Moursund's IT in Education Home Page

Brief Introduction to Problem Solving

"If you cannot solve a problem, then there is an easier problem you cannot solve: find it."

(George Polya: How to Solve It. Princeton, 1945).

Last revised Spring 2002. This is a chapter-length introduction problem solving and roles of computers in problem solving. Click here for a short book on the same topic

Introduction

Problem and Task Team

Domain Specificity

What is a Formal Problem?

Representations of a Problem

Representing Problems Using Computers

Problem Posing and Clarification

Some Problem-Solving Strategies

A General Strategy for Problem Solving

Working Toward Increased Expertise

Transfer of Learning

Project-Based Learning

Summary of Important Ideas

References

Introduction

This document gives a brief overview of the "subject" of problem solving and of roles of Information and Communications Technology (ICT) in problem solving. It is targeted specifically toward preservice and inservice teachers. The ideas from this document can be woven into instruction in almost any curriculum area or methods course.

This document focuses both on solving problems and on accomplishing tasks. We will use the term problem solving to refer to both solving problems and accomplishing tasks. Our goal is to help improve the quality of education that students receive in our educational system.

Our educational system attempts to differentiate between lower-order cognitive (thinking) skills and higher-order cognitive (thinking) skills. In recent years there has been increased emphasis on higher-order skills. In very brief summary, we want students to learn some facts, but we also want them to learn to think and solve problems using the facts.

Often the "thinking" that we want students to do is to recognize, pose, and solve complex, challenging problems. Thus, one of the goals of education is to help students to get better at posing, representing, and solving problem. 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.

Many people have observed that the "every teacher teaches problem solving" is a haphazard approach, and that the result is that students do not get a coherent introduction to problem solving. When a student reaches a specified grade level, can the teacher be assured that the student has learned certain fundamental ideas about posing, representing, and solving problems? Can the teacher assume that a student knows the meaning of the terms problem, problem posing, and problem solving? Can the teacher be assured that the students know a variety of general purpose strategies for attacking problems? In our school system at the current time, the answer to these questions is "no."

Thus, each teacher is left with the task of helping their students master both the fundamentals of problems solving and then the new problem-solving topics that the teacher wants to cover. This document covers the basics (fundamentals) of problem solving. It is designed as a general aid to teachers who need to cover the fundamentals with their students.

Of course, the fundamentals need to be interpreted and presented at a grade-appropriate level. This document does not try to do that. It is left to the individual reader to understand the fundamental ideas and then present them in a manner that is appropriate to their students.

This document places particular emphasis on several important problem-solving ideas:

  1. Posing, representing, and solving problems are intrinsic to every academic discipline or domain. Indeed, each discipline is defined by the specific nature of the types of problems that it addresses and the methodologies that it uses in trying to solve problems.
  2. There are some tools (for example, reading and writing) that are useful in addressing the problems in all disciplines. Information and Communications Technology (ICT) provides us with some new and powerful tools that are useful aids to problem solving in every discipline.
  3. Much of the knowledge, techniques, and strategies for posing, representing, and solving problems in a specific domain requires a lot of knowledge of that domain and may be quite specific to that domain. However, there are also a number of aspects of posing, representing, and solving problems that cut across many or all domains, and so there can be considerable transfer of learning among domains. Our educational system should help all students gain a reasonable level of knowledge and skill in these broadly applicable approaches to problem solving.

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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 document 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 1 illustrates the P/T Team. These concepts are explained in subsequent paragraphs.

Figure 1. People aided by physical and mental tools.

Figure 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. Mental aids supplement and extent capabilities of a person's mind.
  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 flint knife, 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.

ICT is a combination of both mental and physical aids. One way to think about this is the use of computers to automate factory machinery. Such machinery stores/contains a certain type of knowledge, and the machinery can use that knowledge to carry out certain manufacturing tasks. An artificially intelligent, computerized robot provides another example of a combination of mental and physical tools.

