Mathematics teachers, on guard!

Mathematics teachers, on guard!Author(s): BRUCE E. MESERVESource: The Arithmetic Teacher, Vol. 13, No. 6 (OCTOBER 1966), pp. 476-484Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41185427 .

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Mathematics teachers, on guard!* BRUCE E. MESERVE University of Vermont, Burlington, Vermont

Bruce E. Meserve is past president of the National Council of Teachers of Mathematics. He is a professor of mathematics at the University of Vermont.

At has become an NCTM tradition to have a presidential "State of the Council" address immediately preceding our annual business meeting. Last year I described in detail our very healthy, professionally active, financially sound organization. This year the title that I have selected is pur- posely ambiguous. I hope that each of you will find in my remarks several chal- lenges, several alerts for your daily professional activities, and the recognition of several situations that you will want to encounter with great care: Mathematics teachers, on guard! Alert! Watch out!

I also hope that you will find in my remarks renewed assurance that your Na- tional Council of Teachers of Mathematics, through its Board of Directors, numerous committees, many affiliates, and legions of energetic members, is on guard, is alert to the ever-increasing opportunities before us, and is watching out for the professional needs of mathematics teachers.

NCTM administration

You expect your NCTM to have ad- ministrative units such as the Board of Directors, the Executive Committee, and the Washington office.

The Board of Directors is the ultimate source of power in the NCTM. The president, executive secretary, and all others

* An address presented at the Forty-fourth Annual Meeting of the NCTM, New York, New York, April 14, 1966.

who take action in the name of the NCTM or strive to coordinate NCTM activities must be constantly alert to the policies of the Board. Your Board is composed of conscientious and energetic people, but they need your help to function effectively. If you see ways in which your president, executive secretary, committees, or other NCTM personnel could be more effective, it is important that your suggestions reach a Board member or some other person who can see that your suggestions are given due consideration.

You and the NCTM cannot afford to have any known suggestions lost in plati- tudes because "things seem to be going along all right," or even very well. Fre- quently your comments will actually apply to an office rather than to an individual holding that office.

In recent years your Executive Com- mittee has acted in an executive capacity on behalf of the Board when the Board is not in session. For the last few years the NCTM has paid one-half of each president's salary so that he could be released part time from his other duties for work on NCTM activities. The NCTM by-laws have been revised so that you always have either a president and a president-elect, or a president and a past president.

I consider the use of elected officers rather than career employees as official NCTM spokesmen to be one of the great strengths of the NCTM. Presidents come and go. Each president has new ideas and

476 The Arithmetic Teacher



tries them. Then the next president evalu- ates, modifies, or replaces procedures in addition to trying his own new ideas. All of this is done within the framework of policies established by the Board of Di- rectors. This frequent fresh outlook is much harder to obtain, and in some re- spects impractical, in the case of professional staff members in the Washington office. I think that our present policy with its close coordination of the work of the Washington office and the NCTM Execu- tive Committee is excellent and the best available.

Committees

Most of the work of the NCTM is done by its committees. The following committees are primarily concerned with NCTM activities:

Financial Policies Committee Nominations and Elections Committee Plans and Proposals Committee Place of Meetings Committee Program Committee

All other NCTM committees are in one of these four major areas:

Professional relations Curriculum Instructional media Publications

I shall not try to name each committee or

project. Rather, I shall discuss a few of

my concerns in each of these areas, men- tion a few related NCTM activities, and

challenge both you and the various committees to increase your guard in several

ways. Also, I shall broaden our usual concern for professional relations to include public relations.

Public relations

Many people feel that the public relations of mathematics educators during the last ten years have been practically a honeymoon. Some of you have been too close to the firing line to accept such an appraisal completely. However, there is a sense in which this has been true, and each

of us (individually and professionally) has a heavy stake in the manner in which our public relations evolve in the future.

The philosophy of consensus has an impact on each of us. We have a definite stake in public opinion. We need to understand how public opinion is formed. Of course, our knowledge of public opinion is not based upon an exact deductive science. However, there have been several studies of the following five stages in the process of a person's making up his mind on any particular issue:

Awareness Interest Evaluation Trial Decision

Three influences are considered at each stage:

Expert opinion The opinions of local people The individual's personal opinion The role of the expert is very high at the

awareness stage, has considerable impact as the person is becoming interested, has a small influence in the evaluation stage, when most people would rather believe other people than facts, has very little influence on any trials that may be involved, and has practically no influence on the final decision or adoption of the idea.

The role of local opinion is usually based

upon informal contacts and casual comments. These local people play a noticeable role at the awareness stage, a major role at the interest stage, the dominant role at the evaluation stage, and a very important major role at the trial stage, even though a person ultimately makes up his own mind on the matter at hand.

