Three 1987 Math Education Scenarios






 * “Education is a human right with immense power to transform. On its foundation rest the cornerstones of freedom, democracy and sustainable human development.” (Kofi Annan; Ghanaian diplomat, seventh secretary-general of the United Nations, winner of 2001 Nobel Peace Prize; 1938-.)


 * "In a completely rational society, the best of us would be teachers and the rest of us would have to settle for something less, because passing civilization along from one generation to the next ought to be the highest honor and the highest responsibility anyone could have." (Lee Iacocca; American industrialist; 1924-.)

This is Chapter 1.5 (Scenarios from an Information Age School) of the following book:


 * Moursund, D. (1989). Effective Inservice for Integrating Computer-as-Tool into the Curriculum. Eugene, OR: Information Age Education. Available free in PDF from http://i-a-e.org/downloads/doc_download/25-effective-inservice-for-integrating-computer-as-tool-into-the-curriculum.html. Available free in Microsoft Word from http://i-a-e.org/downloads/doc_checkout/26-effective-inservice-for-integrating-computer-as-tool-into-the-curriculum.html. Much of the content given here was written during 1986-1987.

Background
The Computer-integrated Instruction (CII) inservice facilitator is a key educational change agent. The facilitator has knowledge of education and of the computer technology that may be the basis for a major change in our educational system. Moreover, the CII inservice facilitator has access to teachers and has the opportunity to help move them toward technology-based changes in the content they teach.

For these reasons, it is essential that the CII inservice facilitator have a good understanding of what constitutes a good education for life in an Information Age society and for the continued change that students will face throughout their lives. This chapter was originally written specifically to depict possible changes in mathematics education. But, in a larger sense it serves as a metaphor for technology-based changes in our educational system. If you are giving math-oriented inservices, the material in this chapter will be of specific and immediate interest to you. If your interests do not include mathematics, then read this chapter with the idea that it is a model of educational change. Create your own model to fit the areas in which you are doing inservice facilitation.

You will notice that it is expected that the reader understands some of the purposes and underlying concepts of mathematics education. If you are doing inservice designed to impact people who teach mathematics, it is important that you understand mathematics education. It is not enough to just understand the computer tools that you are teaching math educators to use. You need to facilitate them learning to make appropriate use of these tools as they change the mathematics curriculum. To do this, you need to have a good understanding of mathematics education.

Information Age Mathematics Education
It is not obvious what constitutes an appropriate education for life in an Information Age society. This chapter gives three scenarios from mathematics education settings in hypothetical Information Age classrooms of the near future. The chapter begins with a discussion of the goals of mathematics education. The reader will want to examine the scenarios to see how well they reflect the goals. Also, look for how well the scenarios reflect your ideas on what might constitute an appropriate education for life in an Information Age society.

Much of the inservice education that is needed to support computer-integrated instruction needs to be specific to the discipline interests of the participants. It is quite difficult for a person who knows little about mathematics or the teaching of mathematics to present an effective computer inservice for mathematics teachers. If you are thinking about designing and implementing computer inservices for mathematics teachers, then this chapter may prove to be a good test of how well you are prepared. The contents of this chapter might well be assigned reading for secondary school math teachers participating in a computer inservice.

Brief History of this Chapter
The material in this chapter is extracted from a paper that has evolved over a number of years and has been used for a variety of purposed. A brief history of the longer paper follows.

In the fall of 1985 the National Research Council created a Mathematical Sciences Education Board (MSEB). MSEB set as its initial task to make recommendations on precollege mathematics education for 10-15 years in the future. In June 1986 I was asked to submit a position paper discussing possible roles of computers in such a mathematics education system, and I did so in October 1986. Nearly a year later I made use of a modified version of that position paper in a presentation done in a fall 1987 computer education conference in Alberta, Canada. Still later I modified the paper again, to reflect input I received in Alberta and from others who had read the paper. Then the paper was used as a resource and discussion-topic paper in the Computers and Mathematics course taught in the University of Oregon summer session 1987.

MSEB held a working session of 20 mathematics educators during August 10-14, 1987 at the Xerox Training Center in Leesburg, Virginia. The five-person working group I was in focused on possible roles of technology in mathematics education in the year 2000 and beyond. Other members of my working group were Richard Anderson (Louisiana), Gail Burrill (Wisconsin), Margaret Kasten (Ohio), and Robert Reys (Missouri). I used my modified position paper as the starting point for the writing I did during that session. After a number of major additions and revisions, it doubled in length and began to reflect quite a bit of the thinking of our group, as well as some of the ideas of the MSEB. Since that working session I have revised and expanded the paper quite a bit more.

The version of the paper presented here has been revised to fit with the general theme effective inservice.

Introduction
The purpose of this paper is to provide a framework for planning major curriculum content and pedagogy changes designed to improve our mathematics education system. Most educational leaders believe that our precollege mathematics education system in the United States is not as good as it should and could be. They cite as evidence test scores within this country, international comparisons, and a variety of national reports of study groups.

It is worth noting that during the past few years there have been a number of national commissions and other groups that have commented on the total educational system in the United States. Their remarks tend to parallel the remarks found in reports directed specifically at our mathematics education system. The general opinion represented in such reports is that substantial reform is necessary if our educational system is going to adequately meet the needs of our country.

Here are five major factors that suggest change is necessary and improvements are possible in our mathematics education system:

1. The nature of the intended audience of our mathematics education system has changed quite a bit in the past couple of decades. For example, kids in high school now have spent about as much time watching TV as time in school. They have spent their entire lives in the Information Age, while our school system was designed to fit the needs of an Industrial Age society. (According to John Naisbitt, the Information Age officially began in the US in 1956.) This analysis suggests that mathematics education might be improved by moving it more towards the needs of people living in an Information Age society.

One key component of the Information Age is rapidly increasing access to more and more information. A common estimate is that the total accumulated knowledge in mathematics is doubling every ten years. This suggests that information retrieval skills are of increasing value and that math-oriented information retrieval be given increased emphasis in the curriculum.

2. Over the past couple of decades there has been substantial progress in our understanding of teaching theory, learning theory, and cognitive science. Our educational system tends to be slow in translating such theory and research results into practice. While progress is occurring, much remains to be done. Research is continuing at a rapid pace.

Research into cooperative learning and cooperative problem solving strongly supports their potential in education. This suggests that cooperative learning and cooperative problem solving should be given increased emphasis in the mathematics curriculum.

3. Calculators and computers can be used to help students learn mathematics topics. (One of the topics might be to learn to use a calculator or computer to help solve math problems.) The research literature on computer-assisted learning (CAL) is extensive and quite supportive of increased use of CAL. While much of the research in the use of CAL in mathematics focuses on basic skills, the body of literature on uses to improve higher-order skills is growing. There is quite a bit of software designed to enhance higher-order skills.

CAL can make available instruction in individual topics or entire courses that might not otherwise be available to students. It can incorporate pedagogy (for example, sophisticated simulations and motion graphics) that is not readily duplicated without the use of a computer.

4. Computers can be a substantial aid to classroom management and to testing, especially as one works to meet the diverse needs of individual students. Computers can help increase the amount of individualization of instruction in our math classrooms.

Computers also can be an aid to teachers in whole-class presentations and activities. Most math teachers already know how to use an overhead projector. There are now relatively inexpensive devices (about $1,000) that allow the output from a computer to be projected using an overhead projector. Many math teachers will eventually find such a system to be quite valuable in their teaching. The cost of these devices will likely decrease substantially in the next few years.

Substantial progress has occurred in developing computer-based classroom management and record keeping systems. This is called computer-managed instruction (CMI). Computer-assisted learning systems often contain a build-in CMI system. CMI systems also exist that can help track students in their total educational program.

Many teachers have learned to make effective use of an overhead projector, film strips, etc. A computer, or a computer system with a videodisc, is "merely" another instructional medium. But it is a powerful medium that can be especially useful in supplementing traditional standup lecture demonstrations in a math class.

