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Nature of Problem Solving
UNIT 1 NATURE OF PROBLEM SOLVING
Structure
1.0 Introduction
1.1 Objectives
1.2 Types of Problems
1.3 Characteristics of Difficult Problems
1.4 Nature of Problem Solving
1.4.1 The Stages of Problem Solving
1.5 Types of Thinking Involved in Problem Solving
1.5.1 The Kinds of Thinking Processes
1.6 Concept of Insight Problems
1.7 Let Us Sum Up
1.8 Unit End Questions
1.9 Suggested Readings
Problem Solving “Solving a problem means finding a way out of a difficulty, a way around an obstacle, attaining an aim that was not immediately understandable. Solving problem is the specific achievement of intelligence and intelligence is the specific gift of mankind. Solving problems can be regarded as the most characteristically human activity.” George Polya (1962)
1.0 INTRODUCTION
In our day to day life we usually solve problems be it the classroom, family, or workplace. It is nearly inescapable in everyday life. We use problem solving when we want to reach a certain goal, and that goal is not readily available. It involves situations in which something is blocking our successful completion of a task. To study problem solving satisfactorily, a good way will be to start the chapter by solving some problems. Give these ones a try:
Problem1: What one mathematical symbol can you place between 2 and 3 that result in a number greater than 2 and less than 3?
Problem2: Rearrange the letters NEWDOOR to make one word.
Problem3: How many pets do you have if all of them are birds except two , all of them are cats except two, all of them are dogs except two.
There are many different kinds of problems, including many kinds of recreational problems, career and school oriented problems (such as the problem of how to get promoted or the problem of how to study for a test), personal problems (such as the problem of who to marry or whether to have a child), and scientific problems (such as how to find a cure for cancer or how to prove a particular theorem) etc.
We all solve many problems like these or others every day. Problem solving can be as commonplace as finding out how to prepare a meal or as significant as developing a psychological test.
Problem Solving
1.1 OBJECTIVES
After completing this unit, you will be able to :
- Define the basic nature of problem solving;
- Enlist the different kinds of problems;
- Describe types of thinking involved in problem solving; and
- Analyse and explain the Insight Problem Solving.
1.2 TYPES OF PROBLEMS
Problems vary from ill defined to well defined. In a well defined problem such as a mathematical equation or a jigsaw puzzle both the nature of the problem and the information needed to solve it are available and clear. Thus, one can make straightforward judgments about whether a potential solution is appropriate. With an ill defined problem, such as how to bring peace, not only may the specific nature of the problem be unclear, the information required to solve the problem may be even less obvious.
Greeno (1978) suggested one method of classifying well defined problems based on the general kinds of psychological skills and knowledge needed to solve different problems. Typically, well defined problems falls into one of the three categories viz., (i) Arrangement (ii) Inducing Structure and (iii) Transformation.
Solving each of these types of problems requires somewhat different type of psychological skills and knowledge.
i) Arrangement of problems requires that the problem solver must rearrange or recombine elements in a way that will satisfy a certain criteria. Usually, several different arrangements can be made but only one or few arrangements will produce a solution. For example, one can say that Anagram problems and jigsaw puzzles are examples of arrangement problems. ii) In Problems of inducing Structure, a person must identify the existing relationships among the elements presented. He then should construct a new relationship among them, so that the problem could be solved. In such a problem, the problem solver must determine not only the relationships among the structures but also the structure and sise of elements involved. iii) In Transformation Problems, one takes into consideration. An attempt is made to change the initial state to a goal state. The Tower of Hanoi is an example of this kind of problem where the initial state is the original configuration, the goal state is to have the three disks on the third peg, and the method is the rules for moving the disks. According to Greeno 1978 solving transformation problems primarily requires skills in planning based on a method called means end analysis. Means end analysis requires identifying differences that exist between the current state and the goal state and selecting operations that will reduce these differences.
Problem Solving Self Assessment Questions
- Define problems ..............................................................................................................
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- Define problem solving ..............................................................................................................
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- What are the different types of problems? ..............................................................................................................
