Mathematics Framework for 

the 2007 National Assessment 

of Educational Progress 

National Assessment Governing Board 

U.S. Department of Education 
What Is NAEP? 

The National Assessment of Educational Progress (NAEP) is a congressionally mandated project of the U.S. 

Department of Education’s National Center for Education Statistics. It assesses what U.S. students should 

know and be able to do in geography, reading, writing, mathematics, science, U.S. history, the arts, civics, 

and other academic subjects. Since 1969, NAEP has surveyed the achievement of students at ages 9, 13, and 17 

and, since the 1980s, in grades 4, 8, and 12. Measuring educational achievement trends over time is critical to 

measuring progress. 

The 2005–2006 National Assessment Governing Board 

The National Assessment Governing Board was created by Congress to formulate policy for NAEP. Among the 

Governing Board’s responsibilities are developing objectives and test specifications and designing the assessment 

methodology for NAEP. 

Members 

Darvin M. Winick, Chair 

President 

Winick & Associates 

Austin, Texas 

Sheila M. Ford, Vice Chair 

Former Principal 

Horace Mann Elementary School 

Washington, D.C. 

Francie M. Alexander 

Chief Academic Officer, Scholastic, Inc. 

Senior Vice President, Scholastic 

Education 

New York, New York 

David J. Alukonis 

Chairman 

Hudson School Board 

Hudson, New Hampshire 

Amanda P. Avallone 

Assistant Principal and Eighth-Grade 

Teacher 

Summit Middle School 

Boulder, Colorado 

Honorable Jeb Bush 

Governor of Florida 

Tallahassee, Florida 

Barbara Byrd-Bennett 

Executive Superintendent-in-Residence 

Cleveland State University 

Cleveland, Ohio 

Shirley V. Dickson 

Educational Consultant 

Aliso Viejo, California 

Honorable David P. Driscoll 

Commissioner of Education 

Massachusetts Department of Education 

Malden, Massachusetts 

John Q. Easton 

Executive Director 

Consortium on Chicago School Research 

Chicago, Illinois 

David W. Gordon 

Sacramento County Superintendent of 

Schools 

Sacramento County Office of Education 

Sacramento, California 

Kathi M. King 

Twelfth-Grade Teacher 

Messalonskee High School 

Oakland, Maine 

Honorable Keith King 

Member 

Colorado House of Representatives 

Colorado Springs, Colorado 

Kim M. Kozbial-Hess 

Fourth-Grade Teacher 

Hawkins Elementary School 

Toledo, Ohio 

Andrew C. Porter 

Director, Learning Sciences Institute 

Peabody College 

Vanderbilt University 

Nashville, Tennessee 

Luis A. Ramos 

Community Relations Manager 

PPL Susquehanna 

Berwick, Pennsylvania 

John H. Stevens 

Executive Director 

Texas Business and Education Coalition 

Austin, Texas 

Mary Frances Taymans, SND 

Executive Director, Department of 

Secondary Schools 

National Catholic Education Association 

Washington, D.C. 

Oscar A. Troncoso 

Principal 

Socorro High School 

El Paso, Texas 

Honorable Thomas J. Vilsack 

Governor of Iowa 

Des Moines, Iowa 

Honorable Michael E. Ward 

Former State Superintendent of Public 

Instruction 

North Carolina Public Schools 

Jackson, Mississippi 

Eileen L. Weiser 

Member, State Board of Education 

Michigan Department of Education 

Lansing, Michigan 

Grover J. Whitehurst (ex officio) 

Director, Institute of Education Sciences 

U.S. Department of Education 

Washington, D.C. 

Charles E. Smith 

Executive Director 

National Assessment Governing Board 
Mathematics Framework for 

the 2007 National Assessment 

of Educational Progress 

National Assessment Governing Board 

U.S. Department of Education 
National Assessment Governing Board 

Darvin M. Winick 

Chair 

Sheila Ford 

Vice Chair 

Charles E. Smith 

Executive Director 

Mary Crovo 

Project Officer 

NAEP Mathematics Project 

Wayne Martin 

Project Director 

John Olson 

Deputy Project Director 

Linda Wilson 

Consensus Coordinator 

Framework Author 

Developed for the National Assessment Governing Board under contract number 

ED–00–CO–0115 by the Council of Chief State School Officers, with subcontracts 

to the Council of Basic Education and the Association of State Supervisors of 

Mathematics 

For further information, contact: 

National Assessment Governing Board 

800 North Capitol Street, N.W.

Suite 825 

Washington, DC 20002–4233 

www.nagb.org 

September 2006 

For sale by the U.S. Government Printing Office 

Superintendent of Documents, Mail Stop: SSOP, Washington, DC 20402–9328
Mathematics Framework for the 2007 NAEP 

Table of Contents 

Chapter One: Overview of Recommendations.................................................................... 1

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

Context for Planning the Assessment .................................................................................. 1 

The Consensus Approach .................................................................................................... 3 

Achievement Levels............................................................................................................. 4 

Recommendations for the NAEP Mathematics Assessment ............................................... 4 

Chapter Two: Framework for the Assessment ................................................................... 7

Content Areas....................................................................................................................... 7 

Mathematical Complexity of Items ..................................................................................... 8 

Distribution of Items............................................................................................................ 9 

Calculators ......................................................................................................................... 10 

Chapter Three: NAEP Mathematics Objectives............................................................... 11 

Mathematical Content Areas.............................................................................................. 11 

Chapter Four: Mathematical Complexity of Items .......................................................... 35 

Low Complexity ................................................................................................................ 36 

Moderate Complexity ........................................................................................................ 37 

High Complexity................................................................................................................ 40 

Balance of Mathematical Complexity ............................................................................... 42 

Chapter Five: Item Formats ............................................................................................... 43

Multiple-Choice Items ....................................................................................................... 43 

Short Constructed-Response Items.................................................................................... 44 

Extended Constructed-Response Items.............................................................................. 45 

Scoring Extended Constructed-Response Items ................................................................ 46 

Bibliography ......................................................................................................................... 49 

Appendix A: NAEP Mathematics Achievement Level Descriptions............................... 51 

Grade 4............................................................................................................................... 53 

Grade 8............................................................................................................................... 54 

Appendix B: Mathematical Complexity Items Scoring Guide ........................................ 57 

Low Complexity ................................................................................................................ 59 

Moderate Complexity ........................................................................................................ 60 

High Complexity................................................................................................................ 64 

iii 
Mathematics Framework for the 2007 NAEP 

Appendix C: NAEP Mathematics Project Staff and Committees................................... 69 

NAEP Mathematics Project ............................................................................................... 71 

Steering Committee ........................................................................................................... 71

Planning Committee........................................................................................................... 73

Technical Advisory Panel.................................................................................................. 74 

Committee Representation................................................................................................. 74 

Acknowledgments ................................................................................................................ 77 

iv 
Chapter One: Overview of Recommendations 

Introduction 

Since 1973, the National Assessment of Educational Progress (NAEP) has gathered 

information about student achievement in mathematics. The results of these 

periodic assessments, produced in print and Web-based formats, inform citizens 

about the nature of students’ comprehension of the subject, inform curriculum 

specialists about the level and nature of student understanding, and inform policymakers 

about factors related to schooling and its effect on student proficiency in mathematics. 

Based on these surveys of students at the elementary, middle school, and high school 

levels, NAEP has provided comprehensive information about what students in the 

United States know and can do in the area of mathematics, as well as in a number of 

other subject areas. These reports present information on strengths and weaknesses in 

students’ understanding and their ability to apply that understanding in problem-solving 

situations; provide comparative student data according to race/ethnicity, type of 

community, and geographic region; describe trends in student performance over time; 

and report on relationships between student proficiency and certain background 

variables. 

This document contains the framework and a set of recommendations for the NAEP 

2007 mathematics assessment, which will assess student achievement nationally and 

state-by-state, as well as in select urban districts, in grades 4 and 8. It includes 

descriptions of the mathematical content of the test, the types of test questions, and 

recommendations for administration of the test. In broad terms, this framework attempts 

to answer the question, “What should be assessed?” The answer to this question must 

necessarily consider the constraints of a large-scale assessment such as NAEP, with its 

limitations on time and resources. 

It is important to understand that this document does not attempt to answer the 

question, “What should be taught, or how?” This is an assessment framework, not a 

curriculum framework. It was developed with the understanding that some concepts, 

skills, and activities in school mathematics are not suitable to be assessed on NAEP, 

even though they may be important components of a school curriculum (e.g., an 

extended project that involves gathering data, or a group project). 

The Assessment and Item Specifications for the 2005 NAEP Mathematics Assessment, 

a companion document to this framework, gives more detail about the items and 

conditions for the NAEP mathematics assessment. 

Context for Planning the Assessment 

The National Assessment Governing Board, created by Congress in 1988, is 

responsible for formulating policy for NAEP. The Governing Board is specifically 

charged with developing assessment objectives and test specifications, identifying 

appropriate achievement goals for each age and grade, and carrying out other NAEP 

policy responsibilities. 

1 
Mathematics Framework for the 2007 NAEP 

Congress also authorized the trial state assessment program in 1988. This voluntary 

program for gathering state-level achievement data in mathematics began with the 1990 

assessment at grade 8, and with the 1992 assessment in grade 4. National data at grades 

4, 8, and 12 were also gathered during those years. 

In preparation for the 1990 mathematics assessment, a contract was awarded to the 

Council of Chief State School Officers (CCSSO) to design a framework for the 

assessment. The CCSSO project gave special attention to the formal state objectives and 

frameworks for mathematics instruction. State-, district-, and school-level objectives 

were considered, as well as the frameworks on which previous NAEP mathematics 

assessments had been based, and a draft version of the National Council of Teachers of 

Mathematics’ (NCTM’s) Curriculum and Evaluation Standards for School 

Mathematics. The result was a “content by mathematical ability” matrix design that was 

used to guide both the 1990 and 1992 mathematics assessments conducted by NAEP at 

the national and state levels. 

To prepare for the next NAEP mathematics assessment, in 1991 the Governing Board 

awarded a contract to College Board to develop the assessment and item specifications 

for the 1994 mathematics assessment. The College Board project had two primary 

purposes: (1) to recommend a framework in terms of describing what students know and 

can do in mathematics, and (2) to develop specifications for the assessment items, 

specifically the mix of item formats, the item distribution for mathematics content areas, 

and the conditions under which items are presented to students (e.g., use of 

manipulatives, use of calculators, and length of time to complete tasks). 

Taking into account the three NAEP achievement levels adopted by the Governing 

Board—Basic, Proficient, and Advanced—the College Board effort resulted in the 

“content areas by mathematical abilities by mathematical power” grid that was used to 

guide the development of the 1996 and 2000 national and state mathematics assessments 

conducted by NAEP. 

Before the reauthorization of the Individuals with Disabilities Education Act (IDEA) 

in the mid-1990s, participation in NAEP of students with disabilities and English 

language learners was limited, with schools excluding them from the assessment in 

accordance with criteria provided by the program at that time. For the 1996 NAEP 

assessment, modifications were made in two areas: inclusion criteria were revised to 

make them more clear, more inclusive, and more likely to be applied consistently across 

jurisdictions in the United States; and a variety of assessment accommodations was 

offered for the first time. 

In 1998, NAEP began new trend lines for some subject areas that presented data 

reported for the first time on samples that included accommodated special-needs 

students. This trend toward more inclusive assessments has also led to a closer look at 

how test items are developed and assessments designed. 

Over the years, NAEP has implemented procedures that are designed to make items 

more accessible for a variety of special-needs students. The NAEP mathematics 

framework enables the continuation of this best practices approach, and the assessment 

and item specifications document (Assessment and Item Specifications for the 2005 

2 
Mathematics Framework for the 2007 NAEP 

NAEP Mathematics Assessment) provides clear and detailed guidance on how to 

develop a more inclusive assessment. 

The long-range plan established by the Governing Board called for the development 

of a new mathematics framework to be used as the basis for the 2005 national and state 

assessments of mathematics. Prior to initiating work on a new framework, the 

Governing Board conducted a series of public meetings to gather input to serve as the 

basic purpose(s) for the new framework. At the public meetings, a strong concern was 

raised by states about the need to continue the short-term trend lines for the nation and 

the states. Many states were using the short-term trend lines as an independent monitor 

of their standards-based reform efforts. 

Several other concerns were also raised: 

• Had computational skills been overlooked by the reform efforts because of the 

emphasis placed on higher order skills by most state mathematics standards and 

the work by NCTM? 

• Were U.S. students capable of handling a more rigorous curriculum, especially 

algebra and geometry, at grade 8? 

• What should our mathematical expectations be for grade 12 students? 

Based on the input received, the Governing Board awarded a contract to CCSSO in 

September 2000 to update the mathematics assessment framework used for the 1996 and 

2000 assessments. The revisions were to address the concerns raised at the public 

meetings while maintaining the short-term trend lines in grades 4 and 8 that began with 

the 1990 mathematics assessment. 

It is within this context that the recommendations contained in this document were 

developed. 

