Curriculum Focal Points

Curriculum Focal Points

for

Prekindergarten

through

Grade 8 Mathematics

A Quest for Coherence

Curriculum Focal Points

for

Prekindergarten

through

Grade 8 Mathematics

Curriculum Focal Points

for

Prekindergarten

through

Grade 8 Mathematics

A Quest for Coherence

The National Council of Teachers of Mathematics, Inc.

1906 Association Drive, Reston, VA 20191-1502

(703) 620-9840; (800) 235-7566; www.nctm.org

Library of Congress Cataloging-in-Publication Data

National Council of Teachers of Mathematics.

Curriculum focal points for prekindergarten through grade 8 mathematics :

a quest for coherence / National Council of Teachers of Mathematics.

p. cm.

Includes bibliographical references.

ISBN 0-87353-595-2

1. Mathematics—Study and teaching—United States—Evaluation. 2.

Curriculum evaluation. I. Title.

QA13.N365 2006

372.7043—dc22

2006019201

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing

vision, leadership, and professional development to support teachers in ensuring mathematics learning

of the highest quality for all students.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherenceis

official position of the National Council of Teachers of Mathematics as approved by its Board of Directors,

April 2006.

Permission to photocopy limited material from

Curriculum Focal Points for Prekindergarten through

Grade 8 Mathematics: A Quest for Coherence

is granted for educational purposes. Permission must

be obtained when content from this publication is used commercially, when the material is quoted in

advertising, when portions are used in other publications, or when charges for copies are made. The

use of material from

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics:

A Quest for Coherence

, other than in those cases described, should be brought to the attention of the

National Council of Teachers of Mathematics.

Printed in the United States of America

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Table of Contents

Preface ................................................................................................................................................................................

vii

Acknowledgments ............................................................................................................................................................ix

Introduction ........................................................................................................................................................................1

1. Why Identify Curriculum Focal Points? ................................................................................................................3

2. What Are Curriculum Focal Points?

.......................................................................................................................5

3. How Should Curriculum Focal Points Be Used? ..................................................................................................7

4. How Do the Curriculum Focal Points Relate to

Principles and

Standards for School Mathematics

? ........................................................................................................................

5. Curriculum Focal Points for Mathematics in Prekindergarten

through Grade 8 ........................................................................................................................................................10

Curriculum Focal Points and Connections for Prekindergarten.............................................................

Curriculum Focal Points and Connections for Kindergarten..................................................................

Curriculum Focal Points and Connections for Grade 1 ............................................................................13

Curriculum Focal Points and Connections for Grade 2............................................................................

Curriculum Focal Points and Connections for Grade 3............................................................................

Curriculum Focal Points and Connections for Grade 4............................................................................

Curriculum Focal Points and Connections for Grade 5............................................................................

Curriculum Focal Points and Connections for Grade 6............................................................................

Curriculum Focal Points and Connections for Grade 7............................................................................

Curriculum Focal Points and Connections for Grade 8............................................................................

Appendix............................................................................................................................................................................21

A Comparison of the Curriculum Focal Points and Connections with the

Expectations of the Content Standards in

Principles and Standards for

School Mathematics

References..........................................................................................................................................................................41

As states and local school districts implement more rigorous assessment and accountability systems,

teachers often face long lists of mathematics topics or learning expectations to address at each grade level,

with many topics repeating from year to year. Lacking clear, consistent priorities and focus, teachers stretch to

find the time to present important mathematical topics effectively and in depth.

The National Council of Teachers of Mathematics (NCTM) is responding to this challenge by presenting

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence.

Building

Principles and Standards for School Mathematics

(NCTM 2000), this new publication is offered as a start -

ing point in a dialogue on what is important at particular levels of instruction and as an initial step toward a

more coherent, focused curriculum in this country.

