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Mathematical Preparation
of Elementary School Teachers

Issues and Recommendations





A report prepared in 1994 for the Presidents and Executive Directors of the American Mathematical Association of Two-year Colleges (AMATYC), the American Mathematical Society (AMS), the Mathematical Association of America (MAA), the National Council of Teachers of Mathematics (NCTM), and the Society for Industrial and Applied Mathematics (SIAM).

Making A Commitment
Fulfilling the Commitment Extending the Commitment




A child's mind is a fire to be ignited,
not a pot to be filled.


Experts agree: Excitement about learning mathematics is an important goal of school education. Dozens of reports have made hundreds of recommendations about how this goal might be achieved. Although many individuals and institutions have begun to implement these recommendations, their efforts have not yet had a significant nationwide impact on teacher education programs.

This brief paper suggests ways to transform these individual projects into a national movement. It has been prepared at the request of the Presidents and Executive Directors of five major mathematics professional societies in order to articulate an agenda for these societies to help improve the mathematical preparation of elementary school teachers. It is intended:

Elementary school is important to children both for their cognitive development and as a foundation for further education. For children with weak academic support at home, the primary responsibility for providing this foundation rests on elementary classroom teachers. Society now expects that schools will successfully prepare all students to meet national standards in all subjects.

Many observers fear that the forthcoming standards in different school subjects will, when taken as a whole, be overly ambitious for students, surpass what teachers know, and exceed what parents believe to be essential. Although these standards often exhibit the ambitions of disciplines vying for center stage in the education reform movement, they tend to be consistent in their emphasis on active learning and in their constructivist perspectives. The pressure of multiple standards ensures that teacher preparation programs are and will remain in constant flux.

Preparation for elementary school teaching is a life-long activity, encompassing teachers' experiences as K-12 students, as undergraduates, and as professionals who learn from experiences throughout their careers. The formal teacher preparation program-- post-secondary but pre-service--occupies a relatively short but crucial part of this experience. The focus of this report, reflecting the missions of the professional societies to whom it is addressed, is on just one component of teacher preparation: the mathematical preparation of prospective elementary school teachers.

Prospective elementary school teachers encounter mathematics in several different contexts: content courses, usually offered by departments of mathematics; methods courses, usually offered by faculty with appointments in education departments; and experiences in school classrooms, supervised by practicing elementary school teachers. Each encounter should offer solid mathematics and model sound pedagogy, and all should work together to provide a consistent view of mathematics. The special focus of this paper is on one leg in this triad--what goes on under the jurisdiction and responsibility of departments of mathematics whose members form the constituencies of the mathematics professional societies. However, in order to ensure the success of teacher preparation programs, it is necessary that firm linkages be established and maintained between all three components of the prospective teacher's mathematical preparation.

A coordinated effort, led by the mathematics professional societies and focused on promoting successful teacher preparation programs, can break what some critics have described as the "cycle of failure" in mathematics teaching. It can also encourage more college and university faculty to make teacher preparation a higher priority in their own professional lives. Thus it is especially timely for professional societies to undertake a special initiative to bring about much-needed improvement in the mathematical preparation of elementary school teachers.

The following recommendations build on a history of public statements and address unresolved issues and emerging ideas in a context that is within the authority and mission of the mathematical professional societies:

Make a Commitment:..The mathematics professional societies should develop and make public a consensus statement regarding the critical importance of the mathematical preparation of elementary school teachers.

Fulfill the Commitment:..The mathematics professional societies should develop a coordinated program of activities and publications to support their members in providing outstanding mathematical education to prospective elementary school teachers.

Extend the Commitment:..The mathematics professional societies should work both at the national level and with their state and local affiliates to develop strategies for engaging and influencing educational policy.





Making A Commitment

Make a Commitment:..The mathematics professional societies should develop and make public a consensus statement regarding the critical importance of the mathematical preparation of elementary school teachers.

A well-publicized consensus statement by the professional societies would provide a visible public commitment that can create a platform for further action by the entire mathematical community. A basis for that consensus statement can be found in recommendations contained in reports on the mathematical preparation of elementary school teachers. These recommendations, summarized below, suggest considerable agreement on the requisite characteristics of strong programs:

  1. Mathematics departments should take seriously the challenges and obligations of courses intended for prospective elementary school teachers. Too often, "mathematics for elementary teachers" is a neglected component of a mathematics program, scorned by senior faculty and assigned to teachers with least seniority or without appropriate expertise. To restore vitality to mathematics education, these courses should be viewed instead as cornerstones of a department's program--courses with the capability of doing the greatest long-term good.

