
A superbly trained runner can run a mile in under four minutes. With the aid of a bicycle, many people can traverse a mile in much less time than that. With a car, still less time is required. If the problem is to get from Point A to Point B rapidly, then running, using a bicycle, using a car, or using an airplane might all be alternatives to consider. The most appropriate tool will depend on a number of different conditions, including the distance between Point A and Point B. Donald Norman (1993) argues that much of the "smartness" and overall performance capabilities of people is due to the tools that humans have invented. Humans have developed aids to their physical capabilities (physical artifacts) and aids to their mental capabilities (cognitive artifacts). The knowledge and skills to use these tools greatly enhances one's abilities to solve problems and create products. This chapter discusses tools as a resource in problem solving. The main focus is on computerastool. Calculator ExampleThe history of the abacus goes back at least 5,000 years. The abacus is such a good aid for doing arithmetic that its use is still taught in some countries. Moreover, the abacus provides a good model for learning and understanding number systems and arithmetic. Thus, bead frames are often used as a learning aid in elementary school classrooms. The handheld electronic calculator is a relatively new aid to computation. A calculator can be thought of as a limitedpurpose computer. By 1980, handheld calculators were inexpensive enough to come into widespread use by the general public. The National Council of Teachers of Mathematics recommended their routine use in schools. However, parents, teachers, and the overall educational system have resisted this change in our educational system. This section gives some insights into the nature of the arguments for and against the introduction of calculators into schools. The goal is to help you gain increased insight into both roles and controversy of making use of technology in education. Since handheld calculators have become cheaply and readily available, people have argued about whether students should be allowed to use them. There is widespread agreement that students need to understand the concepts underlying the use of arithmetic to represent and solve problems. Thus, the argument is on what tools students should be allowed to use as they actually carry out the processes involved in arithmetic computation. Should students learn to do paperandpencil arithmetic before they are allowed to use calculators? The process of paperandpencil multiplication or long division is rather complex, relative to the cognitive abilities (the innate, natureprovided mathematical/logical intelligence) that most people have. Our education system provides students with several hundred hours of instruction and practice in multiplication and long division. Still, many students achieve only a modest level of speed and accuracy at these tasks. That is, it takes a great deal of time and effort to develop a minimal amount of speed and accuracy at doing paperandpencil arithmetic operations. That is not the case when a calculator (rather than paper and pencil) is the tool. It takes only a short amount of time to learn to use a calculator. In many cases, a calculator is much faster than paperandpencil calculation. And, for many students, a calculator is more accurate than paperandpencil calculation. The previous paragraph contains three key ideas. The first two ideas have to do with time. First, there is the time needed to learn to use a tool to do arithmetic. It takes far more time to learn to do paperandpencil calculations than it does to learn to do calculatorassisted calculations. Second, there is the time needed to actually carry out calculations. In many cases, a calculator is faster than paper and pencil. The third key idea is accuracy. The human mind is not particularly good at carrying out detailed processes that require great accuracy. We may set the passing mark at 75% on a paperandpencil computation test. We would surely set a much higher passing make if the student is using a calculator. That is, the calculator has changed contemporary standards for accuracy in doing arithmetic. [[Moreover, it takes practice to maintain one's computational speed and accuracy. Many adults find that they seldom need to carry out multidigit multiplication or division in their everyday lives. More over, if a problem situation that includes such calculations does arise, a calculator is apt to be readily available. From the adult's point of view, the goal is to solve the problemand doing a computation rapidly and accurately is a critical part of the task.]] Concept Versus ProcessIn doing arithmetic, there is a substantial difference between the concept and the process. When do you divide two numbers? What is accomplished by dividing two numbers? What kinds of problems are solved by division? These are questions about the concept of division. Notice that these concept questions are independent of the process being used to do the division. The process might be carried out mentally, using pencil and paper, using a slide rule, or using a calculator. By and large, the human mind is much better at learning concepts than it is at learning [[detailed, multistep, repetitive]] processes and then developing speed and accuracy in carrying out the processes. That is