Computer-based, adaptive tests, are gradually being developed by the Educational Testing Service and other groups. Such tests adjust the questions being presented on the basis of student responses, thus more quickly arriving at a solid estimate of student performance in a specified area.

5. Calculators, computers, and other related technology have become more and more available as aids to productivity and problem solving in our society. Our current mathematics curriculum largely ignores possible impacts of computer-related technology on content. Perhaps the classical examples are the use of quite inexpensive calculators to do arithmetic calculations and the use of computers (or, more sophisticated calculators) to graph data and functions. Widespread implementation of even just these two types of aids to problem solving would have a significant impact on the mathematics curriculum.

This paper focuses largely on the last of the five factors listed above, the computer as tool. This is also called computer-integrated instruction. However, the other four factors are also given serious consideration.

Nine Overriding Goals in Math Education
This section suggests nine overriding goals that can be used when examining an existing or proposed mathematics education system. The first six are goals for students to achieve, and the educational system should be designed to provide students good help in achieving these goals. The seventh goal specifically mentions technology. While the computer is important both in shaping mathematical content and in pedagogy, it is clearly not the central theme or purpose in mathematics education. The eighth indicates that our mathematics education system needs to be concerned with preserving itself. The last goal is for teachers.

G1. Reasoning. The goal is that in a mathematical context students can argue, conjecture, validate, prove, follow proofs and logical arguments, etc.

G2. Mental mathematics. Within the framework of the mathematics that students have studied, they can:


 * a. Mentally solve "simple" problems. What is simple will, of course, vary with the student. But, for example, most students can learn to do one digit addition and multiplication; to mentally decompose a modestly complex geometric figure into component parts (for example, note that a kite-shaped figure can be decomposed into two triangles); to mentally collect terms in an algebraic expression; to transfer simple mental counting and computational skills to real world situations such as dealing with money; to visualize graphs of simple functions; etc. Here we set as a goal that students have "number sense," as well as "Mathematics sense" at a level appropriate to the math that they have studied.


 * b. Mentally estimate answers to problems of a considerably greater complexity than those under (a) above. Mental estimation in arithmetic, for example, builds on having good mental computational skills. Mental graphics allows one to visualize the shape of a function or the graph of some data.


 * c. Have a reasonably well developed "mathematical intuition" on the correctness of proposed results. Have a sense for what they know and don't know, or what is known and not known within a framework of the mathematics they have studied. Consider two examples. First, I ask you for President George Bush's phone number. You might respond, "I don't know, but I can probably look it up in a Washington DC phone book or ask the operator." Next I ask you for President George Washington's phone number. You probably laugh and indicate that phones did not exist when he was president, or that he has been dead for a long time.

G3. Valuing. Mathematics is part of our history and culture. It is a human endeavor that is fun and exciting for many people. The goal is to have students value and appreciate mathematics and their ability to know and do mathematics at the level to which they have studied it. Students should have good self-esteem, and taking math classes should not damage that self esteem.

G4. Problem solving. Learning theorists talk about transfer of learning, and the ideas of near transfer and far transfer. Suppose that a student uses beans and bean sticks to add 8 and 13. Then it is probably a near transfer for the student to add 8 pennies and 13 pennies. It is a further transfer for an 8-year-old child to determine his/her age in 13 years, or a 13-year-old child to determine his/her age in 8 years. Problem solving involves the transfer of knowledge and skills. The further the transfer and the larger the number of steps required in the process, the more difficult the task tends to be. The goal is for students to learn to solve math-oriented problems that are solvable within the mathematics they have studied.

The innate ability to transfer learning to new problem situations varies tremendously among people. But appropriate education can increase this ability. Thus, there must be a major emphasis in mathematics education to teach for transfer of problem-solving skills. (Another way of saying this is that there should be a decrease in emphasis on lower-order skills and some of the time saved should be used to increase emphasis on higher-order skills. Some of the time saved could also be given over to increased emphasis on topics such as informal geometry, probability, and statistics that are not currently given enough emphasis.)

Problem solving is a rather general goal. It subsumes the following two subgoals.


 * G4a. Data analysis and representation. The goal is for students to learn mathematics needed to deal with data. This includes such things as to extract information from data, represent data graphically or in appropriate tables, use data as an aid to solving problems, appropriately tabulate statistical data, perform simple statistical computations, interpret statistical results, etc.


 * G4b. Problem representation. Mathematics provides vocabulary and notation for the representation of a wide range of problems. The goal is that students can use the mathematics they have studied to represent real world problems. We call this mathematical modeling, and it should be given considerably greater emphasis in the curriculum.

G5. Communication. The goal is for students to be able to speak mathematics and understand spoken mathematics; to read and write mathematics; and to do math-oriented information retrieval. Our current mathematics education system is particularly weak in helping students to learn to retrieve math-oriented information, so this area needs special attention.

G6. Study and learning skills. The goal is for students to develop study skills appropriate for learning mathematics and to learn how to learn mathematics. (Research supports the value in teaching study skills.)

G7. Technology. The goal is for students to learn to do mathematics in the type of environment they are most apt to encounter after they leave school. This means that the mathematics education system must consider the full range of environments, from the unaided human brain to a highly computerized environment.

Our understanding of transfer of learning suggests that if we want students to function well in a particular environment, we should educate them in that environment. Thus, if we want students to learn to function well in an environment in which computers are routinely used as an aid to problem solving, we should educate them in an environment in which computers are routinely available and used as aids to problem solving.

G8. Producing mathematics leaders. As we work to improve our mathematics education system, we need to pay special attention to students who have particularly good mathematical ability. The goal is to foster this ability and to help these students develop a strong interest in mathematics. The future of mathematics education depends on having a continuing supply of very competent mathematicians and mathematics education leaders.

G9. Teachers' role. The goal is for teachers to adequately and appropriately facilitate students in G1-G8 above. Research suggests that it is helpful if teachers role model the behaviors they want their students to learn. Thus, one specific goal here is for teachers to learn to role model learning and doing mathematics in an environment that includes calculators and computers.

Scenario 1 (A Third Grade)
This is the first of three scenarios reflecting ideas on the mathematics curriculum of the year 2000. This scenario represents a third grade classroom in the year 2000. Other scenarios in this chapter give glimpses into possible futures of middle school and high school.

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It is the year 2000, a little before 10:00 in the morning, and you are visiting a third grade classroom. As you enter the school building it reminds you of when you were in school. Not surprising, since you attended this same school twenty years ago. New school buildings are somewhat rare.

You have asked the teacher to tell you when the math period would be. The teacher hedged the answer, indicating that students may be doing math types of things at almost any time of the day. However, at 10:00 in the morning most of the class is typically engaged in a math type of activity.

As you walk into the classroom you notice that there are a number of computer display screens and keyboards, several with groups of 2-4 students around them. You do a rapid mental estimation which suggests that there is roughly one computer work station for every three students in the class.

You notice the teacher working with a group of students. The students are practicing mental computation. A student has posed the problem of finding the sum of 23 and 18. As you mentally try to visualize these two numbers lined up vertically in your head, you hear several students and the teacher respond with an answer of 41. Terry, one of the students in the group, explains one way to do it. "I think of a reference number that is near 23 and 18, and which is easy to work with. I use 20, since I can easily see that 20 plus 20 is 40. But 23 is 3 more than 20, and 18 is 2 less than 20. So, I need to add 1 to 40 to get the answer." The teacher says, "I did it a little different. I saw that 23 was the bigger number, so I moved some of it to the 18. That is, I changed the problem to 19 and 22, and then to 20 and 21. Then I could see that the answer was 41." Another student, Pat, says, "I remembered that 18 and 18 are 36. I then counted on from 36 as I went up from 18 to 23.