The individual's own views and uncer- tainties play a very small role at the awareness stage, an increasing but very minor role at the stages of interest and evaluation, and still a minor role at the trial stage, even though the person completely controls the decision.

Last month at the San Diego meeting

October 1966 471



of the NCTM, Francis Mueller gave,a very impressive documentation for the point of view that the era of the dominant role of the expert in mathematics education, and especially in new mathematics curricula, is just above over. Ten years ago articles in popular magazines and newspapers picked up themes of interest and hope - "Look what they are teaching in mathematics (even though I do not understand it)." The public was becoming aware of, interested in, and hopeful regarding the new mathematics programs. During the past year attitudes reflected in popular news media indicate that we are approaching, or are in, the stages of evaluation and trial. Probably many of you already sense this in your own communities. Different communities, however, may be at very different stages in their consideration of new mathematics curricula.

I call your attention to the general change from the stages of awareness and interest to the stages of evaluation and trial because you need to be alert to the changing sources of authority. The importance of the views of the experts seems to be rapidly decreasing. The dominant factors at our present stage are probably the views of your local leaders of all sorts. What- ever philosophy you may have regarding the usefulness and desirability of new approaches to the teaching of mathematics, you should realize that the jury before which your case is being tried is changing. The experts are being replaced by respected local citizens, and you should present your case accordingly.

Increase yoyr efforts to interest your local civic leaders in what is happening in mathematics education. Help them understand the real advantages of our new procedures. Don't try to impress them with your facility with terms or symbols that they don't understand. Your future rests in their hands. They deserve an op- portunity to understand what is going on.

There are many things that you can and should be doing locally. The NCTM can help, but outside experts are suspicious

characters. You need to present your own case, even though you seek outside advice and witnesses. Here are two suggestions:

1. Bring in interesting and informative mathematicians and mathematics educators who can speak effectively, who can develop a good rapport with parents in your local PTA, and who can help parents understand what you are trying to do. The NCTM has done very little in this area. Explicit suggestions will be very welcome. Last year the Delegate Assembly adopted a reso- lution that a study be made by the NCTM Board of Directors on the establishment of a Committee on Adult Education and Public Relations, for the purpose of dis- seminating information to the interested lay public. The Board approved and asked several of our committee chairmen for suggestions. This week the Board has decided that adult education is an appropriate activity for the NCTM and has started on its detailed search for ways of becoming a constructive influence in this area.

2. Help your teachers and colleagues to understand what the new approaches to the teaching of mathematics are, and how to use these approaches effectively in their classrooms. This is not a matter simply of adopting a new textbook. No matter how good the book is, teachers need, and can be most effective with, materials that they thoroughly understand and can teach with confidence. In other words, I firmly believe that some teachers are more effective with so-called obsolete textbooks that they understand than with new texts that they do not trust. The problem also cannot be solved by sending one or two teachers to an institute or in-service program. Enough teachers must be influenced to affect the outlook of all teachers. I personally believe that this can best be done by in-service programs sponsored by local schools or school districts. Many suggestions for such programs will be found in the booklet on the in-service education of elementary school teachers that has just been completed by the NCTM In-




service Education Committee and will be published late next fall. For the secondary school teachers there is a tendency to rely upon courses offered by colleges and universities. I personally feel that this cannot be effective without at least an aggressive campaign to alert the college mathematicians to the real needs and problems of teachers.

The special problems of administrators are considered in the booklet Administra- tive Responsibility for Improving Mathe- matics Programs, which was developed by a joint committee of the NCTM, the American Association of School Adminis- trators, the National Association of Second- ary-School Principals, and the Association for Supervision and Curriculum Develop- ment.

In the general area of public relations you should consider your local mathematics and other professional organizations, the cooperation of the NCTM Committee on Affiliated Groups, your local and regional affiliates of the NCTM, your state NCTM representatives, the activities of the NCTM Committee on International Mathematics Education (including their exhibit of foreign textbooks here at this meeting), the activities of the NCTM Committee on Re- lations with Industry, and the many representatives of the NCTM who work with other professional organizations.

The NCTM is rapidly coming of age in its working with other professional groups. The NCTM has been an enthusiastic sup- porter of the Conference Board of the Mathematical Sciences (an organization composed of the major professional organizations in the mathematical sciences) and is now represented on its Executive Committee. The president of the Mathe- matical Association of America and your president recently sent out a joint letter to approximately three thousand teachers of mathematics in junior and community colleges. In these and many other ways the NCTM. is striving to serve, and to improve its public relations with, all people interested in mathematics education.

Cooperation with parents

Before turning to the area of curriculum, I shall suggest a specific approach for working with parents. First, parents must be helped to realize that in our modern approach to the teaching of mathematics we are not trying simply to develop a new dogma or a new cult. Rather, we are concerned with and are seeking (1) concepts that provide students with the greatest possible mathematical insights and power, (2) a variety of teaching techniques that have been proved effective, and (3) nota- tions and definitions that will enhance the student's understanding and facilitate his use of the concepts.

After parents have gained this insight into our activities, we need to help parents understand that we have a basic goal of teaching students to think creatively in their dealings with quantitative data and geometric figures. Finally, we need to help parents realize that there are ways in which they can help their children with their homework. I shall describe my personal philosophy on this in some detail, since, as I have just noted, the future of our work depends upon our success in working with parents and other members of the community.

We do not know exactly what problems our children will face ten years from now; thus it is futile to teach students only memorized procedures for dealing with known problems. Rather, we must teach them to use the basic tools that we have and to apply these tools to a wide variety of problems. This use of knowledge on the uncharted road ahead is so closely related to creative thinking that I shall use the four steps in creative thinking to provide the framework for my comments:

First there is preparation. What is the problem?

Then there is incubation. What facts are known? What elements are involved? How can the elements be represented? What relations exist among the elements?

The next step is the critical one, illumina-

October 1966 479



tion. After we meditate upon the relations that exist, a plan of action appears promising. Usually we can find a structure for the problem in relation to other facts and results. Frequently there is an estimate of the expected result or answer.

Then there is verification. The con- jectured plan of action is tried out; the expected result is proved or shown to be false. The problem may be solved, or shown to be impossible (and in that sense solved); or we may discover that a new plan of action will be needed, and so re- turn to the stages of incubation and illumination.

These four stages - preparation, incubation, illumination, and verification - are the four stages of creative thinking. In our schools and in many other aspects of our society we need to take still another step, communication. We need to be able to report to others what we have done and the results that we have obtained.

What are we as teachers doing and what can parents do to help students in the stage of preparation? How do we help students understand problems? Many parents think of mathematics books as having one or two sample problems worked out and then a set of exercises that may be worked like those in the samples; then, more samples and exercises, more samples and exercises, and so forth. They think of a classroom situation in which the teacher shows, or tells, the students what to do and the students try to do it. Many school administrators are telling their teachers to stop their "show and tell" acts. Students need to learn to read problems, to translate verbal problems into mathematical statements, to select important information from a paragraph. Many scientists tell us that as soon as they understand what the problem is that they are working with, the rest is usually easy. Thus, we try to make it necessary for the students to read to find out what the problem is. Admittedly, we have trouble getting the students even to read the instructions to see what they are asked to do. However, teachers and

parents can often be very helpful by asking students what the problem is: "Tell me in your own words what you are looking for."

The role of a modern approach to the teaching of mathematics is particularly noticeable in relation to the incubation stage. We used to skip this stage entirely by giving the student a rule that told him exactly what to do. Now we try to stimu- late his own thinking with questions such as these:

What is the problem about? What elements are involved? What facts do you already know about

these elements? Can you represent the elements in other

ways, such as by a figure or by mathematical symbols?

The examination of the facts that are known depends not only on the facts given in the statement of the problem but also on those known by the student. For example, consider the question: Are two lines that are perpendicular to a given line parallel?

You and I might draw a figure, remember vaguely some proof that we once studied, and reply "Yes." We have made a good use of another representation of the elements involved in the question, but this representation may not be good enough in a modern approach to the teaching of mathematics. We live in a three-dimensional world, and we must teach our youngsters to think in both two- and three- dimensional spaces: that is, both on a plane and in space. Let's look at some other figures. Whether we think of spokes of a wheel or of a TV antenna, we may visualize




the lines as intersecting lines. Thus we now realize that in space the lines may intersect. We might also think of the spokes of two wheels on the same axle, sketch a figure

for this situation, and notice that in space the two lines may be skew lines - that is, lines that are not intersecting and are not

parallel - in other words, two lines that are not on the same plane.

Our own incubation of the problem is beginning to be helpful. The light is beginning to dawn. We are approaching the stage of illumination. We are beginning to have some suspicions about the problem. The two lines appear to be parallel if they are both on a plane with the given line; the two lines may be either intersecting lines or skew lines when they are not both on a plane with the given line. These expecta- tions can be verified and we can com- municate our findings:

If two lines are perpendicular to the same given line and are on a plane with that given line, then they are parallel. If two lines are perpendicular to the same given line and are not on a plane with that given line, then they may be either intersecting lines or skew lines.

