The mathematics consultant: A suggestion for improving the elementary mathematicsprogramAuthor(s): WILLIAM R. ASTLESource: The Arithmetic Teacher, Vol. 9, No. 4 (APRIL 1962), pp. 203-205Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186619 .

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The mathematics consultant A suggestion for improving the elementary mathematics program

WILLIAM R. ASTLE Western Illinois University, Macomb, Illinois

Professor Astle is a member of the department of mathematics, Western Illinois University.

Uuring the past few years there has been a considerable change in the overall aca- demic objective of elementary-school edu- cation. The elementary school is no longer a terminal institution. It is emerging as a place where children are prepared for more advanced study in the secondary school. What are the implications of this new objective for elementary mathe- matics? A professor of mine once said the objective in a mathematics class was to learn mathematics. With this statement I must agree. However, we surely can't teach all of mathematics in one class. It is this author's opinion that we must de- velop a minimum program of mathe- matics education for everyone. The objec- tive of this program would be to provide each citizen with the mathematical knowl- edge to live effectively in our "mathe- matized" culture. For the average (in abil- ity to learn mathematics) student this program would be of ten years duration; the slower and faster students would reach this minimum goal in more than or less than ten years respectively. I do feel that those faster students who achieve this goal in seven, eight, or nine years should be re- quired to continue studying mathematics at least until the end of the tenth year in school. I do not intend to investigate the content of such a program, but rather the implications of such a program for ele- mentary schools.

On the basis of the above discussion it is clear that the mathematics program of the first six grades is in no sense terminal. Every student will be continuing his study

of mathematics until the minimum goal is achieved. Some children, an ever in- creasing number, will be going into some phase of mathematics as a career. Some children will be destined to use mathe- matics in the physical sciences and en- gineering. The social sciences and business administration will also draw a large number of mathematically trained people. The other children, not necessarily a ma- jority, will need mathematics to make in- telligent decisions and to understand the increasingly technical culture surrounding them. The key concepts and skills (be- cause of the difficulty in comprehending them) necessary for these groups of "fu- ture citizens' ' would be discussed near the end of the program. This does not mean that they cannot be introduced earlier, but rather that they should not be for- malized (i.e., verbalized) earlier. It is for this reason that elementary mathematics will have an essentially new role. The task of the elementary program will be to sup- ply the skills, concepts, and language nec- essary for a clear and correct development of these more difficult topics in the second- ary school. In other words, the purpose of elementary mathematics is to enable the understanding of more advanced, more difficult, and more useful mathematics in the secondary program. We must realize that even considering a program of this nature hinges on two vital points:

1 Every student will stay in school through Grade 10 (or until completion of the minimum program).

April 1962 203

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2 We will have teachers capable of han- dling a program of this nature in the ele- mentary schools. Let us consider the latter. Unfortu-

nately, we must face up to the fact that the typical elementary-school teacher is not prepared to handle a program of the type I have suggested. They just don't know enough mathematics. Well, just how much should they know? They should know, at the minimum, the content of our ten-year minimum program. Above this, a thor- ough and careful knowledge of numbers, systems of numeration, structural prop- erties of numbers, Euclidean geometry, and an informal knowledge of sets and logic would be desirable. There are cer- tainly other topics which would prove val- uable, among them probability, statistics, elementary analysis, number theory, and some non-Euclidean geometry. These lat- ter, however, are not essential. I think it is somewhat unrealistic to expect every elementary teacher to go back to school to acquire this background. How then can we improve upon our program? A solution would be having at least one teacher in each elementary school who will act as a "mathematics consultant. " This person would have the usual classroom responsi- bilities but could be relieved of some of the special areas (art, music, physical educa- tion, lunchroom duty, etc.) to assist the other teachers in both mathematics and methods of teaching mathematics. The ac- tivities engaged in by a mathematics con- sultant would include : 1 In-service programs to reteach and ex-

pand mathematical concepts; 2 Providing leadership in curriculum plan-

ning, textbook selection, etc.; 3 Acting as a resource person for the other

teachers; 4 Serving as a feedback mechanism for the

institutions of teacher training; 5 Assisting in the planning and admin-

istering of "in-school" experiments. Some people will object to this proposal

on the grounds that "We've already tried

it (years ago) and it doesn't work." My reply is this. We are entering an era when the mathematics program of the elemen- tary school is no longer terminal. In re- sponse to this new development we must improve the mathematical competency (in matter and method) of our elementary teachers. It is the writer's feeling that mathematics consultants can improve our elementary mathematics. Other people have proposed a departmentalized ele- mentary-school program. I do not intend to investigate the pros and cons of this alternative. I will note, however, that while this might prove an acceptable solu- tion for the mathematics program, it might damage the program in "social develop- ment." This perhaps is one reason the idea of a mathematics consultant is so appealing. It would involve only a small administrative change in our elementary schools. The adoption of any program in- volving mathematics consultants or a de- partmentalized elementary program will obviously require a new breed of elemen- tary teachers, and it is in this respect that our institutions of teacher education enter the picture.

In view of the need to create a new pro- gram of elementary-mathematics educa- tion, teacher training programs will have two major responsibilities: (1) to offer more and better (more appropriate) mathematics for all elementary teachers, and (2) to offer special programs for ele- mentary-mathematics consultants. In re- cent years the first responsibility has been assumed by more and more colleges and universities, while the second has hardly been recognized. In meeting this second responsibility colleges and universities could adopt one of two approaches. They could offer the work a mathematics con- sultant should have as a fifth-year pro- gram or incorporate it into their normal four-year undergraduate curriculum. I doubt if the latter would be satisfactory. There is, of course, the possibility of a combination of these two approaches. In view of the activities engaged in by a

204 The Arithmetic Teacher

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mathematics consultant, it would prob- ably be desirable to utilize teachers with at least two or three years of classroom experience, in which case the college pro- gram could be offered in four consecutive summers or during one academic year. It seems likely that financial support for the participants of such a program would be available. The important question now is what should go into such a teacher educa- tion program? The content of a program of this nature may be divided into three gen- eral groups - subject matter (i.e., mathe- matics), pedagogy (i.e., the psychology of learning), and professionalized subject matter. I have already mentioned the mathematics which a mathematics con- sultant should know. Naturally it should be taught by the mathematics depart- ment. The psychology of learning should be of a useful (not theoretical) nature. The mathematics consultant will need to know how children learn mathematics and to be familiar with current research in this ever- changing field. The professionalized sub- ject-matter courses should include: 1 A history of mathematics education and

an analysis of the forces which can bring about a change in mathematics educa- tion;

2 Specific techniques for handling certain topics in the elementary school;

3 A study of current proposals for cur- riculum change (at all levels) ;

4 A discussion of current and a review of past research in mathematics education;

5 The preparation by the student of a re- source unit on some particular topic from the elementary program.

The professionalized subject matter should be taught by a mathematics edu- cator or a team of mathematicians and educators. It is hoped that the close work- ing relationship necessary to the consult- ant's "feedback" activities would be established with these people. I do feel that many colleges and universities could offer a program of this nature with little change in their current course offerings.

There are several reasons why the in- troduction of mathematics consultants is an appealing idea.

1 It would cost the taxpayer almost nothing and, therefore, probably not meet with resistence from that quarter.

2 It would require minimal changes in elementary-school schedules.

3 The changes necessary in teacher educa- tion programs are small.

In other words, the idea isn't radical enough to stir up "false" opposition. The most important reason for adopting a consultant program of this nature is that it would facilitate a relatively rapid im- provement of the elementary-school pro- gram. Implementation would require a co- operative effort on the part of elementary teachers, elementary administrators, and college teachers. Once we are capable of teaching our children the necessary pre- requisite mathematics, we can look for- ward to a solid and profitable program of mathematics education in the secondary school. Perhaps then, each citizen would have the mathematical knowledge to live effectively in our "mathematized" cul- ture.

"We have no right to ask society and the teach- ers to accept the new simply because it has been proved moderately successful. However, they can be encouraged to accept and try that which has been tested on a sufficient scale and has been found successful and more desirable." - W. D. Wall in New Thinking in School Mathe- matics.

April 1962 205

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