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Notion With the 1989 release of Everybody Counts by the Mathematical Sciences Education Board (MSEB) of the National Research Council and the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM), the "standards movement" in K-12 education was launched. Since that time, the MSEB and the NCTM have remained committed to deepening the public debate, discourse, and understanding of the principles and implications of standards-based reform. One of the main tenets in the NCTM Standards is commitment to providing high-quality mathematical experiences to all students. Another feature of the Standards is emphasis on development of specific mathematical topics across the grades. In particular, the Standards emphasize the importance of algebraic thinking as an essential strand in the elementary school curriculum. Issues related to school algebra are pivotal in many ways. Traditionally, algebra in high school or earlier has been considered a gatekeeper, critical to participation in postsecondary education, especially for minority students.) Yet, as traditionally taught, first-year algebra courses have been characterized as an unmitigated disaster for most students.2 There have been many shifts in the algebra curriculum in schools within recent years. Some of these have been successful first steps in increasing enrollment in algebra and in broadening the scope of the algebra curriculum. Others have compounded existing problems. Algebra is not yet conceived of as a K-14 subject. Issues of opportunity and equity persist. Because there is no one answer to the dilemma of how to deal with algebra, making progress requires sustained dialogue, experimentation, reflection, and communication of ideas and practices at both the local and national levels. As an initial step in moving from national-level dialogue and speculations to concerted local and state level work on the role of algebra in the curriculum, the MSEB and the NCTM co-sponsored a national symposium, "The Nature and Role of Algebra in the K-14 Curriculum," on May 27 and 28, 1997, at the National Academy of Sciences in Washington, D.C. The goals of the symposium were 1. to promote an informed dialogue on issues concerning the K-14 algebra curriculum, such as · necessary changes in the current curriculum · identification of research needed to help build a continuous and effective algebra curriculum ~Hawkins, B.D. (1993.) "Math: The Great Equalizer Equity 2000 and QUASAR, Improving Minority Standing in Gatekeeper Courses." Black Issues in Higher Education, 10(6), 38-41. 2 Steen, L. (1992.) "Does Everybody Need to Study Algebra?" Basic Education, 37(4), 9-13. 1
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2 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM · the role of technology in the teaching and learning of algebra and how technology can enhance the development of algebraic reasoning and conceptual understanding · ways to change the public perception of algebra from a year-long course to a continuous K- 14 mathemat . . Cal experience · issues surrounding all students and the learning of algebra and · determination of the algebraic ideas that are important at different grade levels and the incorporation of these ideas into the curriculum 2. to provide examples of students' algebraic thinking, to synthesize research, and to consider how these factors can impact the reality of algebra in the school curriculum 3. to provide a forum for those involved in algebra-related curriculum projects at the elementary-school, middle-school, secondary-school, and postsecondary school levels to share their visions of curricula, teaching, and assessment 4. to prepare participants to replicate similar symposia on a smaller scale in their local regions. Over 300 invited participants representing all the states attended the symposium, including K-14 mathematics teachers, mathematics supervisors, postsecondary educators, community college mathematicians, collegiate math- ematicians, and representatives of the mathematics associations. This report describes the symposium, including presentations and discussions. To summarize briefly, on the first day, Hyman Bass, Department of Mathematics, Columbia University, opened the meeting with a mathemati- cian's perspective on algebra. He argued for an early emphasis on use of the real number line and the geometric representation of arithmetic operations. He emphasized the significance of the Euclidean algorithm, both as a source of place value representation of numbers and as an illustration of the similarities and differences between integer and polynomial arithmetic. John Dossey, mathematics educator, Illinois State University, then discussed algebra in the past and in the present, including changes outlined in a discussion document that is printed here as Appendix E. Dr. Dossey then presented several models to consider in striving for student understanding of algebra and discussed several goals for school algebra programs. During the balance of the first day of the symposium, participants focused on key elements of algebra instruction by rotating through sessions entitled "Algebra: A K-14 Curriculum Endeavor"; "Technology and Algebraic Reasoning"; and "Algebra: What All Students Can Learn." In these first-day sessions, several speakers presented their view of the issues, and their papers are contained in this record of the proceedings. Participants then worked in groups to discuss questions with respect to the issues raised by the topics. The following questions were among those discussed: · In order to develop coherence and depth in the algebra curriculum across K-14 what transitions do students need to make and over what period of time? how long must students be engaged with an important mathematical idea so that understanding is achieved? · In order to develop criteria or lenses to select curriculum, appropriate pedagogy, and assessment . how should algebraic reasoning be defined? how should state, national, or college placement exams be taken into account? In order to develop long-term support for teachers and Standards-based algebra how much help do teachers need? how do we develop public support for teachers and Standards-based algebra? The second day of the symposium focused on "Algebra in the Curriculum." The following topics were featured functions and relations; modeling; structure; and language and representation. Individuals affiliated with National Science Foundation (NSF)-funded curriculum projects in elementary-school, middle-school, high-school, and postsecondary mathematics gave presentations on these topics. Their papers are included in this record of the proceedings. The afternoon session also brought together participants from common geographic regions to plan local and regional algebra initiatives, which are now under way. Closing remarks were provided by Beverly Williams, Chair of the Task Force that planned the symposium.
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INTRODUCTION Several background readings were provided for participants, and this record of the symposium publishes them in full. Two invited papers were commissioned and are also included, in Appendix D: "Stating the Obvious: Mathematics Course Taking Matters," which raises equity issues around course taking and algebra; and "Algebra, Technology, and a Remark of I.M. Gelfand," which focuses on technology and algebra. A discussion document adapted from work done by the NCTM Algebra Working Group also was sent to participants prior to the symposium and appears here in Appendix E by permission of the NCTM. It is entitled "A Framework for Constructing a Vision of Algebra: A Discussion Document."
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Representative terms from entire chapter: