of Elementary School Teachers

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:

- to synthesize current issues within the ever-changing context of educational reform;
- to marshal the energies of the professional societies on an agenda of action;
- to reach professional leaders who are ready to hear and act on
the message.

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.*

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:

*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.*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.*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.*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.*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.*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.*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.*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.*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.

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:

- Activities at annual and regional meetings designed to promote a sense of community among those who are involved in the mathematical preparation of prospective elementary school teachers.
- Workshops, minicourses, and other opportunities to prepare mathematics faculty and graduate students to teach courses for prospective elementary school teachers.
- Dialogue sustained through newsletters, journals, and e-mail on issues in teaching and learning related to the mathematical preparation of elementary school teachers.
- Creation of an on-line "virtual journal" using "gopher" and "mosaic" to alert individuals to the presence and location of relevant articles published world-wide in current periodicals and to provide timely information about upcoming conferences.
- Stimulation of electronic networks among individuals across the country who are interested in sharing practices and consulting with one another.
- Dissemination of rich examples of promising practice in the mathematical preparation of elementary school teachers.
- Promotion of opportunities for college and university mathematicians to learn first-hand about the classrooms in which prospective teachers will work, about the new curricula that are available for K-6 settings, and about research concerning children's learning.
- Identification of resources that provide examples of interesting mathematics and mathematical activities that can challenge pre- service teachers to think mathematically.
- Support for programs that provide prospective elementary school teachers with mathematics-rich experiences in non- academic settings.
- Discussion of diverse strategies to assess student learning including open-ended questions, group or individual projects, and student portfolios.
- Exploration of programs for specialist preparation in elementary mathematics and science teaching.
- Dissemination of case studies of departments of mathematics working with neighboring school districts to link mathematicians with elementary school children, teachers, and administrators.
- Providing examples of mathematicians and mathematics educators working collaboratively.

- Traditional programs leading to a major in education with minimal coursework in mathematics.
- "Holmes"-type programs in which the undergraduate education major is abolished in favor of traditional subject-matter majors.
- The "Project 30 Alliance" in which liberal arts courses are substituted for education courses in an attempt to enrich the traditional education major.
- "Alternative certification" in which anyone with a university
degree can obtain a teaching certificate through a combination of
supervised teaching and special examinations, often without any
additional mathematics content or pedagogy courses.

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.

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.

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.

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 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.

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.

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.

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:

- Collect, cross-reference, and disseminate information about the range of practice in elementary teacher preparation from state to state;
- Examine the needs and varied responses of industry for a better educated local work force, including examples of where business and education have formed effective alliances to provide needed improvements;
- Provide examples of effective public action in support of sound policies regarding teacher preparation;
- Gather and publicize information about resources for teacher certification initiatives and the work of organizations such as NCATE and NBPTS.

- Battosta, Michael T. "Teacher Beliefs and the Reform
Movement in Mathematics Education."
*Phi Delta Kappan*, Feb. 1994, pp. 462-470. - Cipra, Barry.
*On the Mathematical Preparation of Elementary School Teachers.*Report of a conference held at The University of Chicago, 1991. - Clemens, Herbert. "The Park City Institute: A Mathematician's
Apology."
*Notices of the American Mathematical Society,*Vol. 39, March 1992. - Cooney, Thomas J.
*Teaching and Learning Mathematics in the 1990s.*1990 Yearbook. National Council of Teachers of Mathematics, 1990. - Garfunkel, Solomon A. and Young, Gail S.
*In the Beginning: Mathematical Preparation for Elementary School Teachers.*COMAP, Inc.,1992. *Guidelines for Mathematics Departments at Two-Year Colleges*. The American Mathematical Association of Two-Year Colleges, 1994.*Guidelines for Programs and Departments in Undergraduate Mathematical Sciences.*Mathematical Association of America, 1994.- Hungerford, Thomas W. "An Experiment in Teaching
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Neglected Constituency."
*American Mathematical Monthly*, January 1994, pp. 15-21. - Leitzel, James. R. C. (Ed.).
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*UME Trends*, Vol. 3, December, 1991. - Madison, Bernard L.
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*Counting on You: Supporting Standards for Mathematics Teaching.*National Research Council, 1991. - National Research Council.
*Moving Beyond Myths: Revitalizing Undergraduate Mathematics.*National Academy Press, 1991. *Professional Standards for Teaching Mathematics.*National Council of Teachers of Mathematics, 1991.*Recognition and Rewards in the Mathematical Sciences.*Joint Policy Board for Mathematics, 1994.- Shulman Lee S. "Knowledge and Teaching Foundation of the
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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