The writing team for

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

consisted

of nine members, with at least one university-level mathematics educator or mathematician and one pre-K–8

classroom practitioner from each of the three grade bands (pre-K–grade 2, grades 3–5, and grades 6–8). The

writing team examined curricula from multiple states and countries as well as a wide array of researchers’ and

experts’ writings in creating a set of focal points for pre-K–grade 8 mathematics.

On behalf of the Board of Directors, we thank everyone who helped make this publication possible.

Cathy Seeley

President, 2004–2006

National Council of Teachers of Mathematics

Francis (Skip) Fennell

President, 2006–2008

National Council of Teachers of Mathematics

Preface

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Members of the Writing Team

Jane F. Schielack,

Chair,

Texas A&M University, College Station, Texas

Sybilla Beckman, University of Georgia, Athens, Georgia

Randall I. Charles, San José State University (emeritus), San José, California

Douglas H. Clements, University at Buffalo, State University of New York, Buffalo, New York

Paula B. Duckett, District of Columbia Public Schools (retired), Washington, D.C.

Francis (Skip) Fennell, McDaniel College, Westminster, Maryland

Sharon L. Lewandowski, Bryant Woods Elementary School, Columbia, Maryland

Emma Treviño, Charles A. Dana Center, University of Texas at Austin, Austin, Texas

Rose Mary Zbiek, The Pennsylvania State University, University Park, Pennsylvania

Staff Liaison

Melanie S. Ott, National Council of Teachers of Mathematics, Reston, Virginia

iii

Preface

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Drafts of

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coher-

ence

were shared with a diverse group of mathematicians, mathematics educators, curriculum developers,

policymakers, and classroom practitioners, who provided formal reviews. In addition, many other individu-

als and collegial groups committed to improving pre-K–12 mathematics teaching and learning offered their

perceptions and comments informally. The Board of Directors and the writing team are grateful to all the

reviewers who shared their expertise. The comments of the reviewers do not constitute endorsement of the

final document.

We extend sincere thanks to the following individuals, who offered their insights, perspectives, and

advice in formal reviews of the first draft of

Curriculum Focal Points for Prekindergarten through Grade 8

Mathematics.

Their diverse commentary provided helpful guidance that made the final publication stronger,

clearer, and more meaningful.

David Bressoud, Macalester College, St. Paul, Minnesota

William Bush, University of Louisville, Louisville, Kentucky

Anne Collins, Lesley University, Cambridge, Massachuettes

Joan Ferrini-Mundy, Michigan State University, East Lansing, Michigan

Linda Gojak, John Carroll University, University Heights, Ohio

Jeremy Kilpatrick, University of Georgia, Athens, Georgia

Denise Mewborn, University of Georgia, Athens, Georgia

Anne Mikesell, Ohio Department of Education (retired), Columbus, Ohio

R. James Milgram, Stanford University, Stanford, California

Barbara Reys, University of Missouri–Columbia, Columbia, Missouri

J. Michael Shaughnessy, Portland State University, Portland, Oregon

Norma Torres-Martinez, Texas Education Agency, Austin, Texas

Norman Webb, University of Wisconsin–Madison, Madison, Wisconsin

Barbara G. Wells, University of California, Los Angeles (UCLA), Los Angeles, California

Acknowledgments

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Nancy Acconciamessa

Susan Addington

Richard Askey

Deborah Loewenberg Ball

Thomas Banchoff

Hyman Bass

Michael T. Battista

Gail Burrill

John Carter

Dinah Chancellor

Al Cuoco

Jerome Dancis

Valerie DeBellis

Cathie Dillender

John Dossey

Jerry Dwyer

Karen Fuson

E. Paul Goldenberg

Eric Hart

Wayne Harvey

David W. Henderson

Cheryl Hlavsa

Roger Howe

Susan Hudson Hull

Lisa Kasmer

Catherine Kelly

Cliff Konold

Glenda Lappan

Steve Leinwand

Mary Lindquist

Johnny Lott

Frank Marburger

Robert McIntosh

Gregg McMann

Debbie Nix

Jana Palmer

Caroline Piangerelli

Gerald R. Rising

Joseph Rosenstein

Susan Jo Russell

Yoram Sagher

Kay B. Sammons

Richard Schaar

Janet K. Scheer

William Schmidt

Marjorie Senechal

Nina Shteingold

Dorothy Strong

Maria Terrell

John Van de Walle

Patsy Wang-Iverson

Virginia M. Warfield

Donna Watts

Iris Weiss

Grayson H. Wheatley

W. Stephen Wilson

We also offer thanks to the following individuals, who examined various versions of the manuscript and

commented informally:

Acknowledgments

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

In 1980 the National Council of Teachers of Mathematics (NCTM) published

An Agenda for Action

(NCTM 1980), launching an era of bold professional outreach by describing the shape that school mathemat -

ics programs should take. That publication outlined ten recommendations for K–12 mathematics programs,

focusing on the fundamental need of students to learn how to solve problems. In 1989, the Council published

Curriculum and Evaluation Standards for School Mathematics

(NCTM 1989), expanding these recommenda -

tions into a vision for mathematics teaching and learning in K–grade 4, grades 5–8, and grades 9–12.

Cur-

riculum and Evaluation Standards

provided major direction for states and school districts in developing their

curriculum guidelines.

Principles and Standards for School Mathematics

(NCTM 2000) followed at the turn

of the new century, adding underlying principles for school mathematics and clarifying and elaborating on the

1989 Standards for pre-K–grade 2, grades 3–5, grades 6–8, and grades 9–12.

Principles and Standards for School Mathematics

remains the comprehensive reference on developing

mathematical knowledge across the grades, and the Council continues to produce numerous related publica-

tions and services to support, expand, and illuminate this work.

Curriculum Focal Points for Prekindergarten

through Grade 8 Mathematics: A Quest for Coherence

extends the Council’s leadership of more than twenty-

five years by describing an approach to curriculum development that focuses on areas of emphasis within

each grade from prekindergarten through grade 8.

An approach that focuses on a small number of significant mathematical “targets” for each grade level

offers a way of thinking about what is important in school mathematics that is different from commonly ac -

cepted notions of goals, standards, objectives, or learning expectations. These more conventional structures

tend to result in lists of very specific items grouped under general headings. By contrast,

Curriculum Focal

Points for Prekindergarten through Grade 8 Mathematics

offers more than headings for long lists, providing

instead descriptions of the most significant mathematical concepts and skills at each grade level. Organiz-

ing a curriculum around these described focal points, with a clear emphasis on the processes that

Principles

and Standards

addresses in the Process Standards—communication, reasoning, representation, connections,

and, particularly, problem solving—can provide students with a connected, coherent, ever expanding body of

mathematical knowledge and ways of thinking. Such a comprehensive mathematics experience can prepare

students for whatever career or professional path they may choose as well as equip them to solve many prob-

lems that they will face in the future.

The curriculum focal points presented here offer both immediate and long-term opportunities for im

proving the teaching and learning of mathematics. They provide ideas that may kindle fruitful discussions

among teacher leaders and teachers about areas to emphasize as they consider the developmental needs of

their students and examine a year’s program of instruction. Teachers might also see opportunities to develop

or select lessons that bring together related topics in meaningful contexts to reinforce or extend the most

Introduction

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

important connections, understandings, and skills. The long-term opportunity, however, is for mathemat -

ics leaders at every level to use

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

launch an ongoing, far-reaching, significant discussion with the potential to guide the thinking of the profes -

sion in the development of the next generation of curriculum standards, textbooks, and tests. This work may

assist in the creation and eventual development of new models for defining curriculum, organizing instruc-

tion, developing materials, and creating meaningful assessments that can help students learn critical math-

ematical skills, processes, and ways of thinking and can measure and communicate what students know about

the mathematics that we expect them to learn.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

thus represents an important,

initial step in advancing collaborative discussions about what mathematics students should know and be able

to do. Use the focal points presented here to guide discussions as you review, refine, and revise mathemat-

ics curricula. Take this opportunity to share the best that we know as we work together to produce improved

tools that support our shared goal of a high-quality mathematics education for every student.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

