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Home Page of Problem Solving Book

Introduction to the Book

Chapter 1: Introduction to Problem Solving

Chapter 2: Overview of Resources in Problem Solving

Chapter 3: Intelligence as Resource

Chapter 4: Tools as Resource

Chapter 5

Building on Previous Knowledge

Procedural and Declarative Knowledge

Domain Specificity

Don't Reinvent the Wheel

Declarative and Procedural Knowledge

Studies of Chess Experts

Strategy is a General Plan

General Strategy for Problem Solving

Concepts and Underlying Theory

What is the Right Balance?

Computers--A New Aid to Problem Solving

Activities and Self-Assessment

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 5: Accumulated Knowledge as Resource

You carry around a great deal of knowledge and experience in your mind/body. That is, you have a great deal of experiential intelligence. This has been accumulated over many years. You use much of this knowledge freely and easily--nearly effortlessly. For example, when you are talking, you formulate an idea in your head, translate it into words, and speak the words. [[When you are speaking in your native tonge or in a well-learned second language, no conscious thought is required to move from the thought to the spoken words. This is an automatic, subconscious activity. The Brain Science sopic "mirror neurons" is providing some insight into this aspect of brain functioning.]] When you want to use a fork to get a bite of food from a plate and transfer it to your mouth, little conscious effort is required.

As is suggested by Figure 5.1, problem solving can be viewed as an interaction among the humans working to solve a problem, accumulated knowledge of the human race, and the problem to be solved. (Remember, quite a bit of the accumulated knowledge needed for a particular problem may reside in the mind/body of the problem solver.)

Figure 5.1 Interacting facets of the problem-solving process.

Building on Previous Knowledge

You routinely solve many of the problems that you encounter by just using the knowledge you have accumulated throughout your lifetime. However, you are but one person. Collectively, the human race has accumulated a great deal of knowledge, far more than any one person can learn. Much of this accumulated knowledge is stored in people's heads; however, a great deal is stored in books, films, tapes, paintings, and other artifacts. Now, an increasing percentage of that knowledge is stored in computer systems.

Remember, a computer is both a storage device and a processing device. Thus, a computer is an excellent aid to building on the previous work of other people.

You have heard the expression, "Don't reinvent the wheel." Many of the problems that people want to solve can be solved by finding out how others have solved the same problem in the past, and then doing what they did. Computers are making it easier and easier to find out what others have done when faced with particular problems. The libraries and other information sources throughout the world are being computerized and networked together by the Information Superhighway.

[[It is interesting to think about dividing problems into three categories: 1) Those that can be solved by computers, or where a computer is of great aid in solving the problem; 2) Those that cannot be solved by computers, or where a computer is of little value; and 3) Those in which the value of a computer is not clear. The Venn diagram given below helps to give a picture of the situation. The purpose for raising this issue is that our educational system might want to place some emphasis on having students learn to recognize problem situations in which a computer is apt to be a valuable resource, and ones in which a computer is not apt to be a valuable resource.]]

 

[[Added Figure: Venn diagram of possible usefulness of ICT as a resource.]]

Moreover, software (called groupware) has been developed that helps groups of people work together to solve problems. Progress in computerizing and networking the world is having a profound impact on problem solving.

[[Note also that computers can rapidly and accurately carry out a detailed step by step set of instructions (a computer program). There are many problems that computers can solve. There are a still larger set of problems that computers along with computer-automated machinery can solve. In summary, an increasing amount of human knowledge is being stored in computer systems in a "the machine can do it for you" form. Often it takes relatively little time and effort to learn to make use of this form of previously developed knowledge.]]

Procedural and Declarative Knowledge

Your mind/body "knows" a great deal of information. As Figure 5.2 suggests, this can be divided into two main categories: procedural and declarative. Declarative knowledge can be divided into two categories: episodic and semantic. This section explains each of these types of human knowledge.

Figure 5.2 A model of human memory.

Procedural knowledge is knowledge about how to do things; it is often kinesthetic. Your mind/body has mastered many different physical procedures for accomplishing tasks. For example, when you were a young child, you did not know how to tie your shoes. Now, you tie your shoes without conscious effort. You have mastered this skill; it is stored in your mind/body.

[[Walking provides another good example.]]

Touch typing (touch keyboarding) provides another example of procedural knowledge. Your procedural knowledge can be thought of as a set of BBRs, immediately available to you in problem solving.