one reason why people have developed tools, such as the calculator and computer, to help automate the carrying out of certain types of processes. In any problemsolving situation, you are apt to have to deal with concepts and processes. At the concept level, you work to understand the problem and to pose a clearly defined problem. You work to represent the problem in a manner that may help in solving the problem. In conceptualizing and representing the problem, you make use of cognitive artifactsfor example, counting numbers and precise vocabulary. After you have appropriately understood and represented the problem, you decide on a course of action to follow in solving the problem. You may make use of tools as an aid to carrying out the processes needed to solve the problem. One of the reasons that computers are so important in problem solving is that they are so fast and accurate at carrying out processes. This is causing educators to rethink the curriculum content in many areas. Math education leaders, for example, are now placing much more emphasis on teaching for conceptual understanding. The increased emphasis is on the higherorder thinking skills. Of course, instruction time also needs to be given to learning to use the tools effectively to carry out the processes. We need not restrict ourselves to examples from mathematics. Consider using a word processor and spelling checker when doing writing. The work processor facilitates repeated revision of one's work. The spelling checker helps to decrease the number of misspelled words. The computer printer produces output that, typically, is far easier to read than the handwriting that most people can produce. These computer tools allow increased focus on the content of the writing. [[Or, consider looking up information in a library. At the current time, the Google search engine has indexed over 1.6 billion Web pages. (This may be only one or two percent of the total amount that are actually posted to the Web.) Imagine going into a physical library that contains all of these documents. Suppose that each Web page is listed (perhaps in several different ways) in a card catalogue in the library. What are the chances you could find appropriate information in this library? The point is, we are now highly dependent on the Web and Web browsers for the storage and retrieval of information. The card catalogue was a great tool in its timebut is no longer up to the task.]] Some Problems With ToolsThe previous section discussed handheld calculators. Many people argue that students should master paperandpencil arithmetic before they are allowed to use calculators. Many others argue that there is little value in mastering paperandpencil arithmetic and that students should be taught to use calculators. Such arguments can become quite heated! Notice that in either case, there is an implicit assumption that it is appropriate to have students learn the cognitive artifact we call arithmetic. People for and against the use of calculators agree that students should learn to use arithmetic to represent and solve problems. People for and against the use of calculators agree that students need to understand the underlying concepts in math problems. The argument is about what cognitive artifacts should students use to carry out arithmetic computations. This type of argument raises a number of important issues that need to be examined whenever one is deciding on what tools to learn to use.
These are not easy questions. Try answering them for the issue of learning paperandpencil arithmetic versus learning to use a handheld calculator. Keep in mind that a solar batterypowered handheld calculator now costs about $4. The same questions can be applied to other tools, such as a word processor with spelling checker and laser printer. This is a relatively expensive tool, and often it is not readily available. It takes quite a bit of training and practice to learn to make effective use of a word processor. At the current time, most schools have decided that students should learn both paperandpencil writing and also writing in a wordprocessing environment. Moreover, students have to learn which of these aids to writing is more appropriate for themselves and for various types of writing tasks that they face. In this case, the computer has added to the amount that students have to learn. [[It takes a lot of time and effort to develop good cursive handwriting skills, and to be good at spelling. Some people (for example, people with dyslexia) find this to be particularly difficult, and perhaps impossible. But, good handwriting and spelling are merely aids to effective written communication. In the "real world" the goal is to solve certain types of communication problems by use of writing. A word processor with a spelling checker is a powerful aid to written communicationespecially so for dyslexics and others who have problem with their handwriting and spelling.]] Computer as ToolThe previous section raised a number of questions about learning to use tools as an aid to solving problems. All of these questions can be directed specifically toward computers. Historically, most people thought of computers only as an aid to solving math problems. That line of thinking has gradually changed. Now, people recognize that computers are useful aids to problem solving in all academic disciplines. Over the past 50 years, computers have gotten more and more costeffective. Indeed, the cost of carrying out a given set of steps has gone down by a factor of a million or more since the first few generalpurpose computers were constructed. This means that the computer is becoming a more costeffective aid in problem solving. As with other tools, it is useful to think about the computer tool both from a concept and a process point of view. A computer is an informationprocessing machine. Some key concepts include:
The diagram given in Figure 4.1 represents one way of looking at the computer environment for problem solving. Figure 4.1 Computer environment for problem solving. Computerassisted problem solving begins with recognition of a problem situation and posing a problem (A and B in Figure 4.1). The people in (A) work to represent the problem in a form that allows them to use a computer system (C) to aid in solving the problem. Typically, this representation process and overall solution process requires substantial interaction among A, B, and C. There are a large number of people (D) who have developed the hardware and software that will be used to help to solve the problem. When you use the computer as a tool, you are building on the work of these people. Now, imagine for a moment that the hardware and software developers (D) could read your mind and anticipate your every need. For any problem that you wanted to use a computer to help solve, the hardware and software would exist to accommodate your exact needs. All you would have to do is interact with the hardware and softwarefeed in the data needed for the specific problems that you want to solve. Certainly that would cut down on the effort you would need to put forth learning to use a computer to solve problems and actually using a computer to solve problems. Obviously, the hardware and software developers cannot read your mind and anticipate your every need. Moreover, you are not the only customer for their services. Commercially available hardware and software has been developed to fit the needs of the mass market. If the problems that you want to solve by using a computer are similar to the types of problems that lots of other people want to solve, then it is likely that the commercially available hardware and software will suit your needs. However, if the problems that you want to solve and the tasks that you want to accomplish are out of the ordinary, you may find that the commercially available hardware and software do not specifically fit your needs. This places an increased learning burden on you. You have to learn to represent the problems that you want to solve in a form that fits the available hardware and software. You may have to spend a great deal of time and effort to learn to make effective use of the hardware and software. If the problems that you want to solve are quite out of the ordinary, you may need to learn to write computer programs. Computer programming is discussed in Chapter 9. [[The ideas given above need further development. They are an over simplification. A great many of the problems that one encounters cannot be solved by computers. Consider, for example, the day to day interpersonal problems that you encounter and deal with. BuildingBlock ResourcesThink about a problem that you routinely encounter and solve. For example, perhaps you are a touch typist. Then, you routinely and easily solve the problem of typing words. You can compose at the keyboard, using your conscious thinking efforts to decide on the words you want to write. An automatic part of your brain and body accomplishes the task of getting the words typed. Perhaps you are very good at shooting baskets in basketball. You routinely, easily, and without conscious thought, solve the problem of tossing a basketball through the hoop. The typing and basketball examples illustrate a very important idea about problem solving. If you frequently encounter the same problem, it may be worth quite a bit of study and practice time to develop the knowledge and skills needed to solve the problem easily [[quickly and easily, perhaps with little or no conscious thought]]. It may be possible to develop a high level of automaticity in solving the problem. In this book, we will use the terminology buildingblock resource (BBR) to represent a problem that you are highly skilled in solving or a task that you are highly skilled in accomplishing. The terminology BBR is meant to suggest that such knowledge and skills can be used as building blocks in solving still more complex problems. If typing is a BBR for you, this can help you solve a writing problem. If shooting free throws is a BBR for you, this can help you to solve the problem of being a good basketball player. It is evident that each person has their own set of BBRs. Take a look at the list given in Table 4.1. Which of the problems listed are BBRs for you? Add a few items to the list so that it contains a number of your own BBRs. Fill in the column on the far right. If the item is a personal BBR, make an estimate of how many hours it took you to acquire your current level of knowledge and skills. If an item is not a personal BBR, make an estimate of how long it would take you to make it a personal BBR. The key concept here is that a person can acquire additional BBRs. However, it can take considerable time and effort to do so. The middle column is discussed in the Activities and SelfAssessment section of this chapter.
