Appendix C: Overview of Problem Solving

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Moursund, D.G. (1999). Project-based learning using information technology (Selected Chapters) Eugene, OR: International Society for Technology in Education.

The materials that follow are from a next-to-final version of the above named book.

Preface

Chapter 1: Introduction and a PBL Example

Chapter 2: Overview of IT-assisted PBL

Chapter 3: Some PBL Lesson Topic Ideas

Chapter 4: The Case for PBL

Chapter 7: Assessment in IT-assisted PBL

Appendix C: Overview of Problem Solving

References and Resources

 

Appendix C: Overview of Problem Solving

One of the goals of education is to help students to get better at problem solving. A few schools actually offer specific courses on problem solving. For the most part, however, students learn about problem solving through instruction in courses that have a strong focus on a specific content area. Every teacher teaches problem solving within the specific subject matter areas of their curriculum.

This appendix gives a brief overview of the "subject" of problem solving. The ideas from this appendix can be woven into instruction in almost any curriculum area. IT-assisted PBL provides an excellent environment for teaching these ideas.

Problem and Task Team

Donald Norman is a cognitive scientist who has written extensively in the area of human-machine interfaces. Norman (1993) begins with a discussion of how tools (physical and mental artifacts) make us smart. David Perkins (1992) uses the term "Person Plus" to refer to a person making use of physical and mental tools. In many situations, a person with appropriate training, experience, and tools can far outperform a person who lacks these aids.

In this book, we use the term Problem or Task Team (P/T Team) to refer to a person or a group of people and their physical and mental tools. Figure C.1 illustrates the P/T Team. These concepts are explained in subsequent paragraphs.

Figure C.1. People aided by physical and mental tools.

Figure C.1 shows a person or a group of people at the center of a triangle of three major categories of aids to solving problems and accomplishing tasks:

  1. Mental aids. Even before the invention of reading and writing, people made use of notches on bones and other aids to counting and to keeping track of important events. Reading, writing, and arithmetic are mental aids. These have led to the development of books, math tables, libraries, calculators, computers, and many other mental aids.
  2. Physical aids. The steam engine provided the power that led to the beginning of the industrial revolution. Well before that time, however, humans had developed the stone ax, spear, bow and arrow, plow, hoe, telescope, and many other aids to extend the physical capabilities of the human body. Now we have cars, airplanes, and scanning electron microscopes. We have a telecommunications system that includes fiber optics, communications satellites, and cellular telephones
  3. Education. Education is the glue that holds it all together. Our formal and informal educational systems helps help people learn to use the mental and physical tools as well as their own minds and bodies.

The three supportive components of a P/T Team are dynamic, changing based on research, the develop of new tools, and the steadily increasing information base being developed by researchers throughout the world. This is a challenge to our educational system, because the pace of change in mental and physical aids is so rapid.

People who are skilled at functioning well in a P/T Team environment have a distinct advantage over those who lack the knowledge, skills, and access to the facilities. Such analysis leads to the recommendation that the P/T Team and problem solving should be a central themes in education. IT-assisted PBL provides an excellent environment to work on these two aspects of education.

Problem Solving

Each academic discipline focuses on a category of problems that help to define the discipline and methodologies for solving these problems. Chemistry, history, and mathematics are different disciplines because they address quite different types of problems and have developed quite different methodologies for addressing problems.

While many aspects of problem solving are specific to the academic area (domain) of the problem, there are also many ideas about problem solving that cut across all domains. Thus, with appropriate education and experience, a person can gain some general expertise in problem solving that is useful in addressing any new problem that them might encounter.

How does one learn to think and perform like an artist, mathematician, social scientist, or scientist? It is through explicit instruction and guided practice. Every IT-assisted PBL lesson should be viewed as an opportunity for students to increase their problem-solving expertise both within the domains of the project and across all domains.

The remainder of this chapter focuses on problem solving and accomplishing tasks. (We will use the term problem solving to refer to both solving problems and accomplishing tasks.) It provides information that you can use to help your students get better at problem solving.

What is a Formal Problem?

There is a substantial amount of research literature as well as many practitioner books on problem solving (Polya, 1957; Frensch and Funke, 1995; Moursund, 1996).

Problem solving consists of moving from a given initial situation to a desired goal situation. That is, problem solving is the process of designing and carrying out a set of steps to reach a goal. Usually the term problem is used to refer to a situation where it is not immediately obvious how to reach the goal. The exact same situation can be a problem for one person and not a problem (perhaps just a simple activity or routine exercise) for another person.

Figure C.2. Problem-solving process--how to achieve final goal?

Here is 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 a nationwide ad campaign to increase the sales of a set of products that your company produces. The campaign is to be completed in three months, and not to exceed $40,000 in cost. Three months is a time resource and $40,000 is a money resource. You can use the resources in solving the problem, but the resources do not tell you how to solve the problem.

Problems do not exist in the abstract. They exist only when there is ownership. The owner might be a person, a group of people such as the students in a class, or it might be an organization, or a country.

The idea of ownership is particularly important in teaching. If a student creates or helps create the problems to be solved, there is increased chance that the student will have ownership. The type of ownership that comes from a student developing a problem that he/she really wants to solve is quite a bit different from the type of ownership that often occurs in school settings. When faced by a problem presented by the teacher or the textbook, a student may well translate this into, "My problem is to do the assignment and get a good grade. I don't have any interest in the problem presented by the teacher or the textbook." A skilled teacher will help students to develop projects that contain challenging problems, and the problems are ones that the students really care about. In good PBL, students focus on solving the problems and accomplishing the tasks because they are internally, intrinsically motivated.

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.

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.

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 email 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 C.3. 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 previously done. Mental artifacts, such as paper-and-pencil arithmetic, calculators, and computers, may be useful.

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. 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.

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.

Problem Posing and Clarification

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 problem 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-going process as you work to understand and solve a problem.

Problem posing is a very important idea that should be integrated into IT-assisted PBL lesson. It is a component of problem solving that cuts across all discipline areas. Some additional general purposed problem-solving ideas are given in the next two sections.

Some Problem-Solving Strategies

A strategy can be thought of as a plan, a heuristic, a rule of thumb, a possible way to approach the solving of some type of problem. For example, perhaps one of the problems that you have to deal with is finding a parking place at work or at school. If so, probably you have developed a strategy--for example, a particular time of day when you look for a parking place or a particular search pattern. Your strategy may not always be successful, but you find it useful.

Every problem-solving domain has its own strategies. Research suggests:

  1. There are relatively few strategies that are powerful and applicable across all domains. (Breaking a big problem into smaller problems is one of these general-purpose strategies. Doing library research is another general-purpose strategy.) Each subject matter (each domain) has its own set of problem-solving strategies. One needs to know a great deal about a particular domain to be good at solving problems within that domain.
  2. The typical person has few explicit strategies in any particular domain. This suggests that if we help a person gain a few more domain-specific strategies, it might make a significant difference in overall problem-solving performance in that domain. It also suggests the value of helping students to learn strategies that cut across many different domains.

The idea of breaking big problems into smaller problems is called the top-down strategy. The idea is that it may be far easier to deal with a number of small problems than it is to deal with one large problem.

Library research is a type of "ask an expert" strategy. A large library contains the accumulated expertise of thousands of experts.

You have lots of domain-specific strategies. Think about some of the strategies you have for making friends, for learning, for getting to work on time, for finding things that you have misplaced, and so on. Many of your strategies are so ingrained that you use them automatically--without conscious thought. You may even use them when they are ineffective.

The use of ineffective strategies is common. For example, how do you memorize a set of materials? Do you just read the materials over and over again? This is not a very effective strategy. There are many memorization strategies that are better. For example, a simple strategy is pausing to review. Other strategies include finding familiar chunks, identifying patterns, and building associations between what you are memorizing and things that are familiar to you.