The teacher sends the students off to work together, requesting that they continue to give each other two digit addition problems to do mentally. The teacher suggests that if they have a disagreement on an answer, they may want to check it out on a calculator. You notice that there are a number of calculators readily available to students.

You ask the teacher what is going on at the computer workstations. The teacher responds that each student or group of students is likely working on something different. For example, Tom is working alone, using an "old fashioned" drill program on single digit arithmetic computation facts. You watch as Tom runs through a mixed list of addition, subtraction, multiplication, and division exercises, completing them at the rate of about one every two seconds. You notice that when Tom makes a mistake the machine provides the correct answer and shortly later presents the same exercise again.

After about a minute or so the computer changes the presentation of the problems. It shows a rectangular pen filled with sheep in orderly rows, and asks how many sheep are in the pen. As you watch the computer presents a number of picture-based problems that are solvable by mental single digit arithmetic. After another two minutes, the computer switches to money-based drill exercises.

At the next computer workstation you see three students playing a game that involves finding a lost treasure. They are looking in a castle that has many rooms on each of several floors. Frequently the clues direct them to retrace their path, move in a specified direction a certain distance, or to go to a specific room. One student is taking notes, and all three seem to be discussing the various options at particular decision points. The teacher explains that this computer game is designed to promote cooperative problem solving. (It's hard for one student to keep the necessary record, make the decisions, run the computer, and detect his/her mistakes all at the same time). The game is also designed to help improve spatial orientation, record keeping, and following directions.

At still another computer station you see four students working together. They are playing a business simulation game. Each student is one of the partners in this fruit juice stand business. As they play the game they have to make decisions about how to spend their time and money. How much time should be spent on painting signs? How much fruit juice should they have available, and what should they charge? The goal is to make as much profit as possible. But all four students must agree on each decision before it is entered into the computer. When there is a disagreement, the students must work together until they agree.

At still another computer station you see a student working with some sort of program that allows the student to write, draw pictures, and work with databases. The teacher indicates that the student is using LogoPS that was an outgrowth of the Logo software of the 1980s. In essence, it incorporates a word processor, a database system, and other problem-solving software into the "classical" Logo for microcomputers.

As you move away from the computers you almost trip over a group of students who are repeatedly throwing pairs of dice and recording the results on paper. The students indicate that the goal is to throw the dice 300 times and to see how it comes out. Sue has conjectured that the low numbers (2 and 3) will beat the high numbers (11 and 12). Tom has conjectured that there will be more sixes than anything else. Cathy has conjectured that there will be more even numbered answers than odd numbered answers. Karen has estimated that the four of them will complete the task in less than 10 minutes, and she is keeping one eye on the clock. She hopes that there will be enough time to do it all over again before it is time to do writing. She wants to write about how it comes out. (The teacher indicates that the students will be doing writing as soon as math is over. Often they are asked to write about what they are doing during other parts of the day, such as what they are doing in math.)

Before leaving the third grade classroom you get a chance to talk with the teacher. You ask how it is possible to keep track of what all the students are doing, and how it fits together in a curriculum. The teacher points to a computer, to a stack of activity recording sheets, and to a cabinet of materials. "The cabinet is full of manipulatives—for example, lots of sets of dice, bean sticks, 100s boards, tiles, spinners, timers, and other manipulatives. My computer keeps detailed records for each student. The students work together in groups of four, although one or two students will often split off from a group for a day. A group of four has considerable responsibility for itself and for its individual members. But each student has at least one individual session with the computer each week. In this session the computer asks a lot of questions about what the student has been doing. It is sort of an interactive diagnostic test. The computer system offers suggestions on what the student might work on, and it gives me a detailed print out."

"I realize it all sounds quite complicated, but actually it is easy. Each day each group of four knows what it and its members are to be working on. Partly it is their own choice, partly the computer suggests what they might do, and partly I tell them what to do. When they use a computer it keeps track of what they are doing. When they do off-machine activities, I have them fill out these activity sheets. I feed that information to the computer, so it has a record of what the students are doing."

Needless to say, I was impressed! But I wondered about testing. "How do you give tests on all of this?"

The teacher indicated that no formal pencil-and-paper math tests are given in the third grade. The computer is gathering formative data whenever the student uses the computer. Since each student has at least one individual computer session per week, quite a bit of formative data is gathered. In addition, the teacher observes what the students are doing, and spends a lot of time working alongside the students. The role modeling is another important idea in math education. "It's fun—I get to do what the kids do, and I often learn new things or new ways of looking at the math I learned while I was in school."

I thanked the teacher, indicating once more that I was impressed by the changes from when I was in school. "Math looks like a lot of fun. Maybe if we had had these things while I was in school, I would have liked math."

Two Key Computer-Related Questions
Scenario 1 is all based on ideas and technology that currently exist. While computers play an important role, the human element dominates. Education is a human endeavor. In order to do mathematics it is necessary to carry in one's head a great deal of understanding about mathematics.

However, it is clear that computers and related technology can play an important and increasing role in doing mathematics. Thus, we can think about a person:


 * Doing mathematics making use only of his/her brain.


 * But also making use of conventional aids such as book, pencil and paper, protractor, straight edge and compass, etc.


 * But also making use of inexpensive and easily portable electronic aids such as a handheld solar powered calculator.


 * But also making use of microcomputers (which may or may not be easily portable, but then again, they might be portable).


 * But also making use of access to mini or mainframe computers, networked computer systems, telecommunications, large databases, etc.

In all of this we also have the issue of computer-assisted learning. Thus, two key computer technology-related questions have arisen in mathematics education.

1. How should the content of mathematics education be changed to reflect the availability and capability of computers, calculators, and related aids to problem solving? This question focuses on:


 * a. Use of calculators and computers as tools to help solve problems.


 * b. Changes in the curriculum content, such as increasing the emphasis on exact and approximate mental math, geometry, statistics, and discrete mathematics, while decreasing the emphasis on paper-and-pencil computation and symbol manipulation, and rearranging the order of presenting various topics.

2. Can calculators, computers, and computer-related technology help improve pedagogy in our mathematics education system? This question focuses primarily on use of computers as an aid to learning mathematics, or on CAL in its broadest possible definition. For example, use of a calculator as a manipulative in learning counting would be considered as CAL in this broad definition. But the question also deals with the use of a teacher-controlled computer with a display that can be viewed by the whole class. A Level 1 videodisc system (no computer, and the system may be under teacher control) is also included.

These are difficult questions and cannot be fully addressed in a paper of this length. However, the discussions, scenarios, recommendations, and appendices that follow provide a solid indication of some possible answers.

Computer Facilities: Hardware and Software Considerations
In planning for instructional use of computers in mathematics education, it is helpful to have some model of computer availability and capability in mind. The creation or selection of a model is a challenge, since both computer availability and capability are changing very rapidly. Almost every week one is apt to encounter news of a new product that is significantly better than the product it competes with. Over the past 30 years, progress in computer hardware has led to a price to performance gain by a factor of 10 roughly every seven years. There is good reason to believe this will continue for at least 14 more years. (The article Personal Workstations Redefine Desktop Computing by Jeffrey Bairstow on pages 18-23 of the March 1987 issue of High Technology discusses this in detail.)

People doing long-range planning for mathematics education should not dwell unduly on inadequacies of current computer capabilities and student access to these systems. Rather, they should assume that eventually every student will have easy access to a very powerful computer system. The time frame necessary for making significant changes in our mathematics education system is sufficiently long so that during the same time frame computers will become readily available to all students (and others, such as workers and people in their homes) who have need to use them.

