Curriculum Focal Points - A Quest for Coherence
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Curriculum Focal Points
for Prekindergarten through Grade 8 Mathematics
A Quest for Coherence
Curriculum Focal Points Prekindergarten through for Grade 8 Mathematics
Curriculum Focal Points Prekindergarten through for Grade 8 Mathematics A Quest for Coherence
Copyright © 2006 by
The National Council of Teachers of Mathematics, Inc.
1906 Association Drive, Reston, VA 20191-1502
(703) 620-9840; (800) 235-7566; www.nctm.org
All rights reserved
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 Coherence is an
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
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? . .......................................................................................................................8
5. Curriculum Focal Points for Mathematics in Prekindergarten
through Grade 8 . .......................................................................................................................................................10
Curriculum Focal Points and Connections for Prekindergarten..............................................................11
Curriculum Focal Points and Connections for Kindergarten...................................................................12
Curriculum Focal Points and Connections for Grade 1.............................................................................13
Curriculum Focal Points and Connections for Grade 2.............................................................................14
Curriculum Focal Points and Connections for Grade 3.............................................................................15
Curriculum Focal Points and Connections for Grade 4.............................................................................16
Curriculum Focal Points and Connections for Grade 5.............................................................................17
Curriculum Focal Points and Connections for Grade 6.............................................................................18
Curriculum Focal Points and Connections for Grade 7.............................................................................19
Curriculum Focal Points and Connections for Grade 8.............................................................................20
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
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics 
Preface 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
on 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 Francis (Skip) Fennell
President, 2004–2006 President, 2006–2008
National Council of Teachers of Mathematics National Council of Teachers of Mathematics
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics vii
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
viii Preface
Acknowledgments 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
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics ixWe also offer thanks to the following individuals, who examined various versions of the manuscript and
commented informally:
Nancy Acconciamessa John Dossey Steve Leinwand Richard Schaar
Susan Addington Jerry Dwyer Mary Lindquist Janet K. Scheer
Richard Askey Karen Fuson Johnny Lott William Schmidt
Deborah Loewenberg Ball E. Paul Goldenberg Frank Marburger Marjorie Senechal
Thomas Banchoff Eric Hart Robert McIntosh Nina Shteingold
Hyman Bass Wayne Harvey Gregg McMann Dorothy Strong
Michael T. Battista David W. Henderson Debbie Nix Maria Terrell
Gail Burrill Cheryl Hlavsa Jana Palmer John Van de Walle
John Carter Roger Howe Caroline Piangerelli Patsy Wang-Iverson
Dinah Chancellor Susan Hudson Hull Gerald R. Rising Virginia M. Warfield
Al Cuoco Lisa Kasmer Joseph Rosenstein Donna Watts
Jerome Dancis Catherine Kelly Susan Jo Russell Iris Weiss
Valerie DeBellis Cliff Konold Yoram Sagher Grayson H. Wheatley
Cathie Dillender Glenda Lappan Kay B. Sammons W. Stephen Wilson
Curriculum Focal Points for Prekindergarten through Grade
Acknowledgments
8 MathematicsIntroduction 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
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 to
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 Mathematics1 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-
Why Identify sional Standards for Teaching Mathematics (NCTM 1991), and Assessment Standards for School Mathematics
Curriculum Focal (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-
Points? 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? As 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).
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 & Data Analysis, Total Number
Operations Geometry Measurement Algebra Prob & Stat of LEs
California 16 11 4 7 5 43
Texas 15 7 3 4 3 32
New York 27 8 10 5 6 56
Florida 31 11 17 10 20 89
Ohio 15 8 6 6 13 48
Michigan 37 5 11 0 3 56
New Jersey 21 10 8 6 11 56
North Carolina 14 3 2 3 4 26
Georgia 23 10 5 3 4 45
Virginia 17 8 11 2 3 41
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 Mathematics2 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
What Are to educated citizens, and they provide the foundations for further mathematical learning. Because the focal
Curriculum 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
Points? 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.
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 Mathematics3 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
How Should at the national, state, and local levels. As a result of the wide variation in the placement of topics in current
Curriculum Focal 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
Points Be Used? 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.
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics 4 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-
How Do the ence on the development of mathematical content and processes across the grades. Since the publication of
Curriculum Focal this influential work in 2000, ideas like coherence, focus, high expectations, computational fluency, represen-
tation, and important mathematics have become regular elements in discussions about improving school
Points Relate mathematics, and thinking about these ideas has evolved considerably. As the next step in devising resources
to to support the development of a coherent curriculum, NCTM now offers a new publication, with a set of cur-
Principles 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-
and Standards tent through the processes of problem solving, reasoning and proof, communication, connections, and repre-
for School sentation. Although some of these processes may be evident in the descriptions of particular focal points, this
Mathematics? 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
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematicsand 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 5 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
Curriculum in these content areas should incorporate—
Focal Points for • the use of the mathematics to solve problems;
Mathematics in • 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
Prekindergarten among, and communicate about the ideas within and outside of mathematics.
through Grade 8 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.