Table 4.1 Some possible BBRs. It is evident that having a large repertoire of BBRs can be quite useful in solving problems and accomplishing tasks. Thus, quite a bit of formal education is designed to help students gain BBRs. However, it often takes a lot of time and effort to gain a reasonable level of expertise in an area. Thus, it can take a great deal of time and effort to build a wide range of BBRs. Time&emdash;not neural intelligence&emdash;is the limiting factor. ComputerBased BuildingBlock ProblemsComputers can carry out many tasks rapidly and accurately. For example, a computer can do a spell check. It can alphabetize a list. It can sort addresses by ZIP code. It can rotate a threedimensional drawing of a house, showing the different perspectives. It can change the color of an object in a picture. It can calculate the energy efficiency of a building design. Think of these tasks as BBRs. It is easy for a person to learn to use a computer to accomplish such tasks. By learning to use a computer, a person can quickly acquire a large number of BBRs. This certainly adds a new dimension to education and to developing expertise in solving problems. Computerbased BBRs can be the basis for significant changes in our educational curriculum. Consider, for example, the time and effort it takes for a student to learn to represent a set of data using a pie chart (a circle graph). The student needs to know how to compute percentages, how to compute percentages of 360 degrees, and how to use a protractor. Moreover, consider how long it takes to actually accomplish such a task for a set of data. But, a modern spreadsheet contains provisions for converting a table of data into a pie chart or into other graphical representations. This is accomplished as quickly and easily as a word processor can spell check a paragraph or alphabetize a list. Both the time to learn to accomplish a pie charting task and the time to actually accomplish such a task are quite short. Chunking Using BBRsA person's shortterm memory is quite limited in terms of the number of items of information that it can simultaneously deal with. It is often suggested that for a typical person, the number of pieces of information that shortterm memory can deal with is about 7 ± 2. Thus, a typical person can go to a phone book, look up a seven digit phone number, and remember it long enough to dial it. However, it is evident that people can learn to actively think about far more complex situations than is suggested by the number 7 ± 2. Paper and pencil are important aids to this shortterm memory limitation. Another big help is chunking. I work at the University of Oregon. All phone numbers at the University of Oregon begin with the three digits 346. I live in Oregon. At the current time, all of Oregon, with the exception of the Portland area, has the long distance prefix 541. Thus, any University of Oregon phone number is (541) (346) followed by four digits. That is, it is two "chunks" and four individual digits. The two chunks are "State of Oregon prefix" and "University of Oregon prefix." The total, six pieces of information, is something that I can deal with. One of the reasons that BBRs are so important is that they are chunks. When I am thinking about how to solve a complex problem, I can think in chunks that are BBRs for me. That relieves my shortterm memory from having to deal with small details that would quickly overwhelm its capacity. [[There are thousands of programmers who are writing computer programs to solve quite specific problems or accomplish quite specific tasks. Thus, there is a steadily increasing number of computerbased BBRs available. Many of these computer programs make use of artificial intelligence. Thus, the programs have a certain level or type of intelligence. In many different problemsolving situations, these BBRs are quite useful aids to problem solving.]] Education for Conceptual UnderstandingThis chapter began with a discussion of handheld calculators. The handheld calculator example was used to help point out that there is a difference between a concept (such as division) and a process (actually doing division). Educators and others are not in full agreement about how much educational emphasis should be placed on understanding concepts and how much emphasis should be placed on learning processes. Interestingly, such disagreements tend to focus on relatively simple BBRs such as the four basic arithmetic operations. There is much less disagreement on more complex BBRs. For example, almost all handheld calculators have a squareroot key. The square root of 9 is 3 because 3 x 3 = 9. (For mathematically inclined people, 3 is also a square root of 9.) An eightdigit calculator indicates that the square root of 2 is 1.4142135. It is possible to calculate the square root of a number by various paperandpencil processes. The process that used to be taught in high school algebra courses is more complex than long division. This has been dropped from the curriculum in most algebra coursesand replaced by having students use a calculator or a computer. The concept of square root is relatively easy to learn. It is very easy to acquire the calculation of a square root as a BBR if one has a calculator available. But, this still leaves a person with no understanding of how it is possible for a calculator to determine the square root of a number. This raises the question of whether it is appropriate for students to acquire the square root BBR with no conceptual understanding of how a calculator is able to accomplish such a task. This question has no simple answer. Moreover, the question is made more difficult as complex tools, such as the computer, come into routine use in education. Should a computer system be viewed as a "black box" that magically carries out assigned tasks? What level of understanding should students have of the software they are using? Is it alright to use software that rotates threedimensional objects, but have no idea of the underlying mathematics used in representing and rotating threedimensional objects? [[Or, let's take a more concrete example. An architect can design a building. Software now exists that can turn the building design into a virtual reality. A person