In a modern approach to the teaching of mathematics we consider problems such as the one that I have just described to you, both at an intuitive level such as we have used, with the figures providing the verification, and at a formal level in the deductive course in geometry. The ne- cessity for distinguishing between plane and space geometry is basically a need for recognizing the mathematical system or domain in which we were working. In many schools still other domains are considered. For example, it is not unusual to

consider the geometry of points and great circles on a sphere as a mathematical model of the geometry on the surface of the earth. In this geometry, as you may visualize in terms of meridians perpendicular to the equator, any two lines that are perpendicular to a given line intersect at two points, the two poles of the given line.

The emphasis upon creative thinking places a modern approach to the teaching of mathematics in a central and basic position in the school program. Consider the effectiveness of these five stages throughout the school program:

Preparation - reading for understanding Incubation - restating in one's own

words, comparing with other facts that have been accumulated as general background information

Illumination - finding patterns that lead to a guess or estimate of the results to be obtained

Verification - checking that our expected results are correct in the system in which we are working

Communication - making a concise statement of the results obtained

Parents may ask what has happened to the drill and skills that they associate with their own study of mathematics. We should admit that these are still needed, that we now look for a variety of procedures for motivating practice in using mathematical concepts, and that when modern textbooks fail to include sufficient computational work, teachers must correct this deficiency. Many parents are deeply

October 1966 481



interested in what is happening in mathematics programs. Tell them. Reassure them. Help them understand that by questioning their children (whether in true or feigned ignorance of what is going on) they can greatly enhance their children's pro- gress without destroying the exploration and discovery approaches that we are frequently trying to use. Emphasize that questioning, not telling, is the key to effective assistance.

Curriculum

The NCTM created a Curriculum Com- mittee one year ago and is still in the process of structuring its approach to this very important and, in a sense, all-en- compassing area. Our present efforts are primarily in the areas of research, mathematics for the non-college-bound, and computer-oriented mathematics.

For several years the NCTM has sought ways of improving the quality of research in mathematics education. An NCTM booklet illustrating some promising prac- tices in mathematics education and dis- cussions of research in this field is essentially complete and will be published next winter. This booklet provides a start in the right direction. However, with massive federal funding of Educational Research Informa- tion Centers, with at least one other wide- spread information exchange project (EDUCOM) emerging on the horizon, with large universities actively interested in such projects, and with still larger industrial corporations undertaking an active interest in providing materials and services for educational systems, the NCTM and in- deed all mathematics teachers must make special efforts to bring to bear the special needs of students and teachers as plans are established. It seems obvious that many guidelines will be established and that these guidelines should have a sound professional basis rather than a genesis in political ex- pediency, commercial aggressiveness, or poorly designed experiments.

I am convinced that the needs of our profession can be effectively served. My

warning is simply that we must be active in the formulation of plans. If we are not, others will make the decisions, with or without an understanding of the situations that they are trying to serve.

The NCTM Committee on Mathematics for the Non-College Bound is concerned with an area in which guidelines and extensive efforts are needed. A preliminary report of some of the recent activities of this committee is entitled "Helping the Low Achiever to Learn Mathematics." Free copies are available with other NCTM materials at this meeting.

The publication of ten units of material for general students, "Experiences in Mathematical Discovery," will provide in a few months another evidence to the professional community of NCTM interest in this problem.

The role of electronic computers in the mathematics curriculum needs considerable study. There are many evidences regarding things that students can do. Similar studies appear needed as to how much students should do with electronic computers. Pre- sent indications are that in the long run electronic computers will and should be involved in an integral part of our secondary school mathematics program. How- ever, the related technology is being developed very rapidly, and it appears to be sheer folly for any school or school system to try to develop an electronic computer center of its own at this time. The NCTM tried to provide a few warnings in the article "Computers for School Mathe- matics" in the May 1965 issue of The Mathematics Teacher. A booklet on methods of operation is now being polished and should be submitted to the NCTM Publications Committee later this month.

Our aim is to help teachers and administrators understand the various ways of obtaining access to electronic computational facilities and to recognize the advantages of some of the most promising methods of operation. Additional publications are contemplated to illustrate the in- troduction of an algorithmic language and




ways in which electronic computational facilities may be used to increase the effectiveness of the development of topics in the mathematics curriculum. It appears highly probable that computer-oriented mathematics will become an integral part of the secondary school mathematics curriculum.

Instructional media

Computational facilities will also serve as instructional aids. In this context they should be considered as one part of very comprehensive instructional systems in- volving teachers, textbooks, films, film- strips, other projectuals, programmed materials, and computer-assisted instruction.