The Consensus Approach 

CCSSO established a steering committee, representative of national policy 

organizations, mathematics associations, research mathematicians, business and 

industry, and educators, to develop policy recommendations for the mathematics 

assessment and to guide the direction and scope of the project. A planning committee of 

mathematics educators, mathematicians, curriculum supervisors, and teachers was 

established to draft the content of the framework, working within the policy 

recommendations established by the steering committee. Care was taken with both 

committees to ensure that the diversity of opinion regarding mathematics issues was 

represented and reflected. To reach meaningful consensus, all points of view were heard 

and considered. Such consensus has been the goal of this project. 

A technical advisory panel, composed of university professors, state testing 

specialists, and measurement experts from private research organizations, was 

established to consider the policy recommendations and to draft content from the 

perspective of whether it would threaten NAEP’s ability to continue the short-term 

mathematics trend lines that began in 1990. 

3 
Mathematics Framework for the 2007 NAEP 

The steering committee and planning committee both began their work with a review 

of the framework used in 1996 and 2000. A discussion of the current debates in 

mathematics education was also part of their meetings. In their deliberations, the 

committees considered state mathematics content standards and frameworks, new 

standards prepared by NCTM, reports from the Third International Mathematics and 

Science Study, reports from the Achieve Project, and a 2001 report issued by the 

National Research Council of the National Academy of Sciences. In addition, input was 

provided by mathematics teachers and supervisors as well as others through CCSSO’s 

partners, the Council for Basic Education and the Association of State Supervisors of 

Mathematics. 

The suggested revisions contained in this framework are intended (1) to reflect recent 

curricular emphases and objectives; (2) to include what various policymakers, scholars, 

practitioners, and interested citizens believe should be in the assessment; (3) to maintain 

the short-term trend lines in grades 4 and 8 that began with the 1990 mathematics 

assessment to permit the reporting of changes in student achievement over time; and (4) 

to include objectives that are more clear and more specific for each grade level. 

Achievement Levels 

The NAEP Mathematics Framework was considered in light of the three NAEP 

achievement levels: Basic, Proficient, and Advanced. Basic denotes partial mastery of 

prerequisite knowledge and skills that are fundamental for proficient work at each 

grade. Proficient represents solid academic performance for each grade assessed. 

Students reaching this level have demonstrated competency over challenging subject 

matter, including subject-matter knowledge, application of such knowledge to realworld situations, and analytical skills appropriate to the subject matter. Advanced

represents superior performance. 

These levels are intended to provide descriptions of what students should know and 

be able to do in mathematics. Established for the 1992 mathematics scale through a 

broadly inclusive process and adopted by the Governing Board, the three levels per 

grade are a major means of reporting NAEP data. The updated mathematics framework 

was developed to ensure congruence between the achievement levels and the test 

content. See appendix A for the NAEP Mathematics Achievement Level Descriptions. 

Recommendations for the NAEP Mathematics Assessment 

As a result of analysis and review, the steering committee and planning committee 

endorsed the following recommendations for the NAEP mathematics assessment: 

1. Content Areas 

The NAEP mathematics assessment should be based on essentially the same five 

content areas as the 1996 and 2000 assessments: (1) Number Properties and Operations, 

(2) Measurement, (3) Geometry, (4) Data Analysis and Probability, and (5) Algebra. 

Details about each of these content areas can be found in chapter three. 

4 
Mathematics Framework for the 2007 NAEP 

2. Mathematical Complexity of Items 

The second dimension of the framework, formerly known as “mathematical abilities,” 

should be refined to describe the level of mathematical complexity that an item demands 

of a student. Each level describes the degree of procedural knowledge, conceptual 

understanding, problem solving, reasoning, or communicating required to respond to an 

item at that level. Further description of how this revision relates to the former 

framework is presented in chapter two, and the levels are described at length in chapter 

four. 

3. Distribution of Items 

The percentage of items allotted to each of the five content areas should remain the 

same at grade 4 as called for in the framework for the 1996–2005 assessments. The 

percentages for grade 8 should be altered somewhat in the areas of Number Properties 

and Operations and Algebra to reflect the increasing importance of algebraic concepts. 

At grade 12, the percentages should be changed to more closely correspond to the 

mathematics that high school students experience in a typical 3-year sequence of 

courses, with algebra and geometry/measurement forming the anchors, along with an 

increasing emphasis on data analysis and probability. The recommended percentages are 

discussed in chapter two. 

4. Item Formats 

Given that NAEP is a paper-and-pencil test administered in a timed setting (for 

students not receiving accommodations), the assessment should continue to use three 

types of items: multiple choice, short constructed response, and extended constructed 

response. As in the previous framework, approximately half of a student’s testing time 

should be allotted to multiple-choice items, with the remaining half devoted to 

constructed-response items of both types. Further description of the item formats can be 

found in chapter five. For items with multiple-score points, students receive full credit 

only if they fulfill all the requirements of the item and provide the correct solution. 

5. Manipulatives 

The assessment should continue to use reasonable manipulative materials, where 

possible, in measuring students’ ability to represent their understandings and to use tools 

to solve problems. Such manipulative materials and accompanying tasks should be 

carefully chosen to cause minimal disruption of the test administration process. 

6. Calculators 

It is appropriate for some portions (two-thirds) of NAEP at all grade levels to assess 

students’ mathematical knowledge and skills without access to a calculator, but for other 

portions (one-third) of the test to allow the use of a calculator. At grade 4, a fourfunction calculator should be supplied by NAEP with appropriate training at the time of 

administration. Eighth- and 12th-grade students should be allowed to bring whatever 

calculator, graphing or otherwise, they are accustomed to using in the classroom. 

5 
Mathematics Framework for the 2007 NAEP 

The Governing Board determined that eighth-grade students should be provided with 

scientific calculators beginning with the 2005 assessment based on a survey of state 

testing programs. More details and discussion about the calculator recommendations can 

be found in chapter two. 

6 
Chapter Two: Framework for the Assessment 

This chapter further discusses the rationale for recommendations, especially those 

that reflect a change from current policy. 

Content Areas 

Since its first mathematics assessments, NAEP has regularly gathered data on 

students’ understanding of mathematical content. Although the names of the content 

areas in the frameworks, as well as some of the topics in those areas, may have changed 

somewhat from one assessment to the next, there remained a consistent focus toward 

collecting information on student performance in five key areas: 

• Number (including computation and the understanding of number concepts) 

• Measurement (including use of instruments, application of processes, and concepts 

of area and volume) 

• Geometry (including spatial reasoning and applying geometric properties) 

• Data Analysis (including probability, graphs, and statistics) 

• Algebraic Representations and Relationships 

The framework for the mathematics assessment is anchored in these same five broad 

areas of mathematical content: 

• Number Properties and Operations 

• Measurement 

• Geometry 

• Data Analysis and Probability 

• Algebra 

These divisions are not intended to separate mathematics into discrete elements. 