People doing very long-range planning (10 -15 years) for computers in mathematics education might want to assume that something like today's Macintosh 2, IBM PS/2 Model 70, or NeXT computers will be readily available to students. Let's call this a Mathematics Education Computer System (MECS). The needed software and courseware for MECS has four main components. While much of this software and courseware already exists in discrete components, it has not been drawn together in a unified manner. Thus, we should assume that the software and courseware facilities available for the MECS will continue to improve rapidly with time. The four components of this software and courseware are:

1. A mathematical reference library containing the equivalent of many hundred of books. Materials would be available for students at a variety of grade levels and mathematical maturity levels. This library would also contain instructional support materials for teachers, such as back issues of the publications of the NCTM, sample lesson plans, courseware developed by federally-supported projects, etc.

Note that one CD-ROM disc can hold 550 million characters; a thick novel is about a million characters in length. A CD-ROM can also store digitized pictures and diagrams. Thus, the above library can be stored on a modest number of CD-ROMs. (The cost of making a large number of copies of a CD-ROM, once an original has been produced, is under $2 each. A CD-ROM player has only a little greater complexity than a CD audio player. Thus, the price will eventually be in the $200-$300 range or perhaps lower.)

Texts written specifically for access via computer can be interactive. They can make provisions for moving more deeply into a particular topic, or backing off and looking less deeply into parts of it. (Ted Nelson called this concept hypertext when he pioneered it in the late 1960s. Hypertext is now coming into common use, mainly through a piece of software called HyperCard that runs on Macintosh computers.) A whole new style of writing will need to be developed, along with a careful cross-indexing system that helps guide readers through the wealth of available materials.

We already have the concept of dynamic texts. Data in a computerized database can easily be ordered, selected, graphed, etc. to meet one's specific needs. A spreadsheet program can take in data (perhaps from a computerized database), perform a variety of calculations, and display the results in a variety of formats. All of this is supportive of the idea that in mathematics education we need to have students learn to make use of multiple sources of information. The fixed, static printed text that is changed once every six years cannot serve as the dominant basis for an Information Age mathematics curriculum.

It is difficult to appreciate the benefits of having easy access to lesson plans, assignments, worksheets, exams, etc. in a computer readable form. This type of aid to teacher productivity is not yet available to most teachers. The effort of computerizing all of one's own filing cabinets of such materials is overwhelming. But imagine all of the "neat stuff" that master teachers have accumulated over the years. Then imagine a beginning teacher being provided with a CD-ROM of such materials. This would be a tremendous aid to most teachers.

2. Applications (computer-as-tool). This would include a basic core of general-purpose applications software, such as a two and three-dimensional graphics package, a word processor designed to handle mathematical notation, a general purpose equation solver, a statistical package, spreadsheet, database, and an algebraic symbol manipulation system. It would also contain many hundreds of more special-purpose programs designed to help solve more specific categories of mathematics problems. All of this software will need to be cross-indexed with the reference materials discussed above and with the computer-assisted instructional materials to be discussed next. Eventually all three of these sets of materials will need to be integrated into one comprehensive system.

3. Computer-assisted learning materials covering the K-14 mathematics curriculum. In addition to traditional CAL, this would include simulations that create problem-solving environments, logic proof checkers, and other interactive aids to learning and doing problem solving.

Very roughly speaking, CAL materials can be divided into the categories of "primary" and "supplemental." By primary, we mean materials designed to stand alone and be a primary resource to students studying a certain area. (Typically the CAL materials would be supplemented by a standard text or other print materials.) Much of the CAL math material that now exists was designed for supplemental use. For example, students might use drill and practice in arithmetic materials for ten minutes a day to improve their arithmetic computational skills. Students might use a piece of problem solving software to practice a couple of heuristic methods in problem solving.

Several companies now market primary, full year length CAL math courses for the secondary school math curriculum. In many cases the quality leaves much to be desired. The cost of developing a very high quality primary CAL year length course is probably in the range of $5-$10 million. While the potential seems good, the reality is that few if any really good CAL-based courses exist for precollege mathematics. (Some quite good pieces of courses exist.)

4. Programming languages and aids to computer programming. There are hundreds of programming languages, CAL authoring languages, CAL authoring systems, etc.

The past few years have seen a widening of the gap between "professional level" computer science and computer programming, and "personal" computer programming. It seems clear that a rigorous introduction to computer science and programming in a structured language such as Pascal will not become part of the regular precollege mathematics system. However, all students who can learn the regular mathematics curriculum can easily gain a modest, but highly useful level of personal programming skills. Such programming skills can be used to reinforce math concepts and to add another avenue for mathematical exploration. This has been amply demonstrated by users of Logo in elementary and middle schools and by users of BASIC at a variety of grade levels. (While BASIC is looked down upon with disdain by computer scientists, it will remain as a viable tool of many students and other computer users. Logo seems to be gaining acceptance at the secondary school level.)

It is evident that no precollege students currently have assess to the MECS hardware and software we have described in this paper. However, many scientists and engineers have access to a combination of computer facilities, libraries, and support staff that are roughly equivalent to MECS. By the year 2000 many students will have access to a significant portion of this system. Moreover, mathematics education leaders could set a goal of making MECS available to all students.

You will note that we have not mentioned calculators in this section. A calculator can be viewed as a special purpose, more easily portable, less expensive computer. The capabilities of handheld calculators have continued to grow. Very roughly speaking, the best handheld calculators of today are somewhat equivalent in compute power to low to medium priced mainframe computers of about 25-30 years ago, and this 25-30 year gap is being maintained over time. It seems clear that the handheld calculator will be with us for the foreseeable future. (If we want to be a little science fictionish, eventually the handheld calculator will become a voice input device that is part of the telecommunications system. It will be able to handle "simple" problems using its own compute power, and it will serve as both a telephone and as a terminal to mainframe computers, the Library of Congress, etc. rapid progress in telecommunications technology is contributing to significant progress toward networking the world.)

Accumulated Mathematical Knowledge
Perhaps the single most important idea in problem solving is to build on the previous work of oneself and others. Mathematics, with its careful notation, precise definitions, and formal proofs is well suited to helping people build on the previous work of themselves and others. A student learning to count and to write the numerals is building on the work of those who invented counting and the notation we now use for numerals. (For most purposes, it is a far superior notation than Roman numerals.) Students who have learned how to count can use this skill in solving a wide range of problems.

The accumulated mathematical knowledge of the human race is, roughly speaking, in three general categories of storage and processing "media."

1. Human minds. Note that the human mind is both a storage and a processing medium. (Note the parallel with a computer.)

2. Books, journals, written notes, photographs, paintings, and other passive media that can be repeatedly accessed. Category 2 also includes phonograph records, tapes, movies, videotapes and other dynamic storage media that technological progress has produced in the past century. Still more recently we have magnetic tape, magnetic disk, and laser disc storage systems for computer-readable data.

3. Artifacts that people use to help "do" mathematics. This includes tools such as abacus, slide rule, straight edge and compass, protractor, calculator, and computer. Pencil, paper, chalk and chalkboard can all be included in this category.

A protractor is an excellent example of a mathematical artifact. It embodies substantial mathematical knowledge. Most students can easily learn to make use of some of its capabilities to help solve problems. It is not necessary for a student to fully understand the mathematics embodied in a protractor, nor to understand all of its uses, to begin to make effective use of this tool. A protractor, like many of the other mathematical artifacts, both stores mathematical knowledge and aids in processing or making use of the knowledge.

A mathematics education system is designed to build on the capabilities and limitations of each of the three categories of storage and processing media. Any significant change to one of the categories may lead to a significant change in our mathematics educational system. For example, the development of reading and writing greatly changed Category 2 and certainly led to major changes in both the field of mathematics and in mathematics education. The development of movable type, another major change in Category 2 that eventually greatly increased access to books, changed mathematics education. In Category 3, solar powered handheld calculators have had a significant impact on adults and a more modest impact on our mathematics education system. Gradually the use of calculators has come to be accepted in school mathematics. Recent years have seen significant progress toward allowing use of calculators in statewide and other assessment settings.

