Volume 22 1994-95 Editorial (with Retrospective Comments)
Moursund, D.G. (September 1994). Progress and Evidence in Educational Technology. The Computing Teacher. ISTE.
[Sidebar.] Dave Moursund is Editor-in-Chief of The Computing Teacher and Executive Officer of ISTE. He has been teaching, writing, and speaking in the field of computer technology in education since 1963. In this new column, he offers answers to questions from readers.
Q. You have been involved in the field of computers in education/or a long time. What is the best thing that you have seen happen?
A. Computers have become sufficiently inexpensive and user friendly so they can be used by children. Children can learn many things about computers faster and better than adults. I believe that if adults (including teachers) provide appropriate guidance and don't get in the way too much, the long-term results will be a major improvement in our educational system. This is because computers help to create a rich, learn-by-doing environment in which children can explore, pursue their own interests, and take a greater ownership of their own learning.
Q. What are some of the areas where progress has been slower than you would have expected?
A. I will give two examples. First, my doctorate is in mathematics. I am sorely disappointed in the progress that has occurred in this field. Thirty years ago it was clear to me that the basic nature of mathematics education should be drastically changed by computers. The National Council of Teachers in Mathematics has strongly supported such changes. For example, the 1989 NCTM standards call for extensive use of computers. But relatively little change has occurred in the classroom. My feeling is that many math educators "just don't get it."
I also feel that the teacher education system has done poorly in rising to the challenge of computer technology. Most newly-graduated teachers are computer-illiterate relative to the standards needed for working with current educational uses of computers.
Q. Can you provide solid evidence that computers make a positive difference in education?
A. This is a difficult question mainly because people are not satisfied with the nature of the three-part answer that I like to give.
It is common to divide instructional uses of computers into three categories. First, there is teaching about the field of computer and information science. Computer and information science has emerged as an important discipline. I don't believe that anybody doubts the need to have computer facilities for use in such teaching or the major strides that this academic discipline has made.
Second, there is the use of computers in instructional delivery-often called computer-assisted instruction. CAI includes drill and practice, tutorials, and simulations. In recent years, most CAI development has been focused in hypermedia environments that make use of graphics, text, sound, color, and animation. There has been a large amount of research on CAI. The meta-studies (the studies of studies) provide solid evidence of the effectiveness of CAI in many different settings.
The third main use of computer technology in education is tool uses of computers. This includes word processor, database, spreadsheet, graphics, telecommunications, and so on. Many people who ask the original question are really only interested in computer as a tool.
A general answer is to note that such tool uses of computers have become commonplace in a wide range of job settings. Moreover, it takes quite a bit of education, training, and experience to become an effective user of these computer tools. Thus, the case for such computer use in schools is simple. These computer tools are a new "basic." Schools need to help students gain basic skills in reading, writing, arithmetic, speaking, listening, and tool : use of computers as an aid to their becoming fully functioning, educated adult in our Information Age society.
However, even this answer is not very satisfying to many people who ask the original question. They want still more specific answers. "Show me that if I teach my students to use a word processor, they will be better writers." There has been quite a bit of research on this specific topic. For the most part, the results are inconclusive. However, this is no longer a particularly interesting question. It is now common in both higher education and in job settings to expect people to use a word processor. At the same time, we still expect people to communicate effectively in writing. Effective communication is the major issue. The tool-whether it be a ball point pen or a word processor-is not the primary determinant of writing quality.
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Q. What is the Information Superhighway? Is it just hype, or should schools be doing something about it?
The Information Superhighway is a combination of "hype" and a profound component of the Information Age.
It is appropriate to think of any type of a computer network as an information highway Timeshared computing makes use of computer terminals that are located some distance from the computer. Timeshared computing was already well established when the programming language BASIC was being developed about 30 years ago.
The storage capacity and speed of computers has grown immensely over the past few decades. This has made it desirable to increase the speed of the networks connecting computers. Thus, year after year, people have been working to increase the capabilities of the information highways. Breakthroughs such as communication satellites and fiber optics have made major contributions.