Rather, they are intended to provide a helpful classification scheme that describes the 

full spectrum of mathematical content assessed by NAEP. Classifying items into one 

primary content area is not always clear cut, but doing so brings us closer to the goal of 

ensuring that important mathematical concepts and skills are assessed in a balanced 

way. 

At grade 12, the five content areas are collapsed into four, with geometry and 

measurement combined into one. This reflects the fact that the majority of measurement 

topics suitable for 12th-grade students are geometrical in nature. 

It is important to note that certain aspects of mathematics occur in all of the content 

areas. The best example of this is computation. Computation is the skill of performing 

operations on numbers. It should not be confused with the content area of NAEP called 

Number Properties and Operations, which encompasses a wide range of concepts about 

7 
Mathematics Framework for the 2007 NAEP 

our numeration system (see chapter three for a thorough discussion). Certainly the area 

of Number Properties and Operations includes a variety of computational skills, ranging 

from operations with whole numbers to work with decimals and fractions and finally 

real numbers. But computation is also critical in Measurement and Geometry, such as in 

calculating the perimeter of a rectangle, estimating the height of a building, or finding 

the hypotenuse of a right triangle. Data analysis often involves computation, such as 

calculating a mean or the range of a set of data. Probability often entails work with 

rational numbers. Solving algebraic equations usually involves numerical computation 

as well. Computation, therefore, is a foundational skill in every content area. 

Mathematical Complexity of Items 

The framework used for the 1996 and 2000 NAEP included three dimensions: 

mathematical content, mathematical abilities, and power (reasoning, connections, and 

communication). That framework was intended to address the primary need of ensuring 

that NAEP assessed an appropriate balance of mathematical content and at the same 

time assessed a variety of ways of knowing and doing mathematics. The abilities and 

power dimensions were not intended for reporting, but rather to provide for a wide 

range of mathematical activity in the items. 

That framework had many laudable features. Notions of conceptual understanding, 

procedural knowledge, and problem solving sent a strong message about the depth and 

breadth of engaging in mathematical activity. The dimensions of mathematical power 

gave further emphasis to the idea that certain activities cut across content areas. At the 

same time, there was an acknowledgement that the dimension of mathematical abilities 

proved somewhat difficult for experts to agree on, relying as it does on inferences about 

students’ approaches to each particular item. 

The intent of this current framework is to build on the previous one, retaining its 

strengths while addressing some of its weaknesses. The purpose remains the same: to 

ensure that NAEP assesses an appropriate balance of content as well as a variety of 

ways of knowing and doing mathematics. The major change is to create a second 

dimension of the framework based on the properties of an item, rather than on the 

abilities of a student. Mathematical complexity of an item answers the question, “What 

does the item ask of the students?” 

Each level of complexity includes aspects of knowing and doing mathematics, such 

as reasoning, performing procedures, understanding concepts, or solving problems. The 

levels are ordered, so that items at a low level would demand that students perform 

simple procedures, understand elementary concepts, or solve simple problems. Items at 

the high end would ask students to reason or communicate about sophisticated concepts, 

perform complex procedures, or solve non-routine problems. Ordering of the levels is 

not intended to imply a developmental sequence or the sequencing in which teaching or 

learning occurs. Rather, it is a description of the different demands made on students by 

particular test items. See chapter five for further discussion of the levels of mathematical 

complexity. 

8 
Mathematics Framework for the 2007 NAEP 

Distribution of Items 

The distribution of items among the various mathematical content areas is a critical 

feature of the assessment design, as it reflects the relative importance and value given to 

each of the curricular content areas within mathematics. As has been the case with past 

NAEP assessments in mathematics, the categories have received differential emphasis at 

each grade, and the differentiation continues in the framework for this assessment. Table 

1 provides the recommended balance of items in the assessment by content area for each 

grade (4, 8, and 12) in this assessment. Note that the percentages refer to numbers of 

items, not the amount of student testing time (see chapter five for recommendations on 

item formats and student testing time). 

Table 1. Percentage Distribution of Items by Grade and Content Area 

Content Area (2005) Grade 4 (%) Grade 8 (%) Grade 12 (%) 

Number Properties and Operations 40 20 10 

Measurement 20 15 30 

Geometry 15 20 

Data Analysis and Probability 10 15 25 

Algebra 15 30 35 

The percentages in grade 4 are not different from those recommended in the previous 

framework. Change was not recommended, as these numbers continue to be a 

reasonable reflection of the relative weights of each of these content areas at that grade 

level. 

At grade 8, new number concepts occur in the form of more advanced work with 

properties and operations on rational numbers (fractions and decimals) and more 

sophisticated work in number theory. However, much of the work in numbers happens 

in the context of other content areas, such as Measurement and Data Analysis and 

Probability. Grade 8 also has an increased emphasis on informal algebraic concepts and 

on Geometry. Therefore, the percentages show a slight increase in Algebra and a 

corresponding slight decrease in Number Properties and Operations at grade 8 compared 

with the percentages of the previous framework. 

More students are taking higher levels of mathematics in high school. According to 

data from the 2000 NAEP, approximately 79 percent of 12th-grade students have taken 

the equivalent of 2 years of algebra and 1 year of geometry. Because NAEP needs to 

assess the full range of content in mathematics, the percentages for 12th grade have 

been adjusted, with primary emphasis on Geometry/Measurement and Algebra. At this 

grade level, the majority of measurement topics are geometrical in nature and the 

distinction between Geometry and Measurement becomes blurred, so this framework 

calls for these two content areas to be combined into one. The percentages also show the 

increased importance of Data Analysis and Probability in the secondary school 

9 
Mathematics Framework for the 2007 NAEP 

curriculum. The majority of work in Number Properties and Operations is done in the 

context of the other content areas, so it receives a decreased emphasis. 

Calculators 

At each grade level, approximately two-thirds of the assessment measures students’ 

mathematical knowledge and skills without access to a calculator; the other third of the 

assessment allows the use of a calculator. The assessment contains blocks for which 

calculators are not allowed, and calculator blocks, which contain some items that would 

be difficult to solve without a calculator. The type of calculator students may use on a 

calculator block varies by grade level, as follows: 

• At grade 4, a four-function calculator is supplied to students, with training at the 

time of administration. 

• At grade 8, a scientific calculator is supplied to students, with training at the time 

of administration. 

No items at grade 8 will be designed to provide an advantage to students with a 

graphing calculator. The estimated time required for any item should be based on the 

assumption that students are not using a graphing calculator. 

In determining whether an item belongs on a calculator block or a non-calculator 

block, the developer should consider the technology available to students and the 

measurement intent of the item. The content and skills being assessed should guide 

whether an item should be considered calculator active and whether it should go in a 

calculator block. For example, a multiple-choice item asking students to select the graph 

that represents a given equation should be on a non-calculator block, if the intent of the 

item is to measure students’ ability to recognize the graph of a given equation and if the 

equation is in a form that is readily entered into a graphing calculator to obtain a graph 

(for example, y = –x

2

). 