  2. Prospective elementary school teachers need to learn a broad range of elementary mathematics from an advanced perspective. Elementary school teachers need to know (and to teach) much more than arithmetic. In order to help their students gradually develop abilities in abstract thinking, K-6 teachers themselves need to be comfortable with abstraction, generalization, and "symbol-sense." At the same time, in order to provide their students with substantive examples of the mathematics used in life and work, prospective teachers also need opportunities to apply elementary mathematics to problem- solving in realistic situations.

  3. Mathematics courses for prospective elementary school teachers should do more than cover a list of topics; they should help future teachers make sense of mathematics. Elementary school teachers need a deep and robust understanding of the nature of mathematical thinking. Prospective teachers especially need to reflect on their experiences as students--what they have learned about the nature of mathematics and about the process of learning. They must come to understand that mathematics is about ideas, not just procedures, and that learning requires extensive engagement with those ideas. Then they must learn how to listen for and interpret students' mathematical ideas.

  4. Mathematics departments should provide prospective teachers with extensive opportunities to reflect on the important connections among content, pedagogy, and learning. In addition to teaching mathematics and modeling appropriate pedagogy, the mathematical component of the undergraduate program for prospective elementary school teachers should provide opportunities for students to reflect on their personal experiences in learning mathematics and to place those experiences in a broad professional context. Faculty in mathematics and mathematics education should work collaboratively to achieve these results.

  5. All college and university mathematics teaching should model the pedagogy that will be expected of future teachers. Prospective teachers need extensive opportunities to construct for themselves the mathematics they will be teaching. "Enriched" courses that merely inject technology and hands-on activities into traditional courses consistently fail to penetrate prospective teachers' fundamental image of mathematics as a collection of answer-getting rituals. It is important that these courses connect students' hands-on experiences with the mathematics those experiences represent. All courses for prospective teachers should be designed to organize students' mathematical experiences in ways that help develop the habits of mind of those who use mathematics in their life and work.

  6. College courses for prospective teachers should illustrate the way mathematics is practiced. Mathematics in practice uses technology, collaboration, communication, and exploration. Too often college and university faculty teach as they were taught when they were students rather than as their students will be expected to teach when they become teachers. The mathematical preparation of prospective teachers should enable them to implement an important goal of school mathematics--to prepare students to use mathematics at work and in their lives.

  7. Colleges and universities need to provide all prospective elementary school teachers with significant opportunities to learn how to teach children of diverse racial, ethnic, and linguistic backgrounds. The reality of today's classrooms in the United States is that they are multicultural, multiracial, and multilinguistic. Since effective mathematics learning arises from meaningful contexts, prospective teachers need opportunities to learn multiple contexts in which to make mathematics significant to their students. Therefore it is especially important that mathematics faculty participate fully in opportunities offered by their institutions to learn about diverse teaching and learning strategies.

  8. Courses for prospective elementary school teachers should include significant coverage of the contributions to mathematics of diverse cultural and ethnic groups. Not all mathematics was discovered by any one culture or gender, but the dominance of one perspective in most presentations of mathematics tends to exclude women and people of different cultures from the community of mathematical scholars. Especially since schools in the United States are so multicultural, it is vitally important that prospective teachers become fully aware of the universal character of mathematics and the influence of various cultures on its evolution.

  9. College and university mathematicians need to develop effective working relationships, based on mutual respect, with those who have a stake in school mathematics. Effective programs to prepare elementary school teachers require collaboration among mathematics educators, mathematicians, education faculty, and school teachers. Such collaboration should extend also to non-educators--business and civic leaders, parents and taxpayers.





Fulfilling the Commitment

Recommendation: The mathematics professional societies should develop a coordinated program of activities and publications to support their members in providing outstanding mathematical education to prospective elementary school teachers.