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- What is meant by difficult problems? ..............................................................................................................
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- What are the characteristics of difficult problems.? ..............................................................................................................
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Nature of Problem Solving
1.4 NATURE OF PROBLEM SOLVING
The nature of human problem solving methods has been studied by psychologists over the past hundred years. There are several methods of studying problem solving, including introspection, behaviourism, simulation, computer modeling and experiment.
Beginning with the early experimental work of the Gestaltists in Germany and continuing through the 1960s and early 1970s, research on problem solving was typically conducted in relatively simple, laboratory tasks that appeared novel to participants (e.g. Mayer, 1992). In these tasks, they had clearly defined optimal solutions which were solvable within a relatively short time frame, and researchers could trace participants’ problem-solving steps, and so on.
The researchers made the underlying assumption that simple tasks such as the Tower of Hanoi captured the main properties of “real world” problems, and that the cognitive processes underlying participants’ attempts to solve simple problems were representative of the processes engaged in when solving “real world” problems. Thus researchers used simple problems for reasons of convenience.
Simple laboratory based tasks can be useful in explicating the steps of logic and reasoning that underlie problem solving. However, they omit the complexity and emotional valence of “real-world” problems. In clinical psychology, researchers have focused on the role of emotions in problem solving, demonstrating that poor emotional control can disrupt focus on the target task and impede problem resolution (Rath, Langenbahn, Simon, Sherr, & Diller, 2004).
Human problem solving consists of two related processes, viz.,(i) problem orientation, and (ii) problem-solving skills which, if While problem orientation deals with the motivational/attitudinal/affective approach to problematic situations, the problem solving skills deal with the actual cognitive behavioural steps. If cognitive skills are successfully implemented, it will lead to effective problem resolution.
Problem solving is a mental process and is part of the larger problem process that includes problem finding and problem shaping. Considered the most complex of all intellectual functions, problem solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills.
Problem solving occurs when an organism or an artificial intelligence system needs to move from a given state to a desired goal state. Problem solving is one of the many forms that thinking or cognition may take.
By cognition one refers to the mental representation of information which can be manipulated and used to solve problems. Problem solving is a major cognitive behaviour.
Problem solving becomes necessary when an individual wants to reach a goal but that goal is not easily available. It is a state in which there are some obstacles to reach the goal. Various psychologists have defined problem solving.
Nature of Problem Solving
- What are the two processes in human problem solving. ..............................................................................................................
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- Describe the stages of problem solving. ..............................................................................................................
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- What are the four molar components of any problem solving activities? ..............................................................................................................
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- What is meant by “Problem Space”? ..............................................................................................................
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1.5 TYPES OF THINKING INVOLVED IN
PROBLEM SOLVING
According to Wertheimier (1959) effective problem solving requires:
i) Productive thinking
ii) Being sensitive and open to structural requirements
iii) Going beyond the knowledge learnt from previous problem solving tasks
For productive thinking there is a need to have a grasp of the general principles that apply in the particular problem situation. Since individuals do have a tendency to reproduce thinking appropriate for other situations, they need to think beyond
Problem Solving (^) that solution and look for unique solutions. It is important to keep in mind the
structure of the problem without which solutions may not come about.
1.5.1 The Kinds of Thinking Processes
The kinds of thinking processes involved in problem solving are:
- Analytical Thinking
- Synthetic Thinking.
In analytic thinking, there is nothing more in solution than in the premise. For example if the problem is a simple question like “how many doors are there in your house”, then the answer is simple counting of the doors and adding it up. There can be no other answer and there can also be no other solution.
In contrast, Synthetic Thinking does not contain the conclusion in the premise itself because the solution is not needed in the construction of the mental object. For example, we know that 2 is a divisor of 4, 4 is a divisor of 8, and 2 is also a divisor of 8. In general, it is true that a divisor of a divisor of a number is a divisor of that number.
Such solutions are best reached by constructing mental model like images like number lines.
The importance of synthetic thinking is that you can get out more than you put into it.
After you construct a mental model, you can see relationships that were not evident before you constructed it. Seeing these new relationships is what comprises problem solving through synthetic thinking.
