Natural Sciences and Mathematics || Academic and Professional Preparation of Secondary School Mathematics Teachers

Academic and Professional Preparation of Secondary School Mathematics TeachersAuthor(s): Phillip S. Jones and Arthur F. Coxford, Jr.Source: Review of Educational Research, Vol. 34, No. 3, Natural Sciences and Mathematics(Jun., 1964), pp. 322-333Published by: American Educational Research AssociationStable URL: http://www.jstor.org/stable/1169408 .

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CHAPTER VI

Academic and Professional Preparation of Secondary School Mathematics Teachers

PHILLIP S. JONES and ARTHUR F. COXFORD, JR.

THE PAST three years, a period of turmoil and experimentation in the content and teaching of secondary mathematics, were also characterized by recommendations for the improvement of teacher training. Many of these recommendations were based upon extended discussions involving persons who were well informed about problems of teaching mathematics, about the training of mathematics teachers, and about characteristics of secondary school mathematics programs. Although many of the studies and reports to be cited in this chapter cannot be described as research, they are so important in characterizing current trends in the training of mathematics teachers and in suggesting needed research that no further apology for their inclusion will be made.

Mathematics Teacher Training Recommendations

The Mathematical Association of America, Committee on the Under- graduate Program in Mathematics (CUPM) (1960), published a report containing recommendations of its Teacher Training Panel. This report apparently stimulated great interest. An account by Hendrix (1962) of follow-up activity in Illinois indicated that there had been several changes both in college course offerings and in state and institutional certification programs. She did express some doubt about the complete feasibility of the recommended two-year college mathematics program for those elementary school teachers who are classified as Level I. However, substantial in- creases in the amount of mathematical background needed by elementary school teachers seems to be accepted as necessary and forthcoming in view of both the renewed stress upon teaching for understanding and the mathematical nature of the content of new programs for the elementary schools.

In the previously mentioned CUPM report, stress was placed upon the need for mathematics courses specifically designed for teachers-especially those teachers at Level I. However, Smith (1963) noted that, of the teacher training institutions that he surveyed, 56 percent had no courses especially designed to familiarize prospective teachers with the mathematical content of new curricular materials and 77 percent had no special courses or sections for teachers returning to do graduate work in mathematics. These findings are not surprising in view of Schumaker's (1961) survey of trends in teacher education in mathematics during 1920-58. The data in the latter

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survey clearly indicated that changes in teacher education came slowly and, unfortunately, tended to follow rather than to precede changes in the secondary school curriculum. This latter conclusion suggests, as do other factors such as the continuing program of institutes sponsored by the Na- tional Science Foundation (NSF), that rather substantial and continuing changes in both preservice and in-service teacher training programs are, and will be, taking place. The recommendations in the CUPM report, which dealt largely with the mathematical training needed by today's teachers, were also concerned with the need for teachers to be familiar with curricular and pedagogical ideas.

The National Association of State Directors of Teacher Education and Certification (NASDTEC) (1961) carried out a study concerned with teacher preparation and certification. The publication furnished a philo- sophical framework together with concrete recommendations for a bal- anced teacher training program. In such a program, recognition is given to the prospective teacher's need for (a) general education, (b) familiarity with curricular and pedagogical problems, and (c) extensive as well as intensive knowledge of subject matter. This knowledge should extend not only to the level that his students will achieve in subsequent courses but also to the point at which he himself can continue in graduate study in his major field and simultaneously have some knowledge of related fields.

These reports of CUPM and NASDTEC devoted no space to the person who teaches mathematics as a second subject. The problems that are related to the administrative necessity for having such persons were documented by Conant (1963) and by Obourn and Brown (1963). Obourn and Brown's data, gathered in 1961, show that one-seventh of the nation's 118,000 instructors of high school mathematics taught only one mathematics class. These teachers as well as many others with split schedules were, in general, inadequately prepared. Using data from a NASDTEC study reported by Viall (1962), Conant (1963) observed that mathematics is in a poorer position than is any other science, except physics, with re- spect to the percentage of classes taught by inadequately prepared teachers, especially in grades 7 and 8.

Obourn and Brown's data showed that this condition is not limited to small schools, since 20.2 percent of the one-class teachers in their sample were in large high schools. Their data also bore out Conant's statement that there are significant regional differences in teacher qualifications. Conant proposed that mathematics teachers with less than 9 hours of college mathematics should be assigned to other classes and suggested that this course of action would be facilitated by consolidation of small schools. He suggested that teachers with between 9 and 17 hours of college mathematics, with teaching experience, and with scientific background have potential that should be developed through new teacher training programs which allow for the fact that these teachers are not initially qualified for regular subject matter graduate work.