The National Council of Teachers of Mathematics produced

Principles and Standards for School Math-

ematics

(NCTM 2000) to update and extend the recommendations for learning and teaching mathematics

that had appeared in

Curriculum and Evaluation Standards for School Mathematics

(NCTM 1989),

Profes-

sional Standards for Teaching Mathematics

(NCTM 1991), and

Assessment Standards for School Mathematics

(NCTM 1995).

Principles and Standards

enunciated the Curriculum Principle, which states, “A curriculum is

more than a collection of activities: it must be coherent, focused on important mathematics, and well articu-

lated across the grades” (p. 14). Specifically, “a well-articulated curriculum gives teachers guidance regarding

important ideas or major themes, which receive special attention at different points in time. It also gives guid-

ance about the depth of study warranted at particular times and when closure is expected for particular skills

or concepts” (p. 16).

This definition of curriculum articulation echoes a central question that occupies state and local leaders

in mathematics education:

What mathematics should be the focus of instruction and learning at particular

grade levels of the pre-K–12 educational system?

Principles and Standards

states, “Those who design cur -

riculum frameworks, assessments, instructional materials, and classroom instruction based on

Principles and

Standards

will need to make their own decisions about emphasis and order” (p. 31).

Curriculum Focal Points

for Prekindergarten through Grade 8 Mathematics

provides one possible response to the question of how to

organize curriculum standards within a coherent, focused curriculum, by showing how to build on important

mathematical content and connections identified for each grade level, pre-K–8.

Inconsistency in the Placement of Topics by Grade Level in

U.S. Mathematics Curricula

Analysis of curricula of countries participating in the Third International Mathematics and Science

Study (TIMSS [1997]; now known as the Trends in International Mathematics and Science Study) led to the

familiar description of school mathematics in the United States as “a mile wide and an inch deep” (Schmidt,

McKnight, and Raizen 1997). In addition, research on the curricular expectations of states and school systems

across the country indicates inconsistency in the grade placements of mathematics topics, as well as in how

they are defined and what students are expected to learn.

State and local districts, with varying resources for providing leadership in mathematics education, have

been working fairly independently to develop student learning expectations, as required by the federal law No

Child Left Behind

(2002). The result has been a wide variety of mathematics curriculum standards, with little

consensus on the placement or emphasis of topics within specific grade levels (Reys et al. 2005). For example,

in a study of the mathematics curriculum standards of ten states (Reys et al. 2006), the total number of grade-

level expectations in mathematics for grade 4 ranged from 26 to 89 (see table 1).

Why Identify

Curriculum Focal

Points?

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Table 1. Number of Fourth-Grade Learning Expectations (LEs) per State by Content Strand

(from Reys et al. 2006, p. 20)

Number &

Operations

Geometry

Measurement

Algebra

Data Analysis,

Prob & Stat

Total Number

of LEs

California

Texas

New York

Florida

Ohio

Michigan

New Jersey

North Carolina

Georgia

Virginia

The Importance of Curricular Focus in Mathematics

Many factors have contributed to the need for a common mathematical focus for each grade level,

pre-K–8. These include the increased emphasis on accountability testing, high levels of mobility of both

students and teachers, and greater costs of curriculum development. A focused, coherent mathematics cur-

riculum with a national scope has the potential to ease the impact of widely varying learning and assessment

expectations on both students and teachers who relocate. In addition, a focused curriculum would allow

teachers to commit more time each year to topics receiving special emphasis. At the same time, students

would have opportunities to explore these topics in depth, in the context of related content and connected ap-

plications, thus developing more robust mathematical understandings.