Mathematics faculty in colleges and universities bear primary responsibility for the mathematical preparation of elementary school teachers, but they often work in isolation, lacking suitable infrastructure to strengthen their professional engagement with this undertaking. This is a need the professional societies are especially constituted to meet, both through cooperative and coordinated activities and through special initiatives addressed to their own members. Strategies could include:

Many of these strategies are currently employed either in college and university programs or in activities of professional societies, but often their focus is on in-service teacher education. Responses to the special needs of pre-service education--the main focus of this recommendation--can build on experiences gained through these in- service programs by appropriate extension and adaptation. Such activities will take place in a variety of contexts ranging from higher education policy to the design of individual courses.

Higher Education Context

Whereas formerly most elementary school teachers were educated through a relatively predictable and standardized education major, today there are many conflicting and constantly changing models for teacher preparation programs: In many states, the majority of students who become elementary school teachers begin their postsecondary education in two-year colleges where they take some or all of their required mathematics credits. Even though they may not think of themselves as teacher- preparation institutions, these two-year colleges represent an entry point to careers in elementary education for many students, especially minority students.

Although standards for mathematics content for teacher preparation are explicated in A Call for Change, the diverse and ever- changing variety of teacher preparation programs may allow prospective teachers to avoid the breadth of mathematics recommended in that document. Moreover, as other disciplines argue effectively for the inclusion of courses in their areas, mathematics requirements may be diminished to accommodate crowded programs. It is important, therefore, that mathematicians play a critical role in developing and implementing sound educational programs for prospective elementary school teachers.


Departmental Context

Full recognition of the importance of elementary school teacher preparation will require explicit broadening of the mission of mathematics departments in most postsecondary institutions, and full engagement of the faculty in those departments. In institutions with programs to prepare elementary school teachers, mathematics departments must recognize their role in the mathematical preparation of these teachers and their responsibility to provide continuing resources (seminars, Internet access, consulting support, summer institutes) to support their graduates and other teachers in neighboring communities. In many cases, this may require enlarging the department's mission and securing additional resources. Mathematical preparation of teachers doesn't end with their college courses.

A recent report by the Joint Policy Board for Mathematics (JPBM) has launched a vigorous campaign to broaden the basis for recognizing and rewarding mathematics faculty. This effort includes recognition of the importance of program development, teaching, and scholarship associated with the mathematical preparation of teachers. Where these changes are implemented, faculty who teach courses for prospective elementary school teachers will more readily secure the time, opportunity, and resources needed to focus on this kind of work. Especially in times of limited budgets, departments can make a strong statement of support for these efforts by giving priority to the special resources needed by those who teach prospective teachers.

As mathematics departments wrestle with the challenges of improving teacher preparation, the professional societies can provide needed stimulation by providing information about programs that work. In addition to strengthening routine courses, certain evolving areas require special attention:

Learning from Research. Faculty teaching mathematics to prospective teachers need to know what research says about children's learning of mathematics. They also need to incorporate the results of that research into the courses they teach, which is not an easy matter. It is not enough to explain the results of current research literature on how children learn mathematics to prospective teachers or merely to ask them to read research reports. Prospective teachers need opportunities to experience for themselves the principles embodied in that research.

Mathematics in Practice. Teachers need real-world experiences of the practice of mathematics and science in order to portray accurately the nature of these disciplines. All too often, teachers enter their careers without ever having experienced any work situation other than education--first as students, then as teachers. To understand the ways mathematics is used, it is important for prospective teachers to have internship-like opportunities in real work sites. Departments can work with local employers to create internships for prospective teachers just as they now do for students who are interested in careers in business and industry.

Supporting Multicultural Education. Teacher preparation programs are beginning to address the crucial need to prepare teachers for multicultural, multiethnic and multilinguistic classrooms by developing courses in multicultural education. Yet most mathematics programs for prospective elementary school teachers have only tenuous links to these generic courses, largely for lack of appropriate historical and cultural materials suitable for elementary school mathematics instruction. Thus, prospective teachers have few opportunities to see mathematics as a multicultural activity, and to overcome the hidden racial and class biases of those who have not had a chance to live and work in multicultural environments. Professional societies can help mathematics departments by gathering and disseminating materials appropriate to this particular need.