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REVIEW OF EDUCATIONAL RESEARCH

Although Conant conceded a need for physics and chemistry teachers to be prepared in more than one field, he felt that a comparable type of preparation was neither feasible nor necessary for mathematics teachers. Prior to the appearance of Conant's book, the American Association for the Advancement of Science, Cooperative Committee on the Teaching of Science and Mathematics (1960), conceded a need for some persons to teach mathematics as a second subject and suggested a minimum program for the preparation of such persons. In other respects, this report of the Cooperative Committee has much in common with the CUPM and NASDTEC reports.

Comparative Mathematics Education

The increased general interest in all aspects of internationalism, the exchanges of teachers and lecturers, and popular criticisms of American education in comparison with European programs have served to stimulate a number of studies of foreign teacher training programs. In all these studies, such as the ones by Vogeli and Lindquist (1962) concerning Rus- sia, by Polya (1960) regarding Switzerland, and by Bunt (1963) about the Netherlands, it appeared that the training of teachers in the upper levels and in the European schools of higher quality has been often substantially better than or at least equal to that recommended for CUPM Levels III and IV-levels that the best-trained teachers in American high schools equal or, in many instances, exceed. However, Bunt was concerned that the Netherlands may be losing needed teachers because of too abstract an approach to mathematics in the teacher training program. Dodd (1963) recorded an interesting development in which the English Mathematical Association has been awarding a "diploma" to those teachers passing examinations on mathematics which they have studied; he also pointed out that many of the teachers in the United Kingdom's "modern schools," to which 75 percent of the pupils in "maintained schools" (publicly supported schools) are allocated, are either "nongraduates" or "nonspecialists." Ke- meny (1963) emphasized that teacher training is a general problem among many if not all the nations that submitted reports to the International Committee on Mathematical Instruction.

Certification and Preservice Programs

Recommendations for changes in the content training of mathematics teachers evolved from the efforts of NASDTEC members to furnish guid- ance to state certification directors on the basis of data from national surveys and in terms of certain perceived needs. NASDTEC also motivated the initiation of several local and regional surveys concerning not only practices in certification but also teacher training programs in college

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mathematics. Two illustrative studies are the study conducted by Ford (1962) for the Southern Association and another carried out by Valsame (1961) for North Carolina. As in the case of the previously cited paper by Smith (1963), the results of these two studies indicated that new teachers entering the profession (a) had neither much familiarity nor much experience with new secondary school materials and (b) had neither the amount nor the kind of training in subject matter that had been recommended either by any of the previously cited national reports or by the various projects concerned with secondary school texts and curricular development.

However, the beginnings of improvement were evident in the actions of several certification groups and local professional organizations. These varied changes lead one to expect continuing diversity both in college programs and in certification requirements. One also may anticipate an increased stress both on subject matter and on experience with new materials in elementary and secondary curriculums. The New York State Board of Regents was reported as doubling minimum subject matter requirements for teachers entering high schools in 1963 (American Mathematical Monthly, 1961). In addition, Hendrix (1962) and Zant (1961) noted efforts of regional, professional, and semiofficial committees to influence statewide programs and requirements.

Edson and Buchta (1961) for the University of Minnesota and Bryant, Kelley, and Flanders (1963) for the University of California described interesting new cooperative undergraduate programs in mathematics and education. The former program, which is in addition to the previously existing program in education, leads to degrees in both education and liberal arts. A graduate has both a teacher's certificate and the prerequisites for graduate study in mathematics. The University of California program involves an interesting "interne" experience, special sections of mathematics classes, a particular degree, and specifically chosen visiting profes- sors. These programs suggest and embody some experimentation that is related to the long-discussed five-year programs.

The needs for such programs stem (a) from the gap between the level of knowledge of entering freshmen and that needed by today's beginning teachers-documented by Sandefur (1962)-and (b) from academic programs that by choice concentrate initially on subject matter and postpone professional training largely to the fifth year or to the fifth and sixth years. Woods (1961) studied the content needed in a fifth year for Missouri teachers. The American Mathematical Monthly (1963) published a description of the new two-year master's degree program for mathematics teachers at the University of Chicago. Extended discussions of variations in five- year programs can be located in the additional references enumerated in the next two lines.

Additional References: Cartwright (1961); Harap (1961); Jarvi (1961); Kazamias (1961).