In other words, one is synthesizing the available information and facts to derive new solution. This is also termed as developing insight.
Newell is one of the most influential cognitive psychologists who made computer stimulation approach to the study of problem solving.
Newell stated that the goal is to construct a mental model. From this model one will find answers to a problem by inspecting that model itself. To do this, one writes parts of the problem mentally on the model. Once the model has been constructed one can read the results of what has been written. It is important to note that in order to read these results one needs the “mind’s eye”.
The mind’s eye has traditionally been a controversial issue in cognitive psychology. Another word for it is “homunculus” meaning “little man in the head”. Most cognitive psychologists disapprove of this concept of Mind’s eye on the premise that it reflects nonscientific theories of behaviour that were largely based on soul.
Self Assessment Questions
- What types of thinking are involved in problem solving? ..............................................................................................................
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Problem Solving (^) An insight is something that happens to the person, not something that a person
decides to have. In insight solution, the problem is solved by the sudden illumination characteristic of insight.
Insight is preceded by a gradual process whereby relevant parts of the problem are identified. However, solvers may not be aware that this process is leading toward an insight.
Another aspect of insight problems is that the source of difficulty in some insight problems is the inability to see that something you already know is needed for the solution. Hints given within the context of the problem are fairly effective in facilitating subsequent insight As the Gestalt psychologists often observe, people are generally not efficient at realising that a new problem can be solved with information already at their disposal.
People differ in their ability to select information that is relevant to the problem at hand. This ability to discover what is essential about situation is important as well as the ability to remember information that is relevant to the problem. This ability is called sagacity. Sagacity differs from learning in that it involves a sensitivity to detail, a discernment of what is important in a situation. Sagacity is the ability to see into the situation and to discriminate the important aspects of it.
The format in which the information is presented makes a difference in insightful problem solving. That is, one can give the information in a puzzle format or in a declarative format. The information given in the puzzle format leads the subject to discriminate the relevant information better than when the information is in declarative format. This is so because, the puzzle format leads the subject to process the relevant information in a way that makes it accessible for later use. On the other hand the declarative format leads to the acquisition of the relevant information, but in a way that makes it possible for the person to see its relevance for subsequent problem solving.
Select a word that can be interpreted in different ways. For example, the word lake can refer to a frozen or unfrozen body of water. Most people interpret the word to refer to an unfrozen body of water. A riddle can be constructed by requiring the problem solver to come up with the less accessible meaning in order to make sense of what is being described. If the subject is presented with a clue that the stone rested on the surface of the lake for 3 months, after which it sank to the bottom some 10 meters below., this would provide the solution that lake here refers to frozen one for 3 months and then running water lake afterwards.
1.7 LET US SUM UP
Problem solving is an important part of thinking processes. In general a problem is any kind of conflict or difference between one situation and another that one wishes to to produce, that which is the goal.
Problem solving typically involves three major stages: preparation, production of solutions and evaluation of solutions that have been generated.
Problem solving can be done either with or without insight and utilises analytical, synthetic thinking along with metacognition.
This it can be said that, like many other cognitive processes, problem solving is^ Nature of Problem Solving a dynamic and complex process and involves many different types of thinking processes depending on the nature of the problem.
1.8 UNIT END QUESTIONS
What are the various categorisations of problems? Can you categorise your day to day problems into these categories?
What is the role of various types of thinking involved in solving diferent kinds of problems?
Describe the four molar stages of problem solving.
Compare and contrast the role of productive thinking and structurally blind thinking in problem solving.
Emotional states can affect many cognitive processes. What can be the affect of these various emotional states on problem solving?
Critically discuss the concept of insight problems and insightful solutions.
Solutions involving insight and solutions without involving insight – Differentiate.
1.9 SUGGESTED READINGS
Feldman, R. S. (2008). Essentials of Understanding Psychology. New Delhi: Tata McGraw Hill.
Sternberg, R.J. (2009). Applied Cognitive Psychology: Perceiving, Learning, and Remembering. London: Cengage
Galotti , K.M. (2008). Cognitive Psychology In and Out of the Laboratory. Canada: Nelson Education.