Many people now have a "gig" drive on their personal computers. A gigabyte of storage is 1,024 megabytes of storage. At one time people thought of a gigabyte as an overwhelming amount of data. If you are talking strictly about text, then a gigabyte is approximately 1,000 full length novels.
However, suppose that you are talking about graphics. How much storage does one high resolution color picture take? Suppose that the picture is eight inches by 10 inches, that it is stored at a resolution of 1,200 dots per inch, and that it is represented using 256 colors. If no data compression is used. this one picture requires about 112 million bytes of storage. A gigabyte of storage is filled up by about 10 of these pictures.
Suppose that you wanted to send a gigabyte of data between two computers by use of a telephone line and modem.
In the early days of computer networking, the information highway often had a speed limit of 300 baud (30 characters per second). To transport a gigabyte of data at this speed would take about 10,000 hours, or well over a year.
We now have much faster modems, and the telephone lines that are in place can handle much higher data rates. Many people make use of a modem that operates at a speed of about 1,400 bytes per second. At this speed, it takes only about 200 hours to transfer a gigabyte of data. Clearly, that is still far to long a time. The postal service or a courier service can do it faster and cheaper.
What is needed, of course, is a data transfer rate that is consistent with the storage capacity of modem computers. If you are working with color pictures where a single document is more than a tenth of a gigabyte, you would like to be able to send that document to another computer in a "reasonable" amount of time, such as a second or a few seconds.
If an information highway can handle a data rate of a few thousand bytes per second, what do you call a system that can handle 10 million or 100 million bytes per second? The term "Information Superhighway" comes from such thinking.
Of course it is hype to suggest that a sudden breakthrough has occurred or that such a breakthrough is needed. Rapid progress has been occurring for years and such progress will continue to occur for years to come. However, the term Information Superhighway helps to focus attention on the cumulative effect of this rapid change.
An Information Superhighway changes the nature of how teams of people work together to solve problems. Consider a team of people who are working together on a computer-based project, but the team members are scattered throughout the country. Their computers are connected by an Information Superhighway This means that as they work together on a projectperhaps one involving high resolution color graphical datathe time delays caused by transmission of shared data are minimal. For example, one of the sites may have sophisticated scientific instrumentation that is making measurements or collecting photographic data that needs to be analyzed by the other sites. The results of their analysis are fed back to the data-gathering site (in "real time") to guide further data gathering.
Now, what does all of this mean for education? The underlying ideas of the Information Superhighway are of teams of people working together to solve problems and accomplish tasks that involve huge amounts of data. The data may be stored at a variety of sites. The Information Superhighway allows the team members to be located throughout the country. A person may be a member of several different teams. Teams are created and disbanded as the need arises.
Education needs to prepare students for problem working in such environments. Thus, students need to learn to work with problems that are represented using large amounts of data. They need to learn to work in teams, where the members of the team are separated by a distance.
With a little ingenuity, this type of situation can be created using a low-speed local area network within a school building or even within a classroom. For example, one member of the team may be gathering data in a science lab while a different team member is using a scanner to generate graphical data. Other members of the team may be in other parts of the building, retrieving information from databases, analyzing the data, and directing the efforts of the student working in the lab and the student using the scanner. Communication among team members is done strictly via computer. The goal is for the team to accomplish a prescribed task and produce appropriate reports on the results.
[Send your questions for this column to The Computing Teacher, ISTE, 1787 Agate Street, Eugene, OR 97403-1923; fax 503/346-5890; E-mail firstname.lastname@example.org. You may E-mail Dr. Moursund directly at moursund@oregon. uoregon. edu.]
Sidebar: Dave Moursund is Editor-in-Chief of The Computing Teacher and Executive Officer of ISTE. He has been teaching, writing, and speaking in the field of computer technology in education since 1963. In this new column, he offers answers to questions from readers.