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In-Service Training

In-service education is a continuing need for all mathematics teachers who are faced with increasing pressure to use new methods and materials. This need is particularly striking for those teachers whose preservice training was inadequate for their jobs. Retraining is vital for many teachers, especially since the majority of teachers in mathematics earned degrees before 1957.

The needs for retraining are being met in various ways. In a question- naire survey of 417 individuals in colleges, in state departments of education, and in school systems, which was described by Syer (1961) in a publication prepared by Brown and Snader (1961), the retraining procedures most commonly provided for mathematics teachers were found to be (a) college-sponsored programs-on-campus and extension courses, institutes sponsored by the colleges or NSF, and college consultative services-and (b) programs reported by state departments and school systems-reading by individual teachers and formal and informal discussion groups, with or without outside aid; conferences and lectures; curriculum planning; and cooperative enterprises with colleges. Examples of in-service programs initi- ated by large cities were cited, and the importance of further advances in the use of television, films, and traveling consultants for the smaller school systems was noted.

Since its activity began in 1953 with two summer institutes, NSF has be- come increasingly involved in the training of teachers in mathematics and science. In 1960, NSF sponsored 212 summer institutes, 32 academic-year institutes, and 136 in-service institutes in which mathematics teachers were included. The stated objective of the NSF institutes was to improve the mathematics education of students by upgrading the mathematical competence of their teachers. The magnitude and the importance of this opera- tion have served to stimulate the initiation of several research studies that were planned to evaluate the extent to which the NSF institutes accom- plished their objectives.

These studies fell into three basic categories. The first category of evaluative investigations included those presenting analyses of the opinions of participants regarding the institute and its personal effects. Heideman (1962), Ostlund (1962), and Wiersma (1962) found that the participants reacted favorably to the institutes, to the curriculums and staffs, and to both the physical and the social atmosphere. They indicated that "basic" courses constituted the most valuable aspect of their experience (Heide- man, 1962 ). Wiersma and Heideman reported that the members augmented their mathematical knowledge, increased their confidence in presenting mathematics and in attempting new methods, extended their professional awareness, and developed their occupational potential. Ostlund indicated that gains in salaries were obtained by 42 of the 71 respondents. 326

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The second category of evaluative studies was concerned with analyses of the effect of NSF institutes on courses taught by participants. Ostlund (1962) reported 111 changes in classroom procedures by 71 participants. These changes included (a) increase in the variety of topics taught, (b) encouragement of students to display initiative, and (c) additional activi- ties. Connellan (1962), Whitaker (1961), and Wiersma (1962) corrob- orated these types of changes. Connellan observed that, subsequent to their institute experience, teachers relied less on textbooks than formerly and consulted college texts and experimental programs for content; on the other hand, nonparticipants who tended to be fearful of modern mathematics cus- tomarily employed textbooks whose emphases were judged to be conven- tional or traditional. However, Whitaker noted that summer-institute participants who had had a superior mathematics background tended not to use experimental materials in class, whereas more than one-half of the less adequately prepared and less experienced teachers did employ experimental materials.

The third category of evaluative studies analyzed the previous training of NSF institute participants. Whitaker (1961) ascertained that members of summer institutes were a select group, since (a) more than 50 percent held master's degrees and (b) more than 40 percent had 30 or more hours of college mathematics. Moreover, the participants were sensitive to current changes; approximately 75 percent had introduced or expanded their treatments of such topics as sets and numeration systems. Whitaker also called attention to the need for more institutes for the less well-prepared teachers. In the discussion of this problem at a conference sponsored by the Mathematical Association of America, Committee on Institutes (1961), it was recommended that a system of progressive institutes be be- gun. The conference also produced rather detailed recommendations for qualifications of the staff and for conduct of all institutes.

What may be said of the future? Research showed that, although NSF institutes were effective, the individuals reached were the better-prepared teachers. As noted earlier, Conant urged the continuation of institutes, but stressed the need for special programs for poorly prepared teachers. In such planned offerings, the teacher can be led from his present position to one commensurate with the demands of the materials that he will most likely teach. Increased local and regional efforts are needed and must be encouraged.

In addition to a description of these needs, a variety of suggestions for fulfilling them was developed in an extensive report of a conference on in- service education of high school mathematics teachers (Brown and Snader, 1961). Pertinent to fulfilling these same needs was a publication by Snader (1962) that was concerned with the leadership of state supervisors of mathematics.