10 Curriculum Focal Points for Prekindergarten through Grade 8 MathematicsCurriculum 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.
Prekindergarten Curriculum Focal Points Connections to the Focal Points
Data Analysis: Children learn the foundations of data
Number and Operations: Developing an understanding of whole numbers, including analysis by using objects’ attributes that they have
concepts of correspondence, counting, cardinality, and comparison identified in relation to geometry and measurement (e.g.,
Children develop an understanding of the meanings of whole numbers and recognize the number of size, quantity, orientation, number of sides or vertices,
color) for various purposes, such as describing, sorting,
objects in small groups without counting and by counting—the first and most basic mathematical
or comparing. For example, children sort geometric
algorithm. They understand that number words refer to quantity. They use one-to-one correspondence
figures by shape, compare objects by weight (“heavier,”
to solve problems by matching sets and comparing number amounts and in counting objects to 10 and “lighter”), or describe sets of objects by the number of
beyond. They understand that the last word that they state in counting tells “how many,” they count to objects in each set.
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. Number and Operations: Children use meanings of
numbers to create strategies for solving problems and
responding to practical situations, such as getting just
Geometry: Identifying shapes and describing spatial relationships enough napkins for a group, or mathematical situations,
Children develop spatial reasoning by working from two perspectives on space as they examine the such as determining that any shape is a triangle if it has
shapes of objects and inspect their relative positions. They find shapes in their environments and exactly three straight sides and is closed.
describe them in their own words. They build pictures and designs by combining two- and three- Algebra: Children recognize and duplicate simple
dimensional shapes, and they solve such problems as deciding which piece will fit into a space in a sequential patterns (e.g., square, circle, square, circle,
puzzle. They discuss the relative positions of objects with vocabulary such as “above,” “below,” and “next square, circle,…).
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. They identify measurable attributes such as length and weight and
solve problems by making direct comparisons of objects on the basis of those attributes.
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics 11Curriculum 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.
Kindergarten Curriculum Focal Points Connections to the Focal Points
Data Analysis: Children sort objects and use one or
Number and Operations: Representing, comparing, and ordering whole numbers and more attributes to solve problems. For example, they
joining and separating sets might sort solids that roll easily from those that do not.
Children use numbers, including written numerals, to represent quantities and to solve quantitative Or they might collect data and use counting to answer
such questions as, “What is our favorite snack?” They
problems, such as counting objects in a set, creating a set with a given number of objects, comparing
re-sort objects by using new attributes (e.g., after sorting
and ordering sets or numerals by using both cardinal and ordinal meanings, and modeling simple
solids according to which ones roll, they might re-sort
joining and separating situations with objects. They choose, combine, and apply effective strategies for the solids according to which ones stack easily).
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: Children integrate their understandings of
geometry, measurement, and number. For example, they
understand, discuss, and create simple navigational
Geometry: Describing shapes and space
directions (e.g., “Walk forward 10 steps, turn right, and
Children interpret the physical world with geometric ideas (e.g., shape, orientation, spatial relations) walk forward 5 steps”).
and describe it with corresponding vocabulary. They identify, name, and describe a variety of shapes, Algebra: Children identify, duplicate, and extend simple
such as squares, triangles, circles, rectangles, (regular) hexagons, and (isosceles) trapezoids presented number patterns and sequential and growing patterns
in a variety of ways (e.g., with different sizes or orientations), as well as such three-dimensional shapes (e.g., patterns made with shapes) as preparation for
as spheres, cubes, and cylinders. They use basic shapes and spatial reasoning to model objects in their creating rules that describe relationships.
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. They 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.
12 Curriculum Focal Points for Prekindergarten through Grade 8 MathematicsCurriculum 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.
Grade 1 Curriculum Focal Points Connections to the Focal Points
Number and Operations and Algebra: Developing understandings of addition and subtrac- Number and Operations and Algebra: Children use
tion and strategies for basic addition facts and related subtraction facts mathematical reasoning, including ideas such as commu-
tativity and associativity and beginning ideas of tens and
Children develop strategies for adding and subtracting whole numbers on the basis of their earlier work ones, to solve two-digit addition and subtraction
with small numbers. They use a variety of models, including discrete objects, length-based models (e.g., problems with strategies that they understand and can
lengths of connecting cubes), and number lines, to model “part-whole,” “adding to,” “taking away from,” explain. They solve both routine and nonroutine
and “comparing” situations to develop an understanding of the meanings of addition and subtraction problems.
