Problem Solving Book Home Page

Introduction to the Book

Chapter 1

Some Examples of Problems

What is a Formal Problem?

Representations of a Problem

Roman Numeral Example

Representing Problems Using Computers

Poorly Defined Problems and Problem Posing

Difficult and Unsolvable Problems

Persistence and Motivation

Working Toward Increased Expertise

Activities and Self-Assessment

Chapter 2: Overview of Resources in Problem Solving

Chapter 3: Intelligence as Resource

Chapter 4: Tools as Resource

Chapter 5: Accumulated Knowledge as Resource

Chapter 6: Education and Training as Resource

Chapter 7: A Computer System

Chapter 8: Personal Productivity Tools

Chapter 9: Computer Programming

Chapter 10: Final Remarks

References and Resources

Search Engine in Lieu of Index

Chapter 1: Introduction to Problem Solving

In your daily life you routinely encounter and solve problems. You pose problems that you need or want to solve, you make use of available resources, and then you solve the problems. Some categories of resources include: the time and effort of yourself and others; tools; information; and money.

Some of the problems that you encounter and solve are quite simple. For example, you are getting dressed and you need to tie your shoes. For most people, this is a relatively simple task. Many people find that putting together a color-coordinated outfit is a more challenging task.

[[It is not easy to learn to tie shoelaces. And, some people have handicapping conditions that add to the difficulty--perhaps making it impossible to learn to solve this problem. Shoes with Velcro fasteners may prove to be an appropriate accommodation. There are a wide range of handicapping conditions. A person who is color blind may have difficulties in solving the "color-coordinated outfit" problem.]]

People who do research on problem solving tend to differentiate between simple problems and more complex problems. Often they focus their research on how people learn to solve and actually solve complex problems (Frensch and Funke, 1995).

The computer is a resource--a versatile tool--that can help you solve some of the problems that you encounter. A computer is a very powerful general-purpose tool. Computers can solve or help solve many types of problems. There are also many ways in which a computer can enhance the effectiveness of the time and effort that you are willing to devote to solving a problem. Thus, it will prove to be well worth the time and effort you spend to learn how to make effective use of this tool.

Some Examples of Problems

In this book we will use the terms accomplish a task and solve a problem interchangeably. We begin with a simple exercise.

Think about some of the tasks you have accomplished recently--some of the problems you have solved. Make a written list of six such problems. For example, did you eat breakfast this morning? If so, you solved a breakfast problem. Did you pay some bills? If so, you solved a bill-paying problem. Did you use a telephone to talk to a person? If so, you accomplished a distance-communication task. Did you settle an argument with a friend? If so, you solved an interpersonal problem.

Clearly, some problems are more difficult than others. Think about the problems on your list. Which were really easy? Which were more difficult? Label your problems using a three-point scale of low, medium, and high difficulty. Think about what makes one problem more difficult than another. Think about how the same problem that may be easy for you may be difficult for another person, or vice versa. Give some examples that illustrate why this is the case.

[[You know, of course, that reading an academic book is not like reading a novel. When reading an academic book, you will greatly increase your learning if you frequently pause and reflect about what you are reading. Probably you have heard about constructivism--a learning theory about how people construct their own knowledge. As you pause to reflect about what you are reading, think about how it fits in with what you already know. Think about what seems relevant to you, personally. Think about how you might make use of the information that you are reading.]]

Let's carry this thinking exercise one step further. For each problem on your list, make a note of the main resources used to solve the problem. Perhaps you solved some of the problems purely by the use of your mind and body. For other problems, you may have obtained help from friends. For still other problems you may have used tools, such as a telephone, calculator, or car.

Table 1.1 contains a sample of the type of list you may have created. By doing this exercise, you have reminded yourself of three facts:

  1. You routinely encounter and solve a wide range of problems.
  2. The problems that you solve in your everyday life vary in difficulty, and the level of difficulty varies from person to person.
  3. As you solve problems, you make use of resources such as:
    • mental and physical resources of yourself and others;
    • tools;
    • information resources such as books and videos;
    • money, to acquire other resources.