Some learners are good at inventing strategies that are effective for themselves. Most learners can benefit greatly from some help in identifying and learning appropriate strategies. In general, a person who is a good teacher in a particular domain is good at helping students recognize, learn, and fully internalize effective strategies in that domain. Often this requires that a student unlearn previously acquired strategies or habits.

Strategies can be an IT-based PBL lesson topic within any subject that you teach. Individually and collectively your students can develop and study the strategies that they and others use in learning the subject content area and learning to solve the problems in the subject area. A whole-class project in a course might be to develop a book of strategies that will be useful to students who will take the course in the future.

A General Strategy for Problem Solving

Here is a general six-step strategy that you can follow in attempting to solve almost any problem. This six-step strategy is a modification of ideas discussed in Polya (1957). Note that there is no guarantee of success. However, this six-step strategy might get you started on a pathway to success.

  1. Understand the problem. Among other things, this includes working toward having a clearly defined problem. You need an initial understanding of the Givens, Resources, and Goal. This requires knowledge of the domain of the problem.
  2. Determine a plan of action. This is a thinking activity. What strategies will you apply? What resources will you use, how will you use them, in what order will you use them? Are the resources adequate to the task?
  3. Think carefully about possible consequences of carrying out your plan of action. Place major emphasis on trying to anticipate undesirable outcomes. What new problems will be created? You may decide to stop working on the problem or return to step 1 as a consequence of this thinking.
  4. Carry out your plan of action. Do so in a thoughtful manner. This thinking may lead you to the conclusion that you need to return to one of the earlier steps. It is this reflective thinking that leads to increased expertise.
  5. Check to see if the desired goal has been achieved by carrying out your plan of action. Then do one of the following:
    • If the problem has been solved, go to step 6.
    • If the problem has not been solved and you are willing to devote more time and energy to it, make use of the knowledge and experience you have gained as you return to step 1 or step 2.
    • Make a decision to stop working on the problem. This might be a temporary or a permanent decision. Keep in mind that the problem you are working on may not be solvable, or it may be beyond your current capabilities and resources.
  6. Do a careful analysis of the steps you have carried out and the results you have achieved to see if you have created new, additional problems that need to be addressed. Reflect on what you have learned by solving the problem. Think about how your increased knowledge and skills can be used in other problem-solving situations. (Work to increase your reflective intelligence!)

This six-step strategy for problem solving is worth memorizing. One of goals in teaching problem solving is to have all students memorize this strategy and practice it so that it becomes second nature. This will help to increase your students' expertise in solving problems. Many of the steps in this strategy require careful thinking. However, there are a steadily growing number of situations in which step 5 can be carried out by a computer. The person who is skilled at using a computer for this purpose may gain a significant advantage in problem solving, as compared to a person who lacks computer knowledge and skill.

Working Toward Increased Expertise

One of the goals in IT-assisted PBL is to help students increase their expertise at solving problems. People can get better at whatever they do. A person can get better at a sport, at a hobby or craft, or in an academic field. A person's level of expertise can increase through learning and practice. A person who is really good at something relative to his/her peers is considered an expert.

It is important to distinguish between having some level of expertise and being an expert. The word expertise does not mean any particular level of ability. For anything that you can do, you can imagine a scale of performance that runs from very low expertise to very high expertise. When a person has a high level of expertise in some particular area, we call this person an expert. Bereiter and Scardamalia (1993) contains an excellent summary of research about expertise.

Figure C.4. An "expertise" scale.

Research on expertise indicates that it takes many years of study, practice, and hard work for a person to achieve their full potential in any particular area of expertise. For example, consider any one of the eight areas of intelligence identified by Howard Gardner. If a person is naturally talented in one of these areas and works really hard for 10 to 15 years within that specific area, they are apt to achieve world class status in that area. It is a combination of talent and hard work over many years that allows a person to achieve their full potential.

Transfer of Learning

Transfer of learning deals with transferring one's knowledge and skills from one problem-solving situation to another. You need to know about transfer of learning in order to help increase the transfer of learning that your students achieve.