Mathematics Specialists. Many observers have urged that the United States adopt a model of specialist teachers in elementary school, and many districts have been experimenting for some time with various roles for specialists. Magnet programs, building and district specialists, and paired teaching (e.g., language arts and science-mathematics) all fall within the general scope of such specialist programs. Yet there is no common understanding within the mathematics community about the appropriate preparation of mathematics (or mathematics-science) specialists for elementary school, nor has there been much work done on developing courses especially suitable to this goal. What do specialists need in way of preparation that generalists do not also need? Surely the answer is not just more courses suitable to high school or college teachers. To permit exploration of this idea, the community needs better information about experimental programs, as well as serious dialogue about how to approach elementary school mathematics from an advanced perspective.

Evaluating Programs. Assessing program effectiveness is crucial to achieving quality. Assessment is especially important and delicate in situations in which approaches to teacher preparation are exploratory or part of special curriculum development projects. Most mathematics faculty know very little about program evaluation or classroom-based research. Increased knowledge about these areas would better position mathematics faculty to respond to questions raised by the public about the status of progress toward the national goal of improving mathematics education. It would also enable mathematics faculty to take leadership roles within their own communities when issues arise about mathematics education reform.

Professional Context

The variety of current courses and the conflicting recommendations for change in teacher preparation programs can be resolved only through dialogue among elementary school teachers and administrators, mathematics educators, and college and university mathematicians. Increased dialogue will make all those involved think more deeply about the broad context of mathematics education, as well as about their own work.

However, many barriers to effective communication still divide these different constituencies. Much of the literature of mathematics education is written in a language that mathematicians find difficult to understand, and most articles about mathematics are written in ways that are not useful to teachers and mathematics educators. Faculty at two- and four-year colleges rarely talk with one another about matters of teacher preparation, even though many prospective teachers complete half their post-secondary education (and often all their mathematics credits) in two-year colleges. Effective programs for preparation of elementary school teachers also will require on- going substantive contacts between college faculty and elementary school teachers. Professional societies can help by using sessions at meetings and articles in journals to break down the barrier of jargon that impedes effective communication on issues involving mathematics education.

Mathematics faculty who teach courses for prospective elementary school teachers often have inadequate experience and understanding of how children learn mathematics. As often as not, they generalize unwarrantedly from experience with their own or their friends' children and thus fail to recognize the enormous diversity in how children construct mathematical knowledge. Yet each year scores of faculty and graduate students are asked to take on the assignment of preparing elementary school teachers--an assignment for which they have no preparation and for which there are virtually no programs to help provide necessary background.

The overwhelming need of faculty who teach courses for prospective elementary school teachers is for strategies to enable students to think mathematically. Yet none of the channels of information to which mathematics faculty normally turn provide adequate information. Often, only one person on each campus teaches the courses for elementary school teachers, so their only sources of collegial support are individuals in similar circumstances on other campuses. Resources that would be useful include surveys of relevant educational research, examples of challenging mathematical topics set in a context appropriate for elementary school, samples of curriculum materials, and information about teacher preparation programs that exemplify research-based recommendations. Professional societies can play a unique and valuable role in linking individuals on different campuses to create a nation-wide focus on this issue.

Course Context

Courses designed to prepare elementary school teachers typically seek to achieve one or more of the following broad objectives:
  • Competence: In-depth introduction to the mathematics of a standards-based elementary school education, including arithmetic, geometry, probability, algebra, modeling, and data analysis.
  • Exploration: Reflective experience in thinking mathematically and in constructing one's own mathematical knowledge. Emphasis is on the nature of mathematical inquiry, not on the content of mathematics.
  • Understanding: Broad survey of the big ideas and unifying themes of mathematics so as to reveal the subject as a whole and thereby to appreciate the foundation being laid during elementary school.
  • Ideally, these goals should be integrated into all mathematics courses for prospective elementary school teachers because they are, fundamentally, the goals that all elementary school children should achieve.

    Instructional strategies for these courses should address these three goals for mathematics, should model good pedagogy, and should employ assessment strategies related to the goals of the course. This last is especially important since prospective teachers must explicitly learn how to assess their students' mathematical knowledge in terms of competence, exploration, and understanding. One challenge for mathematics faculty teaching prospective teachers is to find ways to assess student learning, especially among students with non- traditional backgrounds or whose understanding of mathematics may not be revealed through traditional testing. Ordinary tests often fail to measure students' real skills; not even the experts quite know how to do it right.