Moursund, D.G. (October 1994). What Computer Should I Buy? The Computing Teacher.
Q. What computer should I (we) buy?
A. Parents and educators asking the question often expect me to tell them the exact machine that is the best buy to fit their needs. However, I am seldom able or willing to do this.
There is no one best computer system for all needs. (Suppose there were. Don't you think that enough people would have discovered this so that all other computer companies would have gone out of business?)
The question about what computer to buy serves as an opening to a dialogue between a consultant and a client. How does a professional consultant handle such a task?
In the initial interaction, the consultant attempts to determine what the client knows about computer systems and their uses. The consultant determines whether the client wants a "turnkey" solution or wants to be seriously involved in helping to formulate an answer to the question.
In a turnkey approach, the client hires people to analyze the problem; make appropriate decisions; acquire the needed hardware, software, and support materials; and train the staff. If the overall choice and implementation of computer systems proves to be less than adequate, the client can blame the consultant.
In the client education approach, the consultant serves in the role of a personal tutor and a source of information. The consultant helps the client to lern enough to guide the decision making and implementation process. The client ends up with a great deal of knowledge and personal involvement in the overall process. If the net result is not satisfactory, it is the client who must assume responsibility.
You can see why I have difficulty responding to the "What computer should I buy?" question. I am not being asked to either deliver a turnkey system or to do client education. Thus, for the most part I do not attempt to provide an answer.
However, there is one set of circumstances in which I do provide some specifics. If parents indicate that the goal is to provide a computer for their child to use for academic purposes, the answer tends to be simple. I tell the parent to find out what the school is using. Get hardware and software that is compatible with what the school is using and is somewhat close to the best that the school has available. If the school is hopelessly out of date, I tell the parent to look at other schools and the school their child will attend in the future.
I also provide a few general guidelines. Think in terms of the computer system serving your child for about four or five years. Think about repair and upgrade services over this four to five year period. Will the manufacturers and vendors still be in business? Think about who will help your child to learn. Appropriate learning help and guidance is the key to making a worthwhile investment in technology.
Moursund, D.G. (October 1994). What Are Your Thoughts on Donated Equipment? The Computing Teacher.
Q. People have proposed that businesses should donate old computer equipment to schools. This is being done in California on a wide scale basis. What do you think about this idea?
A. For the most part, I think this is a poor idea. The companies making the donation feel that the equipment is no longer cost effective for their own use. They feel that it is well worth the hardware, software, and training costs to install newer equipment.
One obvious educational problem is the limited capabilities of such donated equipment relative to the demands of the newer software and courseware. Another is the reliability and maintenance problem.
There are many hidden costs in this computer donation program. For example, many school board members and school administrators think that the key issue is the student to computer ratio. If the ratio can be improved by donated equipment, they may feel that the problem is solved. They no longer need to be concerned with needed permanent changes in the budget to support up-to-date computer-related technology
However, there is a deeper issue. Recently I was talking with an executive from General Motors. At one time GM used to donate old equipment to technical institutes. These technical institutes ended up with lots of five year old equipment. They argued that training and education in this environment was totally inappropriate for preparing students to move into the types of job openings that GM and other companies have available. GM agreed with this argument. It dropped its program of donating out of date equipment and began donating state of the art equipment.
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Moursund, D.G. (December/January 1994-95). How Can We Make Donations Pay Off? The Computing Teacher. ISTE.
Q: My organization is considering a multi-year project of donating a lot of resources to schools that serve disadvantaged students. The goal is to help increase the level of computer-oriented technology literacy of a very large number of disadvantaged students throughout the United States. What are your recommendations?
Wouldn't it be wonderful if educators got a request like this every month? We should rejoice at corporations or government agencies offering to lend a hand. However, this request also forces us to understand the scale of the challenges facing education, and the difficult choices we need to make in allocating resources-even large donations.
The person submitting this question did not provide a definition of "disadvantaged students" and did not suggest how much money the organization might have available. Perhaps the organization is a private foundation or a potential new federal program.