10 
I 

Chapter Three: NAEP Mathematics Objectives 

n order to describe the specific mathematics that should be assessed at each grade 

level, it is necessary to organize the domain of mathematics into component parts. 

This is accomplished by using the five content areas, as described in chapter two. 

Though such an organization brings with it the danger of fragmentation, the hope is that 

the objectives and the test items built on them will, in many cases, cross some of the 

boundaries of these content areas. 

One of the goals of this framework is to provide more clarity and specificity in the 

objectives for each grade level. To accomplish this, a matrix was created that depicts the 

particular objectives that are appropriate for assessment under each subtopic. Within 

Number, for example, and the subtopic of Number Sense, specific objectives are listed 

for assessment at grade 4 and grade 8. The same objective at different grade levels 

depicts a developmental sequence for that concept or skill. An empty cell in the matrix 

is used to convey the fact that a particular objective is not appropriate for assessment at 

that grade level. 

In order to fully understand the objectives and their intent, please note the 

following: 

• Further clarification of some of these objectives, along with some sample items, may 

be found in the companion document, Assessment and Item Specifications for the 

2005 NAEP Mathematics Assessment. 

• While all test items will be assigned a primary classification, some test items could 

potentially fall into more than one content area or more than one objective. 

• When the word “or” is used in an objective, it should be understood as inclusive; that 

is, an item may assess one or more of the concepts included. However, all concepts 

described should be measured across the full range of the assessment. 

• These objectives describe what is to be assessed on the 2007 NAEP. They should not 

be interpreted as a complete description of mathematics that should be taught at these 

grade levels. 

Mathematical Content Areas 

NUMBER PROPERTIES AND OPERATIONS 

Numbers are our main tools for describing the world quantitatively. As such, they 

deserve a privileged place in this framework. With whole numbers, we can count 

collections of discrete objects of any type. We can also use numbers to describe 

fractional parts and even to describe continuous quantities such as length, area, volume, 

weight, and time, and more complicated derived quantities such as rates, speed, density, 

inflation, interest, and so forth. Thanks to Cartesian coordinates, we can use pairs of 

numbers to describe points in a plane or triples of numbers to label points in space. 

11 
Mathematics Framework for the 2007 NAEP 

Numbers let us talk in a precise way about anything that can be counted, measured, or 

located in space. 

Numbers are not simply labels for quantities. They form systems with their own 

internal structure. The arithmetic operations (addition and subtraction, multiplication 

and division) help us model basic real-world operations. For example, joining two 

collections, or laying two lengths end to end, can be described by addition, while the 

concept of rate depends on division. Multiplication and division of whole numbers lead 

to the beginnings of number theory, including concepts of factorization, remainder, and 

prime number. Besides the arithmetic operations, the other basic structure of the real 

numbers is ordering, as in which is greater and lesser. These reflect our intuitions about 

the relative size of quantities and provide a basis for making sensible estimates. 

The accessibility and usefulness of arithmetic are greatly enhanced by our efficient 

means for representing numbers: the Hindu-Arabic decimal place value system. In its 

full development, this remarkable system includes decimal fractions, which let us 

approximate any real number as closely as we wish. Decimal notation allows us to do 

arithmetic by means of simple, routine algorithms, and it makes size comparisons and 

estimation easy. The decimal system achieves its efficiency through sophistication, as 

all the basic algebraic operations are implicitly used in writing decimal numbers. To 

represent ratios of two whole numbers exactly, we supplement decimal notation with 

fractions. 

Comfort in dealing with numbers effectively is called number sense. It includes firm 

intuitions about what numbers tell us; an understanding of the ways to represent them 

symbolically (including facility with converting between different representations); the 

ability to calculate, either exactly or approximately, and by several means (mentally, 

with paper and pencil, or with calculator, as appropriate); and skill in estimation. The 

ability to deal with proportion, including percents, is another important part of number 

sense. 

Number sense is a major expectation of the 2007 NAEP. At fourth grade, students are 

expected to have a solid grasp of whole numbers, as represented by the decimal system, 

and to have the beginnings of understanding fractions. By eighth grade, they should be 

comfortable with rational numbers, represented either as decimal fractions (including 

percents) or as common fractions. They should be able to use them to solve problems 

involving proportionality and rates. Also in middle school, number should begin to 

coalesce with geometry via the idea of the number line. This should be connected with 

ideas of approximation and the use of scientific notation. Eighth graders should also 

have some acquaintance with naturally occurring irrational numbers, such as square 

roots and pi. By 12th grade, students should be comfortable dealing with all types of 

real numbers. 

12 
Mathematics Framework for the 2007 NAEP 

Number Properties and Operations 

1) Number sense 

GRADE 4 GRADE 8 GRADE 12 

a) Identify the place value and 

actual value of digits in whole 

numbers. 

a) Use place value to model and 

describe integers and decimals. 

b) Represent numbers using models 

such as base 10 representations, 

number lines, and twodimensional models. 

b) Model or describe rational 

numbers or numerical 

relationships using number lines 

and diagrams. 

c) Compose or decompose whole 

quantities by place value (e.g., 

write whole numbers in expanded 

notation using place value: 342 = 

300 + 40 + 2). 

d) Write or rename whole numbers 

(e.g., 10: 5 + 5, 12 – 2, 2 x 5). 

d) Write or rename rational 

numbers. 

d) Write, rename, represent, or 

compare real numbers (e.g., π, 

2 , numerical, relationships 

using number lines, models, or 

diagrams. 

e) Connect model, number word, or 

number using various models and 

representations for whole 

numbers, fractions, and decimals. 

e) Recognize, translate between, or 

apply multiple representations of 

rational numbers (fractions, 

decimals, and percents) in 

meaningful contexts. 

f) Express or interpret numbers 

using scientific notation from 

real-life contexts. 

f) Represent very large or very 

small numbers using scientific 

notation in meaningful contexts. 

g) Find or model absolute value or 

apply to problem situations. 

g) Find or model absolute value or 

apply to problem situations. 

h) Interpret calculator or computer 

displays of numbers given in 

scientific notation. 

i) Order or compare rational 

numbers (fractions, decimals, 

percents, or integers) using 

various models and 

representations (e.g., number 

line). 

j) Order or compare whole numbers, 

decimals, or fractions. 

j) Order or compare rational 

numbers including very large and 

small integers, and decimals and 

fractions close to zero. 

j) Order or compare real numbers, 

including very large or small real 

numbers. 

continued on page 14 

13 
Mathematics Framework for the 2007 NAEP 

Number Properties and Operations (continued)