Computers impact each of the three storage media. First, consider the human mind. We now have very good research evidence that computer-assisted learning can help many students learn certain aspects of mathematics significantly faster and better as compared to traditional modes of instruction. Moreover, complete courses can be delivered by CAL, providing good quality learning opportunities that might not otherwise be available to students. Finally, CAL allows increased individualization of instruction, with students working on materials appropriate to their levels and moving at paces appropriate to their abilities.

One of the major goals of education is to help students become independent, lifelong learners. Most students never achieve this goal, especially in mathematics. CAL holds the potential for a shift of responsibility for learning mathematics more toward the student. CAL can provide good and immediate feedback on how well one is doing on a set of material. Students can learn to evaluate their own performance and begin to accept more responsibility for their own learning. This may contribute to helping the students to become independent, lifelong learners.

Category 2 contains both passive storage media such as books, and dynamic storage media such as phonograph records. It is evident that computers provide a new passive storage medium. Computers provide for the storage of a large amount of information in a small space. The previously mentioned CD-ROM is just 14 cm in diameter and the thickness of a phonograph record. But it can store 550 million characters—the equivalent of about 500 thick novels. (Imagine holding the equivalent of an entire elementary school library in the palm of one hand!) Moreover, computer technology facilitates easy access to remotely located databanks. We are moving toward the time when the entire United States Library of Congress is on line and readily available to people who need such access to information.

Computers provide a new type of dynamic storage, an interactive type of storage that is unlike anything we have had before. This is discussed more in the Category 3 discussion.

Category 3, artifacts, contains tools that aid one in doing mathematics. We now have the possibility of students growing up with the computer tool. It seems evident that growing up in a MECS environment will shape students' minds in a manner quite a bit different from what occurs in a non-computer environment. For example, consider computer graphics. Without computers it takes considerable effort and training for a student to represent data or functions graphically. Even a single, crude sketch of a function or a set of data can easily take minutes to produce. Animation is quite difficult to depict in hand drawn sketches. With MECS, graphing a function or a set of data becomes a "primitive" that is usually accomplished in less than a second of computer time after the task has been specified. Students using MECS can create graphical representations of data at a younger age than they can without this tool.

Or, consider solving equations (polynomial, non-polynomial, linear systems, nonlinear systems, etc.). The value of computers is obvious. Many time consuming and tedious tasks become primitives, routinely accomplished both rapidly and accurately by the computer, as one works to solve mathematics problems.

Or, consider linear programming and nonlinear programming. Students can learn to use these tools for mathematical modeling long before they can learn the underlying theory of solving such problems. Computers are already routinely used by all people who solve such problems.

The above analysis illustrates the most obvious ways in which computers impact the storage of accumulated mathematical knowledge. But there is still another, even more important idea. Computers represent a new, dynamic way to store some of the processes of applying human knowledge. In essence, a computer system is a medium combining the second and third storage categories. An application program designed to solve a particular category of problem both stores human knowledge on how to solve the problem and directs hardware to carry out the steps to solve the problem.

Research and development in artificial intelligence are gradually producing computer systems that capture some of the problem-solving capabilities of human experts. Progress of this sort tends to be cumulative. Thus, more and more mathematical problems will be solvable by merely telling the problems to a computer. This topic deserves a much more detailed treatment than we can provide in the limited space available here. Over the long run, progress in artificial intelligence may well change the basic nature of mathematics education. Students will grow up in an environment in which they learn to communicate with a computer system (by voice and keyboard) that has immense mathematical knowledge and ability to solve mathematical problems.

The most important idea in this section on Accumulated Mathematical Knowledge is that a computer can be used to retrieve information and procedures telling how to solve a problem, and it can also execute the procedures both rapidly and accurately. In essence, this adds a new dimension to mathematics education. This will be made clearer in the next section.

A Simple Model of Mathematical Problem Solving
In this section we present a simple-minded model of problem solving in mathematics. (In essence, this is the standard four-part Polya model that math education leaders have been supporting for years.) The purpose is to point out the main places where the MECS will impact people who use mathematics to solve problems. A secondary purpose is to suggest some possible major changes in emphasis in various parts of the mathematics curriculum.

1. Understand the problem. This may require making use of reference materials, and MECS will be useful. But to a large extent, understanding a problem requires drawing on one's total knowledge, asking probing questions, and interpreting problem situations in light of human values. It is a human endeavor, drawing heavily on the total interdisciplinary knowledge and skills of the problem solver. Often it requires good interpersonal communication skills.

A key point is that the typical "real-world" mathematical problem is interdisciplinary in nature. One must know both about the disciplines of the problem and about mathematics to understand such a typical real world problem. Currently, many academic disciplines such as the social studies make minimal use of mathematics in their curricula. MECS provides tools that could change that. Increased application of mathematics throughout the school curriculum would make a significant contribution to mathematics education.

2. Develop a mathematical model of the problem. To a large extent, mathematical modeling is an intellectually challenging human endeavor, drawing upon one's total knowledge of mathematics, the disciplines and specific nature of the problem at hand, and experience in mathematical modeling. The MECS may be useful for information retrieval (for example, retrieving models that might be appropriate), drawing graphs and other pictures, word processing, etc.

MECS changes the range and nature of models available. Students can learn to use linear and nonlinear equation models, linear and nonlinear programming, etc. without knowing how to solve such problems using by-hand methods. Models can be used which require exhaustive search of rather large solution spaces. Statistical models can be used which require extensive computations or exhaustive searches. Graphical models can be used, since two and three-dimensional graphing is easily accomplished by computer. MECS has the compute power and graphical capability to do animation and color graphics.

3. Solve the mathematics problem developed in the previous step. Quite likely the MECS can do this or can make a significant contribution in doing this. Often this step is somewhat mechanical, and it is the step most conducive to being automated. (When secondary school math teachers are asked to examine the curriculum they teach, they typically estimate that between 60% and 80% of the curriculum focuses on this step.)

4. Interpret the results in light of the original problem. Return to Step 1 as needed. This mathematical "unmodeling" and interpretation process has the same characteristics as Step 2. It is a human endeavor requiring good understanding of the original problem and good thinking skills.

Even this simple model of mathematical problem solving makes clear that mathematics is and will remain a human endeavor. This model, and the discussion of the Accumulated Mathematical Knowledge, make it clear that one must "know" a lot of mathematics in order to "do" mathematics. But the doing of mathematics is highly dependent on the tools available and how well one has learned to use the tools. That is, learning to do mathematics is inextricably interwoven with learning to use the tools available to mathematicians.

Educators talk about a concept called "the teachable moment." Imagine a person working to solve a mathematics problem but not having the knowledge and/or skills needed to handle some aspect of the problem. We can imagine that the person might move from a problem-solving mode into a CAL study mode to learn some aspect of the problem, and then back into a problem-solving mode. This would be taking full advantage of a teachable moment. It represents a significant change in mathematics education that could help narrow the gap between learning mathematics and doing mathematics.

But there are two other key ideas evident from this simple model of problem solving. One is the idea of information retrieval. For many reasons we currently do a relatively poor job in helping students learn to use mathematics reference materials. The availability of MECS could (would) provide a strong incentive to make significant changes in this aspect of mathematics education. An increased emphasis on information retrieval in the mathematics curriculum would help move math education into the Information Age.

The second major possible change comes from Step 3 above. Computers can execute algorithms quickly and accurately. The basic nature of the human brain is that it is not good at exact memorization and at doing repetitive tasks requiring extreme accuracy. It "forgets," or becomes bored, or just plain makes an occasional error.

The types of abilities that lead to excellence in doing repetitive computations or symbol manipulations seem only vaguely related to the higher-order, problem-solving skills that we want students to gain through their mathematical studies. Indeed, it could well be that the emphasis on developing such skills is one of the roots of the "I can't do math and I don't like math." outcome that is so frequent in our mathematics education system.