Main Resources



Contents of refrigerator, microwave oven



Bus, my feet

Communication with a friend


Telephone, my voice

Written communication with an irate customer


Stored data on the transaction, careful thinking, advice from my boss, computer

Obtain needed information about a particular company


Several friends, reference books



Meditation, aspirin

Table 1.1 Problems recently encountered and solved.

What is a Formal Problem?

There is a substantial amount of research literature on problem solving. Many textbooks and popular press books discuss problem solving (Polya, 1957; Frensch and Funke, 1995; Peters, 1994).

Researchers and writers use a somewhat common set of vocabulary as they talk about problem solving. Problem solving consists of moving from a given initial situation to a desired goal situation. A different way of saying this is that problem solving is the process of designing and carrying out a set of steps to reach a goal.

Figure 1.1 Problem-solving process--from initial situation to final goal.

In this book, we will use a formal definition of the term problem. You (personally) have a problem if the following four conditions are satisfied:

  1. You have a clearly defined given initial situation.
  2. You have a clearly defined goal (a desired end situation).
  3. You have a clearly defined set of resources that may be applicable in helping you move from the given initial situation to the desired goal situation. There may be specified limitations on resources, such as rules, regulations, and guidelines for what you are allowed to do in attempting to solve a particular problem.
  4. You have some ownership--you are committed to using some of your own resources, such as your knowledge, skills, and energies, to achieve the desired final goal.

These four components of a well-defined problem are summarized by the four words: givens, goal, resources, and ownership.

People often get confused by the resources part of the definition. Resources do not tell you how to solve a problem. Resources merely tell you what you are allowed to do and/or use in solving the problem. For example, you want to create an ad campaign to increase the sales of a set of products that your company produces. The campaign is to be nationwide, to be completed in three months, and not to exceed $40,000 in cost. All of this fits under resources. You still have to figure out how to create the ad campaign.

This definition emphasizes that you, or some entity, have a problem. Problems do not exist in the abstract. They exist only when there is ownership. The owner might be a person, or it might be an organization, or a country. However, the emphasis in this book is on problems that you, personally, may encounter and want to solve.

There are many other definitions of problem. The book edited by Frensch and Funke (1995) begins with an analysis of a number of these definitions. The analysis is done from a point of view of which definitions seem most useful to researchers doing research on problem solving. The book itself focuses on a limited range of problems, called complex problems. Complex problem solving is defined to be the type of thinking that occurs to overcome barriers between a given state and a desired goal state by means of behavioral and/or cognitive, multi-step activities. The given state, goal state, and barriers are complex and may change dynamically during the problem-solving process. This definition fits well with many real world problems, such as those faced by high-level decision makers.

[[The following material about solving complex problems is quoted from: Complex Problem-Solving Tutor in Mathematics [Online]. Accessed 10/27/01:
Complex problem-solving (CPS) is a relatively new educational reform. It recognizes that students require explicit instruction on complex, real-life problem solving across the curriculum. For years, the assumption was held that teaching secondary students to solve basic technical mathematics problems would implicitly enable them to learn thinking skills needed to solve real-life problems. Unfortunately, many students are not able to successfully acquire or apply the implicitly received instruction on problem solving to real-life problems. Even with explicit "back-to-basics" instruction to recognize various mathematical problem types, students often struggle with determining relevancy and transferring the implicitly taught "thinking steps" to solve problems encountered in life. To provide for explicit instruction on mathematics problem-solving thinking processes, a team involving teachers from the school district, and researchers from CITL, designed a six-step problem-solving model to serve as the framework for instruction provided by the project team's web-based mathematics software tutor. The tutor will serve dual roles for both introducing and modeling to teachers and students a model for using complex problem-solving thinking processes to address real-life problems. The existing project work in the area of mathematics will also be extended to the science curriculum.

Use of the search terms "solving complex problems" or "complex problem solving" with the Google search engine will provide you with a number of good leads to information in this area.]]

Representations of a Problem

There are many different ways to represent a problem. A problem can be represented mentally (in your own mind), orally, in writing, on a computer, and so on. Each type of representation has certain advantages and disadvantages.