 2) Estimation 

GRADE 4 GRADE 8 GRADE 12 

a) Use benchmarks (well-known 

numbers used as meaningful 

points for comparison) for whole 

numbers, decimals, or fractions in 

contexts (e.g., 1⁄2 and .5 may be 

used as benchmarks for fractions 

and decimals between 0 and 

1.00). 

a) Establish or apply benchmarks 

for rational numbers and common 

irrational numbers (e.g., π) in 

contexts. 

a) Establish or apply benchmarks 

for real numbers in contexts. 

b) Make estimates appropriate to a 

given situation with whole 

numbers, fractions, or decimals 

by: 

• knowing when to estimate, 

• selecting the appropriate type 

of estimate, including overestimate, underestimate, and 

range of estimate, or 

• selecting the appropriate 

method of estimation (e.g., 

rounding). 

b) Make estimates appropriate to a 

given situation by: 

• identifying when estimation 

is appropriate, 

• determining the level of 

accuracy needed, 

• selecting the appropriate 

method of estimation, or 

• analyzing the effect of an 

estimation method on the 

accuracy of results. 

b) Make estimates of very large or 

very small numbers appropriate 

to a given situation by: 

• identifying when estimation 

is appropriate or not, 

• determining the level of 

accuracy needed, 

• selecting the appropriate 

method of estimation, or 

• analyzing the effect of an 

estimation method on the 

accuracy of results. 

c) Verify solutions or determine the 

reasonableness of results in 

meaningful contexts. 

c) Verify solutions or determine the 

reasonableness of results in a 

variety of situations including 

calculator and computer results. 

c) Verify solutions or determine the 

reasonableness of results in a 

variety of situations including 

scientific notation, calculator, and 

computer results. 

d) Estimate square or cube roots of 

numbers less than 1,000 between 

two whole numbers. 

d) Estimate square or cube roots of 

numbers less than 1,000 between 

two whole numbers. 

continued on page 15 

14 
Mathematics Framework for the 2007 NAEP 

Number Properties and Operations (continued) 

3) Number operations 

GRADE 4 GRADE 8 GRADE 12 

a) Add and subtract: 

• whole numbers, or 

• fractions with like 

denominators, or 

• decimals through hundredths. 

a) Perform computations with 

rational numbers. 

a) Perform computations with real 

numbers including common 

irrational numbers or the absolute 

value of numbers. 

b) Multiply whole numbers: 

• no larger than two-digit by 

two-digit with paper and pencil 

computation, or 

• larger numbers with use of 

calculator. 

c) Divide whole numbers: 

• up to three-digits by one-digit 

with paper and pencil 

computation, or 

• up to five-digits by two-digits 

with use of calculator. 

d) Describe the effect of operations 

on size (whole numbers). 

d) Describe the effect of multiplying 

and dividing by numbers 

including the effect of 

multiplying or dividing a rational 

number by: 

• zero, or 

• a number less than zero, or 

• a number between zero and 

one, 

• one, or 

• a number greater than one. 

d) Describe the effect of multiplying 

and dividing by numbers 

including the effect of 

multiplying or dividing a real 

number by: 

• zero, or 

• a number less than zero, or 

• a number between zero and 

one, or 

• one, or 

• a number greater than one. 

e) Provide a mathematical argument 

to explain operations with two or 

more fractions. 

f) Interpret whole number operations 

and the relationships between 

them. 

f) Interpret rational number 

operations and the relationships 

between them. 

g) Solve application problems 

involving numbers and 

operations. 

g) Solve application problems 

involving rational numbers and 

operations using exact answers or 

estimates as appropriate. 

g) Solve application problems 

involving numbers, including 

rational and common irrationals, 

using exact answers or estimates 

as appropriate. 

continued on page 16 

15 
Mathematics Framework for the 2007 NAEP 

Number Properties and Operations (continued) 

4) Ratios and proportional reasoning 

GRADE 4 GRADE 8 GRADE 12 

a) Use simple ratios to describe 

problem situations. 

a) Use ratios to describe problem 

situations. 

b) Use fractions to represent and 

express ratios and proportions. 

b) Use proportions to model 

problems. 

c) Use proportional reasoning to 

model and solve problems 

(including rates and scaling). 

c) Use proportional reasoning to 

solve problems (including rates). 

d) Solve problems involving 

percentages (including percent 

increase and decrease, interest 

rates, tax, discount, tips, or 

part/whole relationships). 

d) Solve problems involving 

percentages (including percent 

increase and decrease, interest 

rates, tax, discount, tips, or 

part/whole relationships). 

5) Properties of number and operations 

GRADE 4 GRADE 8 GRADE 12 

a) Identify odd and even numbers. a) Describe odd and even integers 

and how they behave under 

different operations. 

b) Identify factors of whole numbers. b) Recognize, find, or use factors, 

multiples, or prime factorization. 

b) Solve problems involving factors, 

multiples, or prime factorization. 

c) Recognize or use prime and 

composite numbers to solve 

problems. 

c) Use prime or composite numbers 

to solve problems. 

d) Use divisibility or remainders in 

problem settings. 

d) Use divisibility or remainders in 

problem settings. 

e) Apply basic properties of 

operations. 

e) Apply basic properties of 

operations. 

e) Apply basic properties of 

operations. 

f) Explain or justify a mathematical 

concept or relationship (e.g., 

explain why 15 is an odd number 

or why 7–3 is not the same as 3– 

7). 

f) Explain or justify a mathematical 

concept or relationship (e.g., 

explain why 17 is prime). 

f) Provide a mathematical argument 

about a numerical property or 

relationship. 

16 
Mathematics Framework for the 2007 NAEP 

MEASUREMENT 

Measuring is the process by which numbers are assigned in order to describe the 

world quantitatively. This process involves selecting the attribute of the object or event 

to be measured, comparing this attribute to a unit, and reporting the number of units. For 

example, in measuring a child, we may select the attribute of height and the inch as the 

unit for the comparison. In comparing the height to the inch, we may find that the child 

is about 42 inches. If considering only the domain of whole numbers, we would report 

that the child is 42 inches tall. However, since height is a continuous attribute, we may 

consider the domain of rational numbers and report that the child is 413/16 inches tall (to 

the nearest 16th of an inch). Measurement also allows us to model positive and negative 

numbers as well as irrational numbers. 

This connection between measuring and number makes measuring a vital part of the 

school curriculum. Measurement models are often used when students are learning 

about number and operations. For example, area and volume models can help students 

understand multiplication and the properties of multiplication. Length models, 

especially the number line, can help students understand ordering and rounding 

numbers. Measurement also has a strong connection to other areas of school 

mathematics and to the other subjects in the school curriculum. Problems in algebra are 

often drawn from measurement situations. One can also consider measurement to be a 

function or a mapping of the attribute to a set of numbers. Much of school geometry 

focuses on the measurement aspect of geometric figures. Statistics also provides ways to 

measure and to compare sets of data. These are some of the ways in which measurement 

is intertwined with the other four content areas. 