The concept of an "inverted" curriculum has arisen from the type of analysis given in this section. In essence, the use of a computer to execute algorithms facilitates teaching students to use a computer to solve certain categories of problems without teaching them either the underlying theory or how to do the computations by hand. We currently have little research to help us understand possible effects of using a computer-based inverted curriculum. But there are quite a few non-computer-based somewhat analogous situations in our current curriculum.

The protractor was emphasized in earlier in this paper because it illustrates some of the inverted curriculum ideas. Similarly, we teach grade school students to make use of a zero and a decimal point; both of these represented major breakthroughs in mathematics, and their underlying theory is well beyond students who are first learning their use. The ideas of a function and of functional notation are introduced rather early in our mathematics curriculum. These are deep mathematical concepts, perhaps only fully understood by people who have both good mathematical ability and who study the subject for many years.

Note that "cookbook" statistics and other math-application courses existed before computers became available to students. The use of computers in such courses has been common for many years. In many ways a cookbook statistics course represents a type of inverted curriculum.

We have not discussed possible applications of artificial intelligence in mathematics education. The MECS we have described is powerful enough to execute the artificial intelligence software that currently exists or is under development. More and more problems will be solvable by merely accurately specifying (describing) the problem to a computer. The computer will interact with the problem poser to assist in this accurate specification process. The potential impact on mathematics education is not clear.

One of the early attempts to apply artificial intelligence ideas to arithmetic instruction was the program Buggy developed by John Seely Brown at Zerox's Palo Alto Research Center. The goal was to develop a program that could detect and classify student subtraction errors, and then provide appropriate remediation. The program wasn't as useful as might have been expected, because of the nature of the human mind. Students tend to make random errors. At one moment they will demonstrate that they can perform a certain type of computation, and a few minutes later they will fail in an attempt to do a nearly similar computation. It seems clear that we need a learning theory that better reflects the frailties of the human mind.

Scenario 2: Middle School
12 September 1999

To Whom It May Concern:

I have been informed that I have been nominated for Teacher of the Year and that I should write a letter supporting this nomination. I am embarrassed to write about myself, but here goes!

I am 61 years old and have been teaching for 32 years. I have three children and five grandchildren. I began as an elementary school teacher in 1960. About fifteen years ago I decided to take my present position, which is teaching all of the middle school students (grades 6-8) in a small rural school. Our school has four teachers, covering grades K-12.

I graduated from college in 1960, which certainly seems like a long time ago. My major was elementary education and I specialized in reading. I have always enjoyed books, and I am good at looking up information in a library. I focused on primary school education because I wasn't sure I could handle the math in the upper grades.

My first teaching assignment was in second grade. I stayed at that level for several years. Then I attended a math workshop that placed special emphasis on use of manipulatives. For the first time I began to understand that math was more than just doing arithmetic, and that math could be fun. I immediately changed my math curriculum to reflect what I had learned. I think we used a book called Math Their Way.

During the next dozen years I taught at most of the elementary school grade levels, but with several years off to have children. I learned quite a bit about science and math, but I continued to focus mainly on language arts. It has always seemed clear to me that reading and writing are at the very core of education. I taught all of my students to have good library-use skills. Even when I was teaching math, I emphasized learning to read the math book.

In the mid 1970s I attended a National Science Foundation inservice that focused on use of calculators and computers. Well, we certainly didn't have any computers in our school—indeed, the only calculators were in the main office. But I bought an electronic calculator and began to experiment with it in my fifth grade class. I let students use it to check answers. Also, students could play with it as a reward for getting their math assignments done quickly and neatly. They had fun making up problems so that when the calculator display was turned upside down it spelled out a word. But, all in all, I was not impressed by such silly uses of this machine.

In 1980 I managed to talk my principal into buying a classroom set of calculators. We got solar powered calculators, and they cost about $25 apiece. I guess that was when I really began to get interested in math. I was teaching sixth grade then, and my students had already had quite a bit of instruction in paper and pencil arithmetic. I decided to let them use calculators whenever they liked, and I began to focus on problem solving. I remember that some of the parents got quite unhappy. But there was an article in the Arithmetic Teacher (December, 1980, by Grayson Wheatley) that gave research supporting my position. And then the Agenda for the 80s came out, and it supported my position.

I moved to my present position in 1985. (My husband is the school principal and teaches part time.) This year I have five sixth graders, four seventh graders, and seven eighth graders. Let me tell you about a project we are working on, since it will give you some idea of how I teach. Each year we spend a whole lot of time on just a few projects. Some of the projects, such as the one I will describe, continue year after year.

The project began several years ago when I first learned about acid rain. It seems like acid rain may be damaging the trees and crops in our community. So, we began to talk about this in my classroom. All of my students expressed some interest in this topic, so we decided to build a unit of study around it. We approached it from a problem-solving point of view with all students working together, as we do for almost everything in my classroom.

I know that I am supposed to allocate a certain number of minutes a day to math, science, language arts, physical education, etc. But I just don't follow these rules very closely. (I do make sure that each student gets an hour a day of drill and practice on "basics" on the computers, covering math facts, spelling, vocabulary, geography, and so on. This hones their fundamentals and ensures they will do well on the standardized tests that they have to take.)

The class and I decided to spend our physical education time going for walks in the woods and fields, seeing whether we could detect changes that might be due to acid rain. The kids began to gather tree leaf samples, as well as samples of various crops. They thought that maybe we would be able to see a change from one year to the next.

We used our computer to search periodicals for articles about acid rain. It seems like this is a problem going back to the 1980s, so we looked up and read a number of old magazine articles. One of the things that we learned is that the Canadian and US governments have been arguing about whose industry was causing acid rain, so I had my students begin to read about this. Each student had to write a paper on how different countries resolve such issues.

We learned a lot about the industrial revolution, competition among companies and countries, and how hard it is to figure out who is to blame. I had my students study the rapid growth of manufacturing during the industrial age and write reports on what they were learning. They had some trouble understanding the big numbers used to describe company sales and profits. So, we spent quite a bit of time on economics and how companies work to make a profit. We made use of a business simulation game—the kids played it for several weeks.

Meanwhile, we had all of the stuff they collected in the woods and fields. We decided it would be a good idea to measure the tree leaves and to find their areas. But it soon became evident that there is no simple formula for the area of a leaf, and that each tree had leaves that differed widely in size and shape. This led us into studying some statistics. Soon I had all of the students attempting to gather a "random sample" of leaves from various trees. We build templates for measuring the length and width of a leaf. Students learned to find area by tracing a leaf on graph paper, and then counting the number of squares. We recorded our data in a computer database. We used the computer to calculate means and other statistics. We also printed out graphs relating length to area, width to area, tree type to average area, etc.

A neat thing happens when you have students of several grade levels working together. The kids that are good at something help the others. When they can't help each other, then I get involved in providing the help. But usually I let them muddle around, trying to figure it out for themselves. We have a lot of good computer-assisted learning materials. The older kids often direct the younger kids to CAL materials that they found particularly useful. In some ways this combination of older kids and computers is like having a half dozen teacher's aides.

It turns out that lots of people are interested in acid rain. We sent away for a kit that allows us to measure the acidity of rain. We built a weather station and spent quite a bit of time studying weather. We set up rain gauges in a whole bunch of places, since there is quite a bit of variation in our region. This way each student was responsible for maintaining one rain gauge, and reading it each time it rained. We got a contest going, to predict how many cm of rain we would have in each of the months remaining in the school year. I showed students how to look up rainfall data from previous years for our part of the state. They used data from the past 20 years to help them make their estimates. Interestingly, although they all had the same data, they all came up with different estimates. We spent quite a bit of time discussing this.

But we had to do something with all of that data we were gathering, so we got involved with the computer again. We decided that we wanted a program that allowed us to type in the data from all of the rain gauges, and that would print out a map showing this data. We also wanted the program to calculate the total amount of rain that had fallen in the circle that is three kilometers in radius and centered on our school. I usually have a couple of students who are good at programming. Three of them worked together to make a program that takes in the data, prints out a map, and calculates total rain. The first year they did this they entered it in a science contest and won second prize. I was really proud of them!