Transfer of learning is commonplace and often done without conscious thought. For example, suppose that when you were a child and learning to tie your shoes, all of your shoes had brown, cotton shoe laces. You mastered tying brown, cotton shoe laces. The next year you got new shoes. The new shoes were a little bigger, and they had white, nylon shoe laces. The chances are that you had no trouble whatsoever in transferring your shoe-tying skills to the new larger shoes with the different shoe laces.

The goal of gaining general skills in the transfer of your learning is easier said than done. Researchers have worked to develop a general theory of transfer of learning--a theory that could help students get better at transfer. This has proven to be a difficult research problem.

At one time, it was common to talk about transfer of learning in terms of near and far transfer. This theory of transfer suggested that some problems and tasks were so nearly alike that transfer of learning occurred easily and naturally. This is called near transfer. Other problems and tasks required more concentrated effort and thinking for transfer to occur. This is called far transfer.

We know that near and far transfer occur. But, what is "near" or "far" varies with the person attempting to do the transfer. We know that far transfer does not readily occur. The difficulty with this theory of near and far transfer is that it does not provide a foundation or a plan for helping a person to get better at transfer.

In recent years, the low-road/high-road theory on transfer of learning, developed by Salomon & Perkins (1988), has proven to be more fruitful. Low-road transfer refers to developing some knowledge/skill to a high level of automaticity. It usually requires a great deal of practice in varying settings. Shoe tying and memorized arithmetic facts are both examples of areas in which such automaticity can be achieved and is quite useful.

On the other hand, high-road transfer involves cognitive understanding; purposeful and conscious analysis; mindfulness; and application of strategies that cut across disciplines. In high-road transfer, there is deliberate mindful abstraction of the idea that can transfer, and then conscious and deliberate application of the idea when faced by a problem where the idea may be useful.

For example, consider the strategy of breaking a big problem into smaller components; this is called the top-down strategy. You can learn the name and concept of this strategy. You can practice this strategy in many different domains. You can reflect on the strategy and how it fits you and your way of dealing with the problems you encounter. Similar comments hold for the library research strategy.

Eventually, you can incorporate a strategy into your repertoire of approaches to problem solving. When you encounter a new problem that is not solved by low-road transfer, you begin to mentally run through your list of strategies useful in high-road transfer. You may decide that breaking the problem into smaller pieces would be an effective strategy to apply. Or, you may decide that library research (a Web search) is a good starting point.

Two keys to high-road transfer are mindfulness and reflectiveness. View every problem-solving situation as an opportunity to learn. After solving a problem, reflect about what you have learned. Be mindful of ideas that are of potential use in solving other problems.

Of course, there are a wide range of problems that lie between those easily handled by low-road transfer and those that require the careful, conscious, well-reasoned, mindful approaches suggested by high-road transfer. The previous chapter discussed the many years of hard work required to gain a high level of expertise in a domain. To a large extent, this work results in moving many problems from the middle ground in the domain toward the low-road transfer end of the scale. More and more of the problems that you encounter in the domain are quickly and easily solved, almost without conscious thought and effort.

Summary of Important Ideas

PBL provides an environment that can be used to help students to improve their problem-solving and higher-order thinking skills. Students will make significant progress if:

  1. They have ownership of the problems to be solved and the tasks to be accomplished. They are intrinsically motivated.
  2. The problems to be solve and the tasks to be accomplished are challenging--they stretch the capabilities of the students.
  3. There is explicit instruction on key ideas such as:

    A. Problem posing. Working to achieve a clearly defined problem. As you work to solve a problem, continue to spend time working to define the problem.

    B. Building on the previous work of yourself and others.

    C. Transfer of learning.

    D. Viewing each problem/project as a learning opportunity. As you work on solving a problem, work to learn things that will help you in the future Do metacognition. Do a conscious, considered analysis of the components and the overall process in each challenging problem that you address. This will help you to get better at solving problems.

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