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A DISCUSSION OF THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS' GUIDELINES FOR THE POST-BACCALAUREATE EDUCATION OF TEACHERS OF MATHEMATICS Donald J. Dessart The University of Tennessee, Knoxville In the spring of 1986, John Dossey, the president of the National Council of Teachers of Mathematics (NCTM), appointed a task force to develop guidelines for the post-baccalaureate education of teachers of mathematics (the PBETM guidelines). The task force was charged to develop a document that identifies specifically the desirable compe- tencies to be achieved by the post-baccalaureate study of mathematics and pedagogy and to circulate the draft document to a broad sample of the profession for review and reactions. Crisis in Mathematics Education The results of national and international studies document that the United States faces a crisis in mathematics education. The mathematical and scientific communities are arming to face this crisis. Not only have groups within these communities been concerned with the content of mathematics education in the schools, colleges, and universities, but also they are pursuing actively reforms in teacher education and in the context in which teaching takes place. Such groups include the Mathematical Sciences Education Board of the National Research Council; the Committee on Mathematical Education of Teachers of the Mathematical Association of America; the Commission on Standards for School Mathematics of NCTM; and Project 2061 of the American Association for the Advancement of Science. Against this backdrop of keen interest in reform in mathematics education came two very significant reports: Tomorrow's Teachers (commonly referred to as the Holmes Group Report) and A Nation Prepared: Teachers for the 21st Century (the Carnegie Report). At its initial meeting in St. Louis in December 1986, the PBETM Task Force reviewed the work of these groups and reflected on its implications for the work of the task force. One possible conclusion could have been, "Let's wait for the dust to settle before writing the guidelines!" The task force chose not to wait, but to forge ahead, taking into account the wealth of recommendations available. -81-
-82- Assumptions In developing the PBETM guidelines, certain assumptions were made explicitly by the PBETM Task Force and other assumptions entered the guidelines implicitly. Some of these assumptions are given below: (1) NCTM's 1981 Guidelines for the Preparation of Teachers of Mathematics represent the competencies necessary for initial licensure to teach. The PBETM guidelines were developed assuming that a post-baccalaureate teacher had satisfied the competencies of the 1981 Guidelines. (2) Mathematics specialists are as desirable for the elementary grades as for the secondary schools. Such specialization probably will require some reorganization of the elementary schools, but should not require additional funding to implement when compared to a school staffed only by generalists. (3) Grades K-8 and 7-12 are the most prevalent programs at teacher education institutions and precise grade specializations occur because of experience and practice; therefore, the guidelines reflect these programs. (4) Differentiated staffing is the wave of the future, demanded because of academic and economic reasons. (5) Problem-solving is becoming more and more an attitude or a frame of mind that a mathematics teacher brings to many varied teaching and learning situations. (6) The potential uses of technology to improve mathematics instruction are in their infancy at this time. (7) The competencies recommended in the PBETM guidelines can be gained in a variety of ways, including in-service education, professional meetings, self study, and workshops, as well as formal college courses. Formal college courses provide for in-depth study that is often not possible in other ways. Categories of Mathematics Teachers PBETM guidelines were developed for each of the following categories of teachers. It is felt that these categories are consistent with the trend toward differentiated staffing in the schools as well as being realistic enough to provide guidance for current staffing patterns.