[[Touch keyboarding provides a good topic for discussion. If one obtains a speed of approximately 25 words per minute, this procedural skill will remain with you even if it not used for many years. Your skill will become "rusty," but a little practice will lead to regaining much of your original skills.

On the other hand, touch keyboarding instruction leading to a speed of 15 words per minute does not stay with you. That is, the procedural skill is not sufficiently well developed so that it survives a period of non use (such as over a summer). After a long period of non use, it takes nearly as much time and effort to regain the original speed and accuracy as it did to gain it originally.

This same idea tends to hold true for many different kinesthetic procedural skills. Research in this area helps educators to make informed decisions on design of curriculum to help students gain procedural skills. Currently, much instructional time and student effort is wasted by approaches that do not take into consideration what is known about developing kinesthetic procedural knowledge that will be retained over long periods of little or no use.]]

Declarative knowledge is concerned with the facts that you know. You have memorized a great many facts or pieces of information. For example, you know the alphabet, some telephone numbers, addresses, names of people, dates, and so on. Such declarative information is important to problem solving.

People often differentiate between procedural and declarative knowledge by calling the former "know how" and the latter "know what." Your "know how" and your "know what" [[knowledge]] work together as you solve problems and accomplish tasks.

Declarative knowledge can be divided into two categories. Episodic declarative knowledge is concrete. The knowledge is established when an episode or unusual event occurs. (Think of this as one trial learning.) [[Episodic]] Declarative knowledge is very personal, intimately tied to specific episodes or settings such as: one's first kiss on a date; one's wedding; or a particularly nice vacation event. [[Almost any "first time, memorable event" has a reasonable chance of becoming part of your Episodic Declarative knowledge.]]

Semantic declarative knowledge is more abstract. It tends to be symbolic and context free. You probably do not remember when you first learned the alphabet, the counting numbers, or the chemical formula for water. Semantic declarative memories are relatively easy to create through study, and they are easy to forget.

Procedural knowledge and semantic declarative knowledge take time and effort to acquire. Moreover, a certain amount of forgetting occurs. The adage "Use it or lose it" tends to apply. Procedural knowledge and semantic declarative knowledge differ somewhat in this regard. You are probably skilled in memorizing pieces of information for a test. This is semantic declarative information. Once the test is over, you soon forget what you have memorized. If you find that you need this information a few years later, the task of memorizing it at that later date is not helped much by the earlier memorization process.

However, once you develop a reasonable level of procedural skill in a particular area, such as touch typing or piano playing, quite a bit of this procedural knowledge stays with you for a lifetime. Even if you don't type or play the piano for years, you can quickly regain a reasonable percentage of your original skill. A somewhat different way of thinking about this is that procedural knowledge BBRs tend to last a lifetime.

Our educational system struggles with the question of what knowledge all students should acquire. For example, is there some set of declarative information that every student should be required to learn? Hirsch (1988), whose book was a national bestseller, argues that the answer is "yes" and lists 5,000 "essential names, phrases, dates, and concepts that every American needs to know." From a communication point of view, it is helpful that speakers and listeners share a common core of declarative knowledge. Hirsch argues that this common core of declarative knowledge is a type of cultural literacy.

However, many people are critical of such a simplistic approach to education. Part of the argument against the Hirsch approach is that it does not place enough emphasis on higher-order skills&emdash;thinking and problem solving using one's declarative knowledge. Instead, it fosters a back to basics approach to education in which the goal is to memorize a lot of material.

[[This book places quite a bit of emphasis on having students learn domain specific and domain independent strategies for problem solving. The general idea is that these are procedures that can become part of one's procedural knowledge. If the procedures are practiced expensively over a wide range of problems and over a significant period of time, they will then endure even over long periods on non use.

This book also focuses on the idea that there are many procedures that a computer can carry out. A person can learn the general concepts of such procedures, and lot learn the details. For example, I understand the concept of square root, and I remember that most calculators have a built in square root function. Thus, I have a functional procedural knowledge of square root (I understand both the concept and how to calculate it). The details of how to calculate square root do not disappear through non use.

The topic of computer-based procedural knowledge is discussed later in this book.]]