I went to a conference and found out that there is a computer network of people interested in acid rain. I got our school involved, and I told them about our leaf measurements. We got tied in with several schools in other states and a couple outside the United States. We had them gather data about the tree leaves in their area, and we created a large database with all of that data.

I suppose that project is why I have been nominated to be a Teacher of the Year. We have been working on it for ten years, and it has gotten quite a bit of publicity. I even wrote an article on it, and it got published. Each year my students spend quite a bit of time on this project. We plot multi-year year trends, and we think of new ways to analyze the data. Each year we also think of additional data to gather.

I could go on about other projects, but you have the general idea. We make a lot of use of computers, and I spend a lot of time working with my students. They learn all kinds of things that I don't know much about, since they all get good at looking up stuff in the computer information retrieval system. We learn together, and I feel that is what education is all about.

Sincerely yours,

Mrs. Sally Jones

Scenario 3: High School
3 November 2000

Dear Diary:

I can't tell you how much fun I had today. And I thought it was going to be a bummer!

Today was parent's day at my twin's school. Kay and Ken informed me that if I attended, they wouldn't have to go to school that day. What could I say? Fortunately, they said I didn't have to go to their physical education class. They said that I was too old for gymnastics.

So, off I went, quite prepared to suffer through the day. And, wouldn't you know it, the first class was Second Year Conversational Japanese. I have picked up a couple of words from the kids, but I am not sure what they mean. They are always jabbering to each other, so I guess they have learned a lot.

Well, in I went, and the teacher greeted me in rapid fire Japanese. I mumbled something about Kay and Ken, and hurried to a back corner of the room.

The teacher noticed my discomfort and suggested that I might like to play with the CAL videodisc lessons. The classroom has one MECS per student, each equipped with a videodisc player and earphones. The teacher got me started with lesson 1, and I soon become engrossed. The pictures were amazing, but what was most amazing was the voice input system. The computer system would pronounce a word and display its voice pattern on the screen. Then I pronounced the word, trying to match the voice pattern. The computer provided feedback on how well I was doing, and it even made some suggestions on how to do better! The class period passed quickly, and soon it was time to go on to the First Year Physics class.

I thought I would be more comfortable in physics, since I had that course in high school. But what a change! It was a lab day, and the students were doing an experiment about acceleration. They had a little device that they said was like the auto focus mechanism in a camera. It measured distance quickly enough so that it could give good data on a moving object, such as a falling weight. It fed the data into a computer.

The students then used the computer to fit a curve to the data. They said they were doing a "least squares" fit, and that this made use of calculus and solving linear systems of equations. I asked them if they understood the calculus. They replied that they hadn't studied calculus yet, but that it wasn't necessary to understand calculus in order to understand fitting a function to some data.

By the end of the period, some of the students were beginning to write up their lab report using the word processor on the computer. They explained that they were using an integrated package, so that they could incorporate the experimental data, as well as some graphs produced by the computer. One of the students showed me a computer printout of the data and the function the computer had fit to the data. It looked like a parabola to me.

Third period was Current World Problems. I was a couple of minutes late, since I got lost in the hallways. By the time I got there a couple of the students were reporting on their most recent electronic mail "conversations" with students in Russia. It turned out that each student in the class has an "electronic mail pal" in another country. Part of the required work in the course is to write monthly reports on the ideas discussed with their electronic mail pals.

After a couple of brief reports, the teacher engaged the students in a discussion on where in the world one might most expect to find quite a bit of terrorism. I guess this was a long-term project, since the students seemed to make frequent references to discussions in previous days. It was interesting how they used computers in studying this question.

The students had a computer database listing all countries in the world, with a number of characteristics of each country. For example, the database contained information on population, fertility rate, area, average number of years of schooling, per capita income, form of government, percentage of the population with various religious beliefs, and so on.

Initially the teacher reviewed how one might find relationships between sets of data. The teacher demonstrated use of the computer to graph pairs of data, such as per capita income versus fertility rate. The class made conjectures on what relationships one might expect to find (for example, low income being associated with high fertility or with low life expectancy) and the teacher helped them graphically explore these ideas.

Students were then assigned to work in groups of three, using the MECS in the room. The assignment was to make at least five somewhat related conjectures, test them using graphic techniques, and write a brief report on the findings. The students were to share in developing the conjectures, but each was to write their own report interpreting the results. I could see how this work tied in with making conjectures about factors related to terrorism.

Fourth period was Math, and I was really bushed by then. I don't see how the kids can handle so many hard classes, back to back. I had been looking forward to the math class, since I was a math major my first two years in college. That was before I decided to be a business major and to go into the insurance business.

I noticed that there was a MECS at each student desk. As students came in they immediately flipped on their computers and set to work. I asked the teacher what they were doing. The teacher explained that the first ten minutes of each math period were devoted to playing some simulation game or practicing some basic materials students have studied in the past. This is part of a carefully designed, systematic review and reinforcement schedule which helps improve long term retention of the math students have studied. It also gives students feedback on areas where they need to do more review or further study.

Today's game was a quite old piece of software called Super Factory (from Sunburst Communications). In it students get to see several views of a cube with different pictures on some of the faces. Then they have to direct the computer in creating a cube that looks just like the original. The teacher explained that playing the game helps many students to improve their three dimensional visualization skills.

After ten minutes the teacher flipped the power switch to the student computer display screens, and turned on the power to the classroom computer display. The teacher indicated that the lesson for the day was on use of mental skills and computer graphics to solve equations with one unknown.

The teacher asked for some examples of equations that couldn't be easily solved mentally. Various students provided suggestions, and the teacher typed them into the computer so the equations were displayed on the screen. For example, the students suggested problems such as:


 * 3x^2 - 15x^(1/2) + 6 = 0


 * 4sin(x) - 2x^3 + 5x -12.8 = 0


 * 2^x - 25x + 3 = 0


 * x^(1/2) + x^(1/3) + x^(1/4) - 9 = 0

For each equation, the teacher discussed how one might be able to mentally figure out if there is a solution or more than one solution. For example, on the first equation when x = 0 the function is positive. But when x = 1, the function is negative. So, the equation has at least one root between 0 and 1. [Editor's Note: This assumes that the function is continuous in the interval with end points 0 and 1.]

After an equation was discussed, the teacher had the computer graph it, and then showed how to read off the places where it crossed the x-axis. The teacher also suggested that a problem such as the second one might better be handled by graphing the following two functions, and seeing where they intersect.


 * y = 4sin(x)


 * y = 2x^3 - 5x +12.8

The computer system had a "zoom" capability that allows the teacher to use a mouse to point to a part of the graph, and to have that part be expanded. This can be used to investigate a pair of equations in very fine detail, to see if and where they intersect.

I am afraid that I got carried away, since I raised my hand and was called upon. I said, "All of those examples look too easy, and they certainly aren't the type of problems I have to solve in my insurance business. Why not try a real world problem? For example, suppose that I deposit $800 at the beginning of each year for five years, and I want to have $5,000 at the end of five years. What does the interest rate need to be, if interest is compounded at the end of each year?

The teacher appeared delighted by the question, and said to the class: "Here is a real world problem. How many of you think that you would be able to solve it by the end of the period?" A couple of students thought they might be able to do so, but most indicated they had never seen as problem remotely like that before. Upon further prodding, most indicated that they knew about compound interest, but didn't know a formula for this problem.

The teacher then turned to the chalkboard and began to think out loud about the problem. "Let's use x as the interest rate. If the interest rate were zero, I would only end up with $4,000. That suggests that the problem makes sense. The interest rate needs to be large enough so that all of the interest adds up to $1,000.