In this NAEP Mathematics Framework, attributes such as capacity, weight/mass, 

time, and temperature are included, as well as the geometric attributes of length, area, 

and volume. Although many of these attributes are included in the grade 4 framework, 

the emphasis is on length, including perimeter, distance, and height. More emphasis is 

placed on area and angle in grade 8. By grade 12, volumes and rates constructed from 

other attributes, such as speed, are emphasized. 

Units involved in items on the NAEP assessment include non-standard, customary, 

and metric units. At grade 4, common customary units such as inch, quart, pound, and 

hour and the common metric units such as centimeter, liter, and gram are emphasized. 

Grades 8 and 12 include the use of both square and cubic units for measuring area, 

surface area, and volume; degrees for measuring angles; and constructed units such as 

miles per hour. Converting from one unit in a system to another (such as from minutes 

to hours) is an important aspect of measurement included in problem situations. 

Understanding and using the many conversions available is an important skill. There are 

a limited number of common, everyday equivalencies that students are expected to 

know (see the Assessment and Item Specifications document for more detail). 

Items classified in this content area depend on some knowledge of measurement. For 

example, an item that asks the difference between a 3-inch and a 13⁄4-inch line segment 

is a number item, while an item comparing a 2-foot segment with an 8-inch line segment 

is a measurement item. In many secondary schools, measurement becomes an integral 

17 
Mathematics Framework for the 2007 NAEP 

part of geometry; this is reflected in the proportion of items recommended for these two 

areas. 

Measurement 

1) Measuring physical attributes 

GRADE 4 GRADE 8 GRADE 12 

a) Identify the attribute that is 

appropriate to measure in a given 

situation. 

b) Compare objects with respect to a 

given attribute, such as length, 

area, volume, time, or 

temperature. 

b) Compare objects with respect to 

length, area, volume, angle 

measurement, weight, or mass. 

c) Estimate the size of an object with 

respect to a given measurement 

attribute (e.g., length, perimeter, 

or area using a grid). 

c) Estimate the size of an object with 

respect to a given measurement 

attribute (e.g., area). 

c) Estimate or compare perimeters or 

areas of two-dimensional 

geometric figures. 

d) Estimate or compare volume or 

surface area of three-dimensional 

figures. 

e) Solve problems involving the 

coordinate plane such as the 

distance between two points, the 

midpoint of a segment, or slopes 

of perpendicular or parallel lines. 

f) Solve problems of angle measure, 

including those involving triangles 

or other polygons or parallel lines 

cut by a transversal. 

g) Select or use appropriate 

measurement instruments such as 

ruler, meter stick, clock, 

thermometer, or other scaled 

instruments. 

g) Select or use appropriate 

measurement instrument to 

determine or create a given length, 

area, volume, angle, weight, or 

mass. 

h) Solve problems involving 

perimeter of plane figures. 

h) Solve mathematical or real-world 

problems involving perimeter or 

area of plane figures such as or 

composite figures. 

h) Solve mathematical or real-world 

problems involving perimeter or 

area of plane figures such as 

polygons, circles, or composite 

figures. 

i) Solve problems involving area of 

squares and rectangles. 

j) Solve problems involving volume 

or surface area of rectangular 

solids, cylinders, prisms, or 

composite shapes. 

j) Solve problems involving volume 

or surface area of rectangular 

solids, cylinders, cones, pyramids, 

prisms, spheres, or composite 

shapes. 

continued on page 19 

18 
Mathematics Framework for the 2007 NAEP 

Measurement (continued) 

1) Measuring physical attributes (continued) 

GRADE 4 GRADE 8 GRADE 12 

k) Solve problems involving indirect 

measurement such as finding the 

height of a building by comparing 

its shadow with the height and 

shadow of a known object. 

k) Solve problems involving indirect 

measurement such as finding the 

height of a building by finding the 

distance to the base of the 

building and the angle of elevation 

to the top. 

l) Solve problems involving rates 

such as speed or population 

density. 

l) Solve problems involving rates 

such as speed, density, population 

density, or flow rates. 

m) Use trigonometric relations in 

right triangles to solve problems. 

2) System of measurement 

GRADE 4 GRADE 8 GRADE 12 

a) Select or use appropriate type of 

unit for the attribute being 

measured such as length, time, or 

temperature. 

a) Select or use appropriate type of 

unit for the attribute being 

measured such as length, area, 

angle, time, or volume. 

a) Select or use appropriate type of 

unit for the attribute being 

measured such as volume or 

surface area. 

b) Solve problems involving 

conversions within the same 

measurement system such as 

conversions involving inches and 

feet or hours and minutes. 

b) Solve problems involving 

conversions within the same 

measurement system such as 

conversions involving square 

inches and square feet. 

b) Solve problems involving 

conversions within or between 

measurement systems, given the 

relationship between the units. 

c) Estimate the measure of an object 

in one system given the measure 

of that object in another system 

and the approximate conversion 

factor. For example: 

• Distance conversion: 1 

kilometer is approximately e

of a mile. 

• Money conversion: U.S. dollar 

is approximately 1.5 Canadian 

dollars. 

• Temperature conversion: 

Fahrenheit to Celsius 

d) Determine appropriate size of unit 

of measurement in problem 

situation involving such attributes 

as length, time, capacity, or 

weight. 

d) Determine appropriate size of unit 

of measurement in problem 

situation involving such attributes 

as length, area, or volume. 

continued on page 20 

19 
Mathematics Framework for the 2007 NAEP 

2) Systems of measurement (continued) 

GRADE 4 GRADE 8 GRADE 12 

e) Determine situations in which a 

highly accurate measurement is 

important. 

e) Determine appropriate accuracy of 

measurement in problem situations (e.g., the accuracy of each 

of several lengths needed to obtain 

a specified accuracy of a total 

length) and find the measure to 

that degree of accuracy. 

e) Determine appropriate accuracy of 

measurement in problem 

situations (e.g., the accuracy of 

measurement of the dimensions to 

obtain a specified accuracy of 

area) and find the measure to that 

degree of accuracy. 

f) Construct or solve problems (e.g., 

floor area of a room) involving 

scale drawings. 

f) Construct or solve problems (e.g., 

number of rolls needed for 

insulating a house) involving scale 

drawings. 

g) Compare lengths, areas, or 

volumes of similar figures using 

proportions. 

GEOMETRY 

Geometry began as a practical collection of rules for calculating lengths, areas, and 

volumes of common shapes. In classical times, the Greeks turned it into a subject for 

reasoning and proof, and Euclid organized their discoveries into a coherent collection of 

results, all deduced using logic from a small number of special assumptions called 

postulates. Euclid’s Elements stood as a pinnacle of human intellectual achievement for 

over 2000 years. 