Domain Specificity (Expertise in a Specific Domain)

It is useful to talk about the domain of a particular field of expertise. Some domains are formal academic fields; thus anthropology, biology, chemistry, dentistry, economics, and so on are each a domain. However, there are many more domains that are not formal academic fields. For example, you may have considerable expertise in antique collecting, bird watching, card playing (poker or bridge), driving in congested traffic, and so on.

In this book, we use the word domain to refer to a coherent area in which one [[poses, represents, and]] solves problems and accomplishes tasks. Through study and practice, one's level of expertise in the domain can be increased.

[[A formal academic domain may cut across a number of "traditional" academic subjects. For example, one might be a specialist in physical chemistry, molecular biology, or mathematical physics.Here is a quote from a November 2001 RFP from the National Science Foundation:
Program Title: Quantum and Biologically Inspired Computing (QuBIC)

Synopsis of Program: The program will support interdisciplinary research to improve the fundamental capabilities of computer and information sciences and engineering by incorporating insights from either biological systems or quantum foundations or both. To achieve this improvement, fundamental research is needed towards the unification of information science across computer science, physics, biology, and engineering. Expanding efforts in interdisciplinary research areas that are at the interface of information science and technology with the fields of biology, chemistry, engineering, physics, and computer science will lead to better understanding in all areas of science. This will increase the ability to develop future information technologies that are very critical to the economy and society at the national and international level. Group proposals representing multiple disciplines will receive priority but single investigator proposals that are cross-disciplinary will also be considered.]]

Researchers have carefully examined the nature of increasing one's expertise in a particular domain. A key idea is domain specificity, or domain-specific knowledge. The more knowledge you have about a domain, the better problem solver you tend to be on problems within that domain.

One way to think about domain specificity is from an information-retrieval model of problem solving. A person studying a particular field gradually gains knowledge about each of the major problems within that field. The person gradually memorizes a large number of patterns that correspond to the frequently occurring problems or subproblems of the domain, and also memorizes what to do when each of these patterns is encountered. In essence, the person acquires BBRs. Thus, solving a problem often consists of recognizing that the problem is somewhat familiar, and then retrieving from one's mind the information needed to solve the particular problem.

Or, if the problem is not recognized as a familiar pattern, perhaps it breaks easily into pieces that are recognized. This is one strategy that is useful in many different domains. Each domain has problem-solving strategies that are quite specific to the domain. Through study and practice one learns some of these strategies and becomes skilled in using them.

Knowing a lot about a particular domain is a combination of knowledge, skills, and experience. It can include considerable formal knowledge (book learning), or it may consist entirely of informal knowledge (learn by doing, street smarts). In either case, a lot of time and effort is needed to develop a reasonable level of expertise. For the most part, this time and effort is spent gaining knowledge and skills that other people in the domain have already acquired in the past.

[[The last sentence is a very important point. Most of the learning that a person does is learning that others have already done. We have technology (such as books and other information storage and retrieval systems) for storing data, information, and knowledge. This is helpful to the learner. We also have developed a theory and practice of teaching. A good teacher is a valuable aid to learning. Still, the totality of human knowledge (which continues to grow rapidly) is overwhelmingly vast relative to what a person can learn.

Thus, we can improve education by developing tools (artifacts) that are easy to learn to use and that incorporate knowledge and skills.]]

Learning is an individual process. When you are struggling to learn something, it doesn't help you very much to know that millions of people have had the same learning struggles in the past. Indeed, you may even find this frustrating. You may wonder why you have to spend so much time learning to do something that other people have already learned how to do.

Don't Reinvent the Wheel

One of the key ideas in problem solving is "Don't reinvent the wheel." In many situations it is possible to build on your own previous work and that of others.

The following few sections of this chapter discuss three general categories of knowledge and skills that a person can acquire and build on: declarative and procedural knowledge; concepts and underlying theory; and strategies.

As the diagram in Figure 5.3 suggests, these three categories of knowledge and skills are closely related. In essence, both the concepts and theory category and the strategies category are components of the declarative and procedural knowledge category. However, from a problem-solving point of view, it is useful to discuss them separately.

Figure 5.3 Three general categories of knowledge and skills.

Problem solving makes use of all three categories of knowledge. The specific balance of knowledge that is needed for problems in a particular domain varies both with the domain and with the person working to solve a problem.

