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

Principles and Standards

expects instruction to address in the corresponding grade bands.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Three curriculum focal points are identified and described for each grade level, pre-K–8, along with con -

nections to guide integration of the focal points at that grade level and across grade levels, to form a compre-

hensive mathematics curriculum. To build students’ strength in the use of mathematical processes, instruction

in these content areas should incorporate—

•

the use of the mathematics to solve problems;

•

an application of logical reasoning to justify procedures and solutions; and

•

an involvement in the design and analysis of multiple representations to learn, make connections

among, and communicate about the ideas within and outside of mathematics.

The purpose of identifying these grade-level curriculum focal points and connections is to enable students

to learn the content in the context of a focused and cohesive curriculum that implements problem solving,

reasoning, and critical thinking.

These curriculum focal points should be considered as major instructional goals and desirable learning

expectations, not as a list of objectives for students to master. They should be implemented with the intention

of building mathematical competency for all students, bolstered by the pedagogical understanding that not

every student learns at the same rate or acquires concepts and skills at the same time.

Those who are involved in curriculum planning for grades 6–8 should note that this set of curriculum fo

cal points has been designed with the intention of providing a three-year middle school program that includes

a full year of general mathematics in each of grades 6, 7, and 8. Those whose programs offer an algebra course

in grade 8 (or earlier) should consider including the curriculum focal points that this framework calls for in

grade 8 in grade 6 or grade 7. Alternatively, these topics could be incorporated into the high school program.

Either way, curricula would not omit the important content that the grade 7 and grade 8 focal points offer

students in preparation for algebra and for their long-term mathematical knowledge.

Curriculum

Focal Points for

Mathematics in

Prekindergarten

through Grade 8

Curriculum Focal Points and Connections for Prekindergarten

The set of three curriculum focal points and related connections for mathematics in prekindergarten follow. These topics are the recommended content empha -

ses for this grade level. It is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections,

and designing and analyzing representations.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Prekindergarten Curriculum Focal Points

Number and Operations:

Developing an understanding of whole numbers, including

concepts of correspondence, counting, cardinality, and comparison

Children develop an understanding of the meanings of whole numbers and recognize the number of

objects in small groups without counting and by counting—the first and most basic mathematical

algorithm. Tey understand that number words refer to quantity. Tey use one-to-one correspondence

to solve problems by matching sets and comparing number amounts and in counting objects to 10 and

beyond. Tey understand that the last word that they state in counting tells “how many,” they count to

determine number amounts and compare quantities (using language such as “more than” and “less

than”), and they order sets by the number of objects in them.

Geometry:

Identifying shapes and describing spatial relationships

Children develop spatial reasoning by working from two perspectives on space as they examine the

shapes of objects and inspect their relative positions. Tey find shapes in their environments and

describe them in their own words. Tey build pictures and designs by combining two- and three-

dimensional shapes, and they solve such problems as deciding which piece will fit into a space in a

puzzle. Tey discuss the relative positions of objects with vocabulary such as “above,” “below,” and “next

to.”

Measurement:

Identifying measurable attributes and comparing objects by using these

attributes

Children identify objects as “the same” or “different,” and then “more” or “less,” on the basis of

attributes that they can measure. Tey identify measurable attributes such as length and weight and

solve problems by making direct comparisons of objects on the basis of those attributes.

Connections to the Focal Points

Data Analysis:

Children learn the foundations of data

analysis by using objects’ attributes that they have

identified in relation to geometry and measurement (e.g.,

size, quantity, orientation, number of sides or vertices,

color) for various purposes, such as describing, sorting,

or comparing. For example, children sort geometric

figures by shape, compare objects by weight (“heavier,”

“lighter”), or describe sets of objects by the number of

objects in each set.

Number and Operations:

Children use meanings of

numbers to create strategies for solving problems and

responding to practical situations, such as getting just

enough napkins for a group, or mathematical situations,

such as determining that any shape is a triangle if it has

exactly three straight sides and is closed.

Algebra:

Children recognize and duplicate simple

sequential patterns (e.g., square, circle, square, circle,

square, circle,…).

Curriculum Focal Points and Connections for Kindergarten

The set of three curriculum focal points and related connections for mathematics in kindergarten follow. These topics are the recommended content emphases

for this grade level. It is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections,

and designing and analyzing representations.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Kindergarten Curriculum Focal Points

Number and Operations:

Representing, comparing, and ordering whole numbers and

joining and separating sets

Children use numbers, including written numerals, to represent quantities and to solve quantitative

problems, such as counting objects in a set, creating a set with a given number of objects, comparing

and ordering sets or numerals by using both cardinal and ordinal meanings, and modeling simple

joining and separating situations with objects. Tey choose, combine, and apply effective strategies for

answering quantitative questions, including quickly recognizing the number in a small set, counting

and producing sets of given sizes, counting the number in combined sets, and counting backward.