can "walk through" the building, viewing it from inside. A person can see how the lighting will be at different times of day (sun light changes the situation) and night. Developing such a virtual reality program is a very complex task. Learning to apply the basic software to one's own computerized architectural drawings of a building is a less complex task. Understanding the meaning of walking through a building, viewing it under different conditions, takes almost no effort. It is easy to understand the concepts of the problem to be solved.]] To a reasonable extent, educators agree that students should understand the underlying concepts of both the problems that they are working to solve and the tools and processes being used to solve them. A reasonable level of understanding can be helpful in detecting and correcting errors, and in handling difficulties that arise as you use various aids to solve problems. Let's take a specific example. A computer is an electrically powered machine, operating off of household current or batteries. The same holds true for a calculator, radio, television set, tape recorder, and CD player. One underlying concept for all of these tools is that they will not work without power. Thus, if such a device fails to work, perhaps the first thing to do is to check the on/off switch. Perhaps the second thing to do is to check to see if the device is plugged in and the electric current is working, or if the battery has a charge in it. By understanding the underlying theory of how such devices are powered, a person can detect and correct several common sources of malfunction. An understanding of underlying concepts is very important in detecting and correcting errors or malfunctions. Let's take this example back to calculators. A firstgrade student can learn what sequence of keys to press on a calculator when faced with a calculation such as 285.69 divided by 54.8. The student can memorize how to do this with a high level of accuracy. But, what level of understanding does the student have of decimal notation and of long division? Does the first grader understand what problems can be represented and solved by division? How does the student detect an error, such as not keying in the decimal point? [[And, keep in mind that the keyboards on inexpensive handheld calculators are difficult to use. It is easy to accidentally press a key twice, or not hard enough, when entering a digit. Thus, it is easy to make errors when using a calculator. This is especially true for younger students.]] These same difficulties carry over to computers. It is possible for a person to acquire a large number of BBRs by using a computer. Some may be so intuitively obvious and transparent that a high level of underlying concepts seems inherent to the BBR. For example, you know a great deal about rotating objects and what they look like from different angles. You know a great deal about what objects look like on a television screen. Combining these areas of knowledge gives you good insight as to what to expect when you use a computer program to rotate objects in three dimensions. Contrast this with a sophisticated mathematical technique such as the amortization of a loan. Unless you have had specific education in this area, the chances are that you do not understand the underlying concepts of the problem to be solved. Thus, you have no way of knowing if the mathematical procedure is applicable to a particular problem you want to solve or whether you have made a gross error in entering the problem into the computer. As you learn more about computers, you will continually be faced with the issue of how much understanding you need. How much time and effort are you willing and able to devote to learning underlying concepts? To what extent are you comfortable and satisfied with using computer hardware and software as a black box? Does your previous training and experience help you to detect and correct errors that you make when using computer hardware and software? How important is it that you be able to detect errors? Activities and SelfAssessment1. Go back to the list of problems that you developed in the Activities and SelfAssessment section of Chapter 1. Think of this list in terms of the physical tools that you would use to help solve the problems. Create a threecolumn table to summarize your analysis. Some sample entries are given in Table 4.2.
Table 4.2 Sample list of problems you may have come up with at the end of Chapter 1. 2. There are lots of problems where a computer is of little or no use. Go back to the set of problems that you wrote down in dealing with Table 1.1 at the beginning of Chapter 1. Add a few more problems to the list, with special emphasis on problems where you feel a computer might possibly be a useful resource. Then add a fourth column to the table as illustrated in Table 4.3. Classify each of your problems on a "Potential Computer Usefulness" scale. For example: Figure 4.2 Potential computer usefulness scale.
Table 4.3 Determining how useful a computer is with respect to a problem. 3. Table 4.1 lists a number of possible BBRs. Add some items to this table. Fill in the middle column, analyzing how each of the potential BBRs relates to the seven intelligences in the Howard Gardner list. Also, analyze each potential BBR from the point of view of how much time it would likely take you to acquire a reasonable level of expertise. In this analysis, indicate how you might measure whether you had acquired the level of expertise that you have in mind. 4. How do you feel about learning to do paperandpencil arithmetic versus learning to use a calculator? Analyze your feeling from the point of view of understanding concepts and understanding processes. Some people consider a calculator to be a black box. For you, personally, are the algorithms that you have memorized for multiplication and long division actually black boxes? Extend your discussion to the use of a calculator to calculate square roots. 5. Take a look at the word processor that you use. Does it contain a graphing package? If so, explore use of this package. Discuss how easy it is to learn how to use and how easy it is to use. Compare and contrast this with byhand graphing of data. 