Declarative and Procedural Knowledge

Your mind/body has a great deal of procedural knowledge. You use it as you walk, talk, drive a car, ride a bicycle, write, keyboard, and so on. Procedural knowledge plays an important role in athletics. Athletes practice fundamentals over and over again throughout their careers. These fundamentals are the BBRs necessary to achieve a high level of athletic performance.

In the remainder of this section we focus on declarative knowledge, especially semantic declarative knowledge. Much of this is based on "book" learning. Let's begin with a thought experiment. Suppose that you had memorized the contents of every book and journal that is currently in the United States Library of Congress (a very extensive library) and you could easily recall any of this memorized information whenever you liked. Would this memorized knowledge help you to be an expert problem solver across many different domains?

The basic question is: how important is memorization in solving problems and accomplishing tasks? Researchers on expertise have examined this question in some detail. The answer has several parts:

  1. There are many recurring problems. Memorization of how to solve them can be useful. Moreover, these memorized solutions serve as BBRs. A large repertoire of BBRs can be useful in problem solving.
  2. Real-world problems occur in varying contexts or environments. Often it is difficult to recognize that a particular context or environment is just a slight variation of something that you know. Thus, rote memorization is, for the most part, only useful in problem solving if one has good skills at transferring the memorized information into the actual problem-solving situation. Transfer of learning is discussed in the next chapter.
  3. Many of the problems that people want to solve are unique--they have not occurred frequently enough in the past to warrant their inclusion in books, journal articles, and other reference materials. Thus, rote memory by itself, even with good transfer of learning skills, will not make you an expert at solving the full range of problems in a domain.

A number of different researchers have studied world-class experts. A world-class expert is really good relative to the best in the world. We are used to the idea of world-class athletes. But, the concept of world class exists in all areas of human productivity and problem solving. There are world-class economists, musicians, mathematicians, political leaders, writers, and so on.

The chances are that you are not a world-class expert in any particular [[well studied]] [[The point is, you know more abut being you than anyone else in the world. However, "you" are not a domain that lots of people study.]] domain of problem solving. Unfortunately, for the most part expertise researchers have not studied more ordinary people--people who are not world-class experts in anything--and their efforts to achieve their full potential level of expertise. Thus, you will want to interpret the information that follows in light of your own personal goals.

Researchers have examined world-class experts in diverse areas, such as business, chess, mathematics, music, and physics. Two ideas emerge. First, it takes many thousands of hours of hard work to achieve world-class expertise. Several researchers have indicated that it takes about 10 years of hard work, as well as suitable instruction and coaching. The second important idea is that an essential part of the expertise seems to be that of having committed to memory approximately 50,000 patterns and what to do when one of these patterns (or a slight variation thereof) is encountered. To a large extent, this memorized knowledge is a set of BBRs that is internalized at a subconscious level.

[[Actually, three ideas emerge. The third idea is that it takes a high level of natural talent, in addition to the first two ideas that are discussed.]]

Similar studies in domains that depend quite a bit on procedural knowledge, such as sports, have yielded similar results. World-class gymnasts can be "produced" by careful selection of young children who seem to have a great deal of natural kinesthetic intelligence, and then putting these children through approximately 10 years of intense training.

In summary, research studies of world-class experts suggest that for any person to achieve their full potential level of expertise in some domain, they will need to work quite hard for many years and internalize a great deal of declarative and/or procedural knowledge.

[[Good teaching (coaching) is quite helpful. Research suggests that one--on one teaching (individual tutoring) helps a person to learn better and faster than does conventionally sized classes.]]

The educational implications of this seem clear. Persistence is essential. It takes many years of hard work in a field to develop a high level of expertise. If you want to be good in a field--whether sports or academics--you must be willing to work hard at learning the field. This hard work needs to be continued over a long period of time, enough time to internalize many thousands of patterns and what to do when these patterns are encountered.

Of course, this creates a dilemma. Suppose you want to be good in a number of different fields? Suppose that you want to solve interdisciplinary problems? Suppose that you frequently encounter new problems that are outside the domains that you have studied? Life is not long enough to spend years becoming an expert in a large number of different domains. Time is a limited and limiting resource.

A partial answer to this dilemma is that each of us can work toward a balance of general breadth and specific depth. It is possible to acquire a useful level of knowledge and skills in a large variety of domains. If one of your goals is to acquire a high level of expertise in one or two domains, then the chances are that you will not develop as much breadth as people who don't focus so much attention on narrow domains.