Several projects for in-service training-especially of elementary teachers-in which films and television were used will be noted in the next section. DeVault, Houston, and Boyd (1963) reported a study dealing with

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consultant services. Instruction by television was also an important feature of this investigation. Although visits, group and individual conferences, and demonstrations described by these investigators were all with elementary teachers, the authors' conclusions seem likely to apply to other teachers. Whereas in the over-all study there was no significant difference between the final achievement of groups served by consultants and of those not served, there were some differences within the former group. The amount of consultative help requested diminished with the years of experience of the teacher. Apparently, small group discussions were more use- ful than were individual conferences. Pupil achievement and teacher reac- tion were positively related to the number of classroom demonstrations. The few significant correlations were not high. Although the design of the investigation and the collection of data were difficult and time-consuming tasks, the gap between the need for consultative services and the amount of support given to them is so great that considerably more care and attention to such studies should be given than has been given in the past.

Additional References: Atkins (1960); Baird (1961); Henkelman (1963) ; Parker (1960) ; Pownall (1963).

Media and Methods

There is little question that the growing student population, the even more rapidly growing enrollment in mathematics, the relative decline in personnel highly qualified to participate in the preparation of teachers, and the continuing development of both new mathematics and new curricular materials have created an expanding problem in both preservice and in-service teacher training. Partial solutions are being sought in the use of programed materials, television, and films. The "Continental Classroom" television courses in both modern algebra and probability and statistics were designed with teacher training as one of their objectives. In a report of his experiences as a teacher of the latter course, Mosteller (1963) noted that at least nine school boards gave in-service credit to teachers who followed the televised course. Woodby (1962) described an extensive and continuing project that was planned to train 1,100 elementary school teachers in 90 schools in the structure of the number system through the medium of television. Winsor (1961) demonstrated that off-campus instruction in the teaching of arithmetic through use of television along with some associated seminars was as effective as instruction solely by seminar. This finding, which was consistent with outcomes of previous research investigations concerned with the effectiveness of television, would suggest the efficacy of the current preparation of extensive filmed materials for instructional purposes by at least three groups-the School Mathe- matics Study Group (SMSG), CUPM, and the National Council of Teach- ers of Mathematics. It also would be interesting to have comparative in-

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vestigations of televised and correspondence study, such as those that West (1961) carried on for an in-service course in audiovisual aids.

As yet there have been no reports concerning the effectiveness in teacher training of the extensive films of secondary school teaching of modern mathematics prepared by Rosenbloom (1962). Likewise, at the elementary level, one eagerly awaits the final report concerning the effectiveness of films and tape recordings employed in the Syracuse University and Web- ster College Madison Project, described in an abstract by Davis (1962). To a considerable extent, these two projects as well as several others that were abstracted in a publication of the National Science Foundation (1962) were planned to show (a) the psychological nature of the processes of dis- covery and creativity in learning mathematics and (b) ways in which teachers can develop teaching procedures consistent with the psychology of learning. Considerable study is needed concerning the use of films and of other devices for communicating to teachers an understanding of various teaching procedures.

One of the greatest problems in mathematics teacher education has been the lack of knowledge about the nature and effectiveness of varied classroom methods, not only in relation to the individual student's capacity and maturity but also with reference to the increasingly stressed objectives of developing mathematical maturity, understanding and insight, appreciation, and creativity. Even less is known about how to develop teachers who actually have the capacity to achieve these objectives with their students than is known about the objectives themselves. Some research and development in these areas is under way.

Mayor and others (1962) reported upon experimental investigations that aimed to improve the professional education of science and mathematics teachers. For example, in a study at the University of Delaware, Brown and Henry (1962) sought to train student teachers and their cooperating teachers in the nature of problem solving and in how one makes use of this knowledge in teaching. This pilot project entailed the participation of more than 400 high school students. They expressed pleasure in studying materials that involved them in "thinking about thinking." In addition, they were trained in problem solving through methods incorporating con-

cept development. Fifteen teachers and four student teachers taught these same materials. They first had participated in a 10-week study of problem solving and in a task involving the translation of problems from English to symbols of mathematics and logic.

In a project at the University of Hawaii, Easley (1962) attempted to

develop practical operating procedures that would give first-year education students laboratory courses and actual experience in the use of varied

teaching procedures with students in grades 4-8. Green (1962) described a project at Emory University in which college

seniors were given some experience in assisting with freshman classes before they undertook student teaching. Seeley (1962) explained the purpose of a seminar at the University of Arizona, in which each preservice teacher

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prepared and taught a 50-minute lesson to his fellow students. An analysis of the tape recording of his teaching afforded not only a means for the student to see the importance of diction, careful choice of words, use of visual aids, and logical development but also a basis to evaluate his own voice and speaking mannerisms.