Given enough money, we know how to have a significant positive impact on one school. Examples of this include the Apple Classroom of Tomorrow projects that created high density hardware sites. Other successful projects have focused on empowering teachers, staff development, and curriculum development. The problem is, how can we scale up good projects to a huge number of schools?
Suppose that just 10°o of the schools in the United States are classified as serving a large proportion of disadvantaged students. That would be about 10,000 schools, serving perhaps four to five million students! The combined annual budgets for 10,000 schools is about $25 billion per year. Even a one-percent increase in their budgets would cost $250 million per year. This is far beyond what a private foundation might be able to contribute on an annual basis. While a new federal program of this size might be possible, it is unlikely in today's atmosphere of federal fiscal restraint.
Indeed, $50 million per year-which is about two-tenths of one-percent of the total budgets for these schools-is still a hefty sum for almost all private foundations or for a new federal program. And yet, $50 million is only $5,000 per school per year if the goal is to reach 10,000 schools!
What significant benefits can be realized with $5,000 per school per year? Some key components for computer technology in a school include hardware, software, curriculum materials, knowledgeable staff, and supportive stakeholders (parents, school administrators, school board, etc.) I will examine each of these as possible components of an answer to the question.
The K-12 educational system in the United States is now spending over $2 billion a year on hardware. Because quite a bit of this comes from a variety of state and federal programs, it may well be that the 10,000 schools being discussed in this article are receiving the funds at least in proportion to their numbers. If so, they are spending an average in excess of $20,000 a year per school on hardware.
Computer hardware is only one component of computer technology in education. Thus, even if the project directors decided that hardware was of relatively high priority, probably less than half of the project funds would be devoted to hardware. The result would be a modest blip in the amount of hardware already available in these schools. My personal recommendation is that if we are limited to $5,000 per school per year, little of these new funds should go to hardware.
In contrast to hardware, software and curriculum materials have the characteristic that their retail sales price is a very modest percentage of the cost of producing an extra copy. Suppose, for example, that a 10,000-school project that serves mainly disadvantaged students approached some of the major software and curriculum materials vendors and asked about the price of 10,000 school site licenses.
The project might well be able to make "deals" at just a few cents on the dollar as compared to regular retail prices. The software, instruction manuals, and curriculum materials might be distributed to the 10,000 schools on a CD-ROM with appropriate permission to make the needed number of copies. Local resources and time would be used to make the needed copies and would constitute part of the local contribution to the overall project.
It is also important to recognize that a project of this size could work with other funding agencies such as the National Science Foundation and the Office of Education to develop materials specifically for this project. These two Federal organizations could build this wide scale dissemination into many of the curriculum projects that they fund. In addition, a project of this size could fund some curriculum projects. Such are some of the advantages of an economy of scale.
My conclusion from this type of analysis is that there could be a substantial economy of scale in terms of software and curriculum materials. Software and curriculum materials should be a significant component of a $5,000 per school large-scale project. However, I would still probably restrict hardware, software, and curriculum material expenditures to no more than 20 percent of the total project budget.
Where would other 80 percent go? If I were managing this $5,000/school project, I would put the bulk of my resources into staff development. Next month in this space, I will explain why and how.
[Send your questions for this column to The Computing Teacher, ISTE, 1787 Agate Street, Eugene, OR 97403-1923; fax 503/346-5890; E-mail email@example.com. You may E-mail Dr. Moursund directly at firstname.lastname@example.org.]
Moursund. D.G. (February 1995). The Computing Teacher.
Q: My organization is considering a multi-year project of donating a lot of resources to schools that serve disadvantaged students. The goal is to help increase the level of computer-oriented technology' literacy of a very large number of disadvantaged students throughout the United States. What are your recommendations?
The previous issue of The Computing Teacher discussed the magnitude of the task of trying to reach a significant portion of disadvantaged students. The goal might be to have a significant impact on 10,000 schools in a multi-year project budgeted at $5,000 per school per year.