and strategies to solve such arithmetic problems. Children understand the connections between Measurement and Data Analysis: Children
counting and the operations of addition and subtraction (e.g., adding two is the same as “counting on” strengthen their sense of number by solving problems
two). They use properties of addition (commutativity and associativity) to add whole numbers, and they involving measurements and data. Measuring by laying
create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to multiple copies of a unit end to end and then counting
solve addition and subtraction problems involving basic facts. By comparing a variety of solution the units by using groups of tens and ones supports
strategies, children relate addition and subtraction as inverse operations. children’s understanding of number lines and number
relationships. Representing measurements and discrete
Number and Operations: Developing an understanding of whole number relationships, data in picture and bar graphs involves counting and
including grouping in tens and ones comparisons that provide another meaningful connection
to number relationships.
Children compare and order whole numbers (at least to 100) to develop an understanding of and solve
Algebra: Through identifying, describing, and applying
problems involving the relative sizes of these numbers. They think of whole numbers between 10 and number patterns and properties in developing strategies
100 in terms of groups of tens and ones (especially recognizing the numbers 11 to 19 as 1 group of ten for basic facts, children learn about other properties of
and particular numbers of ones). They understand the sequential order of the counting numbers and numbers and operations, such as odd and even (e.g.,
their relative magnitudes and represent numbers on a number line. “Even numbers of objects can be paired, with none left
over”), and 0 as the identity element for addition.
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.
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics 13Curriculum 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 Connections to the Focal Points
Number and Operations: Developing an understanding of the base-ten numeration system Number and Operations: Children use place value and
and place-value concepts properties of operations to create equivalent representa-
tions of given numbers (such as 35 represented by 35
Children develop an understanding of the base-ten numeration system and place-value concepts (at ones, 3 tens and 5 ones, or 2 tens and 15 ones) and to
least to 1000). Their understanding of base-ten numeration includes ideas of counting in units and write, compare, and order multidigit numbers. They use
multiples of hundreds, tens, and ones, as well as a grasp of number relationships, which they demon- these ideas to compose and decompose multidigit
strate in a variety of ways, including comparing and ordering numbers. They understand multidigit numbers. Children add and subtract to solve a variety of
numbers in terms of place value, recognizing that place-value notation is a shorthand for the sums of problems, including applications involving measurement,
multiples of powers of 10 (e.g., 853 as 8 hundreds + 5 tens + 3 ones). geometry, and data, as well as nonroutine problems. In
preparation for grade 3, they solve problems involving
multiplicative situations, developing initial understand-
Number and Operations and Algebra: Developing quick recall of addition facts and related
ings of multiplication as repeated addition.
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 Geometry and Measurement: Children estimate,
subtraction facts. They solve arithmetic problems by applying their understanding of models of measure, and compute lengths as they solve problems
addition and subtraction (such as combining or separating sets or using number lines), relationships involving data, space, and movement through space. By
composing and decomposing two-dimensional shapes
and properties of number (such as place value), and properties of addition (commutativity and associa-
(intentionally substituting arrangements of smaller
tivity). Children develop, discuss, and use efficient, accurate, and generalizable methods to add and shapes for larger shapes or substituting larger shapes for
subtract multidigit whole numbers. They select and apply appropriate methods to estimate sums and many smaller shapes), they use geometric knowledge and
differences or calculate them mentally, depending on the context and numbers involved. They develop spatial reasoning to develop foundations for understand-
fluency with efficient procedures, including standard algorithms, for adding and subtracting whole ing area, fractions, and proportions.
numbers, understand why the procedures work (on the basis of place value and properties of opera-
tions), and use them to solve problems. Algebra: Children use number patterns to extend their
knowledge of properties of numbers and operations. For
Measurement: Developing an understanding of linear measurement and facility in example, when skip counting, they build foundations for
measuring lengths understanding multiples and factors.
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). They understand linear measure as an iteration of units and use
rulers and other measurement tools with that understanding. They 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).
14 Curriculum Focal Points for Prekindergarten through Grade 8 MathematicsCurriculum 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.