Geometry:

Describing shapes and space

Children interpret the physical world with geometric ideas (e.g., shape, orientation, spatial relations)

and describe it with corresponding vocabulary. Tey identify, name, and describe a variety of shapes,

such as squares, triangles, circles, rectangles, (regular) hexagons, and (isosceles) trapezoids presented

in a variety of ways (e.g., with different sizes or orientations), as well as such three-dimensional shapes

as spheres, cubes, and cylinders. Tey use basic shapes and spatial reasoning to model objects in their

environment and to construct more complex shapes.

Measurement:

Ordering objects by measurable attributes

Children use measurable attributes, such as length or weight, to solve problems by comparing and

ordering objects. Tey compare the lengths of two objects both directly (by comparing them with each

other) and indirectly (by comparing both with a third object), and they order several objects according

to length.

Connections to the Focal Points

Data Analysis:

Children sort objects and use one or

more attributes to solve problems. For example, they

might sort solids that roll easily from those that do not.

Or they might collect data and use counting to answer

such questions as, “What is our favorite snack?” Tey

re-sort objects by using new attributes (e.g., after sorting

solids according to which ones roll, they might re-sort

the solids according to which ones stack easily).

Geometry:

Children integrate their understandings of

geometry, measurement, and number. For example, they

understand, discuss, and create simple navigational

directions (e.g., “Walk forward 10 steps, turn right, and

walk forward 5 steps”).

Algebra:

Children identify, duplicate, and extend simple

number patterns and sequential and growing patterns

(e.g., patterns made with shapes) as preparation for

creating rules that describe relationships.

Curriculum Focal Points and Connections for Grade 1

The set of three curriculum focal points and related connections for mathematics in grade 1 follow. These topics are the recommended content emphases for

this grade level. It is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and

designing and analyzing representations.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Grade 1 Curriculum Focal Points

Number and Operations

and

Algebra:

Developing understandings of addition and subtrac-

tion and strategies for basic addition facts and related subtraction facts

Children develop strategies for adding and subtracting whole numbers on the basis of their earlier work

with small numbers. Tey use a variety of models, including discrete objects, length-based models (e.g.,

lengths of connecting cubes), and number lines, to model “part-whole,” “adding to,” “taking away from,”

and “comparing” situations to develop an understanding of the meanings of addition and subtraction

and strategies to solve such arithmetic problems. Children understand the connections between

counting and the operations of addition and subtraction (e.g., adding two is the same as “counting on”

two). Tey use properties of addition (commutativity and associativity) to add whole numbers, and they

create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to

solve addition and subtraction problems involving basic facts. By comparing a variety of solution

strategies, children relate addition and subtraction as inverse operations.

Number and Operations:

Developing an understanding of whole number relationships,

including grouping in tens and ones

Children compare and order whole numbers (at least to 100) to develop an understanding of and solve

problems involving the relative sizes of these numbers. Tey think of whole numbers between 10 and

100 in terms of groups of tens and ones (especially recognizing the numbers 11 to 19 as 1 group of ten

and particular numbers of ones). Tey understand the sequential order of the counting numbers and

their relative magnitudes and represent numbers on a number line.

Geometry:

Composing and decomposing geometric shapes

Children compose and decompose plane and solid figures (e.g., by putting two congruent isosceles

triangles together to make a rhombus), thus building an understanding of part-whole relationships as

well as the properties of the original and composite shapes. As they combine figures, they recognize

them from different perspectives and orientations, describe their geometric attributes and properties,

and determine how they are alike and different, in the process developing a background for measure-

ment and initial understandings of such properties as congruence and symmetry.

Connections to the Focal Points

Number and Operations

and

Algebra:

Children use

mathematical reasoning, including ideas such as commu-

tativity and associativity and beginning ideas of tens and

ones, to solve two-digit addition and subtraction

problems with strategies that they understand and can

explain. Tey solve both routine and nonroutine

problems.

Measurement

and

Data Analysis:

Children

strengthen their sense of number by solving problems

involving measurements and data. Measuring by laying

multiple copies of a unit end to end and then counting

the units by using groups of tens and ones supports

children’s understanding of number lines and number

relationships. Representing measurements and discrete

data in picture and bar graphs involves counting and

comparisons that provide another meaningful connection

to number relationships.

