Mathematics For All
November 2, 2000
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Mathematics For All
Pre-Kindergarten
Through Grade 12
Standards
Developed for Massachusetts Students
by Massachusetts Educators
A Work in Progress
Why Produce Mathematics For All?
The effort to determine preK-12 learning standards for mathematics in the Commonwealth of Massachusetts has been an arduous journey over the past eight years. With great hope, many individuals have worked with our state’s policy makers to consider, recommend and write portions of Mathematics Frameworks during this period. Under the umbrella of Education Reform, educators have written standards and expectations, and also helped to create the Massachusetts Comprehensive Assessment System. All of this has been done with the singular goal of providing a strong foundation in mathematics for all of the Commonwealth’s children.
During the past year, this process has broken down to the point where the recently approved, revised version of a Mathematics Curriculum Framework no longer represents the viewpoint of mathematics education shared by previous writers or, we believe, the philosophy of the National Council of Teachers of Mathematics (NCTM) as espoused in the April 2000 release of Principles and Standards for School Mathematics. In Mathematics For All, educators from across the Commonwealth offer an alternative set of broad learning standards that, we are fully aligned with NCTM and also reflect curriculum decisions that have been made in Massachusetts’ school districts over the past decade.
We urge our colleagues in Massachusetts to evaluate Mathematics For All, both as a guide to how we educate our children and in order to continue the dialogue to create an exemplary system of mathematics education. A common voice is needed to prevent the production of another generation of mathematics phobics as has been the legacy of times past.
Key features of Mathematics For All:
· There is a common set of Learning Standards and Learning Expectations for all children.
· The focus is on teaching for understanding.
· The Learning Standards are those of NCTM’s Principles and Standards for School Mathematics.
· The breadth within these Learning Standards ensures rich meaning for students of all abilities.
· The Learning Standards are not a list of what is to be tested through a pencil-and-paper format; rather, a variety of assessments should be used to evaluate student learning.
· The Learning Expectations include explorations of concepts that are mastered in later years.
We need the critical feedback of our colleagues in the Massachusetts mathematics education community to strengthen this document and to make it the one reference representing the best thinking of our profession. We encourage and welcome your comments, criticisms and suggestions. Currently, the most effective method to provide feedback is to visit the Massachusetts Educators for Mathematical Excellence (MEME) discussion site on the web at http://mathforum.com/discuss/meme. In future months, writers of Mathematics For All will reach out in a variety of forums that will be advertised on the MEME discussion site or through other means. [We are very grateful to Helen Plotkin and Richard Tchen, creators of MathForum.com and WebCT.com, the e-Learning Hub, who have offered space on their website for Massachusetts educators.]
Points to consider:
· What are the strengths of this document?
· What areas need improvement, and what suggestions do you make?
· Would you be willing to work on revisions?
· Would you like your name to be included as a contributor in the final document?
· Please provide your name, affiliation and e-mail address in the MEME discussion.
Mathematics For All is still very much a “work in progress.” Further editions of Mathematics For All will evolve as we work together to make this the document guiding mathematics instruction in Massachusetts.
Table of Contents
A Message to Readers of Mathematics For All / 2
Table of Contents / 3
Acknowledgements / 4
Preface and History of Development / 5-6
Organization of Mathematics For All / 7
Guiding Philosophy / 8-10
Habits of Mind / 11
Guiding Principles / 12-20
Strand Overviews
Strand 1: Number Sense and Operations / 21-23
Strand 2: Patterns, Functions and Algebra / 24
Strand 3: Geometry / 25
Strand 4: Data Analysis, Statistics and Probability / 26
Strand 5: Measurement / 27
Standards and Learning Expectations
PreK-K / 28-30
Grades 1-2 / 31-34
Grades 3-4 / 35-39
Grades 5-6 / 40-43
Grades 7-8 / 44-48
Grades 9-10 / 49-51
Grades 11-12 / 52
Appendix A: Criteria for Evaluating Instructional Materials and Programs in Mathematics / 53-55
Many contributors to Mathematics for All have worked closely with the Massachusetts Department of Education creating both the 1996 and the 2000 editions of the Mathematics Curriculum Frameworks.
Therefore, similar text occasionally appears in Mathematics for All and the Department’s documents.
Acknowledgements
Mathematics For All was developed by a group of teachers and administrators, mathematicians, university mathematics educators and other community members. Many more people lent their support over the past nine months of work, but we did not want to include their names without their permission. We hope that our list will grow exponentially as Mathematics For All is distributed and critiqued by many more Massachusetts mathematics educators.