A second answer lies in focusing on the reflective intelligence component of intelligence that Perkins (1995) discusses in his book. Reflective intelligence includes strategies that cut across many domains. It includes habits of mind and attitudes that are essential to attaining expertise in any domain. This will be discussed more in the next chapter.

In any event, the tools that humans have produced can be of immense help in developing a reasonable level of expertise in many different domains. Later in this chapter, we will focus on the computer-as-tool strategy in developing broad-based problem-solving skills.

Studies of Chess Experts

A number of studies of world-class experts have focused on chess players. Moreover, there has been a great deal of progress in developing computer programs that can play chess well. It is no longer unusual to have chess tournaments in which humans and computers compete against each other. The studies of human chess players and the work to develop computer programs that play chess well have led to considerable insight into problem solving. This section explores some of the ideas that have come out of this research.

[[In 1997 a chess match was held between Deep Blue (an IBM computer designed specifically to be fast at analyzing chess moves) and Gary Kasparov, the reigning world chess champion. The computer won.]]

Some people have a knack for chess. This may be a combination of spatial and logical/mathematical intelligence. Through many years of hard work and conscious directed effort, they can develop this innate ability into a high level of chess expertise. It is important to emphasis the "hard work and conscious directed effort." Many people play a game such as chess or bridge for years, but don't improve appreciably after the first few years. They do not put in the hard work and conscious directed effort needed to continue their improvement. [[It has been estimated that it takes 50,000 hours of study and practice to become a world class chess player.]]

One of the classical chess studies had both expert and novice chess players look for a few seconds at a collection of chess pieces arranged on a chess board and attempt to memorize their placement. If the pattern of pieces is a naturally occurring one from the middle of a reasonably well-played game, an expert chess player can accurately place more than 90% of the pieces after viewing the board for 10 seconds. A novice player accurately places less than 25% of the pieces under the same conditions. And, if the pieces are merely located at random on the board, both the expert and the novice score at less than 25%. That is, an expert chess player can quickly recognize naturally occurring chessboard positions but is no better than a novice at memorizing random board positions. This is taken as evidence that a mental pattern-matching process is occurring at a subconscious level.

Humans are very good at subconscious pattern matching. You do it when you recognize a familiar face, a familiar voice, or a familiar smell. To-date, humans are far better at such pattern-matching tasks than computer systems. This presents a major challenge to researchers in the field of artificial intelligence.

However, computers are far better at rote memory than humans. Thus, you might think that computers could be very good at chess through rote memory. Just program the computer to play through every possible game of chess and memorize every winning combination of moves.

However, that is impossible. It has been estimated that there are 10**120 (10 to the 120th) different possible games in chess. The number 1 followed by 120 zeros is a very large number! If a computer could memorize a trillion trillion possible board positions every second, it would still take far longer than the age of the universe for it to memorize even a tiny fraction of the possible board positions.

That does not mean that rote memorization is not important in chess. At the beginning of the game, the number of possible moves is limited and these have been studied extensively. Memorizing opening moves is a good way to get better at chess. The same holds for end-game situations.

Rote memory is useful in problem solving in every domain. But, even such a game as chess overwhelms a rote-memory approach to problem solving. Many of the problem-solving domains in the real world are far more complex than chess. And, there are thousands of different domains. Thus, rote memorization is of limited value in gaining increased expertise in problem solving.

Something else is needed in addition to memorization. Part of the answer is the learning of strategies and developing skill in their use. Another part of the answer lies in learning general concepts and underlying theory. These ideas are discussed in the sections that follow.

A Strategy is a General Plan

Figure 5.3, shown previously in this chapter, diagrams three general types of knowledge and skills useful in problem solving. Both strategies and concepts and theory can be thought of as special types of declarative and procedural knowledge.

This and the following section discuss 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&emdash;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 few strategies that are powerful and applicable across all domains. That is, in problem solving, we face the issue of domain specificity. 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.

Do you know some general strategies that are useful in many different problem-solving domains? How about the idea of breaking big problems into smaller problems. This 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. Another strategy is to draw a picture, diagram, or other graphic aid to represent the problem. This is the draw-a-picture strategy.

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&emdash;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 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.