Algebra:

Trough identifying, describing, and applying

number patterns and properties in developing strategies

for basic facts, children learn about other properties of

numbers and operations, such as odd and even (e.g.,

“Even numbers of objects can be paired, with none left

over”), and 0 as the identity element for addition.

Curriculum Focal Points and Connections for Grade 2

The set of three curriculum focal points and related connections for mathematics in grade 2 follow. These topics are the recommended content emphases for

this grade level. It is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and

designing and analyzing representations.

Grade 2 Curriculum Focal Points

Number and Operations:

Developing an understanding of the base-ten numeration system

and place-value concepts

Children develop an understanding of the base-ten numeration system and place-value concepts (at

least to 1000). Teir understanding of base-ten numeration includes ideas of counting in units and

multiples of hundreds, tens, and ones, as well as a grasp of number relationships, which they demon-

strate in a variety of ways, including comparing and ordering numbers. Tey understand multidigit

numbers in terms of place value, recognizing that place-value notation is a shorthand for the sums of

multiples of powers of 10 (e.g., 853 as 8 hundreds + 5 tens + 3 ones).

Number and Operations

and

Algebra:

Developing quick recall of addition facts and related

subtraction facts and fluency with multidigit addition and subtraction

Children use their understanding of addition to develop quick recall of basic addition facts and related

subtraction facts. Tey solve arithmetic problems by applying their understanding of models of

addition and subtraction (such as combining or separating sets or using number lines), relationships

and properties of number (such as place value), and properties of addition (commutativity and associa-

tivity). Children develop, discuss, and use efficient, accurate, and generalizable methods to add and

subtract multidigit whole numbers. Tey select and apply appropriate methods to estimate sums and

differences or calculate them mentally, depending on the context and numbers involved. Tey develop

fluency with efficient procedures, including standard algorithms, for adding and subtracting whole

numbers, understand why the procedures work (on the basis of place value and properties of opera-

tions), and use them to solve problems.

Measurement:

Developing an understanding of linear measurement and facility in

measuring lengths

Children develop an understanding of the meaning and processes of measurement, including such

underlying concepts as partitioning (the mental activity of slicing the length of an object into equal-

sized units) and transitivity (e.g., if object A is longer than object B and object B is longer than object C,

then object A is longer than object C). Tey understand linear measure as an iteration of units and use

rulers and other measurement tools with that understanding. Tey understand the need for equal-

length units, the use of standard units of measure (centimeter and inch), and the inverse relationship

between the size of a unit and the number of units used in a particular measurement (i.e., children

recognize that the smaller the unit, the more iterations they need to cover a given length).

Connections to the Focal Points

Number and Operations:

Children use place value and

properties of operations to create equivalent representa-

tions of given numbers (such as 35 represented by 35

ones, 3 tens and 5 ones, or 2 tens and 15 ones) and to

write, compare, and order multidigit numbers. Tey use

these ideas to compose and decompose multidigit

numbers. Children add and subtract to solve a variety of

problems, including applications involving measurement,

geometry, and data, as well as nonroutine problems. In

preparation for grade 3, they solve problems involving

multiplicative situations, developing initial understand-

ings of multiplication as repeated addition.

Geometry

and

Measurement:

Children estimate,

measure, and compute lengths as they solve problems

involving data, space, and movement through space. By

composing and decomposing two-dimensional shapes

(intentionally substituting arrangements of smaller

shapes for larger shapes or substituting larger shapes for

many smaller shapes), they use geometric knowledge and

spatial reasoning to develop foundations for understand-

ing area, fractions, and proportions.

Algebra:

Children use number patterns to extend their

knowledge of properties of numbers and operations. For

example, when skip counting, they build foundations for

understanding multiples and factors.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Curriculum Focal Points and Connections for Grade 3

The set of three curriculum focal points and related connections for mathematics in grade 3 follow. These topics are the recommended content emphases for

this grade level. It is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and

designing and analyzing representations.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Grade 3 Curriculum Focal Points

Number and Operations

and

Algebra:

Developing understandings of multiplication and

division and strategies for basic multiplication facts and related division facts

Students understand the meanings of multiplication and division of whole numbers through the use of

representations (e.g., equal-sized groups, arrays, area models, and equal “jumps” on number lines for

multiplication, and successive subtraction, partitioning, and sharing for division). Tey use properties

of addition and multiplication (e.g., commutativity, associativity, and the distributive property) to

multiply whole numbers and apply increasingly sophisticated strategies based on these properties to

solve multiplication and division problems involving basic facts. By comparing a variety of solution

strategies, students relate multiplication and division as inverse operations.

