means for reaching these. A frequently cited example is the Tower of Hanoi Problem shown in the figure below (Halpern, 1984, 182-4).¹²

The object is to move three coins of one stack (quarter, nickel, penny, with the quarter on the bottom) from the first site to the third site and restack them in the same order at the third site by using the intermediary site for staging and moving only one coin at a time. For this problem it is best to work in steps. One subgoal is to get the quarter to the third site. Another subgoal is to get the nickel and penny off the quarter so it can be moved. That can be done by moving the penny to the third site, the nickel to the second site, and the penny back to the nickel. This leaves the third site vacant so the quarter can be moved to it. And so forth. The mental operations for many problems can be managed in this way by breaking the problem into subgoals and then working on them one at a time, step by step.

Working backward. The natural question for all of us to ask is, “What should I do first?” With certain kinds of problems, it may be best to ask, “What should I do last?” Next to last? And so on, back to the beginning. The paper and pencil mazes that children enjoy can be more easily solved by starting at the goal and working backward (Halpern, 1984, 184–5).¹³ Most event planning or project management works best by setting the final date and working backward through the supporting logistics.

Split-half method. You can guess the age of anyone under one hundred by asking the subject first if they are under fifty. If the answer is yes, the next question is whether the subject is over twenty five. The next question always splits the remaining amount in half until the answer is found. You will never need to ask more than seven questions to establish someone’s age using this method (Halpern, 1984, 192–3).¹⁴ Some problems can best be solved through a narrowing-down process.

Simplification. Some problems are so complicated, that it is useful to try to suspend the rules temporarily or cut out some of the details. (Wickelgren, 1974, 46–47).¹⁵ Ask yourself, “What is the essence of this problem, the main goal, the central outcome?” Try to reduce the problem to its simplest elements.

Using actual data. Many problems are hard to deal with in the abstract, but they sometimes become more manageable when actual numbers or objects are used (Wickelgren, 1974, 26).¹⁶ Problem solving often involves plugging in the numbers to see whether a proposed solution will or will not work.

Contradiction. One way to eliminate potential solution paths is to see if they are contradictory to the “givens” in the initial state, or incompatible with what might reasonably be expected in the goal state. In some cases an eyeball comparison or an estimate will generate the awareness, “It couldn’t be that!” (Wick-elgren, 1974, 109–10).¹⁷

Graphs and diagrams. Some problems tend to overwhelm us with information. Often it is necessary to put the data into a better semblance of order. This may mean transforming data provided in narrative form into written lists, charts, or diagrams (Halpern, 1984, 167–74).¹⁸ If you get the feeling that this is not the kind of problem you can solve in your head, start putting the information on paper in a way that enables a useful analysis.

Analogy. Perhaps the problem is analogous to another. Two problems that appear quite unrelated may have similarities when compared. Something learned from one problem may be applicable to another (Holyoak and Nisbett, 1988, 82–3).¹⁹ The challenge, of course, is to find and apply good analogies and to be able to see the connections.

WATCHING FOR TROUBLE
Avoiding Pitfalls

There are certain common pitfalls associated with problem solving, things to watch out for and avoid because they can cause trouble.

Misunderstanding the problem. People tend to rush into generating solutions before clearly defining the goal state or carefully analyzing the information provided in the initial state. Good information gets ignored and irrelevant information is regarded as valuable. When people rush into generating solutions to the sock problem, strange things happen. They have trouble seeing that the goal is to get one pair of socks of the same color guaranteed with the fewest picks. They think they need two pairs, or that the goal is to get the owner to reorganize his sock drawer so this problem will not occur. They make up all sorts of worries (are the socks the same size?) and rules (do I have to put each sock back in after I have taken it out?). They want to know if they can look into or dump out the whole drawer, and then sheepishly recognize that if they can, it won’t be a problem anymore. They tend to ignore the most important piece of information in the initial state (that there are only two colors) and they tend to focus on irrelevant and misleading information (that the ratio of one color to the other is four to five). The solution comes fairly quickly when the problem is clearly understood.

Unrecognized presuppositions. Sometimes we bring presuppositions to the problem that greatly limit the solutions we will consider. Notice in this example the narrowing effect of a presupposition (Sanford, 1985, 46)²⁰:

A man and his son were away on a trip. They were driving down the motorway when they had a terrible accident. The man was killed outright but his son was alive, though badly injured. The son was rushed to the hospital and was to have an emergency operation. In entering the operating theater, the surgeon looked at the boy and said, “I can’t do this operation. This boy is my son.” How can this be?