References
Hunt, R. R., & Ellis, H.C. (2006). Fundamentals of Cognitive Psychology. New Delhi: Tata McGraw Hill.
Reed, S.K. (2010). Cognition: Theories and Applications. London: Cengage.
Solso, R.L. (2006). Cognitive Psychology. New Delhi: Pearson Education.
representation or pattern in your head so that concepts are related to each other^ Stages of Problem Solving in the same way that they are related to each other in the original sentence.
In order to create this pattern in your head you must use background knowledge, such as the meaning of the various words in the sentence. In order to understand a problem you must pay attention to the important information in a problem. Furthermore, you must ignore the information that is irrelevant. As soon as the problem solver has decided which information is essential and which can be disregarded the next step is to find a good way to represent the problem. The representation can be done by using symbols, lists, matrices, graphs and visual images.
2.3 GENERATING SOLUTIONS
The next stage is finding variety of potential solutions for the problem. For this one can use one’s past experiences and/or present resources and thinking critically about the problem. Several strategies can be used to solve the problems. Some methods are very time consuming but they will yield an answer. Other methods are less wasteful of time, but they may not produce a solution. These strategies are discussed in another section of the unit.
2.3.1 Judging the Best Among the Likely Solutions
At this stage we must evaluate each alternatives and outcomes of the problem. The objective of this step is to move closer to goal and learn about major obstacles and hidden costs of the particular solution and choose the best solution.
Carrying out the best solution
This stage involves actually working out the best solution. Various strategies can be adopted in reaching the solution. Algorithms and heuristics are the strategies which can be used depending upon the requirements of the problem. These are being dealt in the next section.
Evaluation of solution
In this the problem solver compares the solution with the representation of the problem. The path taken from the initial state to the goal state is checked as to whether it is specified as legal or not. Sometimes the evaluation stage leads to the conclusion that the problem has not been solved adequately, and thus one starts from an earlier stage, depending upon the source of the inadequacy.
2.3.2 Strategies for Problem Solving
An individual uses various kinds of methods or strategies to solve problems. Some strategies when used take a long time but definitely give an answer. On the other hand, there are some other strategies which take lesser time but do not always guarantee success. These strategies can be mainly classified under two categories, viz., (i) algorithm and (ii) heuristics. These two are discussed in the following section.
Algorithm
An algorithm is a specific procedure for solving a type of problem. An algorithm invariably leads to the solution if it is used properly, that is. It is a procedure of
Problem Solving (^) steps that does guarantee a solution if one follows the steps correctly. The rules
of multiplication constitute an algorithm because a correct answer is guaranteed if one follows the rules.
Consider anagram problems in which we try to reorganise groups of letters into words. Examples of algorithms are systematic and unsystematic random search techniques. Unsystematic random search means that we try out all kinds of possible answers but make no attempt to be orderly in our search and keep no record of our previous attempts. As a consequence we may repeat a response that has already proved to be wrong. In systematic random search we try out all possible answers using a specified system. This method is somewhat more efficient than unsystematic random search, but it is highly time consuming when there are many alternative answers. If you are given a 3 letter anagram, YBO, with instructions to unscramble it, proceed with a systematic random search: YOB, BYO, BOY,! However, notice how time consuming it would be to use a systematic random search to solve a longer anagram, such as LSSTNEUIAMYOUL.
Heuristics In this strategy an individual does not go for all the alternatives to solve problems but choose some most relevant alternatives. This is kind of short cut rule which takes lesser time but does not guarantee success. Heuristics are rules of thumb that help us simplify and solve problems.
Heuristics do not guarantee a correct solution to a problem but when they work, they permit more rapid solutions. A heuristic device for solving the anagram problem would be to look for familiar letter combinations. For e.g. how many words can you make from the pool of letters DWARG? In this anagram we can find the familiar combinations dr and gr. We may then quickly find draw, drag, and grad. The drawback to this method, however, is that we might miss some words.
