Problems in the training of elementary school teachers

Problems in the training of elementary school teachersAuthor(s): GAIL S. YOUNGSource: The Arithmetic Teacher, Vol. 13, No. 5 (MAY 1966), pp. 380-384Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187135 .

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Problems in the training of elementary school teachers

GAIL S. YOUNG Tulane University, New Orleans, Louisiana Professor Young is chairman of the department of mathematics at Tulane University. He is also chairman of the Teacher Training Panel of the Committee on the Undergraduate Program in Mathematics.

In the fall of 1965, the Teacher Training Panel of the Committee on the Under- graduate Program in Mathematics com- pleted a series of meetings involving each of the fifty states in which the panel ex- plained its recommendations for the training of elementary teachers and attempted to see the program implemented. The conferences brought together representatives of teacher training institutions, both from the mathematics departments and schools of education, representatives of state departments of education, and representatives of school systems. In most of the conferences, we met great enthusiasm for our program, and local committees were formed to work toward adoption of the recommendations in the colleges and in the state certification requirements. An ac- count of the 1965 meetings may be found in CUPM Report No. 11 [4].*

Earlier, the National Association of State Directors of Teacher Education and Certification and the American Associa- tion for the Advancement of Science made a joint study of training in mathematics and sciences for elementary teaching; both of these groups accepted our recommendations [2].

With all this success and support, how is it possible that I can write an article with this title? Should it not be "solved

problems"? I believe I can show that there are still many problems.

The impact of developing curricula Do we really know what the elementary

teacher should be taught? The CUPM recommendations [3] are that, starting from a minimal base of two years' college preparatory mathematics in the high school, prospective elementary teachers should have a year's work in the number system, a semester's work in intuitive geometry, and a semester's work in algebra. This is our Level I program. (We say further that about 25 percent of the teachers should have the level of preparation, Level II, proposed for the junior high teacher.) We have given detailed outlines for what such courses should be, though emphasizing - and meaning sin- cerely - that these outlines are intended merely to show one way that the proposed material could be covered in the available time»

I should explain now that in this article I am writing my own personal views, and am not expressing the thinking of the Teacher Training Panel or of CUPM. My own view of these recommendations is like my view of the work of SMSG : They are not revolutionary, "modern," or deep. They are merely the best recommendations the Panel could see for preparing students going into elementary education in the time available and with the college

* Numbers in brackets refer to the References at the end of this article.

380 The Arithmetic Teacher



teaching staffs available. The SMSG courses seemed radical to many school teachers when the courses were intro- duced. Similarly, our recommendations seemed strong to many college people. But they are not. One can say that if a teacher does not have this material, he is certainly not prepared to teach in the elementary school, and that is all.

What are the difficulties? First, it must be obvious that the many successful experimental groups, such as those led by Robert Davis, David Page, Patrick Sup- pes, or Paul Rosenbloom, are rapidly changing our entire concept of what can and should be taught to a child. We are becoming aware that we have never faced up to the deep intellectual problems of elementary mathematical education. These problems involve the foundations of mathematics, and the frontiers of educational or developmental psychology. The elementary curriculum may be really altered as a result of their work, say, in twenty years. The nice, bright, 18-year-old girls who like to work with small children and whom we are putting in freshman Level I courses this year will be only 38 then, and I wonder very much how they will cope with these changes. Indeed, the changes may not occur, or may not occur widely, because of the limited training of the teachers. If this were to happen, it would be a tragedy.

It may be possible to select from the work of these groups certain themes that appear sure to form part of the elementary curriculum of the future. I had the op- portunity last year to attend a conference of representatives from many of the leading experimental centers, and could see some of these common themes. For example, each group had some work which could be described as the study of symmetry. Now, symmetry is one of the great themes of mathematics. It occurs in many guises, in geometry, algebra, analysis. Without getting into the question of which particular form of work with symmetry will prevail in the school curriculum, there

could perhaps be enough study of symmetry in the teacher-training program so that in later years the teacher can understand what is being done in a new text. Such a thematic program will be difficult to develop, but it is a problem that must be faced.

One can oversimplify and say that the experimental groups are showing that children can do wonderful things with mathematics, and that we should make it possible by teacher training for more children to have this developmental experience. But there are groups urging drastic changes on the elementary curriculum for social reasons, to fit the child to work in the mathematized society of the near future. To continue to oversimplify, such a group is the Cambridge Conference on School Mathematics [l]. The conference contained several of the leading figures in mathematical education, but it also included many applied mathematicians, engineers, and physicists. Its report gives a picture of what some of the leading mathematicians and users of mathematics think the child must learn before college. I will not attempt to sum- marize, but will merely point out as an example that they advocate introduction of the number line in Grade 1. It is quite clear that the modest CUPM recommendations do not prepare teachers for the Cambridge conference curriculum. The conference is itself sponsoring a session this summer on the teacher-training program for their curriculum, and its report will be eagerly read.

It is hard to resist improvements in education that have the needs of society as the driving force. Indeed, the future of our society and its ability to meet the strains being put on it by our extraor- dinarily rapid technological development may depend as much on the ability of teachers in Grades K-6 to get the students off to a good start in mathematics as any other one thing.

I have been writing as though all the problems involve the curricula of the fu-

M ay 1966 381



ture. However, we now have a number of new series of elementary texts out that are widely used, and one can ask, are our teachers being adequately trained for these? I do not want to go into detail here. I, myself, am quite critical of some of these texts, but I recognize also that even the ones I am most critical of are better than the texts they are replacing. In the space at my disposal now, it would be invidious to single out one or two texts to analyze. Perhaps a fictitious illustration may get my point across, at the risk of being too erudite to be completely in- telligible. Suppose that a textbook author decided that one of the most important properties of the real number system is that the real numbers form an uncount- able set. This is true, and the fact is not usually discussed before a first graduate course in real variables. The author finds that by devoting the entire fifth year to the task, children can be led to understand this " basic fact." Other authors write competing texts, with the same goal for the fifth grade, but with completely different methods of proof. (There are arguments based on properties of in- finite decimals, on the theory of measure, and on topology.)

If this were to happen, bodies recommending teacher-training programs would face two problems. The first is practical, and in the world of real texts it is very much with us. Here are competing texts using widely differing methods of ap- proaching the same material. Even mathematics graduate students usually see only one or two of these approaches. How is it going to be possible in a limited time to train a teacher to understand the material well enough to teach whichever of these approaches chance brings to him?

The other problem is more basic. Such a fifth-grade program would meet uni- versal criticism from the mathematical world. I cannot imagine a research math- ematician who would believe that this result deserved so much emphasis in the

fifth grade. Suppose that, despite such criticism, these texts continued to flourish. Should recommending bodies then draw up programs to prepare teachers to teach material that no member of the body be- lieves should be taught at this level?

The need for more training The discussion of the last section has

been written with the assumption that mathematics would get normally no more than twelve hours of the elementary teacher training program, and that all elementary teachers would teach mathematics. The twelve hours itself is hard enough to get. We are not alone in having new approaches to our subject that require more time of the prospective teacher if the children are to receive the benefit.

Is it possible to add more training in mathematics in a graduate program? Very many graduate schools will resist stren- uously any attempt to call a course that follows twelve hours of undergraduate work a graduate course. Yet with pay in- creases tied not to actual gains in knowl- edge, but to names of degrees, how is it possible to increase the mathematical competency of many teachers except by courses for graduate credit?

I am not competent to pass judgment, but it is my own belief that we will be forced to specialization in very early grades. The arguments for the self-contained classroom are impelling, but if the limitations of time for training and the needs of the future described above are correct, then specialization seems to me the only way out.

Teachers for teachers The greatest problem has not yet been

discussed. It is the question of teachers for the teachers. Let me indicate the size of the problem by some rough estimates. First, there are only about 8,000 full-time teachers of mathematics in four-year colleges and universities in the country, and perhaps another 2,000 in junior colleges. There are about 1,100,000 elementary




school teachers. If each year a 10 percent replacement were required, then we would have to graduate each year 110,000 teachers. If each teacher took two years of mathematics, that would require 220,000 year-course elections. On a basis of 100 students a teacher, 2,200 full-time teachers would be needed. I am sure this estimate is low, but it is bad enough. It is certainly not true that one-fifth of the effort of college mathematics departments is going into teaching courses designed expressly for elementary teachers. That is, the 2,200 teachers required are not in the colleges, but must be added in large part.