Endorsement by Massachusetts’ Mathematics Organizations for Teachers
Dwayne Cameron, President: Association of Teachers of Mathematics in Massachusetts
Claire Zalewski Graham, President: Boston Area Mathematics Specialists
Spike Clancy, President: Association of Teachers of Mathematics in Western Massachusetts
Lyn Heady, President-Elect: Association of Teachers of Mathematics in Western Massachusetts
Ruth O'Malley, President: Association of Teachers of Mathematics in New England
Mathematics For All
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Leadership Endorsement from Past DOE Statewide Mathematics Coordinators
Anne Collins, Boston College (1997-1999)
Peg Bondorew, CESAME, Northeastern University (1993-1995)
Gisele Zangari, Boston University Academy (1995-1997)
Contributors Who Have Worked on Mathematics For All
Mathematics For All
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Jane Albert, Concord Public Schools
Sheldon Berman, Hudson Public Schools
Peg Bondorew, Northeastern University
John Bookston, Boston Public Schools
Nancy Buell, Brookline Public Schools
Michael Bresnahan, Cambridge Public Schools
Arthur Camins, Hudson Public Schools
Ricky Carter, ARC Implementation Center
Maureen Chapman-Fahey, Medford Public Schools
Rose Christiansen, Brookline Public Schools
Spike Clancy, Ludlow Public Schools
Anne Collins, Boston College
Mary Eich, Newton Public Schools
Rebeka Eston, Lincoln Public Schools
Patricia C. Foley, Westborough Public Schools
Thomas E. Foley, Waltham Public Schools
Christine Francis, Concord Public Schools
Maurice Gilmore, Northeastern University
Claire Zalewski Graham, Framingham State College
Carole Greenes, Boston University
Claire Groden, Watertown Public Schools
Barbara Haig, Northborough Public Schools
Maggi Hartnett, Ayer Public Schools
Deborah Hughes Hallett, University of Arizona
James Hamos, Univ. of Massachusetts Medical School
Mary Hogan, Boston College
Neelia Jackson, Boston Public Schools
James Kaput, University of Massachusetts-Dartmouth
Bill Kendall, Braintree Public Schools
Margaret Kenney, Boston College
Mary Jo Livingstone, Weymouth High School
Christopher Martes, MASS
Cliff Martin, Whitman Hanson High School
Joan Martin, Newton Public Schools
William Masalski, Univ. of Massachusetts-Amherst
Nancy McLaughlin, Lawrence Public Schools
Jan Mokros, TERC
Gloria Moran, Bridgewater-Raynham Public Schools
Christine Moynihan, Wayland Public Schools
Blake Munro, Wellesley Public Schools
Nancy Nichols, Saugus Public Schools
Ruth O'Malley, ATMNE
Margaret Riddle, Northampton Public Schools
Jan Rook, Boston Public Schools
Leanna Russell, E. Bridgewater High School
Mary Sapienza, Newton North High School
Paula Sennett, Silver Lake Regional Public School
Debra Shein-Gerson, Brookline Public Schools
Victor Steinbok, Boston University
J. Bryan Sullivan, Hudson Public Schools
Karen Tripoli, Lexington Public Schools
Rhonda Weinstein, Brookline Public Schools
Susan Weiss, Solomon Schechter Day School, Newton
Carolyn Wyatt, Newton Public Schools
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We wish to acknowledge the consistent and substantive support of our friends and colleagues at the Massachusetts Teachers Association, particularly Laura Barrett, Ralph Devlin, and Kathleen Skinner. Their encouragement has kept us going at critical junctures in our process, and their knowledge and resources have been crucial as we move towards truly making Mathematics for All a key document for teachers across the Commonwealth.
Mathematics For All
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Preface and History of Development
Mathematics For All is a framework for mathematics curriculum that was developed by a group of teachers, administrators, mathematicians, university mathematics educators, and other community members who worked under the leadership of a Steering Committee of representatives of the Massachusetts Advisory Council for Mathematics and Science, the Massachusetts Association of School Superintendents (M.A.S.S.), the Association of Teachers of Mathematics in Massachusetts (ATMIM), and the Association of Teachers of Mathematics in Western Massachusetts (MATHWEST). This committee was appointed by the Commissioner of Education in February, 2000, and was given the task of providing recommendations for the draft Massachusetts Mathematics Framework that would ensure mathematical appropriateness and align the draft document with the new National Council of Teachers of Mathematics Principles and Standards for School Mathematics, released in April, 2000. While a revised Mathematics Framework was approved by the Board of Education in July 2000, those of us who have worked with the Department continue to believe that there is a need for a standards document that reflects Principles and Standards for School Mathematics, a national endeavor based on current research in mathematics education. Mathematics For All is based on this as well as other documents that share the vision that all students must have access to high quality mathematics programs that support successful learning of mathematics and help them develop a mathematical sense and intuitive understanding. This document also continues to support the Policy statement on Mathematics and Science Education, adopted by the Massachusetts Board of Education in 1992: Mathematics and science as academic disciplines and tools for problem solving are central to the vitality of the economy and quality of life. They offer students of all ages opportunities to embark on adventures that stimulate the intellect and imagination.