Psychologists have paid more attention to how humans use heuristics than how they use algorithms. Let us look at some of the heuristics as strategies in solving problems. Problem solvers can use one or more of these heuristics as they attack a problem.
Self Assessment Questions
- Define problem and problem solving. ..............................................................................................................
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- Discuss the nature of the problem. ..............................................................................................................
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Problem Solving (^) The process that most of us use to arrive at the solution to such a problem generally
involves what is called generate and test technique. It consists of generating possible solutions and then testing them. For example, for the above problem of generating a list of things to eat or drink beginning with letter c, some of the names that often come to our mind that sound as if beginning with c but which actually do not (like ketchup) and some that start with c but are not names of eatables/drinks (like cable, can) might have come to your mind a s a solution but on thinking you would have discarded these words. This explains the process of thinking of possible solutions (generating) and then seeing if those possibilities met all the criteria (testing).
Generate and test is a technique that loses its effectiveness very rapidly when there are many possibilities and when there is no particular guidance over the generation process. For example, if you forget your ATM pin, the technique might eventually work but your frustration level by that time might exceed beyond your willingness to work on the task. Moreover, if we do not have a way to keep track of the possibilities that we have already tried, along with the ones that we still have to try, we might be further frustrated.
Generate and test can be useful when there aren’t a lot of possibilities to keep track of. If you have lost your keys somewhere between your classroom and college cafeteria the technique can be easily used to help search the keys.
2.3.3.2 Means-ends Analysis
It is a strategy in which the problem solver divides the problem into a number of sub problems, or smaller problems. Each of these sub problems is solved by detecting the difference between the original state and the goal state and then reducing the difference between these two states. The name means ends analysis fits the process, because it involves figuring out the “ends” you want and then figuring out what “means” you will use to reach those ends. Every day we all solve problems by using means ends analysis. Means ends analysis involves subgoals to eliminate the difference between the current state and the condition for applying desired operator. Our original state at one point is an equation 2s – 10 + 5 = s + 5 + 8 and our goal state is an equation with a single s alone one side. We reduced the difference between the two states by adding +5 to each side of the equation and subtracting s from each side of the equation.
This strategy repeatedly compares states and seeks operators, establishing subgoals and finding ways to reach the sub goals, all on the way to finding a path to the final goal.
2.3.3.3 Backward Search
With the backward search heuristic, the problem solver starts at the goal state. Sometimes it is useful to start at the goal state of a problem and attempt to work backward to the initial state. In solving a paper-pencil maze, it may be easier to see the correct path by starting at the end. Working backwards can be a very useful heuristic, particularly for problems that contain a uniquely specified goal state. For example, a backward search would be ideal for a maze with many paths out of the beginning point yet only one path leading form the goal.
The reason working backward helps lies in the subgoals that one begins to see^ Stages of Problem Solving by starting with the final goal. Once the problem solver can envision a string of subgoals projecting backward from the goal state, then going about solving the subgoals in a forward direction can be readily accomplished. Working backward is only viable when the goal state is uniquely well defined (Wickelgren, 1974).
2.3.3.4 Planning Strategy
In this heuristic, the problem is divided into two simple aspects and complex aspect. First, the simple aspect of the problem is solved leaving behind the complex one, and thereafter the complex aspect is taken and is solved. It is particularly useful if the aspects that are ignored can easily be worked into the solutions to the complicated problems.
One common kind of planning strategy is analogy. In analogy, a solution to an earlier problem is used in order to help with an original problem. The current problem is solved on the basis of past experience of solving similar problems. This heuristic looks for similarities between a current problem and one solved in the past. Try to solve the anagram BODUT. By thinking of words D, O, U will simplify the problem information. The usefulness of the planning strategy depends on the extent to which the solution to the simple problem will be helpful when the complex problem is considered.
2.3.3.5 Thinking Aloud
Ericsson and Simon referred to the method of thinking aloud as a Method for Studying Human Problem Solving is referred too as concurrent verbalisation , that is verbalisation of information at the time the subject is attending to it. This is to be distinguished from retrospective verbalisation, in which the subject is asked about cognitive processes that occurred at an earlier point in time.