The boy can be the son if one sheds the presumption that a surgeon must be a male; the female surgeon could indeed be the boy’s mother.

Functional fixedness. When objects and concepts are defined in a rigid way, it is sometimes difficult to think of using them in any other way. A screwdriver, we know, is for twisting in screws, and it is hard to think of using a screwdriver for anything else because that is what it is for. If we stop thinking of it as a “screwdriver” and call it a “gadget,” that rethinking may open up many more possibilities for its use (Glucksberg, 1988, 225).²¹ The same is true for abstract concepts such as “cost” or “profit” or even “learning.”

FROM PROBLEM SOLVING TO DECISION MAKING
Another Model

As problems are solved and solutions are generated, someone must make decisions about whether the solutions are actually good solutions. As with problem solving, decision making can boggle the mind. It is important to follow a rational decision-making model and to evaluate regularly the results that the model is producing.

The classic mental model for decision making has ten key steps (Halpern, 1984, 1–5)²²:

1. Determine values. Begin at the end of the model and work backward. Values, in this instance, are statements of what has worth or utility. What values will this decision address?

2. Determine outcomes. What outcomes will fulfill the values? Outcomes are the specific results the decision will produce. Most decisions involve several desirable outcomes that reflect more than one set of values.

3. Weight the outcomes. Even if several outcomes are possible, they may not all be equally desirable. Outcomes can be ranked or assigned numerical weights that reflect importance. The weighting process establishes the relative importance of outcomes.

4. Generate options. Options can be plans, scenarios, products, services, or personnel. Options often grow out of problem solving. Not all decisions have good options, but a large number of suitable options improves the likelihood of good results.

5. Identify attributes of options. Options have attributes. For example, in a personnel decision, the candidates (options) usually have differing strengths and weaknesses (attributes). Having certain attributes may compensate for lacking others. What attributes of the options are most desirable?

6. Match attributes to outcomes. In a rational decision the attributes of the options are matched to the weighted characteristics of the outcomes.

7. Make a choice. After considering how various options will produce desired outcomes, make a choice and frame the choice as a decision rule; that is, as a recommendation.

8. Cast the choice as a probability and consider the consequences. Because no one knows how a decision will turn out, recommendations are usually presented as predictions cast as probabilities. What chance does this decision have for success? What is your willingness to bet on it?

9. Predict the likelihood of outcomes. In most decision making, people think the work is done when they have identified options. The hardest work may come in predicting outcomes. What risk is involved in this decision? What could happen?

10. Align the steps. Check the alignment of values, outcomes, and attributes of options involved in the actual choices. Do all the pieces fit together?

The steps of the decision-making model form a circle. The process starts with values and it ends up with values. One phase leads to another, but eventually the decision comes down to this: Which options with what attributes are most likely to result in the desired outcomes and values?

MORE TROUBLE
More Pitfalls

Like problem solving, decision making has its pitfalls. Here are some to avoid.

Wishful thinking. Sometimes known as the Pollyanna Principle, wishful thinking is the tendency to overestimate the chances of being successful. Wishful thinkers tend to overvalue the attributes of particular options, exaggerate the way options will attain outcomes, and overproject the probability of positive results (Halpern, 1984, 221–22).²³

Entrapment. Decisions exist within the context of other decisions, and one decision, especially a bad decision, can affect another. Sometimes previous decisions have turned out badly, and because the decision has already cost a great deal in time, money, and effort, the decision maker is trapped in that previous decision and may make still another bad decision to try to save the first (Halpern, 1984, 222ff).²⁴ The best way to avoid this pitfall is to view each decision separately and approach it on its own terms.

Trade-offs. Trade-offs occur when we are willing to give up one outcome for another or compensate one attribute with another. Trade-offs are sometimes necessary, but having made one compromise, there is a tendency to make others. Sometimes it is necessary to draw the line with regard to what will and won’t be given up in a decision (Baron, 1994, 346–49).²⁵

Gambler’s Fallacy. In so-called wheel-of-fortune games, there is a tendency to say “Number seven hasn’t come up lately, so it is about time for it.” In a truly random situation, however, every number on the wheel has a chance of coming up on every spin because the wheel has no memory (Halpern, 1984, 123²⁶; Baron, 1994, 229–30²⁷). In workplace decisions, probable outcomes are not likely to be random, like spinning a wheel, but people may still think, “We’re due,” “It’s our turn,” or “We’ve had our share of bad luck.” Irrational behavior quickly intrudes on the rational decision-making process.