-83- (1) Mathematics Specialists in Grades K-8 (2) Mathematics Specialists in Grades 7-12 (3) Generalists with Mathematics Education Concentration in Grades K-8 (4) Generalists with Mathematics Education Concentration in Grades 7-12 (5) Generalists in Grades K-8 (6) Mathematics Supervisors in Grades K-12 (7) Mathematics Supervisors in Grades K-8 Each of these categories is defined more fully in the PBETM guidelines. The PBETM Guidelines The guidelines consist of two parts: (a) a listing of the competencies to be attained, and (b) descriptions of formal college courses to help attain the competencies. Guidelines are developed for each category of teachers given above. The competencies fall into five broad classes: (a) mathematics, (b) problem-solving, (c) methods and materials, (d) technology, and (e) mathematics education. To illustrate these competencies, each of the five classes will be described briefly for the "Mathematics Specialist in Grades K-8." A complete listing of the competencies for all categories of teachers is in the guidelines. The mathematics competencies consist of understanding and being able to explain simple number theoretic concepts, elements of probability and statistics, geometrical notions, algebraic properties of the real numbers, topics from discrete mathematics, the logic and reasoning underlying the mathematics of these grades, and events from the history of mathematics. Problem-solving encompasses skill in reading and writing mathematics, problem-posing as well as problem-solving, and the ability to explain mathematical applications, including mathematical modeling. Problem-solving is pervasive and not limited merely to solving certain types of problems.
-84- Competencies related to methods and materials include knowing and being able to locate varieties of commercial and teacher-made materials, the abilities to evaluate textbooks and computer software, knowing methods and materials for teaching reluctant learners as well as gifted and talented students, and understanding and being able to apply various techniques of tests and measurements. Technological competencies include the abilities to understand and write programs in a structured language, such as Logo, Basic, or Pascal, and being able to implement appropriate uses of computers, calculators, and other technology in the classroom. The PBETM Task Force feels that the technological competencies will be those subject to the most change in future years, and that technology should be used to improve, but never replace, mathematical instruction. Mathematics education competencies dwell more on the wider professional role of the mathematics specialists. They include understanding and being able to apply principles of curriculum planning, development, and evaluation in mathematics; understanding the psychological principles underlying the learning and teaching of mathematics at these grade levels; being cognizant of current research in mathematics education; and being familiar with current issues, such as those arising from international studies, the recommendations of professional associations, and current national trends in education. The formal college courses recommended to help post-baccalaureate teachers attain these competencies are: (a) Problem-Solving in K-8 School Mathematics, (b) Technology in K-8 School Mathematics, (c) Probability and Statistics for K-8 School Teachers, (d) Topics in Discrete Mathematics for K-8 Teachers of Mathematics, (e) Geometry for K-8 Teachers of Mathematics, (f) Models of Teaching and Instructional Strategies for Grades K-8, (g) Tests and Measurements, (h) Psychology of K-8 School Mathematics, (i) Current Curricular, Teaching, and Research Issues in Grades K-8, and (j) Diagnosis and Remediation of Learning Difficulties. Responsibilities of Professionals The PBETM guidelines also include a discussion of suggestions for college and university faculties, local and district administrators, members of professional associations, mathematics teachers, and
-85- parents to implement the guidelines. The guidelines are meaningless unless they can be translated into practice by those most affected by and concerned with quality mathematics education. Summary The PBETM Task Force believes very firmly that good mathematics teachers at all levels need sound backgrounds in both mathematics and pedagogy. High competence in these two areas is necessary for excellence in teaching. Furthermore, teachers need the active support of school administrators, fellow teachers, parents, and students to attain the competencies recommended. This support should include encouragement, released time, and financial assistance. The PBETM guidelines represent a model that is recommended to achieve excellence. The task force believes strongly that it is not the only model, and that the PBETM guidelines should never be used to inhibit or impede creative thought in the search for excellence in mathematics teacher education. [Copies of NCTM's PBETM guidelines can be obtained from: National Council of Teachers of Mathematics 1906 Association Drive Reston, VA 22091 (703) 620-9840]