Concurrent verbalisation relies on short-term memory whereas retrospective verbalisation relies on long-term memory.
When subjects think aloud, they put into words a process that normally takes place nonverbally. This provides a description of the subject’s solution process. A verbal description so obtained is called a protocol. Although there may be omissions in these protocols, they still contain a great deal of useful information.
NeweIl (1977) recommended a series of steps in order to clarify the protocol.
1 st^ Step: First, the protocol needs to be divided into phrases, which are descriptions of single acts.
2 nd^ Step: Seconcd, the experimenter constructs a problem behaviour graph, a concrete description of the way in which the subject moves around in the problem space. This description can be used as the basis for a production system designed to model the subject’s behaviour.
Although concurrent verbalisation is widely used, there is evidence suggesting that it may interfere with some aspects ‘of the problem-solving process. For example, Schooler, Ohlsson, and Brooks (1993) did a study in which subjects were interrupted while trying to solve an insight problem.
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- What is involved in “generate and test” technique? ..............................................................................................................
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- What do you understand by the term “Means Ends “ analysis? ..............................................................................................................
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- What is backward search in problem solving? ..............................................................................................................
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- Discuss “thinking aloud” as a method for studying human problem solving. ..............................................................................................................
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2.4 CREATIVITY AND PROBLEM SOLVING
As discussed in Unit1, many of the problems are called insight problems, and are believed to have a solution that comes from insight. The process by which insight comes is not well understood. However it is strongly believed that creativity plays a vital role in insight and vice versa.
Although the term is difficult to define precisely, many psychologists believe that creativity has to do with appropriate novelty, that is, originality that suits some purpose (Hennessey & Amabile, 1988).
Problem Solving (^) Appropriate ideas that lack novelty are mundane; conversely original ideas that
do not address some problem in a useful way are bizarre. Cognitive scientists refer to creativity as a combination, or recombination, of knowledge, information or mental representations. In other words the person who has created it knows about it, and also has depicted it.(Dartnall, 2002). Creative problem solving forms the essence of insight problem solutions. Insight seems to involve at least three separate problem-solving skills (Davidson, 1986; Davidson & Sternberg,
and these are (i) selective encoding (ii) selective combination (iii) selective comparison. These are discussed below.
Selective Encoding. When we try to solve a new problem, we are often overwhelmed by large amounts of useless information. An insight arises when we determine which information is relevant for further consideration. Sir Alexander Fleming discovered penicillin through selective encoding. Fleming’s attempts to grow bacteria had been disrupted by the appearance of a mold in his culture. Rather than dismissing this mold as irrelevant to his original experiments, he realised that the mold’s bacteria has a killing ability that was relevant to a greater medical concern, that is the need for antibiotics to kill disease-causing organisms.
Selective Combination: When we have a problem to solve, we often have all the pieces of the solution, but we do not know how to put them together. An insight arises when we discover a novel way of combining the elements of the solution. For example, Darwin’s construction of the theory of evolution involved selective combination. The various facts about natural history had been available to him for a long time. What Darwin did was to combine this information into an innovative and coherent theory.
Selective Comparison. When we solve a problem, we often use a model solution that we encountered in the past. Insight occurs when we discover that a more novel comparison leads to unforeseen consequences. To give an example, Kekule’s discovery of the structure of the benzene ring involved selective comparison.
In a dream Kekule saw a snake curl back on itself and catch its own tail. Upon waking, he realised that this was the structure he had been seeking. His insight was to discern the underlying similarity between his dream image and the benzene ring.
As the above examples suggest, insight is an important aspect of the creative process. Even though the laboratory puzzles and problems that we have been discussing in this Unit as examples bear little resemblance to the kinds of problems that Fleming, Darwin, and Kekule worked on, they all involve insight in that a solution suddenly comes to mind. They also share another property, that is, the person begins in the dark, not knowing what to do. What factors promote creative solutions to important problems?
2.4.1 Stages Involved in Creative Discoveries
Many creative discoveries seem to occur in these four stages:
- Preparation
- Incubation