In a survey of employees of forty-seven educational agencies—those responsible for improving cur

riculum and instruction in their states—85 percent of the respondents indicated that “national leadership is

needed to assist in future articulation of learning expectations in mathematics, particularly from national pro-

fessional organizations of mathematics teachers (K–12 and university) and mathematicians” (Reys et al. 2005,

p. 17). This publication addresses that need.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Curriculum focal points are important mathematical topics for each grade level, pre-K–8. These areas

of instructional emphasis can serve as organizing structures for curriculum design and instruction at and

across grade levels. The topics are central to mathematics: they convey knowledge and skills that are essential

to educated citizens, and they provide the foundations for further mathematical learning. Because the focal

points are core structures that lay a conceptual foundation, they can serve to organize content, connecting

and bringing coherence to multiple concepts and processes taught at and across grade levels. They are

indispensable elements in developing problem solving, reasoning, and critical thinking skills, which are

important to all mathematics learning.

When instruction focuses on a small number of key areas of emphasis, students gain extended experience

with core concepts and skills. Such experience can facilitate deep understanding, mathematical fluency, and

an ability to generalize. The decision to organize instruction around focal points assumes that the learning

of mathematics is cumulative, with work in the later grades building on and deepening what students have

learned in the earlier grades, without repetitious and inefficient reteaching. A curriculum built on focal points

also has the potential to offer opportunities for the diagnosis of difficulties and immediate intervention, thus

helping students who are struggling with important mathematics content.

What characteristics qualify a concept or topic to be a curriculum focal point? For inclusion in

Curricu-

lum Focal Points for Prekindergarten through Grade 8 Mathematics

, a focal point had to pass three rigorous

tests:

•

Is it mathematically important, both for further study in mathematics and for use in applications in

and outside of school?

•

Does it “fit” with what is known about learning mathematics?

•

Does it connect logically with the mathematics in earlier and later grade levels?

A curriculum focal point may draw on several connected mathematical content topics described in

Principles

and Standards for School Mathematics

(NCTM 2000). It should be addressed by students in the context of the

mathematical processes of problem solving, reasoning and proof, communication, connections, and represen-

tation. Without facility with these critical processes, a student’s mathematical knowledge is likely to be fragile

and limited in its usefulness.

A complete set of curriculum focal points, situated within the processes of mathematics, can provide an

outline of an integrated mathematics curriculum that is different from the outline created by a set of grade-

level mastery objectives or a list of separated content and process targets. In contrast with grade-level mastery

objectives, which can be interpreted as endpoints for learning, curriculum focal points are clearly areas of

emphasis, calling for instruction that will help students learn content that gives them a foundation for increas-

ing their understanding as they encounter richer and more challenging mathematics.

What Are

Curriculum Focal

Points?

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Instruction based on focal points would devote the vast majority of attention to the content identified for

special emphasis in a grade. A curriculum for pre-K–8 based on a connected set of such focal points could

provide a solid mathematical foundation for high school mathematics.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

highlights important math-

ematics at particular grade levels for the use of those who are responsible for the development of mathemat-

ics curricula, standards, and assessment—primarily the mathematics education leaders and policymakers

at the national, state, and local levels. As a result of the wide variation in the placement of topics in current

mathematics curricula (Reys et al. 2006), the grade-level designations of particular curriculum focal points in

this publication may not match the placement of the corresponding content in an existing curriculum. This

publication is presented as a framework on which the next generation of state and district-level mathematics

curricula might be built. Organizational strategies that embody the field’s best thinking, such as these focal

points, can serve as a catalyst to curriculum development and positively influence the design of materials for

instruction and assessment.

The set of curriculum focal points described here represents an attempt to provide curriculum developers

with a clear organizational model for establishing a mathematics curriculum from prekindergarten through

grade 8 by identifying for each grade level important content that can build connected and integrated math-

ematical understanding. The curriculum focal points and their accompanying “connections” to related content

outline instructional targets for a basic, integrated, grade-by-grade framework for a coherent mathematics

curriculum.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

does not specify instructional

approaches for the implementation of the suggested curriculum focal points. Its presentations of the focal

points include neither suggestions for tools to use in teaching nor recommendations for professional develop-

ment in content or pedagogy. The focal points cannot be used alone as lesson plans. Nor do they answer the

question, “What should I do in class on Monday?” Nevertheless, the curriculum focal points identified here

should be of considerable interest to teachers and other practitioners, as well as curriculum developers and

policymakers.