Let me discuss frankly one barrier. If I were chairman of mathematics at X State University, formerly X State Teachers College, I could see where I could get the necessary staff for Level' I training; that is, I could raid the neighboring high schools for good teachers with recent summer institute experience. But it is not at all certain that I would or could do this. In the first place, my administration, spurred by accrediting agencies, would be after me to raise the percentage of Ph.D. 's in the department. In the second place, the very change of name of the school implies that more attention must be paid to developing a pregraduate program for majors and a master's degree program. It is really necessary to have more staff with advanced training, and hopefully Ph.D.'s. My major energies would be-" directed at getting such a staff, and I would not be anxious to bring in less well-trained staff if it endangered my getting Ph.D. 's - which it would. In the third place, I would be swamped. The number of majors alone will probably have quadrupled in the past five or six years, and I would be busy try- ing to get sections of college algebra, analytic geometry, and calculus taught. To take on at the same time the getting of new staff just for elementary teachers - there are already enough headaches.

This is not all there is to it, of course. The institute-trained high school teacher is trained for a specific purpose and does

not have special competency in the Level I area. What is needed is personnel trained for this particular task.

I have been estimating a need to increase the college mathematics staffs by 20 percent for proper elementary teacher- training. This large percentage change is, however, rather small in absolute size. What would be the cost of providing 2,000 persons with a year of special training? To begin with, let us support each of these with $10,000, a total of $20,000,000. The 2,000 would make eighty classes of twenty- five, and suppose each of the eighty classes were to be taught by three graduate teachers, each teaching one course to the group. The school will make a profit if it is given $20,000 for each of the teachers, or $60,000 a group, a total of $4,800,000 for teaching. But let us make that $10,000,000 as long as we are being generous. This makes a total cost of $30,000,000. This is the cost of one mile of the extension of the Massachusetts Turnpike into downtown Boston. It is a great deal of money by some standards, but a trivial amount by others.

I do not present this as a plan, but as an indication that some of our problems that seem at first utterly impossible to solve really may not be if one thinks in the proper financial terms.

Summary Despite the tremendous changes in

mathematics training for elementary teachers of the past few years, we are by no means at an end. I have discussed three areas only, the impact of developing curricula, the difficulty of getting more than twelve hours of training, and the difficulty of staff. Since I limited myself to pre-service training, I could avoid the problem of in-service training for the present teachers. Very bold measures may be needed to solve that problem, and it would require another article to cover it.

I have not mentioned what may be one of the most hopeful signs because I do not have quantitative data, and that is

May 1966 383



the improvement of mathematical competency in the freshmen due to the success of the new school curricula. It must be clear that I do not regard this as providing a way of cutting down on the twelve- hour requirement. Rather, it may permit getting in enough more mathematics to give us a chance to meet successfully the new demands.

Perhaps what I want to say most in this article is that the problems of elementary school mathematics are very difficult and very challenging. The entire elementary program has been neglected as compared to the secondary program. I believe this neglect is ending, but there is plenty of work to be done by our best minds, and the problems of teacher education are by no means the easiest ones in the field.

References 1 Goals for School Mathematics y The Report

of the Cambridge Conference on School Mathematics (Boston: Houghton Mif- flin Co., 1963).

2 Guidelines for Science and Mathematics in the Preparation Program of Elemen- tary School Teachers (NASDTEC- AAAS, 1963).

3 Recommendations for the Training of Teachers of Mathematics (CUPM, 1961). (Available from CUPM, P.O. Box 1028, Berkeley, California. See also The Arithmetic Teacher, VII [December, 1960], 421-425.)

4 Ten Conferences on the Training of Teachers of Elementary School Mathe- matics, CUPM Report No. 11 (CUPM, 1965). (Also available from CUPM.)

Experienced teachers needed to serve as Peace Corps volunteers in Ethiopia

Two hundred volunteer teachers of English, history, geography, math, science, industrial arts, and commercial subjects have been re- quested by the Government of Ethiopia to teach in established junior and secondary schools throughout the country. In addition to classroom teaching, they will be involved in extracurricular activities, counseling, curriculum development, and community projects.

An eight- to ten-week training program will begin in June, 1966, at the University of Cali- fornia at Los Angeles. Upon successful comple- tion of the training period, volunteers may depart for overseas immediately, or if they have already signed a teaching contract, they may fulfill this obligation and depart for Ethiopia during the summer of 1967.

Ethiopia's educational system is faced with a dual problem - an extreme shortage of teachers and a pressing need for professional, experienced educators who can upgrade the total system. The purpose of this program will be to assist the country in solving both of these problems simultaneously.

For further information on the experienced teachers program for Ethiopia, write Education Desk, Peace Corps, Washington, D.C. 20525.




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Problems in the training of elementary school teachers