Some of the needed components for successful use of technology in schools include hardware, software, curriculum materials, stakeholder support, and staff development. In the previous article, I indicated that I would use only about 20 percent of the $5,000 per year for hardware, software, and curriculum materials. I would focus the rest of the resources on obtaining continuing support from the various stakeholder groups (parents, school administrators, school boards, local businesses) and on staff development.
Let me talk about the stakeholder approach first. To a large extent, education is political. There are many different stakeholder groups that have powerful voices in our educational system. They need to be fully represented in school technology planning, and it is essential that they be provided with information that will help them to make good decisions. The goal is to enlist these groups as full partners working to improve the educational system,
ISTE has published a book, The Technology Advisory Council: A Vehicle for Improving Our Schools. It discusses the creation of a Technology Advisory Council (TAG) and the types of work that a TAG can do. I would put resources into the creation and continued work of a TAG for each school involved in the project. This does not take much moneyprobably S500 per school per year would suffice. Quite likely, these funds can be leveraged into projects such as creating school-business partnerships that have a high return on the dollars that are invested.
This leaves the issue of staff development. We have about $3,500 per year left, and I would put that into creating and supporting the development of a two to three-person team of classroom teachers at each school. These School Technology Leadership Teams would be committed to:
I would allocate approximately $2,000 per year to each Leadership Team, The Team could use these funds as they see fit-for a phone line, to attend conferences, to acquire software. They might develop a Future Technology Educators club. Students in the club would help teachers and fellow students make effective use of educational technology.
The remainder of the funds-approximately $1,500 per school, or $15,000,000 for the 10,000-school project-would be used to provide centralized services and as general overhead on the project. These might include:
So my short answer to our potential educational donor is: You could make a significant contribution to technology-supported education with a large-scale project involving $5,000 per school per year. However, the money would need to be carefully allocated, with emphasis on staff development leveraged through a Technology Leadership Team in each school.
[Send your questions for this column to The Computing Teacher, ISTE, 1787 Agate Street, Eugene, OR 97403-1923: fax 503/3465890: e-mail email@example.com. You may e-mail Dr. Moursund directly at firstname.lastname@example.org.]
Moursund, D.G. (March 1995). The Home-School Connection. The Computing Teacher.
Q: What future roles will computers play in the school-home connection?
The same question can be asked about reading, writing, and arithmetic. Over the past 25 years, there has been considerable research showing the academic advantages that children gain if they grow up in a home in which parents routinely model, and help their children acquire, these basic skills. Computer-related technology is providing the underpinnings of some new "basics." This technology is of rapidly growing importance, and it seems clear that it will stand the test of time.
Technology, Now and Just Ahead
Near the end of 1994, Business Week published a special issue titled "21st Century Capitalism," exploring the directions that world business and trade may take in the next decade. This issue is rich in data. For example:
Business Week predicts continued rapid increases in the speed, internal memory, and external storage capacity of microcomputers. There will be continued rapid growth in installation of fiber optic cables, and the carrying capacity of these cables will grow very rapidly There are a number of predictions about how educated people from throughout the world will compete for high tech jobs as global trade rapidly expands and as the Information Superhighway becomes a reality.
Two conclusions seem evident to me. First, school systems throughout the world will be hard pressed to deal with the continued rapid progress in microcomputers and the Information Superhighway Second, there will be rapid expansion in worldwide competition for well-educated employees who function comfortably and effectively in a high-tech world.
The School-Home Connection
What does all of this have to do with the school-home connection? Here are three ideas:
Millions of students are learning more about and from computers at home than in school. Although we lack adequate long-term research on outcomes from this home computer environment, it seems likely that it will produce a substantial advantage for these students. This provides the basis for two recommendations:
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Moursund, D.G. (April 1995). Learning software, constructing knowledge. Learning and Leading with Technology. Eugene, OR: ISTE.