To achieve the best results with students when teaching for the depth, understanding, and proficiency

sought by the curriculum focal points, teachers themselves will need a deep understanding of the mathemat-

ics and facility with the relationships among mathematical ideas. Thus, effective instruction built on the cur -

riculum focal points requires in-depth preparation of preservice teachers and ongoing professional develop-

ment for in-service teachers.

How Should

Curriculum Focal

Points Be Used?

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Principles and Standards for School Mathematics

(NCTM 2000) describes the foundational mathematical

ideas on which the focal points in

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

rest and toward which they direct students’ learning.

Principles and Standards

remains the definitive refer -

ence on the development of mathematical content and processes across the grades. Since the publication of

this influential work in 2000, ideas like

coherence, focus, high expectations, computational fuency, represen-

tation,

and

important mathematics

have become regular elements in discussions about improving school

mathematics, and thinking about these ideas has evolved considerably. As the next step in devising resources

to support the development of a coherent curriculum, NCTM now offers a new publication, with a set of cur-

riculum focal points and connections for mathematics education in prekindergarten through grade 8.

Principles and Standards

includes a thorough discussion of the necessity for learning mathematical con-

tent through the processes of problem solving, reasoning and proof, communication, connections, and repre-

sentation. Although some of these processes may be evident in the descriptions of particular focal points, this

new publication primarily targets

content

. Its presentation of curriculum focal points assumes that the mathe-

matical processes described in

Principles and Standards

will be implemented in instruction that requires

students to discuss and validate their mathematical thinking; create and analyze a variety of representations

that illuminate the connections within the mathematics; and apply the mathematics that they are learning in

solving problems, judging claims, and making decisions.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

identifies three focal points at

each grade level. Each set of three focal points, together with the integrated content taught in the context of

the processes, should encompass the major portion of instruction at that grade level. The presentations of the

focal points for each grade level also identify “Connections to the Focal Points” in a column at the right. These

connections serve two purposes:

1. They recognize the need for introductory and continuing experiences related to focal points identified

for other grade levels.

2. They identify ways in which a grade level’s focal points can support learning in relation to strands that

are not focal points at that grade level.

The “Connections to the Focal Points” column for each grade level brings in other important topics in

meaningful ways. For example, the grade 2 “Connections” highlight the fact that the Measurement Focal Point

for grade 2 (“Developing an understanding of linear measurement and facility in measuring lengths”) includes

work with applications and models using the shapes from the Geometry Focal Point for grade 1 (“Composing

and decomposing geometric shapes”). At the same time, students in grade 2 continue to use vocabulary and

spatial reasoning that will be essential for learning the content specified in the Geometry Focal Point for grade

3 (“Describing and analyzing properties of two-dimensional shapes”). Because a curriculum that is integrated

How Do the

Curriculum Focal

Points Relate

Principles

and Standards

for School

Mathematics

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

and internally connected in this way uses related concepts and skills to support and enrich one or more focal

points at a grade level, it has the potential to maximize students’ learning.

Each focal point in this publication takes its name from the content strand or strands to which it relates in

Principles and Standards for School Mathematics.

Many focal points relate to more than one content strand,

highlighting the integrated nature of the curriculum focal points. That single focal points are often described

with a combination of items from different content strands in

Principles and Standards

reflects the fact that

Principles and Standards

itself presents “a connected body of mathematical understandings and competencies

… rather than a menu from which to make curricular choices” (p. 29). Color-coded comparison charts in the

appendix illustrate the extent to which the curriculum focal points and their connections include content that