    Prospective teachers need to experience mathematics as their students will (or should), in an atmosphere that encourages and rewards exploration. Moreover, elementary school teachers often will be expected to integrate the teaching of mathematics with other subjects, especially science and social studies. Thus they need deep knowledge of the mathematics they will teach in elementary school, experience in making connections between different areas of mathematics, and broad understanding of the ways mathematics is used to solve real life problems. They should have frequent opportunities to explore significant mathematics--both abstract and applied--in contexts that are meaningful to them as adults. Their engagement with ideas of interest to adults will model the process that young children go through as they too pose and solve complex problems within their own spheres of interest.

    Yet many mathematics courses that colleges and universities designate to meet the requirements for prospective elementary school teachers reflect a pattern of thoughtlessness, if not disdain, for the important mathematics that these teachers really need to learn. The collegiate view of mathematical sophistication is to climb the algorithmic ladder that reaches from arithmetic to calculus. This is totally opposite to the NCTM Standards' view of elementary school mathematics as rich in horizontal linkages, mathematical modeling, active discovery, and opportunities for sense-making. All too often current courses for prospective elementary school teachers, driven by a text or syllabus to cover too many topics too rapidly, merely convince anxious students that they don't know mathematics, don't like mathematics, and really don't want to learn mathematics.

    Several very different patterns prevail in providing the mathematical content knowledge for prospective elementary school teachers:

    Mathematics for Elementary School Teachers. A traditional 1-3 course sequence offered from standard textbooks at institutions with sufficient enrollment to maintain special courses in this area.

    Mathematics for Liberal Arts Students. In institutions with insufficient enrollments to warrant special courses, a variety of regular courses are allowed to count as the mathematics content credits for an elementary school teaching certificate.

    Variations on Algebra. Many institutions allow credits from the standard pre-collegiate algebra sequence to meet the mathematics content requirement for prospective elementary school teachers.

    This variety represents uncertainty in the community about whether the mathematics that prospective teachers study should be a review of the mathematics they will teach or a strategy to help them become mathematical thinkers. MAA, NCTM, and NCATE consistently recommend that all prospective elementary school teachers take several courses in mathematics that prepare them to teach mathematics in a manner consistent with the NCTM Standards. Large institutions provide special courses to meet this goal; smaller institutions must often use general courses for dual purposes. Courses in mathematical modeling, problem solving, and finite mathematics are especially suitable for prospective elementary school teachers. The traditional collegiate mathematics curriculum, linearly ordered and based on gathering algorithmic skills, while possibly unsuitable for all students, is especially unsuitable for prospective elementary school teachers.

    The role of technology is another area of uncertainty and controversy. This ambivalence, especially concerning calculators in elementary school, is often an impediment to integrating technology into mathematics courses taken by prospective teachers. Since the NCTM Standards advocate extensive use of calculators throughout all grade levels, prospective elementary school teachers need to be able to confidently integrate the use of calculators in their own classes in meaningful ways that enhance student learning. They must also be prepared to explain the value of calculators to interested and anxious parents. Therefore they must be proficient calculator users themselves, confident in their judgment of appropriate uses of calculators as aids in mathematical problem-solving.

    Student Context

    Important goals for the undergraduate mathematics program for prospective elementary school teachers are to help those students develop positive attitudes about mathematics as a discipline and to create excitement about learning mathematics. The program, likewise, should foster in future teachers beliefs about mathematics that will enable them to help children learn what is mathematically sophisticated, efficient, and elegant. However, in creating these programs, mathematics faculty have to be sensitive to the needs, interests, and backgrounds of students entering teacher preparation programs.

    Like many other students, prospective elementary school teachers often have weak mathematics backgrounds and high levels of math anxiety when they enter college. Unlike many other students, however, elementary school teachers will use mathematics throughout their careers: they will teach mathematics to future generations of children and will have a significant impact on their students' understanding and attitudes. So it is especially important that college mathematics courses for prospective elementary school teachers build on what students know, recognize the reality of anxiety-induced inhibitions, and enhance students' self-confidence as potential learners of mathematics. For some, especially those with particularly weak high school mathematics backgrounds, it may take longer to achieve the expectations of A Call for Change. Mathematics departments need to find flexible means of accommodating the anxieties and varied backgrounds of students while maintaining high program standards.