A DISCUSSION OF THE MATHEMATICAL ASSOCIATION OF AMERICA'S GUIDELINES FOR THE POST-BACCALAUREATE EDUCATION OF TEACHERS OF MATHEMATICS Calvin T. Long Washington State University There has been close liaison between the National Council of Teachers of Mathematics (NCTM) and the Mathematical Association of America (MAA) in producing the various guidelines of the last several years, so it is a pleasure for me to have the opportunity to make this presentation about our work. Our hope is that what we have produced in the guidelines is worthwhile and useful and will not be allowed to fade into oblivion. Our efforts go back a long way to the old Committee on the Undergraduate Program in Mathematics (CUPM) and to various subpanels of that group, mainly the panel on teacher training, which produced its first report in 1961. Since that time, there has been a steady outpouring of reports from that committee. Finally, within the last couple of years, the Committee on the Mathematical Education of Teachers (COMET), which was a subpanel of CUPM, was deemed to be of sufficient importance that it was constituted as a separate commit- tee, independent from CUPM, with the responsibility of recommending MAA stances with regard to teacher training and teacher education. COMET represents a group of people concerned about the present state and the possible future of mathematics education in America who have a strong desire to improve the system and make it one of the very best. This attitude is a reflection of our parent organization, the MAA, and it has been delegated to us now as a full-time standing committee of the MAA. Our charge is that of formulating policy and guiding MAA in activities in this important area of teacher prepa- ration. While I am reporting here on COMET's new Guidelines for the Continuing Mathematical Education of Teachers, such a report would be incomplete if I did not mention first the most recent report of the MAA panel on teacher training, entitled Recommendations on the Mathematical Preparation of Teachers. I also should mention a parallel document prepared by NCTM, entitled Guidelines for the Preparation of Teachers of Mathematics. Both of these documents deal with initial licensure to teach and describe the necessary ingredients for programs to train or prepare new teachers. They are rather prescriptive, and spell out with some care what we feel ought to be included in those training programs. -87-
-88- With regard to the MAA Recommendations and the NCTM Guide- lines for undergraduate preparation, it should be noted that they are examples of significant cooperation between these two organiza- tions. Both organizations were consulted in the preparation of both documents. I also serve on the NCTM task force to develop guidelines for the post-baccalaureate education of teachers of mathematics (the PBETM guidelines), and there is NCTM representation on the COMET committee. Another example of the meaningfulness of this cooperation is the fact that the NCTM guidelines were endorsed formally by the board of governors of the MAA and this endorsement, along with that of other groups, helped to provide the momentum that caused the NCTM guidelines to be accepted as accreditation standards by the National Council of Accreditation for Teacher Education-- a most important step. There is no doubt that this cooperation, and that of the entire mathematics education community, will continue to be significant and important in the coming days of structural and curricular flux in American education that we all anticipate. That this is recognized by the leadership of both NCTM and MAA is evidenced by the fact that we have this significant involvement of members of both organizations on the two committees we have been talking about. Now let me turn to the COMET Guidelines. In the first place, they are addressed to school administrators to assist in planning in-service programs for teachers, programs which we feel are ex- tremely important for maintaining the vitality of teachers. They also are addressed to college administrators and departments of mathematics and mathematics education to assist in the design of teacher-training programs. They are addressed as well to teachers and district supervisors, to national and state governmental agen- cies, to professional societies, to regional and local educational organizations, and to everyone concerned with improving the effec- tiveness of mathematics education in America today. The Guidelines contain suggestions for in-service programs for all teachers, for master's degree programs for elementary school teachers, elementary school mathematics specialists, coordinators of elementary school mathematics programs, teachers of middle school and junior high school mathematics, teachers of high school mathematics, and for the graduate education of mathematics supervisors. While changes in teacher training are taking place in response to such studies as the Holmes and Carnegie reports, it must be noted that the present guidelines or recommendations are designed with current in-service teachers in mind and are intended to provide these teachers with a fresh look at both mathematics and pedagogy. The changes resulting from the Holmes and Carnegie reports may well necessitate the reworking of both the existing undergraduate Recom- mendations and the new Guidelines. At the same time, the changes needed may be relatively minimal and the existing guidelines
-89- should help in the structuring of programs consistent with the Holmes and Carnegie recommendations. For example, in connection with the valid concern that we not lose hard-won ground, we feel that any undergraduate major program devised for elementary school teachers in accordance with the Holmes recommendations should include the three or four undergraduate mathematics courses spelled out in the MAA Recommendations and the NCTM Guidelines. While endorsing the Holmes recommendation that the elementary mathematics major be abolished, we agree wholeheartedly that substitution of precalculus, calculus, or liberal arts mathematics courses for the courses in our existing recommendations would be totally unacceptable. Since our report runs to over 90 pages, let me highlight several of its points and leave you to read the rest of the document in de- tail. The first thing I would like to observe is that, as of this writing, it is a draft. We expect to have the guidelines completed in early 1988. However, if you have strong feelings about any sec- tion of the Guidelines, suggested references, or other comments, I would be pleased if you would write them out carefully and send them to me at the Department of Mathematics at Washington State University. Secondly, these guidelines are much less prescriptive and more suggestive than the 1983 recommendations of the MAA. Types of programs, ways of meeting the needs of teachers, and general mathe- matical content are proposed. These are intended to suggest prom- ising approaches, but they may need to be modified to meet local situations. Thus, these guidelines indicate directions for continued program development rather than giving fine detail for present or future programs. Thirdly, one of the strong points of the guidelines is the very extensive bibliography. However, you will note that we have few or no references for some courses. Obviously, some of the courses suggested depend on current literature. Others still need refer- ences; if you know of any, please send me detailed, complete citations. Fourthly, one recommendation in the guidelines is that states should adopt licensing standards for mathematics supervisors. At present, few states have such criteria. We feel that they are definitely needed since few states have such criteria and because the choice of supervisors in many districts seems to be made on grounds other than demonstrated skill in mathematics education. This is all I really want to say about the guidelines. They are much too long to discuss in detail here. However, there are several other things I would like to mention. From time to time, COMET has been asked by the MAA to prepare position papers on various issues, such as the Holmes and Carnegie reports, for example. These position papers have been submitted to the MAA board of governors and now represent the official position of the MAA. I would like to mention three such papers.
-90- The first, the MAA Statement on Retraining of Mathematics Teachers, concerns the training of teachers from other disciplines to teach mathematics. By the way, this issue has been dealt with also by NCTM. I quote this not primarily because of that particular issue, but to indicate further the philosophy of the group that prepared the guidelines: Students at all grade levels deserve to be taught by fully qual- ified teachers. Yet many students are currently being taught mathematics by teachers with insufficient preparation. To teach any subject successfully, teachers must not only know the subject well, but also like it to the point of enthusiasm. Because of the importance of mathematics to the futures of all students, the problem of inadequately prepared teachers is of great national importance. For this reason, the Mathematical Association of America has adopted the following principles regarding the prepa- ration and retraining of mathematics teachers: (1) Students at all levels should be taught mathematics by teachers who enjoy mathematics and whose training meets or exceeds professional standards. These standards are presented in two documents: Recommendations on the Mathe- matical Preparation of Teachers (1983), available from the Mathematical Association of America, 1529 Eighteenth Street, N.W., Washington, DC 20036; and Guidelines for the Prepa- ration of Teachers of Mathematics (1981), available from the National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091. (2) Teachers who lack adequate preparation should be provided with sufficient in-service training to provide them with both competence and confidence. The mathematical maturity needed by teachers with little or no mathematical background can normally be achieved only after in-service programs involving several years of carefully designed sequential courses. Short courses and workshops designed to meet emergency shortages are at best temporary expedients; they can provide teachers with stimulation, perspective, and incentive for further training, but they alone do not pro- vide adequate preparation for the teaching of mathematics. (3) Institutions of higher education should work coopera- tively with school districts and government agencies to address the mathematics teacher shortage. High-quality training and retraining programs consistent with the Recommendations and Guidelines cited above, must be established in all regions of the country. Funding for these programs must include appropriate financial incentives for teachers and teacher candidates.
