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Colección: La Educación
Número: (119) III
Año: 1994

Comparison of Recommendations and Standards

The set of standards with strong implications for the preservice preparation of teachers of mathematics that are explicated in the Professional Standards for Teaching Mathematics (NCTM 1991) and MAA’s A Call For Change (Leitzel) parallel a number of recent, more general recommendations for teacher preparation (Carnegie Forum, Holmes Group, NBPTS, NASDTEC, NCATE). In this section, we discuss first the recommendations in these more general documents, and then we compare the NCTM and MAA recommendations with them.

A Common Ground

There is general agreement among the Carnegie Forum, the Holmes Group, NBPTS, NASDTEC, and NCATE in their assessment of the components required for the preparation of teachers. All agree that there are three such components, which the Holmes Group identifies as “... subject matter knowledge, systematic knowledge of teaching, and reflective practical experience” (Holmes Group 62).

Where the three newer sets of recommendations and the NASDTEC Standards differ from the NCATE Standards is in the degree of emphasis. While NCATE allocates a portion of a prospective teacher’s undergraduate program to “...appropriate depth and breadth in an integrated course of study that is offered by faculty in the liberal arts and other general studies” (NCATE 48), as well as to achieving “... a high level of academic competence and understanding in the areas in which they plan to work or teach” (NCATE 49), it is difficult to see how NCATE’s recommendations, which also contain professional studies and clinical field experience components, place the same degree of emphasis on a broad and deep background in the liberal arts and sciences. Both the Carnegie Forum and the Holmes Group go so far as to recommend the abolition of the undergraduate major in education. So intense is their belief that the preparation of teachers should include that broad liberal arts background coupled with a deep knowledge of the subject matter that will be taught.

NASDTEC delineates eight standards for the mathematics specialty area:

1. Demonstration of competence in several areas of mathematics;
2. Demonstration of competence in standard mathematics vocabulary, symbols, and methods of proof;
3. Demonstration of competence in the use of appropriate models and manipulatives;
4. An understanding of the intellectual, historical, and philosophical nature of mathematics;
5. Methods of applying mathematical principles to other disciplines, and the relationship of mathematics to social conditions through technology;
6. Demonstration of competence in the selection and creation of appropriate mathematical models to solve applied problems;
7. Demonstration of competence in the use of calculators and computers in mathematical applications and problem solving; and
8. Demonstration of competence in the use of an appropriate computer language to write programs. (NASDTEC 61)

NBPTS, itself an outgrowth of the Carnegie Forum’s recommendations, notes that while a fourth-grade teacher’s depth of knowledge in a particular content area need not approximate that of the high school specialist in that area, fourth-grade teachers must appreciate the richness and complexity of a subject, “if they are to help their students develop higher-order thinking skills—the hallmark of accomplished teaching at any level” (NBPTS 20). While the NASDTEC Standards explicate some specific mathematics content recommendations, the NCATE Standards merely indicate that a criterion for compliance is: “The guidelines and standards of professional learned societies are used in the development of an appropriate sequence of courses for each specialty area” (NCATE 49).

The Carnegie Forum, the Holmes Group, and NBPTS are also in general agreement regarding the “systematic knowledge of teaching” that preservice teachers need to acquire. NBPTS perhaps states it best, noting that proficient teachers must have knowledge of general and subject-specific methods for teaching and evaluating student learning; knowledge of students and human development; skills in effectively teaching students from racially, ethnically, and socioeconomically diverse backgrounds; and the skills, capacities and dispositions to employ such knowledge wisely in the interests of students. (NBPTS 13)

There is nothing here that is incompatible with the NASDTEC and NCATE Standards.

While the means by which preservice teachers are to acquire these “skills, capacities and dispositions” (NBPTS 13) are not directly addressed in the NBPTS recommendations, the implication seems to be that they spring, at least in part, from effective and worthwhile field experiences. Such field experience, whether called clinical experience (Holmes Group, NASDTEC, NCATE) or residency in a school (Carnegie Forum), retain a prominence in the preservice preparation of teachers.

Finally, implicit in the recommendations of the Carnegie Forum, the Holmes Group, NASDTEC and NBPTS is evidence of the development of what might be termed a reflective practice. Such a notion seems clearly to underlie such statements as “Teachers Think Systematically About Their Practice and Learn From Experience” (NBPTS 14), “... focusing clinical experience on the systematic development of practice ...” (Holmes Group 64) or “[teachers shall] ...assess their own teaching effectiveness and professional growth needs” (NASDTEC 13).

Thus, we have some common ground among standards and recommendations for the preservice preparation of prospective teachers. Simply put, “teachers need a command of the subjects they teach, a sound grasp of the techniques of teaching those subjects, information about research on teaching, and an understanding of children’s growth and development and of their different needs and learning styles” (Carnegie Forum 71).

Clearly a desired by-product of such professional development is the simultaneous development of a reflective practice. A necessary condition to the restructuring of teacher education programs along the lines proposed seems to be greater collaboration and cooperation between schools and colleges and universities on the one hand, and between education faculty and arts and sciences faculty on the other. These constructs undergird the messages in each of these documents.