Number and Operations:

Developing an understanding of fractions and fraction

equivalence

Students develop an understanding of the meanings and uses of fractions to represent parts of a whole,

parts of a set, or points or distances on a number line. Tey understand that the size of a fractional part

is relative to the size of the whole, and they use fractions to represent numbers that are equal to, less

than, or greater than 1. Tey solve problems that involve comparing and ordering fractions by using

models, benchmark fractions, or common numerators or denominators. Tey understand and use

models, including the number line, to identify equivalent fractions.

Geometry:

Describing and analyzing properties of two-dimensional shapes

Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and

connect these attributes to definitions of shapes. Students investigate, describe, and reason about

decomposing, combining, and transforming polygons to make other polygons. Trough building,

drawing, and analyzing two-dimensional shapes, students understand attributes and properties of

two-dimensional space and the use of those attributes and properties in solving problems, including

applications involving congruence and symmetry.

Connections to the Focal Points

Algebra:

Understanding properties of multiplication and

the relationship between multiplication and division is a

part of algebra readiness that develops at grade 3. Te

creation and analysis of patterns and relationships

involving multiplication and division should occur at this

grade level. Students build a foundation for later under-

standing of functional relationships by describing

relationships in context with such statements as, “Te

number of legs is 4 times the number of chairs.”

Measurement:

Students in grade 3 strengthen their

understanding of fractions as they confront problems in

linear measurement that call for more precision than the

whole unit allowed them in their work in grade 2. Tey

develop their facility in measuring with fractional parts of

linear units. Students develop measurement concepts and

skills through experiences in analyzing attributes and

properties of two-dimensional objects. Tey form an

understanding of perimeter as a measurable attribute and

select appropriate units, strategies, and tools to solve

problems involving perimeter.

Data Analysis:

Addition, subtraction, multiplication,

and division of whole numbers come into play as students

construct and analyze frequency tables, bar graphs,

picture graphs, and line plots and use them to solve

problems.

Number and Operations:

Building on their work in

grade 2, students extend their understanding of place

value to numbers up to 10,000 in various contexts.

Students also apply this understanding to the task of

representing numbers in different equivalent forms (e.g.,

expanded notation). Tey develop their understanding of

numbers by building their facility with mental computa-

tion (addition and subtraction in special cases, such as

2,500 + 6,000 and 9,000 – 5,000), by using computational

estimation, and by performing paper-and-pencil

computations.

Curriculum Focal Points and Connections for Grade 4

The set of three curriculum focal points and related connections for mathematics in grade 4 follow. These topics are the recommended content emphases for

this grade level. It is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and

designing and analyzing representations.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Grade 4 Curriculum Focal Points

Number and Operations

and

Algebra:

Developing quick recall of multiplication facts and

related division facts and fluency with whole number multiplication

Students use understandings of multiplication to develop quick recall of the basic multiplication facts

and related division facts. Tey apply their understanding of models for multiplication (i.e., equal-sized

groups, arrays, area models, equal intervals on the number line), place value, and properties of opera-

tions (in particular, the distributive property) as they develop, discuss, and use efficient, accurate, and

generalizable methods to multiply multidigit whole numbers. Tey select appropriate methods and

apply them accurately to estimate products or calculate them mentally, depending on the context and

numbers involved. Tey develop fluency with efficient procedures, including the standard algorithm,

for multiplying whole numbers, understand why the procedures work (on the basis of place value and

properties of operations), and use them to solve problems.

Number and Operations:

Developing an understanding of decimals, including the connec-

tions between fractions and decimals

Students understand decimal notation as an extension of the base-ten system of writing whole numbers

that is useful for representing more numbers, including numbers between 0 and 1, between 1 and 2,

and so on. Students relate their understanding of fractions to reading and writing decimals that are

greater than or less than 1, identifying equivalent decimals, comparing and ordering decimals, and

estimating decimal or fractional amounts in problem solving. Tey connect equivalent fractions and

decimals by comparing models to symbols and locating equivalent symbols on the number line.

Measurement:

Developing an understanding of area and determining the areas of two-

dimensional shapes

Students recognize area as an attribute of two-dimensional regions. Tey learn that they can quantify

area by finding the total number of same-sized units of area that cover the shape without gaps or

overlaps. Tey understand that a square that is 1 unit on a side is the standard unit for measuring area.

Tey select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that

involve estimating or measuring area. Students connect area measure to the area model that they have

used to represent multiplication, and they use this connection to justify the formula for the area of a

rectangle.

Connections to the Focal Points

Algebra:

Students continue identifying, describing, and

extending numeric patterns involving all operations and

nonnumeric growing or repeating patterns. Trough

these experiences, they develop an understanding of the

use of a rule to describe a sequence of numbers or

objects.