Grade 3 Curriculum Focal Points Connections to the Focal Points
Number and Operations and Algebra: Developing understandings of multiplication and Algebra: Understanding properties of multiplication and
division and strategies for basic multiplication facts and related division facts the relationship between multiplication and division is a
part of algebra readiness that develops at grade 3. The
Students understand the meanings of multiplication and division of whole numbers through the use of creation and analysis of patterns and relationships
representations (e.g., equal-sized groups, arrays, area models, and equal “jumps” on number lines for involving multiplication and division should occur at this
multiplication, and successive subtraction, partitioning, and sharing for division). They use properties grade level. Students build a foundation for later under-
of addition and multiplication (e.g., commutativity, associativity, and the distributive property) to standing of functional relationships by describing
multiply whole numbers and apply increasingly sophisticated strategies based on these properties to relationships in context with such statements as, “The
solve multiplication and division problems involving basic facts. By comparing a variety of solution number of legs is 4 times the number of chairs.”
strategies, students relate multiplication and division as inverse operations. Measurement: Students in grade 3 strengthen their
understanding of fractions as they confront problems in
Number and Operations: Developing an understanding of fractions and fraction linear measurement that call for more precision than the
equivalence whole unit allowed them in their work in grade 2. They
develop their facility in measuring with fractional parts of
Students develop an understanding of the meanings and uses of fractions to represent parts of a whole, linear units. Students develop measurement concepts and
parts of a set, or points or distances on a number line. They understand that the size of a fractional part skills through experiences in analyzing attributes and
is relative to the size of the whole, and they use fractions to represent numbers that are equal to, less properties of two-dimensional objects. They form an
than, or greater than 1. They solve problems that involve comparing and ordering fractions by using understanding of perimeter as a measurable attribute and
models, benchmark fractions, or common numerators or denominators. They understand and use select appropriate units, strategies, and tools to solve
models, including the number line, to identify equivalent fractions. problems involving perimeter.
Data Analysis: Addition, subtraction, multiplication,
Geometry: Describing and analyzing properties of two-dimensional shapes and division of whole numbers come into play as students
construct and analyze frequency tables, bar graphs,
Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and
picture graphs, and line plots and use them to solve
connect these attributes to definitions of shapes. Students investigate, describe, and reason about problems.
decomposing, combining, and transforming polygons to make other polygons. Through building,
drawing, and analyzing two-dimensional shapes, students understand attributes and properties of Number and Operations: Building on their work in
two-dimensional space and the use of those attributes and properties in solving problems, including grade 2, students extend their understanding of place
applications involving congruence and symmetry. 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). They 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 for Prekindergarten through Grade 8 Mathematics 15Curriculum 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.
Grade 4 Curriculum Focal Points Connections to the Focal Points
Number and Operations and Algebra: Developing quick recall of multiplication facts and Algebra: Students continue identifying, describing, and
related division facts and fluency with whole number multiplication extending numeric patterns involving all operations and
nonnumeric growing or repeating patterns. Through
Students use understandings of multiplication to develop quick recall of the basic multiplication facts these experiences, they develop an understanding of the
and related division facts. They apply their understanding of models for multiplication (i.e., equal-sized use of a rule to describe a sequence of numbers or
groups, arrays, area models, equal intervals on the number line), place value, and properties of opera- objects.
tions (in particular, the distributive property) as they develop, discuss, and use efficient, accurate, and Geometry: Students extend their understanding of
generalizable methods to multiply multidigit whole numbers. They select appropriate methods and properties of two-dimensional shapes as they find the
apply them accurately to estimate products or calculate them mentally, depending on the context and areas of polygons. They build on their earlier work with
numbers involved. They develop fluency with efficient procedures, including the standard algorithm, symmetry and congruence in grade 3 to encompass
for multiplying whole numbers, understand why the procedures work (on the basis of place value and transformations, including those that produce line and
properties of operations), and use them to solve problems. rotational symmetry. By using transformations to design
and analyze simple tilings and tessellations, students
Number and Operations: Developing an understanding of decimals, including the connec- deepen their understanding of two-dimensional space.
tions between fractions and decimals Measurement: As part of understanding two-
Students understand decimal notation as an extension of the base-ten system of writing whole numbers dimensional shapes, students measure and classify
angles.
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 Data Analysis: Students continue to use tools from
greater than or less than 1, identifying equivalent decimals, comparing and ordering decimals, and grade 3, solving problems by making frequency tables,
estimating decimal or fractional amounts in problem solving. They connect equivalent fractions and bar graphs, picture graphs, and line plots. They apply
decimals by comparing models to symbols and locating equivalent symbols on the number line. their understanding of place value to develop and use
stem-and-leaf plots.
Measurement: Developing an understanding of area and determining the areas of two- Number and Operations: Building on their work in
dimensional shapes grade 3, students extend their understanding of place
value and ways of representing numbers to 100,000 in
Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify various contexts. They use estimation in determining the
area by finding the total number of same-sized units of area that cover the shape without gaps or relative sizes of amounts or distances. Students develop
overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. understandings of strategies for multidigit division by
They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that using models that represent division as the inverse of
involve estimating or measuring area. Students connect area measure to the area model that they have multiplication, as partitioning, or as successive subtrac-
used to represent multiplication, and they use this connection to justify the formula for the area of a tion. By working with decimals, students extend their
rectangle. 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.
16 Curriculum Focal Points for Prekindergarten through Grade 8 MathematicsYou can also read