From a personal or ownership point of view, you first become aware of a problem situation in your mind and body. You sense or feel that something is not the way that you want it to be. You form a mental representation, a mental model, of the problem. This mental model may include images, sounds, or feelings. You can carry on a conversation with yourself--inside your head--about the problem.

[[Emotions play a significant role in identifying, posing, representing, and solving problems. In recent years, a number of people have done research on the topic Emotional Intelligence. For example, see Emotions and Emotional Intelligence [Online]. Accessed 10/27/01:

Frequently, a mental model representation of a problem may suffice for solving the problem. You can think about the problem, consider alternatives inside your mind, and decide on a course of action. You may reformulate the problem, deciding on a goal that seems more appropriate. You can then use your mind/body to carry out actions to solve the problem.

For example, your stomach is generating hunger pangs and you sense that you are hungry. You have an "I am hungry" problem. You begin to give conscious thought to the problem. You remember that you ate just two hours ago. You remember that you are trying to watch your weight. You consciously integrate the "I am hungry" information with these other pieces of information. Perhaps you decide that a drink of water is the appropriate course of action.

Mental representations of problems are essential. You create and use them whenever you work on a problem. But, problems can be represented in other ways; for example, you might represent a problem with spoken words and gestures. This could be useful if you are seeking the help of another person in dealing with a problem. The spoken words and gestures are an oral and body language model of the problem. Think about your level of expertise in oral communication. Do you know some people who are particularly good at oral communication? Can you think of ways to increase your level of expertise in this area?

[[The assumption is that readers of this book would like to increase their expertise in solving problems. Metacognition and reflection about your personal problem-solving successes and failures are useful aids to improving your problem-solving expertise. See, for example:

Developing Metacognition. ERIC Digest. (1990) [Online]. Accessed 10/27/01:

Metacognition [Online]. Accessed 10/27/01:

Schoenfeld, A.H. (1987) What's all the fuss about metacognition? [Online]. Accessed 10/27/01:

You might represent a problem using pencil and paper. You could do this to communicate with another person or with yourself. Writing and drawing are powerful aids to memory. You probably keep an address book or address list of the names, addresses, and phone numbers of your friends. Perhaps it contains additional information, such as e-mail addresses, birthdays, names of your friends' children, and so on. You have learned that an address book is more reliable than your memory.

There are still other ways to represent problems. For example, the language and notation of mathematics are useful for representing and solving certain types of problems. For example: A particular type of carpet costs $17.45 per square yard--how much will the carpeting cost for two connecting rooms? One room is 16 feet by 24 feet, and the other room is 12 feet by 14 feet.

Figure 1.2 Two rooms to be carpeted.

Conceptually, the problem is not too difficult. You can form a mental model of the two rooms. Each room will be covered with carpet costing $17.45 per square yard. So, you need to figure out how many square yards are needed for each room. Multiplying the number of square yards in a room by $17.45 gives the cost of the carpet for the room. Add the costs for the two rooms, and you are done.

Note that this is only one of the many possible ways to conceptualize this problem. You may well think of it in a different way.

The field of mathematics has produced the formula A = LW (Area equals Length times Width). It works for all rectangular shapes. Making use of the fact that there are three feet in a yard, the computation needed to solve this problem is:

Answer = $17.45 (16/3 x 24/3) + $17.45 (12/3 + 14/3)

Perhaps you can carry out this computation in your head. More likely, however, you will use pencil and paper, a calculator, or a computer.

There are two key ideas here. First, some problems that people want to solve can be represented mathematically. Second, once a problem is represented as a math problem, it still remains to be solved.

Over the past few thousand years, mathematicians have accumulated a great deal of knowledge about mathematics. Thus, if you can represent a problem as a math problem, you may be able to take advantage of the work that mathematicians have done before. Cognitive artifacts, such as paper-and-pencil arithmetic, calculators, and computers, may be useful.

[[The math example mentions the "language and notation of mathematics." Mathematics is a type of language. An excellent reference on this idea is:
Logan, Robert K.: Sixth Language : Learning a Living in the Internet Age. ON, Canada: Stoddard Publishing Company.

Logan's book also argues that writing, science, computing, and the Internet are all "languages." They are all general purpose aids to the brain in representing and solving problems.]]