The first Massachusetts Mathematics Framework, Achieving Mathematical Power, adopted in June 1996, was based upon two reform initiatives in Massachusetts, the Education Reform Act of 1993 and Partnerships Advancing the Learning of Mathematics and Science (PALMS). PALMS is the Statewide Systemic Initiative, a collaborative effort jointly funded by the National Science Foundation and the Commonwealth of Massachusetts, which began in 1992. Of the seven initial goals for this initiative, the first was to develop, disseminate, and implement curriculum frameworks in Mathematics and Science & Technology. With the passage of the Massachusetts Education Reform Act in June 1993 and additional funding from the U.S. Department of Education, development of the curriculum frameworks was extended to include grades 9-12 and Adult Basic Education.
The creation of Massachusetts’ first mathematics framework was a collaborative endeavor among members of the Framework Development Committee--teachers, school and district administrators, mathematicians, college faculty, parents, and representatives of business and community organizations across the state. A majority of the members were classroom teachers with extensive experience teaching mathematics at elementary, middle, and high school levels. Committee meetings were convened to consider each draft framework from the standpoints of clarity, accessibility, consistency, pedagogy, mathematical correctness, and alignment with the Massachusetts Common Core of Learning and the National Council of Teachers of Mathematics Curriculum and Evaluation Standards.
The core concept of Achieving Mathematical Power was that students develop mathematical power through problem solving, communication, reasoning and connections. The mathematics framework was more than a collection of concepts and skills. For each individual it involved methods of investigating and reasoning, means of communication, notions of context, and development of personal self-confidence. The framework provided quality and equity for all learners. The Guiding Principles and Habits of Mind of this framework outlined ways in which this could become a reality.
The Mathematics Content section presented an outline upon which district and school curricula, instruction, and assessment could be developed. Examples of student learning, vignettes, models, diagrams, and graphics contextualized and enhanced the standards. The goals for all learners in the Massachusetts mathematics framework were that they value mathematics, become confident in their ability to know and to do mathematics, become mathematical problem solvers, and learn to reason and communicate mathematically. The document presented here, Mathematics For All, is an attempt to build on the strengths of Achieving Mathematical Power and continue the Vision espoused by PALMS:
"All Massachusetts students will receive a high quality, hands-on education in mathematics and in science & technology that empowers them to be productive, problem-solving citizens and workers. Partnerships among businesses, institutions of higher education, policy makers, governmental agencies, cultural institutions, teachers and families will create a rich learning environment and provide a lasting foundation for continual improvement.
Challenging standards for content, teaching methods and equity defined in statewide curriculum frameworks will guide district practice. Learning will be active, built on discovery and reflection and a variety of assessments will test for understanding. New teachers will enter the profession with a solid grounding in mathematics and science content and in effective strategies for engaging a diversity of learners. Experienced teachers will continually deepen their knowledge and professional skills. PALMS will be the vanguard of education reform in Massachusetts." (www.doe.mass.edu/palms)
In November 1998, the Department of Education convened a Revision Committee of mathematics teachers, mathematicians, and university mathematics educators to examine Achieving Mathematical Power and to make recommendations that would provide additional guidance to the school districts. Their recommendations included more specificity for the content standards in terms of grade level spans, as well as clarification of many of the original standards, to help guide the ongoing development of the Massachusetts Comprehensive Assessment System. The standards developed by the Revision Committee formed the basis for the additional work of the group of teachers and administrators, mathematicians, university mathematics professors, and other community members who have developed Mathematics For All.
In April 2000, the National Council of Teachers of Mathematics released the Principles and Standards for School Mathematics,[1] a document that builds on and extends the original NCTM Standards documents. Educational research served as a basis for many of the proposals and claims made in the Principles and Standards about what is possible for students to learn about certain content areas at certain levels and under certain pedagogical conditions. The philosophy in Mathematics For All is directly aligned with that in the Principles and Standards, as well as the Vision espoused by PALMS. Indeed, the writers of Mathematics For All believe that Massachusetts’ standards should be the NCTM Learning Standards in the Principles and Standards for School Mathematics. All these efforts call for a common foundation of mathematics to be learned by all students. Mathematics For All is the result of educators’ zeal to provide that foundation.