A General Strategy for Problem Solving

Here is a general six-step strategy that you can follow in attempting to solve almost any problem. 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 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. Try using it with the problems that you encounter. Eventually it will become second nature. You will probably find that learning and using this strategy improves your overall ability to solve the types of problems you encounter in your everyday life.

Understanding of Concepts and Underlying Theory

Figure 5.3 given earlier in this chapter diagrams the three general types of knowledge and skills useful in problem solving. In this section, we will discuss concepts and theory.

Although concepts and theory are intertwined, it is often possible to have a useful understanding of a concept without knowing the underlying theory. Einstein's theory of relativity involves complex mathematics and physics. Only a modest number of people in the entire world have a good understanding of this underlying theory. But, you probably understand the concept that no object can move faster than the speed of light. Perhaps you understand the concept that the mass of an object increases as its velocity increases.

For a more mundane example, consider a computer model of a building, where the computer program allows you to view the building both from its outside and from its inside, from any angle. The concept is easy enough to understand. Such a computer program may be easy to use. However, it is a real challenge to develop such a computer program. This area of problem solving is part of the field called virtual realities.

In academic fields, the general concepts and underlying theory tend to be well developed. Within academic domains, a lot of research has focused on the importance of understanding underlying concepts and theory as an aid to problem solving. The results support the importance of understanding concepts and theory.

High-school geometry provides a good example. It is possible to memorize the proofs of a large number of theorems. Many people manage to pass a geometry course by using this approach. However, this approach is nearly useless when one is asked to deal with problems that they have not encountered before. In addition, research suggests that this approach is not of lasting value. The memorized proofs are soon forgotten and the student is left with little real knowledge of geometry that can be applied to problem solving in math, the other sciences, engineering, and so on.

Outside of academics, domains vary considerably in the nature and extent of the underlying concepts and theory. Often the underlying concepts are intuitive and may be difficult or perhaps impossible to put into words.

For example, a skilled craftsperson or artist produces a remarkable product. Can this person put into words the method of making a beautiful work of art? What are the underlying concepts or theories that make this product so outstanding? Can these concepts be stated as a set of rules that others can learn to follow?

A skilled soccer player has a remarkable sense of space, of being in the right place at the right time, and of making the right plays. This person has high levels of kinesthetic and spatial intelligence. Is there an underlying theory&emdash;the fundamental concepts of soccer&emdash;that this person has learned and follows? Can this theory be represented and taught in a manner to help others develop a high level of expertise in playing soccer?

What is the Right Balance?

Suppose that you have selected a domain in which you intend to develop an increased level of expertise. How much of your learning time and effort should you put into increasing your declarative and procedural knowledge? How much should you put into learning new strategies? How much should you put into learning concepts and underlying theory?

These questions do not have a simple answer. The answers vary both with the domain and with the person working to gain increased expertise in the domain. Thus, it is possible that two people will each have a high level of expertise in a domain, but will have different profiles of knowledge and skills in the domain. The time and effort required to gain the high level of expertise may vary considerably. This may be dependent on the learner's level of neural intelligence, especially as it relates to the specific domain. It may also be quite dependent on the person's experiential intelligence and reflexive intelligence.

This is a challenge to educators and to learners. A teacher or coach may have a great deal of general knowledge about how people learn and develop increased expertise in a domain. However, the teacher or coach cannot know your particular set of knowledge and skill attributes as well as you yourself can know them. Thus, knowledge of self (intrapersonal intelligence) is very important. Ultimately, you are responsible for your own learning.

Computers--A New Aid to Problem Solving

We conclude with a brief discussion of computers and how they relate to the other main themes of this chapter.

The original computers were designed to help do math calculations. Now, of course, computers are a versatile tool, useful in many different domains. In many domains, computers strongly affect the balance of knowledge, strategies, and concepts needed for a high level of performance. The later chapters of this book discuss these ideas in detail. Here is a brief overview.

  1. In terms of declarative and procedural knowledge:
    • Computers are powerful aids to the storage and retrieval of information. In many cases, using a computer to store and retrieve information is an effective substitute for memorizing the information. You can "look it up" using a computer instead of looking in your memory.

      [[People talk about "just in time" education. An example is provided by memorizing lots of "stuff" well in advance of when you will need to use it, versus memorizing it or looking it up at the time you need to use it. Improvements in ICT-based information retrieval systems make it easier to look up information just when you need it, rather than memorizing it well in advance.]]