Roman Numeral Example

Each way of representing a problem [[and the notation used]] has certain advantages and certain disadvantages. A problem may be very difficult when represented one way and very easy when represented another way. Thus, a person who is skilled at representing problems in a number of different ways is likely to be a better overall problem solver than a person who has only a few ways to represent problems.

An interesting example of how problems may be represented in two different ways is provided by Roman and Arabic numerals. Each is an adequate representational system for counting to a modest level. It may be easier to learn to write/count I, II, III than it is to write/count 1, 2, 3. But Arabic numerals and the positional notation system are much better for dealing with large quantities. Arabic numerals are far superior for multiplication, division, and working with fractions.

Figure 1.3 Representing computational problems using Roman numerals and Arabic numerals.

This is a very important idea. Through many hundreds of years of experimentation--trial and error, careful thinking--our current base 10 Arabic numeral system was developed. Our base 10 Arabic numeral system is a cognitive artifact. When you learn it and use it, you are building on centuries of work. You did not have to reinvent this number system; it was taught to you when you were a child.

Fractions are another cognitive artifact. You may think that it is easy to determine that 1/2 + 1/6 = 2/3. However, this task was beyond the abilities of all but the most educated people 2,000 years ago. The contemporary standards for expertise in arithmetic computation have gone up considerably during the past 2,000 years!

Interestingly, the advent of calculators and computers--and their increased use in schools--will likely lead to a decrease in paper-and-pencil computational skills but increased standards for correctness in doing computations. Contemporary standards change with changes in tools.

[[The National Council of Supervisors of Mathematics endorsed school use of calculators in 1979. The National Council of Teachers of Mathematics made a similar endorsement in 1980. The NCTM Standards call for substantial use of both calculators and computers.

I find it interesting to see the resistance of elementary school teachers to the use of calculators. Many still strongly believe that one must first master pencil and paper arithmetic algorithms before they are allowed to use a calculator. Their main argument is that a child must first "understand" computation before being allowed to use a calculator. They define the word "understand" to mean "memorizing a paper and pencil algorithm and developing skill in its use."

To a reasonable extent, the same type of argument is being used in all disciplines when it comes to the use of computers. Thus, schools still spend lots of time and effort helping students learn non-computer methods so that they can compete against computers. From my point of view, much of this time would be better spent if it were applied to learning problem solving and higher-order cognitive skills.]] 

Representing Problems Using Computers

One particularly important feature of a mental model is that it is easily changed. You can "think" a change. This allows you to quickly consider a number of different alternatives, both in how you might solve a problem and in identifying what problem you really want to solve.

Other representations, such as through writing and mathematics, are useful because they are a supplement to your brain. Written representations of problems facilitate sharing with yourself and others over time and distance. However, a written model is not as easily changed as a mental model. The written word has a permanency that is desirable in some situations, but is a difficulty in others. You cannot merely "think" a change. Erasing is messy. And, if you happen to be writing with a ball-point pen, erasing is nearly impossible.

When a problem is represented with a computer, we call this a computer model or a computer representation of the problem. As you proceed in this book, you will explore a variety of computer models. You will see that for some problems, a computer model has some of the same characteristics as a mental model. Some computer models are easy to change and allow easy exploration of alternatives.

For example, consider a document that is represented as a word processor file. It may be easier to revise this document than a paper-and-pencil version of the document. A computer can assist in spell checking and can be used to produce a nicely formatted final product.

[[One can draw a strong parallel between Process Writing (a way to attack a writing problem) and problem solving in a number of other domains. Process Writing has been greatly aided by computers (word processing).]]

In the representation of problems, computers are useful in some cases and not at all useful in others. For example, a computer can easily present data in a variety of graphical formats, such as line graph, bar graph, or in the form of graphs of two- and three-dimensional mathematical functions.

But a computer may not be a good substitute for the doodling and similar types of graphical memory-mapping activities that many people use when attacking problems. Suppose that one's mental representation of a problem is in terms of analogy and metaphor. Research that delved into the inner workings of the minds of successful researchers and inventors suggests this is common and perhaps necessary. A computer may be of little use in manipulating such a mental representation.