Education is big business, and several large business corporations are now very actively interested in education. As teachers and administrators we shall need sound guidelines to keep from being swept off our feet by pedagogically unsound, grand- iose schemes. At the same time we should look forward with confidence to the even- tual development of highly effective instructional systems that are little more than dreams today.

The NCTM is vitally interested in the development of pedagogically sound instructional systems for use at all levels. At the level of materials to help teachers improve their own backgrounds, we co- operated with the Mathematical Associa- tion of America in the development of the film "Mathematics for Tomorrow." Our own series of ten films, "Mathematics for Elementary School Teachers," has just been released to United World Films for distribution. We have also learned that films are not enough. Thus, there will be text materials to accompany this series of films. These text materials will be available in June 1966 from our Washington office. A proposal to extend this series of films is being prepared.

Last December your Board of Directors sent back to the Committee on Films and Television a proposal for films based upon concepts that are appropriate for general

mathematics classes. For each topic there were to be -

1. One or two films on the subject matter.

2. A film showing the topic being taught to a demonstration class.

3. A film clip that teachers could use in introducing the topic in their own classes.

4. Some related text materials.

Your Board felt that there was too heavy a dependence upon films, that the proposal should be reconsidered from a multimedia- instructional-systems approach. We hope to have a revised proposal in the near future.

The members of your Board of Directors are acutely aware of a critical philosophical question: should the NCTM be a producer of materials, or a strictly professional organization concerned with criteria for materials and guidelines for the production and use of materials? This is a difficult and basic question. My personal philosophy is that -

1. The NCTM should serve as a catalyst when possible, a producer when necessary, to make sure that new types of materials are developed in a pedagogically sound manner.

2. Throughout all of its activities the NCTM should maintain a position from which it can provide constructive criticism of all materials.

3. The NCTM has a responsibility provide definitive criteria and sample constructive evaluations based upon pedagogical principles (not just individual preference) for textbooks, programmed materials, and instructional aids of all sorts as background information for the numerous local and regional evaluations that are going on throughout the country.

Publications

Our consideration of NCTM activities has led us directly to the role of our publications. Our journals, yearbooks, and other publications have a very extensive

October 1966 483



impact that is often overlooked. One anec- dote in this connection may be of interest to you. Probably most of you enjoy the cartoon "Peanuts" and were a bit sur- prised at the mathematical insight of the creator of that comic strip when he dealt with mathematical topics several times last winter. I cannot prove that there is any connection between the two events, but I can report to you that the creator of the "Peanuts" cartoon strips ordered a set of the NCTM booklets on "Topics in Mathe- matics for Elementary School Teachers" early last fall.

I have already mentioned several contemplated publications. The publications of the NCTM provide one of its most effective professional services. The circula- tion of these publications has been growing very rapidly. You should be proud of this NCTM service as well as the other aspects of growth of your NCTM. Meetings such as this and the publications of the NCTM provide your best opportunities to share your ideas.

You can set the pace

My aim this morning has been to alert you to the "changing times" in which we are living, to the role of the NCTM in these changes, and to the critical need for your active involvement in these changes. You need to be active on the "home front" with your colleagues and the lay public. You need to be active in making your views known to your NCTM "legislators,"

the members of your Board of Directors. Specifically, I urge each of you to make known your views on such questions as these:

1. Should the NCTM simply list new books as they are published or try to provide reviews that will indicate to teachers where the books can best be used, their good points, and their limita- tions? This might mean two reviews - one of the content of the book when it first comes out and one of its pedagogical usefulness after it has been used in the classroom. In each case it should be possible to make some positive statements as well as to indicate some areas in which certain common goals are not met.

2. Should the NCTM restrict its production activities to experimental materials that others are not ready to undertake, constructive criticisms, and the development of guidelines?

Times are changing. Trends and guidelines in mathematics education will be established. Will they be established upon pedagogically sound principles? You, the members of NCTM (and in so far as I understand the educational climate in which we are living, you alone with your many contacts, activities, and leadership roles in many organizations), can make sure that our changing times reflect sound pedagogical principles and the needs of the citizens of tomorrow.

Mathematics teachers, on guard!

Forty-fifth Annual Meeting - meals for special groups

Groups planning meal functions during the NCTM Annual Meeting in Las Vegas next April should be making arrangements now. A committee has been set up to help make the necessary local arrangements regarding meals for groups desiring such get-togethers. The chairman of this committee is Mrs. Marie Davison. Her address is:

1409 East Wilson Avenue Las Vegas, Nevada 89101




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Mathematics teachers, on guard!