The mental aids and physical aids components of a P/T Team are dynamic, with significant changes occurring over relatively short periods of time. The pace of change of ICT seems breath taking to most people.

On the other hand, our formal educational system has a relatively slow pace of change. This has led to the interesting situation of many preschool children growing up with routine access to mental and physical aids, learning their use through our informal educational system, and then encountering formal education that is woefully inadequate in dealing with such aids. For example, many elementary school students have more ICT knowledge and skill than do their teachers.

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.

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Domain Specificity

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.

Moreover, each academic discipline has a huge amount of accumulated knowledge. A mathematician can spend a lifetime studying a specific subdomain such as algebra, geometry, or statistics, and not fully master just one specific subdomain. A musician cannot hope to gain a high level of expertise in each musical instrument and type of music. Similarly, an artist cannot hope to gain a high level of expertise in each art medium.

Research into problem solving has indicated that one needs considerable domain-specific knowledge and skills to solve pose, represent, and solve problems within that domain. People use the term "domain specificity" when discussing this idea. Thus, it is not surprising that formal education is usually broken up into specific courses that focus on specific components of specific domains. This approach allows courses to be designed and taught by people who are relatively competent in the subject matter domain of the course.

Domain specificity is a major challenge to our educational system. For the most part, "real world" problems cut across different domains. Thus, we teach students in a domain-specific manner and environment, and expect that they will transfer their knowledge and skills to the interdisciplinary problems they encounter outside of school. This is a huge leap of faith, and what actually happens is that relatively few students make such transfers of learning.

Fortunately, the situation is not quite as bleak as it sounds. 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 they might encounter.

Moreover, there is a gradual increasing understanding of how to teach for transfer. That is, progress in the domain of transfer of learning is beginning to provide teachers with specific information about how to teach for transfer.

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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 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). (Some writers talk about having multiple goals in a problem. However, such a multiple goal situation can be broken down into a number of single goal situations.)
  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. If one or more of these components are missing, we call this a problem situation. An important aspect of problem solving is realizing when one is dealing with a problem situation and working to transform that into a well-defined problem.

People often get confused by the resources (part 3) 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 by at least 20% 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. Indeed, the problem might not be solvable. (Imagine an automobile manufacturer trying to produce a 20% increase in sales in three months, for $40,000!)

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. A person may have ownership "assigned" by his/her supervisor in a company. That is, the company, or the supervisor has ownership, and assigns it to an employee or group of employees.

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. Such ownership contributes to intrinsic motivation--a willingness to commit one's time and energies to solving the problem.

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/assigned 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.

Many teachers make use of project-based learning within their repertoire of instructional techniques. Within PBL, students often have a choice on the project to be done (the problems to be addressed, the tasks to be accomplished), subject to general guidelines established by the teacher. Thus, students have the opportunity to have an increased level of ownership of the project they are working on. Research on PBL indicates that this ownership environment can increase the intrinsic motivation of students.

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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 situation. This mental model may include images, sounds, or feelings. You can carry on a conversation with yourself--inside your head--about the problem situation. You begin to (mentally) transform the problem situation into a well-defined 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 representation or 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 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. Indeed, ICT-based computational mathematics is now an important approach in representing and attempting to solve a wide range of mathematics problems.

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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. You can quickly pose and answer "What if?" types of questions about possible alternative actions you might take.

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, suppose the problem that you face (that is, the task you want to accomplish) is writing a report on some work that you have done. You write using a word processor. Thus, you produce a computer model of the report. You know, of course, that a key to high quality writing is "revise, revise, revise." This is much more easily done with a computer model of a report than it is with a paper and pencil model of a report. In addition, 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, metaphor, mental pictures, smells, and so on. 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.