Poorly Defined Problems and Problem Posing

Up to this point, we have used the term problem rather loosely. Many of the things that people call problems are actually poorly defined problem situations. In this case, one or more of the four components of a clearly defined problem are missing. For example, you turn on a television set and you view a brief news item about the homeless people in a large city and the starving children in an foreign nation. The announcer presents each news item as a major problem. But, are these really clearly defined problems?

You can ask yourself four questions:

  1. Is there a clearly defined given initial situation? (Do I really know the facts?)
  2. Is there a clearly defined goal? (Is it really clear to me how I would like things to be?)
  3. Do I know what resources are available to me that I could use to help achieve the goal? In addition, are there rules, regulations, and guidelines that I need to know about as I work to solve this problem?
  4. Do I have ownership--do I care enough to devote some of my own resources? (Am I willing to spend some time on achieving the goal?)

If you can answer "yes" to each of these questions, then you have a formal, clearly defined problem.

Often, your answer to one or more of the questions will be "no." Then, the last question is crucial. If you have ownership--if you really care about the situation--you may begin to think about it. You may decide on what you feel are appropriate statements of the givens and the goal. You may seek resources from others and make a commitment of your own resources. You may then proceed to attempt to solve the problem.

The process of creating a clearly defined problem is called problem posing or problem clarification. It usually proceeds in two phases. First, your mind/body senses or is made aware of a problem situation. You decide that the problem situation interests you--you have some ownership. Second, you begin to work on clarifying the givens, goal, and resources. Perhaps you consider alternative goals and sense which would contribute most to your ownership of the situation.

The result of the problem-posing process is a problem that is sufficiently defined so that you can begin to work on solving it. As you work on the problem, you will likely develop a still better understanding of it. You may redefine the goal and/or come to understand the goal better. You may come to understand the given initial situation better; indeed, you may decide to do some research to gain more information about it. Problem posing is an On-Line process as you work to understand and solve a problem.

Problem posing is a very important idea. It is a particularly personal process, drawing on your full range of capabilities, knowledge, and interests. Often it can be hard work to convert a loosely defined problem situation into a clearly defined problem. Moreover, as you work to solve a problem, you may well decide that you want to change it into a different problem. If you are the one with ownership--if you have posed the problem--then you can modify the problem to fit your interests and needs.

Are you good at problem posing? Are you good at recognizing problem situations and converting them into clearly defined problems? What have you done during the past year to increase your level of expertise in problem posing?

Difficult and Unsolvable Problems

You know that some problems are more difficult than others. Also, you know that a particular type of problem may be quite difficult for you and quite easy for someone else. However, there is one more piece to this puzzle--some clearly defined problems cannot be solved because they have no solution. For example, you are presented with the following problem: "Find a four-letter word that contains all of the vowels." You know that this is an unsolvable problem because there are five vowels.

Here is a slightly more complex example. Suppose you want to solve the simple math problem: "Find two positive odd integers whose sum is an odd integer."

You might begin thinking about this problem by doing a little exploring. A few trials, such as 1 + 1 = 2 (even), 1 + 3 = 4 (even), 3 + 9 = 12 (even) and so on, might lead you to the conjecture that the sum of two positive odd integers is always an even integer. This could lead you to pose a new problem. The new problem would be: "Prove that the sum of two positive odd integers is an even integer." If you solve this new problem, you will have proven that the original problem has no solution.

Proving that a problem has no solution can, itself, be a very difficult task. Thus, one difficulty you face when you're working on a problem and not succeeding in solving it, is determining when to give up. You may give up because the available resources have been exhausted. You may give up because of a conviction that the problem is not solvable with the available resources. And, of course, you may give up upon becoming convinced that the problem is truly unsolvable.

The two examples used in this section are somewhat typical of textbook problems. They are trivial--they pale in significance relative to many real-world problems. Problems or problem situations, such as world peace, the homeless, the hungry, battered children, cancer, and so on, are far more difficult. Many real-world problems have the characteristic that persistent effort can contribute toward making progress on solving the problems, even though no final solution is reached.

Persistence and Motivation

Many real-world problems require a great deal of time and effort to solve. Some may not be solvable with the resources that are available. Some may take many years or many centuries to solve. Persistence is a common trait in successful problem solvers.