Geometry:

Students extend their understanding of

properties of two-dimensional shapes as they find the

areas of polygons. Tey build on their earlier work with

symmetry and congruence in grade 3 to encompass

transformations, including those that produce line and

rotational symmetry. By using transformations to design

and analyze simple tilings and tessellations, students

deepen their understanding of two-dimensional space.

Measurement:

As part of understanding two-

dimensional shapes, students measure and classify

angles.

Data Analysis:

Students continue to use tools from

grade 3, solving problems by making frequency tables,

bar graphs, picture graphs, and line plots. Tey apply

their understanding of place value to develop and use

stem-and-leaf plots.

Number and Operations:

Building on their work in

grade 3, students extend their understanding of place

value and ways of representing numbers to 100,000 in

various contexts. Tey use estimation in determining the

relative sizes of amounts or distances. Students develop

understandings of strategies for multidigit division by

using models that represent division as the inverse of

multiplication, as partitioning, or as successive subtrac-

tion. By working with decimals, students extend their

ability to recognize equivalent fractions. Students’ earlier

work in grade 3 with models of fractions and multiplica-

tion and division facts supports their understanding of

techniques for generating equivalent fractions and

simplifying fractions.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Curriculum Focal Points and Connections for Grade 5

Thie set of three curriculum focal points and related connections for mathematics in grade 5 follow. Thiese topics are the recommended content emphases for

this grade level. It is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and

designing and analyzing representations.

Grade 5 Curriculum Focal Points

Number and Operations

and

Algebra:

Developing an understanding of and fluency with

division of whole numbers

Students apply their understanding of models for division, place value, properties, and the relationship

of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable

procedures to find quotients involving multidigit dividends. Tey select appropriate methods and apply

them accurately to estimate quotients or calculate them mentally, depending on the context and

numbers involved. Tey develop fluency with efficient procedures, including the standard algorithm,

for dividing whole numbers, understand why the procedures work (on the basis of place value and

properties of operations), and use them to solve problems. Tey consider the context in which a

problem is situated to select the most useful form of the quotient for the solution, and they interpret it

appropriately.

Number and Operations:

Developing an understanding of and fluency with addition and

subtraction of fractions and decimals

Students apply their understandings of fractions and fraction models to represent the addition and

subtraction of fractions with unlike denominators as equivalent calculations with like denominators.

Tey apply their understandings of decimal models, place value, and properties to add and subtract

decimals. Tey develop fluency with standard procedures for adding and subtracting fractions and

decimals. Tey make reasonable estimates of fraction and decimal sums and differences. Students add

and subtract fractions and decimals to solve problems, including problems involving measurement.

Geometry

and

Measurement

and

Algebra:

Describing three-dimensional shapes and

analyzing their properties, including volume and surface area

Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of poly-

hedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces.

Students recognize volume as an attribute of three-dimensional space. Tey understand that they can

quantify volume by finding the total number of same-sized units of volume that they need to fill the

space without gaps or overlaps. Tey understand that a cube that is 1 unit on an edge is the standard

unit for measuring volume. Tey select appropriate units, strategies, and tools for solving problems that

involve estimating or measuring volume. Tey decompose three-dimensional shapes and find surface

areas and volumes of prisms. As they work with surface area, they find and justify relationships among

the formulas for the areas of different polygons. Tey measure necessary attributes of shapes to use

area formulas to solve problems.

Connections to the Focal Points

Algebra:

Students use patterns, models, and relation-

ships as contexts for writing and solving simple equations

and inequalities. Tey create graphs of simple equations.

Tey explore prime and composite numbers and discover

concepts related to the addition and subtraction of

fractions as they use factors and multiples, including

applications of common factors and common multiples.

Tey develop an understanding of the order of operations

and use it for all operations.

Measurement:

Students’ experiences connect their

work with solids and volume to their earlier work with

capacity and weight or mass. Tey solve problems that

require attention to both approximation and precision of

measurement.

Data Analysis:

Students apply their understanding of

whole numbers, fractions, and decimals as they construct

and analyze double-bar and line graphs and use ordered

pairs on coordinate grids.

Number and Operations:

Building on their work in

grade 4, students extend their understanding of place

value to numbers through millions and millionths in

various contexts. Tey apply what they know about

multiplication of whole numbers to larger numbers.

Students also explore contexts that they can describe

with negative numbers (e.g., situations of owing money

or measuring elevations above and below sea level.)

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Curriculum Focal Points and Connections for Grade 5

T e set of three curriculum focal points and related connections for mathematics in grade 5 follow. T ese topics are the recommended content emphases for