    Extending the Commitment

    Recommendation: The mathematics professional societies should work both at the national level and with their state and local affiliates to develop strategies for engaging and influencing educational policy.

    Certification standards for elementary education are controlled by state policy, either directly from a central office or indirectly through mandates to local educational agencies or institutions. Mathematicians typically know little about these processes, even though they are responsible for implementing many features of the policies. Issues concerning specialist teachers, state frameworks for mathematics curricula, student testing and promotion policies, local business expectations, teacher rectification, and articulation with higher education frequently flow through state agencies with whom university mathematicians have essentially no significant contact. Mathematics departments need to become informed about and engaged with those state-based organizations that influence mathematics education and in the various large scale reform programs (curriculum projects, regional laboratories, teacher enhancement efforts) and systemic initiatives (state, urban, rural) in their regions.

    The public demand for accountability from the educational system requires methods of evaluation and measurement that will provide parents and employers with meaningful indicators of performance-- both of students and of schools. Mathematicians, mathematics educators, and business leaders need to work together to set performance standards for both skills and understanding that meet legitimate expectations of industry and higher education. The dialogue thus engendered will help insure that students and parents are apprised of expectations, and that schools and teacher preparation programs will have a strong incentive for making the changes necessary to meet those expectations.

    As part of this process, universities, particularly public universities, should become active partners in the political processes--both legislative and executive--through which teacher preparation and school education is regulated and assessed. So too should business and industry. Within the broad general context of educational policy, mathematicians in universities and in industry bear a particular responsibility to monitor and influence those policies that bear on mathematics education. Mathematics needs to have a voice in state and local policies in which the perspective of the schools' clients-- industry and higher education--are strong and clear.

    Some structures to achieve this do currently exist, although their strength and level of activity are highly variable. These include the NSF-supported state, urban, and rural systemic initiatives, the state coalitions for mathematics and science education begun by the MSEB; NSF Collaboratives for Excellence in Teacher Preparation; Eisenhower Partnerships; sections of MAA, and affiliates of AMATYC and of NCTM. With their natural reach into all states through publications and meetings, the professional societies could do much to encourage and coordinate their members' efforts to strengthen the voice of mathematics in local educational policy. They could, for example:

    Conclusion:

    Current thinking about the mathematical preparation of elementary school teachers reveals many possible areas for improvement and suggests important activities that might be put on the agendas of the mathematics professional societies. Among the many challenges the profession faces, the mathematical preparation of elementary school teachers is one of the most important and most urgent. The professional societies can play a special role in providing national leadership to address this challenge. The impact of their efforts will increase to the extent that they can work together to create and implement an efficient agenda for action.


    Resources

    Acknowledgments

    This project was initiated at the request of the Presidents and Executive Directors of the American Mathematical Association of Two-year Colleges, the American Mathematical Society, the Mathematical Association of America, the National Council of Teachers of Mathematics, and the Society for Industrial and Applied Mathematics. Its implementation was made possible through a grant from the Charles A. Dana Foundation, Inc. Preparation of this paper was supported through the active participation of the thirteen members of the Task Force and by dozens of members of the societies who reviewed and commented on earlier drafts.

    Task Force Members

    DEBORAH BALL, Michigan State University, East Lansing, MI
    PATRICIA CAMPBELL, University of Maryland, College Park, MD
    PETER CASTRO, Eastman Kodak Company, Rochester, NY
    JACQUELINE GOODLOE, Burrville Elementary School, Washington, DC
    CAROLE LACAMPAGNE, U.S. Department of Education, Washington, DC
    JAMES LEITZEL, University of Nebraska, Lincoln, NE
    MERCEDES MCGOWAN, William Rainey Harper College, Palatine, IL
    BARBARA SCOTT NELSON, Education Development Center, Newton, MA
    JUDITH ROITMAN, University of Kansas, Lawrence, KS
    SHEILA SCONIERS, University of Chicago, Chicago, IL
    CATHY SEELEY, University of Texas, Austin, TX
    TINA STRALEY, National Science Foundation, Arlington, VA
    PAUL TRAFTON, University of Northern Iowa, Cedar Falls, IA




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    Last Update: 05/25/96