This marked concern for increased and improved preservice and in- service training still leaves to individual opinion the answer to the question concerning just what qualitative and quantitative differences in the learning process are associated with the personality, professional training, and scholarly characteristics of teachers. Very little is really known, despite the great amount of effort that has been expended in educational research during the past few years.

Alpert, Stellwagon, and Becker (1963) reported one particularly significant result of their study of psychological factors in mathematics education. They noted that boys responded in a positive way toward mathematics when their teachers were theoretically oriented and involved in their performance, regardless of the teacher's sex. Additional data concerning the personality characteristics of teachers and their attitudes toward students suggested that further efforts might be advantageously expended in research concerning the selection and training of teachers and that it may even be important to match teachers who display certain characteristics with particular types of programs.

In an evaluation of SMSG materials in terms of student performance, Rosenbloom (1961) found that, except in the twelfth grade, there were only a small number of statistically significant correlations between measures of student achievement and the characteristics of teachers which were evalu- ated. However, there was some indication that effective teachers tended to be flexible, permissive, student centered, and oriented toward letting the class members exhibit initiative. Factors of creativity, critical judgment, and flexibility were judged to be the most important concomitants of teacher effectiveness.

Rosenbloom warned that this concern with personality characteristics did not mean that subject matter competence was not relevant to a teacher's effectiveness. Nevertheless, mere data about credit hours and grades would not measure competence. One should further note that his study was concerned with the teaching of experimental groups through use of SMSG materials. His control classes made use of traditional materials. In this situation, a reasonable prediction would be that earlier subject matter training would be less significant than would the personality characteristics that Rosenbloom studied. Further studies to corroborate and to extend these tentative conclusions are needed-especially when so much effort is being expended on subject matter retraining. Imaginative design of and experimentation with methods for imparting to preservice and in-service teachers such characteristics as a theoretical orientation together with flexibility, critical judgment, and creativity will be eagerly awaited.

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Concluding Statement

Some beginnings are being made in the preparation of new course content materials for teacher training. Several projects in which course materials were prepared and tested were described in a publication of the National Science Foundation (1962).

It is hoped that, during the next few years, investigators will expend systematic efforts that are anchored to a theoretical structure concerning the teaching-learning process in an attempt to evaluate the effectiveness of the many modifications that are being introduced into the academic and

professional training of teachers of mathematics. Additional References: American Mathematical Monthly (1961a, b).

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VALSAME, JAMES. A Study of Selected Aspects of Mathematics Teacher Training in North Carolina as Related to Recent Trends in Mathematics Teaching. Doctor's thesis. Chapel Hill: University of North Carolina, 1961. 289 pp. Abstract: Disserta- tion Abstracts 23: 549; No. 2, 1962.

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BAIRD, GEORGE H. "The Greater Cleveland Mathematics Program." Mathematics Teacher 54: 31; January 1961.

CARTWRIGHT, WILLIAM H. "Fifth-Year Programs in Teacher Education." Journal of Higher Education 32: 297-311; June 1961.

HARAP, HENRY. "Teacher Preparation: 5-Year Programs." School Life 44: 18-21; October 1961.

HENKELMAN, JAMES. "Implementing a New Mathematics Curriculum." Mathematics Teacher 56: 211-17; April 1963.

JARVI, MARJORIE ELIZABETH. A Follow-Up Study of the Carnegie Graduate Fellowship Program at George Peabody College for Teachers. Doctor's thesis. Nashville: George Peabody College for Teachers, 1961. 289 pp. Abstract: Dissertation Abstracts 22: 2706-2707; No. 8, 1962.

KAZAMIAS, ANDREAS M. "The Education of Good Teachers and the Oberlin Master of Arts in Teaching Program." Journal of Teacher Education 12: 205-208; June 1961.

PARKER, ALWIN. A Study of Certain Aspects of Eight National Science Foundation Summer Institutes for High School Science Teachers Conducted in Louisiana. Doc- tor's thesis. Baton Rouge: Louisiana State University, 1960. 76 pp. Abstract: Dis- sertation Abstracts 21: 2537-38; No. 9, 1961.

POWNALL, MALCOLM W. "New York State Workshop for Advanced Placement Mathe- matics Teachers." American Mathematical Monthly 70: 88-89; January 1963.

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