-91- (4) School districts, in concert vith other agencies, should provide teachers at all grade levels vith regular and substantial opportunities for continued growth. As profes- sionals, teachers should work throughout their careers to keep abreast of the latest curricular, pedagogical, and technological advances. A second position paper was addressed to college and university administrations and to departments of mathematics and mathematicians generally. The statement is entitled College and University Respon- sibilities for Mathematics Teacher Education, and I quote: College faculty must become actively involved in the education of teachers if the teaching of mathematics in the schools is to improve significantly. Active leadership and support of college and university mathematicians, mathematics educators, and adminis- trators is essential if our nation is to increase the number of qualified teachers and to strengthen their education. For this reason, the Mathematical Association of America and the National Council of Teachers of Mathematics have adopted the following recommendations for all individuals, in whatever department, who are engaged in teaching mathematics or mathematics education for current or prospective teachers: (1) Colleges and universities should assign significantly higher priority to mathematics teacher education. (2) All individuals who teach pre-service or in-service courses for mathematics teachers should have substantial backgrounds in mathematics and mathematics education appro- priate to their assignments. (3) Mathematics methods courses should be taught by indivi- duals with interest and expertise in school teaching and continuing contacts with school classrooms. (4) All individuals who teach current or prospective mathe- matics teachers should have regular and lively contact with faculty in both mathematics and education departments, e.g., by regular meetings, seminars, joint faculty appointments, and other cooperative ventures. (5) All college and university faculty members who teach mathematics or mathematics education should maintain a vigorous dialogue with their colleagues in schools, seeking ways to collaborate in improving school mathematics programs and in supporting the professional development of mathemat- ics teachers. (6) Faculty advisors should encourage their mathematically talented students to consider teaching careers.
-92- (7) Colleges and universities should vigorously publicize the need for qualified mathematics teachers and strive to interest and recruit capable students into the profession, e.g., by organizing highly visible campus-wide meetings for students to inform them of the opportunities, advantages, disadvantages, and requirements of a career in teaching mathematics. (8) Tenure, promotion, and salary decisions for faculty members who teach current or prospective mathematics teach- ers should be based on teaching, service, and scholarly activity that includes research in mathematics or mathe- matics education. (9) Faculty members in mathematics and in mathematics education who are effective in working with activities in the schools and in the mathematical education of teachers should be rewarded appropriately for this work. (10) All institutions involved in educating mathematics teachers should provide specialized classroom and laboratory facilities equipped with state-of-the-art demonstration materials, calculators, and computers at least comparable to those used in the best elementary and secondary schools so that prospective teachers, like graduates from other profes- sional programs, can be properly prepared for their careers. Finally, I would like to comment on an item in a position paper that we were asked to draw up about the Holmes and Carnegie re- ports --the MAA Statement on the Holmes and Carnegie Reports on Teacher Preparation. Recommendation 7 in that paper reads: Prospective teachers in grades K-8 should major in an academic discipline. Those aspiring to teach at the elementary level should include in their programs of study the mathematics courses recommended in the MAA Recommendations on the Mathematical Preparation of Teachers, and prospective teachers of middle school and junior high school mathematics those specified for that level in the same Recommendations. Beyond this, under- graduate majors in mathematics or combined majors in mathematics and the natural sciences should be developed, especially for prospective elementary school teachers, so that eventually all mathematics in at least grade 3 and beyond is taught only by mathematics specialists. Regarding this point, I want to make the following comment concerning a situation that prevailed about 30 years ago in Mont- gomery County, Maryland, and which I understand prevails to some extent today. A teaching technique used in some of the schools there at that time was to take approximately 80 third- or fourth-grade students and assign them to 3 teachers, in a team approach. One teacher was a specialist in mathematics and science, one in language
-93- arts, and one in social studies. Using ability grouping and teaching only their own specialties, these three teachers were able to meet the students very particularly at their points of greatest need, and they were able to meet them very knowledgeably. To start very young children with a teacher who hates or fears mathematics or arithmetic almost guarantees that we will continue to have students who have picked up the teacher's attitudes and who often become lifelong haters of mathematics. If specialization is anathema to child devel- opment experts, then I suggest that the team-teaching approach is a way around this serious difficulty. [Copies of MAA's guidelines can be obtained from: Mathematical Association of America 1529 18th Street, N.W. Washington, D.C. 20036 (202) 387-5200]