From the Origin provides a forum for lively discussion of issues of importance to the mathematical community. The Michigan Section-MAA Newsletter solicits opinion pieces for publication in this column from anyone in the Michigan mathematical community. In addition, comments on pieces published in earlier issues are welcomed.

Items for From the Origin should be submitted to the editor by the beginning of October to be considered for inclusion in the December issue and by the beginning of February for the April issue. Main opinion pieces should be at most 1800 words long, and responses at most 400. The editors reserve the right to shorten responses, if necessary, in order to fit as many as possible within the available space.

## The National Council of Teachers of Mathematics'

New Principles and Standards for School Mathematics:

Implications and Challenges## by Joan Ferrini-Mundy (MSU)

In April, 2000, the National Council of Teachers of Mathematics (NCTM) released

Principles and Standards for School Mathematics. This was the culmination of a multifaceted, three-year effort to update NCTM's earlier standards documents, and to set forth goals and recommendations for mathematics education in the prekindergarten through grade twelve years. I served as chair of the 26-person Writing Group for this project, and in that role had the chance to interact with mathematicians, teacher educators, classroom teachers, mathematics education researchers, and parents in discussing what these various constituencies feel is best for mathematics education. Here I provide some background about NCTM and the standards, the process of development, ways in which feedback from the mathematics community influenced the document, and thoughts about the implications ofPrinciples and Standardsfor those who are concerned with the teaching and learning of mathematics at the postsecondary level.The NCTM is a 100,000-plus member professional organization of teachers of K-12 mathematics; university faculty in mathematics, mathematics education, and teacher education; state, provincial, district, and national policy makers; administrators; parents; educational researchers; and mathematicians. Serving the U. S. and Canada, NCTM was founded in 1920 as an outgrowth of the Chicago Mathematics Club. NCTM's mission is "to provide the vision and leadership necessary to ensure a mathematics education of the highest quality for all students."

Over the years NCTM has produced various documents intended to guide K-12 mathematics education, including the 1980

Agenda for Action, which called for the inclusion of problem solving, understanding, and applications in school mathematics and was a precursor to the standards documents that followed. NCTM's 1989Curriculum and Evaluation Standards for School Mathematicswas the first contemporary set of subject matter standards in the United States. It was followed in 1991 by theProfessional Standards for Teaching Mathematicsand in 1995 by theAssessment Standards for School Mathematics.These Standards documents have been widely used in various ways. For instance, a total of 49 states now have subject matter standards or frameworks, many of them influenced by the 1989 Standards. Instructional materials have been developed, many with National Science Foundation support, to align with the 1989 Standards, and recommendations for the education of teachers of mathematics have also reflected ideas from the NCTM recommendations.

Despite lack of agreement with the particulars of the ideas of standards, about the impact of standards on practice and policy, or about how the ideas of standards should be interpreted and implemented, one general area of agreement has emerged: The nature and quality of K-12 mathematics education needs improvement, so that more students can be successful in learning significant mathematics with understanding. This improvement requires concerted efforts by a wide variety of individuals and organizations.

## Brief Overview of Principles and Standards for School Mathematics

Principles and Standards for School Mathematicsis a 402-page document, organized into eight chapters. The preface and first chapter introduce the purpose and overall intention of the effort, and chapter 1 portrays "A Vision for School Mathematics". In Chapter 2 a set of six principles is provided, expressing the assumptions that serve as a basis for the recommendations in the document. There are ten standards that describe in a general way the mathematics that students should know and be able to do across the prekindergarten through grade twelve years, and each of these is discussed in Chapter 3. Five of these are considered content standards in that they address mathematical topic areas such as algebra and geometry; five are about mathematical processes, such as problem solving, and reasoning and proof. The following four chapters take up each of the four grade bands: pre-K-grade 2, grades 3-5, grades 6-8, and grades 9-12. There is a section in each chapter on each of the standards elaborating how the recommendations might be enacted in classrooms. For the content standards, more specific grade-band "expectations" are offered. The final chapter discusses how various groups might best work together for the continued improvement of mathematics education. Additional examples are provided in the electronic version ofPrinciples and Standards(available without charge on the NCTM's Web site at http://www.nctm.org).## Developing Principles and Standards

Principles and Standardswas developed by a 26-member team that included teachers, teacher educators, administrators, researchers, and mathematicians with a wide range of expertise. The Writing Group was charged to establish standards that

- build on the foundation of the original Standards documents;
- integrate the classroom-related portions of
Curriculum and Evaluation Standards for School Mathematics,Professional Standards for Teaching Mathematics, andAssessment Standards for School Mathematics;- use four grade bands: prekindergarten through grade 2, grades 3-5, grades 6-8, and grades 9-12.
The Writing Group met for two- or three-week periods in each of the summers of 1997, 1998, and 1999. Occasional two-day meetings were held during the academic years, and the leaders of the grade-band teams and I met intermittently. The development process was designed to be consultative and open, and we produced a draft document that was circulated widely for comment in the 1998-99 academic year. Reviews and reactions to that draft document were collected, synthesized, and analyzed systematically. Using that input, the Writing Group completed a final version of the document that was released in April 2000. The document is available in print and electronic forms. It was distributed as a benefit of membership free of charge to all individual NCTM members.