Q. I think I understand the idea that students construct their own knowledge. How does this fit in with computers in education?
Over the past decade and more, constructivism has drawn the support of many leading educators. The basic idea is that a person builds new knowledge and skills on the knowledge and skills he or she already has. The learner's mind actively constructs knowledge and skills. This is quite different from the "empty vessel" idea in which the learner is merely passively waiting to have new knowledge and skills poured in.
The "empty vessel" theory fits well with a top-down determination of curriculum content and with an industrial age, mass production model of instructional delivery. However, it is increasingly evident that this approach to education does not adequately fit the needs of learners in the Information Age.
Recently I had the time and inclination to learn two new pieces of software Quicken and PowerPoint. Quicken from Intuit, is a home finances package, useful for keeping personal financial records. PowerPoint, from Microsoft, is a desktop presentation package, useful in preparing and doing presentations.
The manual for Quicken is more than 400 pages long, while the manual for PowerPoint is more than 600 pages long. The essence of both manuals is captured by the following quotation from page xi in the PowerPoint handbook: "PLEASE!!! Don't read this book from cover to cover!" Both manuals contain directions for quickly getting started and suggest a learn-by-doing approach.
The assumption is that people acquire one of these pieces of software because they have a task they want to accomplish. Each person has knowledge, skills, previous experiences, and goals that have led him or her to want to learn to use the software. Each learner has a different background and different specific goals. Each learner is intrinsically, internally motivated.
Certainly, the. preceding paragraph describes my situation. My doctorate in mathematics and previous work with spreadsheets helped as 1 approached the learning of Quicken. My experience in using a variety of graphic arts programs and my experience in preparing slides for talks helped me as I approached the learning of PowerPoint.
I found the learning of each piece of software to be a significant challenge. I don't easily learn new pieces of software. However, I am tenacious. Eventually-after about a day on each piece of software-I developed levels of skill and knowledge that suited my immediate needs. In neither case did I really begin to scratch the surface in "covering" the manual. However, in both cases I made significant progress and I laid groundwork for additional learning in the future.
The feedback mechanisms in my learning endeavors were a combination of the computer and myself. I posed questions and tasks for myself. Often I had trouble conveying my ideas to the computer. The error messages and the computer output display provided me with feedback on my lack of understanding. As I progressed, these overt errors decreased. More of the feedback came from within, for example, "Does the graphical layout of this slide really help convey my message?
As an aside, in this overall learning experience, I gained increased insight into the overall process .of learning new pieces of software. I integrated this knowledge with my current and of constructivism and our educational system, These insights and integration of knowledge helped prepare me to write this editorial message.
The lengths of the two manuals I used are similar to the lengths of many textbooks used in high school and college. The complexity of the topics covered in these manuals is comparable to the complexity, of topics covered in high school and college texts.
1 didn't have to take a test of my new knowledge and skills. Rather, the test was whether, I could accomplish the tasks I had in mind when 1 started the learning process. For example, I wanted to prepare the overheads for a talk, and I wanted these overheads to be distinctly better than the overheads that I prepare using a word processor.
Contrast this with what most students face as they first open a text for a course in high school biology, math; or some other course. A prescribed syllabus has been developed by the teacher and people who set curriculum. The curriculum is closely tied to the text, slowly winding its way through as it "covers" the book. There will be assignments. Under the best of circumstances, feedback from these assignments will occur the next day. Often the feedback is merely a check mark or some other indication that the assignment has been recorded in a gradebook. There will be quizzes, tests, final exams, and perhaps competency exams. These will be graded, and a final grade is entered into a permanent record to reflect the student's achievement in the course.
Interestingly-and sadly-there is an increasing number of high school and college courses that use essentially this same "traditional" model of instruction to teach students to use a piece of software. A piece of software becomes the focus of a course, and students are lock-stepped through a curriculum that covers the software.