Your persistence in working on a problem may be determined by what motivates you. Think about intrinsic motivation and extrinsic motivation. In intrinsic motivation, your drive--your push to succeed--comes from within. You are working toward goals that you really want to accomplish. In extrinsic motivation, external factors are acting on you. They are telling you what to do and they are pressuring you to do it. The goals may be set by other people and may not be of any particular interest to you. You may be saying to yourself, "I am doing this to get a good grade. I have no interest in the problem."

Some people are able to have a great deal of persistence based on extrinsic motivation. However, the typical person is apt to have more persistence when driven by a strong intrinsic motivation. Intrinsic motivation and the ownership component in the definition of a problem are closely related.

Working Toward Increased Expertise

Think about some category of problems that you have become good at solving. Perhaps you are a really good housekeeper or a really good teacher. Perhaps you are really good at making friends and working with people. Maybe you are really good at performing music, solving math problems, or reading maps.

[[You might see some similarity between these questions and Howard Gardner's ideas on Multiple Intelligence. People vary widely in their innate abilities within particular areas. By appropriate study and practice you can get better in any field. However, if you are interested in having a relatively high level of expertise--as compared to other people--within some field, you might want to pick a field in which you have a relatively high level of innate ability.]]

At some time in the past, you were just beginning to learn about these types of problems. Gradually your knowledge and skills grew. Your level of expertise in solving the problems increased.

As you look toward the future, do you intend to become still better at solving this category of problems? What are you doing to become more of an expert? Do you just leave it to chance, or are you actively and consciously engaged in increasing your level of expertise?

This book explores a number of ways to get better at problem solving. These suggestions can be applied in almost any problem-solving domain. The goal is to help you increase your level of expertise in whatever areas interest you. The assumption is that you have ownership--that you want to increase your level of expertise in various fields.

One factor in increasing expertise is obtaining appropriate feedback on what you are doing and how well you are doing it. You can provide feedback to yourself--through metacognition and reflective introspection. You can get feedback from a coach, a teacher, or a colleague. In certain types of problem-solving situations, you and a computer working together can provide you with useful feedback.

Another factor in increasing expertise is learning to make effective use of the tools that experts use. The computer is one such tool.

Activities and Self-Assessment

  1. Many people benefit from keeping a journal as they work their way through a book such as this. In the journal, they reflect on ideas that occur to them as they read the book. For example, has it occurred to you that while you were in grade school, you may have spent a great deal of time learning to compete with a calculator or a computer? What are your feelings about this? As you think about problems and problem solving, do you feel adequate or inadequate? After reading Chapter 1, do you feel that you gained anything useful? If so, what? If not, why?

    Start a journal. Make some entries in it each time you read a chapter or part of a chapter. From time to time you may want to go back to earlier entries and write in additional comments.

  2. Think about some general problem-solving idea that you know of that is useful in lots of different settings. For example, if a problem is stressing you, perhaps you have learned to relax, take a few deep breaths, and calm yourself. Where did you learn this idea? Could you help other people learn to do this?
  3. Make a list of problems that you feel are very difficult to solve or are unsolvable. Make sure each is a clearly defined problem. For each, explain why you feel the problem is very difficult to solve or unsolvable.
  4. There have been a large number of research studies on attitudes and how they affect learning, retention, problem solving, and so on. Generally, there is agreement that having a positive attitude is desirable and contributes to learning and the ability to solve problems. Discuss how these research findings relate to the ownership component of a clearly defined problem.
  5. When you are forming a mental model of a problem, is your model mainly in terms of pictures, mainly in terms of sounds, or mainly kinesthetic? For example, do you visualize a problem in your "mind's eye"? There has been quite a bit of research on learning styles. Some people learn better visually, while others learn better orally. Do some introspection about how you think about or represent problems in your head. Think about your learning style. Write a brief report summarizing your findings.
  6. This chapter introduces the idea that most of the things people call problems are actually ill-defined problem situations. It suggests that one key step in problem solving is to develop a clearly defined problem. Suppose you apply this idea to a number of different problem-solving areas. How will this help to increase your expertise as a problem solver?

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