    • Computers and computerized equipment can carry out many different procedures. In many cases, use of computerized or automated procedures is an effective substitute for mind/body procedural knowledge.
  2. It has proven difficult to incorporate strategic knowledge into computers. However, progress has occurred in a number of domains. The field of computer science known as artificial intelligence [[especially, expert systems]] addresses these types of issues.
  3. For the most part, it has proven difficult to incorporate understanding of concepts and underlying theory into computers. To put it bluntly, computers are dumb! Even the best of the artificially intelligent computer systems have almost no understanding of or feeling for what they are doing. Thus, they are not able to detect colossal blunders.

This analysis suggests that there are some things that computers can do well, and there are some things that computers do poorly. To a reasonable extent, the relative strengths of computers correspond to relative weaknesses of people, and vice versa. It is difficult for a person to memorize a book verbatim; but a computer can memorize thousands of books. Humans are not good at repetitious tasks and are prone to error; computers are fast, accurate, and not bored by repetitious tasks.

[[Humans understand what it means to be a human being. They have a high level of Intrapersonal and interpersonal intelligence.]] Humans are good at understanding concepts and using these concepts to make decisions. Humans are good at learning strategies and then making appropriate use of these strategies as they work to solve problems. Humans are better than computers at dealing with concepts and strategies.

As you increase your knowledge of using computers to solve problems, you will gradually develop a repertoire of strategies that will be particularly useful in a computer environment. One good example is the strategy of guess and check (sometimes called trial and error). Guess and check is useful in many domains and is often done without the use of a computer. You have a problem to solve. You make an educated guess at a solution. You then check to see if you have actually solved the problem. Perhaps you use this technique when working on a crossword puzzle or in putting together a picture puzzle.

There are many situations in which a computer is helpful in doing the work of generating a guess. Computer graphics, for example, can make it easy to experiment with a design. You can look at the results of your experiment to see if it solves the problem you had in mind. You and the computer working together become a powerful combination, able to quickly accomplish tasks that neither you nor the computer can do alone.

[[Retrieving information from the Web, using a search engine, can be thought of as a guess and check situation. Both the guessing and the checking require creative intelligence. A person and a computer system working together can easily outdo either one working alone.]]

Another good example comes from the use of spreadsheets. You develop a spreadsheet model of a particular business situation. You can then ask "What if?" questions. A few keystrokes allow you to make changes in the business situation and then to examine the results. Such use of spreadsheets provides a powerful example of making effective use of the strengths of both people and computers.

A third type of example comes from situations in which the computer itself can generate guesses, either in a random or a systematic manner. A computer may be able to check tens of millions of different guesses in a relatively short period of time. This brings a new dimension to problem solving.

Activities and Self-Assessment

  1. Select several different domains in which you have a reasonable level of expertise. For each, make a list of:
    • major types of procedural knowledge and declarative knowledge needed in the domain;
    • useful strategies in solving the problems in this domain;
    • key concepts and underlying theory needed for expertise.
  2. Do some introspection on your relative strengths in declarative and procedural knowledge. Are you better at learning and using declarative knowledge or at gaining and using procedural knowledge? Cite some evidence to support your analysis.
  3. Recall a relatively complex problem that you have solved. Analyze this problem from a point of view of the six-step general problem-solving strategy given earlier in this chapter. Which of the six steps did you follow? What did you do in this overall problem-solving process that might contribute to your experiential intelligence or reflective intelligence?
  4. Make a list of things that you are sure you can do better than a computer. Make a list of things that you feel a computer or computerized equipment can do better than you. Does each individual list contain some common themes? That is, are there some general categories in which you feel you can outperform a computer and vice versa?
  5. What is your domain of greatest expertise? What have you done over the years to increase your expertise in this domain? Make an estimate of how much time you have put in to achieve your current level of expertise in this domain. What approaches are you following to maintain your current level of expertise or to continue to improve it? What are your goals?
  6. One useful technique in problem solving is building on the work of others. A person can become skilled at doing library research on what is known about a particular problem. These library research skills are specific to the domain of library science but are domain independent in the sense that they can be applied across a broad range of academic domains. Analyze your strengths and weaknesses in library research skills. How is this area being affected by computers? Are you comfortable using a computer to do library research? Are your skills equally well developed in searching Internet sources of information?

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