What a waste! The learning environment a student encounters when faced by a new piece of software is a golden opportunity to learn to learn, to gain experience in accepting personal responsibility for learning, and to practice the general ideas of constructivism. It seems obvious to me that schooling should be designed to facilitate students in constructing their own knowledge.
Moursund, D.G. (May 1995) Computers and Mathematics Education. Learning and Leading with Technology.
Calculators and computers are having a significant impact on math education. However. I feel that the impact is smaller than it should be, Math educators need to "think bigger" as they plan for the effective use of calculator and computer technology.
The Essence of Math Education
Mathematics provides vocabulary and notation to represent a wide range of problems. The cumulative mathematical knowledge on how to actually solve math problems is immense. Thus, the essence of math education is to:
People can learn a great deal of useful mathematics with a minimum of formal (in school) instruction, However, formal instruction and disciplined study is needed for most students to make significant progress in working with fractions, algebra, geometry, calculus, statistics, and so on. Math is considered to be such an important subject that most students are required to study this field for 10 or 11 years in the precollege curriculum.
Roles of Computer Technology
Computer technology plays two major roles in math education. First, it can aid in the delivery of instruction. Computer-assisted instruction in mathematics includes drill and practice, tutorials, simulations, and microworlds. All these forms of instruction can help students learn faster and better.
Second, computer technology can facilitate change in what students learn. Fur example, the proliferation of hand-held calculators has lead to the gradual disappearance of bv-hand calculation of square roots from the curriculum.
There has been a tremendous amount of research about CAI over the past 40 years. There have been dozens of metastudies-studies of studies. In brief summary, CAI works. On average, students learn quite a bit faster. They. learn equally well or better, compared to "conventional" instructional methods.
There is a huge and ever increasing amount of mathematics CAI software available commercially There are many different pieces of software that facilitate exploratory activities in graphing, geometry, algebra, and statistics. Many of these pieces of software might be classified as edutainment-certainly they tend to hold students' interest.
I wonder why all students do not have access to such materials at all times when they are learning mathematics.
Computer Technology and Mathematics Content
Our mathematics education system has gradually dropped the bv-hand calculation of square roots as well as interpolation in tables. It has made other modest changes, including an emphasis on the use of hand-held graphing calculators in the upper grades of high school and in college.
The potential of computer-as-tool is tremendous. The following diagram captures the essence of how math is used to solve problems.
The diagram indicates four relatively distinct steps:
The math teachers 1 talk to estimate that about 80 percent of mathematics education time is spent on step 2. This may be an overestimation, because a huge amount of math education time is spent on learning the vocabularv and notation of mathematics. However, it is clear that our mathematics education svstem places far more emphasis on step 2 than it does on the other three steps.
The greatest strength of calculators and computers in mathematics is their abilitv to do step 2. An inexpensive calculator can calculate square roots. A calculator costing less than $ 100 can graph, solve equations, and provide the user with useful help in working with a hundred or more built-in mathematical functions.
The computer-based aids to solving math problems can solve the full range of problems that students typically study up through the first couple of years of college mathematics.
Whv does our mathematics education system spend so much of its instructional effort teaching students by-hand methods of doing what computers can do? Why aren't all students provided with the powerful computer-based mathematical tools?
Mv discussion seems to give equal weight to CAl and the computer-as-tool. However, I believe there is at least an order of magnitude difference in the potential of these two approaches to improving mathematics education. The best of math-oriented CM may well help students learn 30 percent to 50 percent faster. The best of the computer tools may reduce the time unnecessarily spent on by-hand methods by a factor or 10 or more.
The National Council of Teachers of Mathematics has made some progress in supporting the use of calculators and computers. But I believe that the NCTM is thinking too small. Progress toward achieving the full potential of computer technology in mathematics education is proceeding at an agonizingly slow pace.
[Send your questions for this column to Learning and Leading With Technology, ISTE, 1787 Agate Street, Eugene, OR 97403-1923: fax 503/346-5890; e-mail email@example.com. You can e-mail Dr. Moursund directly at firstname.lastname@example.org.]