Misinterpreting trends. Predictions of the outcomes of a decision will sometimes rest on trends. Trends are tricky because they can change as new factors come into play. Trends are sometimes part of larger or related trends. For example, there may be a trend for older women to buy more sneakers, but is that part of a trend toward more exercising in the population, part of a trend toward more informal dress, or are there simply more older women as a proportion of the total population? Decision makers need to be careful in their use of trend data (Halpern, 1984, 23ff).²⁸

LIVING LABORATORIES
Cases and Projects

The key to using mental models for problem solving and decision making is practice. It helps to learn about the steps and to become aware of different mental models, but there is no better way to learn problem solving than by solving problems and no better way to learn decision making than to make decisions. Usually these opportunities are provided through cases or equivalent real-life projects.

The idea of using cases for educational purposes has its modern roots in the case method of Harvard’s School of Business, begun in 1909 (Pigors and Pigors, 1987, 415).²⁹ Originally known as the laboratory method or problem method, the case method gives the learner a chance to “put themselves in the decision maker’s or problem solver’s shoes.” Cases have been called “the corpse for the student to practice on” (Leenders and Irskine, 1973, 110).³⁰

No doubt you have already worked with cases or projects, or perhaps you will seek out this kind of learning. As with the other ways of learning, there are things you can do to maximize your learning when the case method is being used:

• Be able to identify several types of cases and know which types are being used (Pigors and Pigors, 1987, 415–19).³¹ Traditional cases are presented in a written format with extensive background and current information, and may unfold over time through stages. Live cases are presented by a person who represents an organization. This person has lived through the case and makes a live presentation about it. The presenter may return at a later date to guide further discussion. Key incident cases involve a brief live presentation that stimulates questions, the answers to which provide the essential factual basis for the case.

• Read the case carefully or listen closely to the presenter. Get an overall understanding of the case. If the case poses one or more problems, begin to analyze those problems in terms of goal state, initial state, and problem space. Use mental models in the problem space to keep from getting a boggled mind. If the case presents the need for one or more decisions, try to apply the rational decision-making model.

• If the case involves discussion, listen for a while to the chatter. Some participants are bound to jump in with all their biases, assumptions, and gut-level reactions. Others will have important insights. As you listen, try to sort out what you believe are the real problems and decision points, and listen for sense and nonsense in the discussion.

• If there is a facilitator—there probably will be one—notice how the facilitator defines the goals of the case and the expectations for learning. Listen carefully for summaries, hints, changes of direction, suggestions, and feedback. Notice how the facilitator tries to focus the group on what is most important. The facilitator usually won’t try to provide solutions or recommendations, but will provide a setting where participants can do so. The facilitator is like an orchestra conductor who knows the composer and the score (Barnes, et. al., 1994, 46).³² Your role is to find out what instrument you are playing and when to play it. Watch the conductor.

• Get involved. Read the case more than once. Listen carefully. Speak when you can contribute. Volunteer to write a thoughtful response or make an analytical presentation. The opportunity will give you practice in learning to solve problems and make decisions.

• If everything appears to be turning into a muddle, raise some of the basic questions such as: What is the problem here? What would it be like when this problem is solved? What is the goal? What are the givens? What are the values we are trying to address through the outcomes of this decision? Does this decision have any chance of success? What are the risks?

LESSONS LEARNED

Ten Things You Can Do to Maximize Your Learning

1. Avoid the “let-me-out-of-here” approach.

2. Don’t let problem solving and decision making boggle your mind.

3. Use mental models.

4. Look for the goal state, initial state, and problem spaces.

5. Use mental models in the problem space to generate solution paths.

6. Avoid the pitfalls of misunderstanding the problem, unrecognized assumptions, and functional fixedness.

7. To assist in making decisions, use the ten-step rational decision-making model.

8. Make recommendations that project outcomes as probabilities.

9. Avoid the pitfalls of wishful thinking, entrapment, tradeoffs, Gambler’s Fallacy, and misinterpretation of trends.

10. Know what kind of case is being used, what is expected of you, what the facilitator is doing, and what you can do to contribute.