The Recommendations of NCTM and MAA and the “Common Ground”

Against the backdrop of the recommendations of the Carnegie Forum, the Holmes Group, and NBPTS, as well as the historically older Standards established by NCATE and NASDTEC, we now consider the recommendations in NCTM’s Professional Standards for Teaching Mathematics and MAA’s A Call For Change. The discussion will be organized around the major points of the NCTM and MAA recommendations:
  • a more thorough mathematics content background for teachers of mathematics at all levels;
  • a knowledge of students as learners of mathematics;
  • a knowledge of and the ability to use a variety of aspects of mathematics pedagogy;
  • a lifetime of reflective practice; and
  • greater cooperation and collaboration among the providers of mathematics teacher education.
Mathematics content knowledge. NCTM’s and MAA’s call for a more thorough background in mathematics content for prospective teachers of mathematics is in line with the recommendations of the Carnegie Forum, the Holmes Group, and NBPTS; and the NASDTEC and NCATE Standards.

While we appreciate NASDTEC’s attempt to explicate specialty standards for teachers of mathematics, we feel that the standards proposed for mathematics in their 1987 revision fall somewhat short of the expectations of MAA and NCTM. We are in agreement with the NCATE standard for specialty studies requiring that the guidelines of learned societies (e.g. MAA, NCTM) be used in developing the teacher education curriculum.

Our view of the NASDTEC and NCATE standards with regard to elementary and middle school teacher preparation can best be understood against a contrast between a “mathematics teacher” and a “teacher of mathematics.” In our conceptualization, all teachers who teach mathematics are teachers of mathematics. Both MAA and NCTM were careful to discuss standards for “teachers of mathematics.” On the other hand, both NASDTEC and NCATE explicated standards for “mathematics teachers” (i.e., specialists), but were fairly vague regarding mathematics content standards for elementary or middle school teachers who will be “teachers of mathematics.” From this perspective, we recommend that, in their next revision, both the NASDTEC and NCATE Standards for the elementary and middle grades be revised and updated based on the recommendations of MAA and NCTM for the preparation of teachers of mathematics at these levels.

The specific recommendations of the professional organizations for minimum semester hours of the study of mathematics for prospective teachers of mathematics at the various grade levels are clearly an attempt to raise the level of mathematical competence of prospective teachers of mathematics. The specific recommendations for the study of college preparatory mathematics in high school as a prerequisite to teacher education programs is also a clear attempt to raise the quality of teacher education candidates. In both of these areas, MAA and NCTM are in agreement with the Carnegie Forum, the Holmes Group, and NBPTS.

Students as learners of mathematics. NCTM’s and MAA’s recommendations regarding a knowledge of students as learners of mathematics go beyond the recommendations of the other groups by asserting that general principles of teaching and learning do not take into account the special nature of mathematics, nor research on the mathematical thinking of children. In their view, mathematical thinking is an ubiquitous human endeavor, and the fact that children engage in such thinking has special implications for mathematics instruction.

This position is not really at odds with the others. In fact, one might argue that NCTM, in insisting that teachers’ knowledge of students’ mathematics learning be informed by research, is exactly in line with the Carnegie Forum’s suggestion that teachers need information about research on teaching and learning. It clearly is in line with the NASDTEC Standard that “the program shall require study of research about teacher characteristics and behaviors as they affect the learner” (NASDTEC 12).

Mathematics pedagogy. In suggesting that teachers of mathematics, in their preservice preparation, should develop knowledge of and skill in using aspects of mathematics pedagogy, NCTM, in its general documents, is directly addressing the call for the development of a fundamental understanding of the techniques of teaching the various different content areas. In suggesting that research in mathematics education ought to inform the practice of mathematics teaching, NCTM acknowledges the call in these documents for providing preservice teachers with information about research on teaching.

Reflective practice. NCTM’s recommendation that the preservice preparation of teachers of mathematics provide opportunities that might lead to the development of a reflective practice is in line with the recommendations of the Carnegie Forum, the Holmes Group, and NBPTS, as well as the NASDTEC and NCATE Standards.

However, the rhetoric in these latter documents regarding the development of a reflective practice reminds us of the old saw: Everybody talks about the weather, but nobody does anything about it. By enumerating specific ways in which the process of reflectivity might be cultivated, NCTM is clearly “doing something about it!”

Greater cooperation and collaboration. The call for greater cooperation and collaboration, between schools, colleges and universities, and among college and university faculty, appears in all of the recommendations for the improvement of teacher education. NCTM and MAA, in implying such need, are in line with the more general reform documents and the NASDTEC and NCATE Standards.

While NCTM’s suggestion that innovative new collegiate mathematics courses be developed stops short of indicating exactly who should do the developing, we believe that they would subscribe to the MAA position that “college and university faculty, working cooperatively with practicing teachers, can develop...” (Leitzel 40) courses that reflect the what and how of teaching mathematics. This also coincides with the NCATE position calling for “faculty in the professional education unit and faculty who teach the specialty studies from other academic units [to] collaborate in program planning and evaluation of specialty studies” (NCATE 47).

Summary

In this section, we have attempted to synthesize the recommendations of the Carnegie Forum, the Holmes Group, NBPTS, the NASDTEC Standards and the NCATE Standards. We then considered the NCTM and MAA positions, especially as embodied in Professional Standards for Teaching Mathematics, against this background. We find much common ground among the Carnegie Forum, the Holmes Group, NBPTS, NASDTEC and NCATE, as well as a congruence between MAA’s and NCTM’s suggestions and the others.