NCTM is engaged in a series of continuing activities intended to support the dissemination and implementation of the ideas in

Principles and Standards, including a series of Professional Development Academies for K-12 teachers, the Navigations Project, which will supply activities and classroom ideas that are consistent withPrinciples and Standards, and the Illuminations Project, which is a Web-based effort, funded by MCI WorldCom, to provide more sustained examples of standards-based activities and enactments. In addition, NCTM has launched a new committee, the Standards Impact Research Group, that will formulate ways to understanding and studying the impacts of standards.## Implications and Challenges for Mathematics Higher Education

A significant level of controversy and disagreement has developed in the past decade about K-12 mathematics education, much of which has played out very publicly in the national press, over e-mail lists, and via Web sites. These "math wars" have been characterized by derisive critique of various approaches to mathematics education, ranging from the accusations that NCTM promotes "fuzzy math" to the view that critics wish to have only "parrot math" in place in schools. The challenges of understanding the complexity of mathematics teaching and learning in real schools, with the diversity of interests, backgrounds, and inclinations that students bring, are enormous. One major intention for

Principles and Standardsis that this document should serve as a focus for deeper, more thoughtful conversation about the goals and directions for school mathematics, even among groups of people who disagree about the details of its recommendations. The development process that NCTM established forPrinciples and Standardsincluded intensive interaction with a set of 14 "Association Review Groups" (ARGs), representing mathematical professional associations at the national level. Throughout the process of developingPrinciples and Standards, we submitted questions to the ARGs for which we, as a Writing Group, needed advice and input. Our process of systematic synthesis of input then led to principled decisions about the stance we would take on each of the major issues that emerged from the input process. Naturally the positions that we took, on both large and small points, differed from those that some of our reviewers recommended - we could not satisfy everyone and produce a document that would be coherent and consistent. Nonetheless, the process that NCTM established has been lauded by the ARG leaders in a letter that appears inPrinciples and Standards: "With this, the NCTM has established a model, heretofore all too rare, of how to stage civil, disciplined and probing discourse among diverse professionals on matters of mathematics education."Topics that have been central in the "math wars" are addressed in

Principles and Standards- the role of technology in mathematics learning, the importance of computational and procedural fluency, the place of so-called "standard algorithms" for arithmetic computation, and the relationships among procedural and conceptual understanding. For those who teach mathematics at the postsecondary level, I feel thatPrinciples and Standardscould serve as a resource for launching conversations about issues that are also central at this level: Should we be using technology? In what ways? What kinds of entry level knowledge and skill should we expect of incoming students? I know of departments where mathematics education seminars are being initiated to examinePrinciples and Standardsin detail - and I think this will be a productive and interesting activity.I feel that

Principles and Standardshas implications for undergraduate mathematics departments on two major fronts. One has to do with the articulation between high school and first-year college mathematics offerings and expectations. The chapter ofPrinciples and Standardsthat addresses grades 9-12 makes some important recommendations, including that all students should study mathematics for each of the four years they are in high school, and that a single ambitious foundation, or core, of mathematical ideas and applications is intended for all students. The mathematical goals described in this section of the document are ambitious and far-reaching, and are intended to "give all students solid preparation for work and citizenship, positive mathematical dispositions, and the conceptual basis for further study." There is a very strong advocacy for making connections among mathematical ideas, emphasizing the interplay of algebra, geometry, statistics, probability, and discrete mathematics, and employing integrated perspectives and applications. We are well aware that this is an ambitious proposal, and that movement in these directions will be a slow and challenging process. Faculty in university and college mathematics departments could be crucial in enabling these shifts. A first step would be close reading, critique, and discussion ofPrinciples and Standards' Chapter 7. Perhaps groups that involve both high school and undergraduate faculty could be convened to do this work together. As curricula become available that will enable secondary school teachers and programs to move in these directions, college and university mathematics departments can be part of assisting in a smoother transition into undergraduate work, through the rethinking of goals and objectives of first year mathematics courses, perhaps in directions more aligned with what standards recommend; by examining and aligning expectations on placement and diagnostic assessments; and by considering the appropriate role of technology.The second major area of activity that involves undergraduate faculty is in the mathematical education of prospective teachers. The agenda proposed in

Principles and Standardsis quite ambitious mathematically, and to have teachers who could enact it will require some rethinking and reformulation of the mathematical background that teachers experience at the undergraduate level. One major feature of the new Principles and Standards is that each of the ten standards is emphasized across the prekindergarten through grade 12 years. So, for instance, in the prekindergarten through grade two chapter, there is extended discussion about how teachers can introduce algebraic concepts through work with classification, patterns, whole number computation, and functions. To be certain that elementary school teachers have the mathematical preparation to handle such innovations will mean addressing these areas within the undergraduate curriculum in mathematics for teachers.

Principles and Standardsis intended to provide a vision for the improvement of school mathematics education. There is a crucial role in this for those who teach mathematics at the undergraduate level, and I hope thatPrinciples and Standardscan serve as a resource for enabling us all to continue the work of improving mathematics teaching and learning for all students.

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