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Problem Posing and Clarification

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 a foreign nation. The announcer continues with a news item about students in our schools scoring poorly on an international test, relative to those from some other countries. 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? Can I check out the facts through alternative sources that I feel are reliable?)
  2. Is there a clearly defined goal? (Is it really clear to me how I would like things to be? Are there a number of possible goals? Which goal or goals seem most feasible and viable?)
  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 of my own time and money on achieving the goal?)

If you can answer "yes" to each of these questions, then you (personally) 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.

Identifying and posing problem situations, and then transforming them into well-defined problems, are higher-order thinking tasks. These tasks are not adequately addressed in our educational system. To a very large extent, students are asked to work on problems that are posed by the teacher and/or the curriculum materials. The problems tend to be quite limited in scope and typically lack a "real world" quality. Typically students are not asked to explore problem situations such as hunger, homelessness, prejudice, terrorism, and so on. They tend to (incorrectly) "learn" that all problems have solutions, and that they are "dumb" or not working hard enough if they do not find "the solution" to a problem that has been assigned.

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 ongoing process as you work to understand and solve a problem.

Problem posing is a higher-order thinking skill that is an integral component of every domain. Moreover, 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.

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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.) Because each subject matter (each domain) has its own set of domain-specific 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. For example, the task of writing a long document may be approached by developing an outline, and then writing small pieces that fill in details on the outline.

Library research is a type of "ask an expert" strategy. A large library contains the accumulated expertise of thousands of experts. The Web is a rapidly expanding global library. It is not easy to become skilled at searching the Web. For example, are you skilled in using the Web to find information that will help you in dealing with Language Arts problems, Math problems, Science problems, Social Science problems, personal problems, health problems, entertainment problems, and so on? Each domain presents its own information retrieval challenges.

An alternative to using a library in an "ask an expert" approach is to actually ask a human expert. Many people make their livings by being consultants. They consider themselves to be experts within their own specific domains, and they get paid for answering questions and solving problems within their areas of expertise. The "Ask ERIC" system provides a human interface to the ERIC information retrieval system.

The various fields of science share a common strategy called Scientific Method. It consists of posing and testing hypotheses. This is a form of problem posing and problem solving. Scientists work to carefully define a problem or problem area that they are exploring. They want to be able to communicate the problem to others, both now and in the future. They want to do work that others can build upon. Well done scientific research (that is, well done problem solving in science) contributes to the accumulated knowledge in the field.

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. A useful and 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.

Problem-solving strategies can be a 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.

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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. Keep in mind that not every problem is solvable. Also, you may lack the knowledge, skills, time, and other resources needed to solve a particular problem. 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(s) of the problem, which could well be interdisciplinary.
  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. Note that this reflective thinking 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!)

Many people have found that this six-step strategy for problem solving is worth memorizing. As a teacher, you might decide that one of your goals in teaching problem solving is to have all your 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 six-step strategy require careful thinking. However, there are a steadily growing number of situations in which much of the work of step 4 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.

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Working Toward Increased Expertise

One of the goals of instruction in any subject area is to help students increase their expertise at posing, representing, and solving problems in the subject area. 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 to be 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 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 a high level of expertise in an area.

Because it takes so much time, and effort to achieve a high level of expertise in just one narrow field, few people achieve a high level of expertise in multiple fields. For example, consider how few professional athletes perform at a world class level in two different sports. Or, consider the general practitioner versus the specialists in medicine.

One of the successes in the field of Artificial Intelligence has been the development of Expert Systems--computer programs that exhibit a considerable level of expertise in solving problems within a specific domain. In a number of narrowly defined domains, Expert Systems or humans working together with an expert system can perform at a quite high level of expertise. This, of course, has profound implications in education. Suppose a computer program (an Expert System) exists with a specific domain that is being covered in a school curriculum. Now, what do we want students to learn about solving problems in that domain? Do we want students to learn to compete with the Expert System, or learn to work with the Expert System?