Suppose I had the whole $4,000 at the beginning, but it was just invested for 2 1/2 years. An interest rate of 10% would give me more than $1,000 interest. My guess is that the answer will be a little less than 10%.

If I deposit $800 dollars at the beginning of the first year. I will have 800(1+x) dollars at the end of the year. Those original dollars will become 800(1+x)(1+x) by the end of the second year, and 800(1+x)^3 dollars by the end of the third year. Meanwhile, of course, I have the added amount of $800 deposited at the beginning of the second year, and it begins to earn interest. Aha! I am beginning to detect a pattern I am now sure that I can solve the problem."

The teacher then turned on the student computer display screens and indicated which file contained equations to solve using computer graphics. The teacher assigned my problem as extra credit.

Near the end of the period the teacher asked if anyone had been able to solve my problem. Several students indicated they had, and their answers were fairly close together. One student indicated, "I figured out the equation, and it had a bunch of (1+x)s raised to different powers in it. I graphed it, and read off an answer. Then it occurred to me that I could use the computer to simplify all of those powers of (1+x). I used the symbol manipulation program to do it, and I got an ordinary fifth degree polynomial equation. I had the computer graph it, and I got the same answer as before. Then I used the polynomial solver, and the answer was about the same. I am confident that it is right."

Another student indicated that she had tried to look up a formula, but hadn't been able to find one. "I found information about this type of problem. It is called an annuity problem. The computer gave an equation like you started to develop, but it used i instead of x for the interest rate. And there was no formula for finding the answer. I thought that our computer had a formula for just about everything. Did I look in the wrong place?"

The teacher indicated that there aren't any formulas for most problems. "Finding or developing an equation to solve, and having a computer to help do the work, is a more general approach. That is why we are working on general methods for solving equations, such as using computer graphics."

Needless to say, I was impressed! We certainly didn't learn to do things like that when I was in school. As I started to tell what things were like in the "good old days," the bell rang. I played hooky for the rest of the day, since I had to meet a client for lunch. But I'll remember this day for a long time.

Recommendations and Closing Comments
The basic recommendation is that mathematics educators and researchers work to create a MECS mathematics education environment for students. We have described a framework for change, and it can serve as a basis for long-range planning. The following five important steps need to be pursued concurrently and iteratively.

R1. Develop the hardware, software, and courseware of MECS and work to make the entire system cheaply and readily available to students. Begin orienting students to their responsibilities in a MECS learning and work environment.

But note that most of the ideas that we want to teach using MECS can be taught with the types of computers, textbooks, and libraries currently available in most schools. We can begin now, rather than waiting until MECS is available.

R2. Provide appropriate training to existing and new teachers. This will require a massive amount of inservice training as well as changes to our teacher training programs. Increasing, the role of CAL will change the role of teachers—perhaps to more of a mentor or facilitator role.

Most teacher training institutions have made some progress toward providing preservice teachers with a little introduction to computers. But in most cases this instruction is not adequate to prepare teachers to deal with the math curriculum of the year 2000 and beyond envisioned in this paper. The computer needs to be integrated as an everyday tool into a large number of the college classes taken by preservice teachers. Both primary and supplemental CAL needs to be available and routinely used in a variety of these courses.

R3. Begin both the development and the concurrent research on curriculum appropriate to a MECS environment. Be fully aware of the use of MECS as an interdisciplinary tool. Math is important in many fields of study.

The process of research and implementation needs to occur concurrently if the overall task is to be accomplished in a timely fashion. A lot of research and curriculum development has already been done on interdisciplinary aspects of mathematics.

R4. Begin modifying teacher-produced, district-wide, state-wide, and national assessment to reflect and take advantage of a MECS environment.

In many ways, our national assessment instruments drive our mathematics education curriculum. We should move rapidly toward a situation in which both calculators and computers are made available to students during testing.

Perhaps the key idea is that one major goal is to prepare students to do mathematics in the environment they will encounter after leaving school. This environment will include ready access to calculators and computers. Thus, both instruction and testing should (for the most part) be done in an environment of calculators and computers.

R5. Begin working to gain the support of all of the people who must be involved in the changes needed to have mathematics education occur in a MECS environment. This includes students, parents, school board members, teachers, educational leaders, legislators, textbook publishers, etc.

Research on change in education strongly supports the need for long-range planning that involves all of the key stakeholders.

We close this paper with a number of comments related to the ideas presented earlier. Many are points that require additional discussion and/or research.

C1. Computer facilities somewhat equivalent to MECS will increasingly become available to people in business, industry, government, and research. We know quite a bit about transfer of learning. We know that transfer of learning is greatly helped if the learning environment and the applications-of-learning environment are quite similar. This provides a strong argument for integrating the use of MECS into our mathematics education curriculum.

C2. Students vary widely in their mathematical abilities. Mathematics education is designed both to help students to work up to the levels of their mathematical abilities, and to sort out those with greater or lesser abilities. Those with greater abilities are encouraged to seek mathematically oriented careers, while those with lesser abilities are steered in other directions. But the sorting out process is often flawed. For example, students with poor ability to memorize computational and manipulative algorithms and to develop both speed and accuracy in their applications may be discouraged by our current mathematics education system, but we know that many such individuals have great mathematical ability. Education in a MECS environment might be of great help to people with low innate computational skills.

C3. Except in a few physical science courses, most current non-mathematics courses make very little use of mathematics. That is a sad and sorry situation, since mathematics is useful in every discipline. The MECS tool has the potential to change this situation. Curriculum reform is needed in many disciplines.

C4. For many people mathematics is a "game" to be played by certain rules. Thus, use of a calculator is "cheating." It is evident that widespread availability and use of MECS changes the mathematics game. One can expect resistance to such changes. Quite a bit of the resistance will likely come from those currently playing the game quite successfully, including many math teachers. On the other hand, quite a bit of encouragement for the change may come from people who apply math on the job, such as scientists and engineers. For them, math is less a game and more an indispensable tool for solving the problems they encounter on the job.

C5. Our mathematics education system is used to tools such as the compass and protractor. Such tools change very slowly, if at all, during a person's lifetime. Our mathematics education system is not used to rapidly changing tools. Mathematics education, especially at the precollege level, is built on content that may change little during a person's teaching career, and on methodology that changes but little over several decades. Thus, our mathematics education system is basically conservative in nature. This suggests that it will be quite difficult to move this system in the direction of the MECS environment.

C6. Color displays and motion graphics add new dimensions to the tools available to students and teachers. We know little about appropriate uses of such tools. Research is needed.

C7. We have made only brief comment on the teaching of computer programming and computer science. These are topics that are related to change in mathematics education, but are not at its core. Computer science is a discipline that is somewhat distinct from mathematics. However, mathematics educators may decide that it is advantageous for all mathematics students to learn to program. They might decide there should be a computer-programming strand in the mathematics curriculum. That is a good topic for another paper.

Computer science places considerable emphasis on the development and representation of algorithms, on analysis of possible performance of algorithms, on programming algorithms, and debugging programs. All of these ideas are quite mathematical in nature. Studies on factors predicting success in computer programming courses invariably identify mathematical knowledge and ability as key factors. That is, computer science and mathematics are closely related disciplines. Many colleges have chosen to combine these disciplines in a single department.

C8. The ideas proposed in this paper will require many decades to implement. But a significant start can occur in the next ten years. The microcomputers currently available in schools are powerful enough to begin the change to a MECS mathematics education environment.

C9. The proposed changes to the precollege mathematics curriculum will create a major articulation problem with the college curriculum. It is essential that the precollege curriculum revision effort be paralleled by a college mathematics curriculum revision effort.

C10. MECS, and the ideas discussed in this paper, could revitalize mathematics education. It could bring new life and excitement to mathematics students, faculty, researchers and writers.

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This IAE-pedia document was written by David Moursund. Partial funding for preparation of this book was provided by the National Science Foundation grant TEI 8550588.