You may think that such questions do not pertain to our ordinary curriculum. But, one can think of a handheld calculator as having some Artificial Intelligence. More sophisticated calculators can solve a wide range of math problems. A spelling checker in a word processor has a certain level of expertise, as does a grammar checker. The point is, progress in Artificial Intelligence is providing us with powerful aids to problem solving (that is, resources) in many different domains.

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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.

However, there are many transfer of learning situations that are far more difficult. For example, a secondary school math class might teach the metric system of units. From the math class, students go to a science class. Frequently the science teacher reports that the students claim a complete lack of knowledge about the metric system. Essentially no transfer of learning has occurred from the math class to the science class.

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 are so nearly alike that transfer of learning occurs 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.

The theory of near and far transfer does not help us much in our teaching. 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 for most students. 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, keyboarding, steering a car, and memorized arithmetic facts are 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 top-down strategy of breaking a big problem into smaller components. 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. Earlier in this document we 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.

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Project-Based Learning (PBL)

PBL is an individual or group activity that goes on over a period of time, resulting in a product, presentation, or performance. PBL typically has a timeline, milestones, and other aspects of formative evaluation as the project proceeds.

Doing a project and solving a problem have much in common. For example, in PBL a student or team of students typically have considerable latitude in posing the details of what will be accomplished in the project. There are limited resources, such as time. There is a clear goal of a product, presentation, or performance. The student or team of students may well develop a high level of ownership as they work on the project. Thus, any PBL environment is a good environment for teaching problem solving.

PBL shares much in common with Process Writing. In the United States, the roots of Process Writing are often traced back to the Bay Area Writing Project circa 1975. A six step version of Process Writing is:

  1. brainstorming
  2. organizing the brainstormed ideas
  3. developing a draft
  4. obtaining feedback
  5. revising, which may involve going bask to earlier steps
  6. publishing

This list can be viewed as a strategy for accomplishing the task (solving the problem) of doing a writing project.

Summary of Important Ideas

Each classroom teaching situation provides an environment that can be used to help students 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:
    1. 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.
    2. Problem representation (Yarlus and Sloutsky, 2000).
    3. Building on the previous work of yourself and others.
    4. Transfer of learning.
    5. 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.
    6. Roles of Information and Communications Technology in problem solving.

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References

Bereiter, C., & Scardamalia, M. (1993). Surpassing ourselves: An inquiry into the nature and implications of expertise. Chicago and La Salle, IL: Open Court.

Frensch, P. & Funke, J., (Eds.). (1995). Complex problem solving: The European perspective. Hillsdale, NJ: Lawrence Erlbaum Associates.

Moursund, D.G. (1996). Increasing your expertise as a problem solver: Some roles of computers. Eugene, OR: ISTE. Some chapters are available online.

Norman, D. (1993). Things that make us smart: Defending human attributes in the age of machines. Reading, MA: Addison-Wesley.

Perkins, D. (1992). Smart schools: Better thinking and learning for every child. NY: Free Press.

Polya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton, NJ: Princeton University Press.

Problem Posing and Problem Solving: The BioQUEST Library [Online]. Accessed 10/23/01: http://bioquest.org/indexlib.html.

Problem Posing in Academic Writing [Online]. Accessed 10/23/01: http://www.temple.edu/writingctr/cw06001.htm.

Salomon, G., & Perkins, D. (1988, September). Teaching for transfer. Educational Leadership, 22-32.

Yarlus, A.S. and Sloutsky, V.M. (2000). Problem Representation in Experts and Novices: Part 1. Differences in the Content of Representation; Part 2. Underlying Processing Mechanisms.

Part 1 [Online] Accessed 11/2/01: http://www.cis.upenn.edu/~ircs/cogsci2000/
PRCDNGS/SPRCDNGS/posters/yar_slo.pdf.

Part 2 [Online] Accessed 11/2/01: http://www.cis.upenn.edu/~ircs/cogsci2000/
PRCDNGS/SPRCDNGS/PAPERS/SLO-YAR.PDF

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