The 19th century saw a new flowering of geometric thought, going beyond Euclid, 

and leading to the idea that geometry is the study of the possible structures of space. 

This had its most striking application in Einstein’s theories of relativity, which describes 

the behavior of light, and also of gravity, in terms of a four-dimensional geometry, 

which combines the usual three dimensions of space with time as an additional 

dimension. 

A major insight of the 19th century is that geometry is intimately related to ideas of 

symmetry and transformation. The symmetry of familiar shapes under simple 

transformations (that our bodies look more or less the same if reflected across the 

middle, or that a square looks the same if rotated by 90 degrees) is a matter of everyday 

experience. Many of the standard terms for triangles (scalene, isosceles, equilateral) and 

quadrilaterals (parallelogram, rectangle, rhombus, square) refer to symmetry properties. 

Also, the behavior of figures under changes of scale is an aspect of symmetry with 

myriad practical consequences. At a deeper level, the fundamental ideas of geometry 

itself (for example, congruence) depend on transformation and invariance. In the 20th 

century, symmetry ideas were seen to also underlie much of physics, not only Einstein’s 

relativity theories, but atomic physics and solid-state physics (the field that produced 

computer chips). 

20 
Mathematics Framework for the 2007 NAEP 

School geometry roughly mirrors the historical development through Greek times 

with some modern additions, most notably symmetry and transformations. By grade 4, 

students are expected to be familiar with a library of simple figures and their attributes, 

both in the plane (lines, circles, triangles, rectangles, and squares) and in space (cubes, 

spheres, and cylinders). In middle school, understanding of these shapes deepens, with 

the study of cross-sections of solids and the beginnings of an analytical understanding of 

properties of plane figures, especially parallelism, perpendicularity, and angle relations 

in polygons. Right angles and the Pythagorean theorem are introduced, and geometry 

becomes more and more mixed with measurement. The basis for analytic geometry is 

laid by study of the number line. In high school, attention is given to Euclid’s legacy 

and the power of rigorous thinking. Students are expected to make, test, and validate 

conjectures. Via analytic geometry, the key areas of geometry and algebra are merged 

into a powerful tool that provides a basis for calculus and the applications of 

mathematics that helped create the modern technological world in which we live. 

Symmetry is an increasingly important component of geometry. Elementary students 

are expected to be familiar with the basic types of symmetry transformations of plane 

figures, including flips (reflection across lines), turns (rotations around points), and 

slides (translations). In middle school, this knowledge becomes more systematic and 

analytical, with each type of transformation being distinguished from other types by 

their qualitative effects. For example, a rigid motion of the plane that leaves at least two 

points fixed (but not all points) must be a reflection in a line. In high school, students 

are expected to be able to represent transformations algebraically. Some may also gain 

insight into systematic structure, such as the classification of rigid motions of the plane 

as reflections, rotations, translations, or glide reflections, and what happens when two or 

more isometries are performed in succession (composition). 

21 
Mathematics Framework for the 2007 NAEP 

Geometry 

1) Dimension and shape 

GRADE 4 GRADE 8 GRADE 12 

a) Explore properties of paths 

between points. 

a) Draw or describe a path of 

shortest length between points to 

solve problems in context. 

b) Identify or describe (informally) 

real-world objects using simple 

plane figures (e.g., triangles, 

rectangles, squares, and circles) 

and simple solid figures (e.g., 

cubes, spheres, and cylinders). 

b) Identify a geometric object given 

a written description of its 

properties. 

b) Use two-dimensional 

representations of threedimensional objects to visualize 

and solve problems involving 

surface area and volume. 

c) Identify or draw angles and other 

geometric figures in the plane. 

c) Identify, define, or describe 

geometric shapes in the plane and 

in three-dimensional space given a 

visual representation. 

c) Give precise mathematical 

descriptions or definitions of 

geometric shapes in the plane and 

in three-dimensional space. 

d) Draw or sketch from a written 

description polygons, circles, or 

semicircles. 

d) Draw or sketch from a written 

description plane figures (e.g., 

isosceles triangles, regular 

polygons, curved figures) and 

planar images of threedimensional figures (e.g., 

polyhedra, spheres, and 

hemispheres). 

e) Represent or describe a threedimensional situation in a twodimensional drawing from 

different views. 

e) Describe or analyze properties of 

spheres and hemispheres. 

f) Describe attributes of two- and 

three-dimensional shapes. 

f) Demonstrate an understanding 

about the two- and threedimensional shapes in our world 

through identifying, drawing, 

modeling, building, or taking 

apart. 

continued on page 23 

22 
Mathematics Framework for the 2007 NAEP 

Geometry (continued) 

2) Transformation of shapes and preservation of properties 

GRADE 4 GRADE 8 GRADE 12 

a) Identify whether a figure is 

symmetrical, or draw lines of 

symmetry. 

a) Identify lines of symmetry in 

plane figures or recognize and 

classify types of symmetries of 

plane figures. 

a) Recognize or identify types of 

symmetries (e.g., point, line, 

rotational, self-congruences) of 

two- and three-dimensional 

figures. 

b) Give or recognize the precise 

mathematical relationship (e.g., 

congruence, similarity, 

orientation) between a figure and 

its image under a transformation. 

c) Identify the images resulting from 

flips (reflections), slides 

(translations), or turns (rotations). 

c) Recognize or informally describe 

the effect of a transformation on 

two-dimensional geometric shapes 

(reflections across lines of 

symmetry, rotations, translations, 

magnifications, and contractions). 

c) Perform or describe the effect of a 

single transformation on two- and 

three-dimensional geometric 

shapes (reflections across lines of 

symmetry, rotations, translations, 

and dilations). 

d) Recognize which attributes (such 

as shape and area) change or don’t 

change when plane figures are cut 

up or rearranged. 

d) Predict results of combining, 

subdividing, and changing shapes 

of plane figures and solids (e.g., 

paper folding, tiling, and cutting 

up and rearranging pieces). 

d) Describe the final outcome of 

successive transformations. 

e) Match or draw congruent figures 

in a given collection. 

e) Justify relationships of 

congruence and similarity, and 

apply these relationships using 

scaling and proportional 

reasoning. 

e) Justify relationships of 

congruence and similarity, and 

apply these relationships using 

scaling and proportional 

reasoning. 

f) For similar figures, identify and 

use the relationships of 

conservation of angle and of 

proportionality of side length and 

perimeter. 

continued on page 24 

23 
Mathematics Framework for the 2007 NAEP 

Geometry (continued) 

3) Relationships between geometric figures 

GRADE 4 GRADE 8 GRADE 12 

a) Analyze or describe patterns of 

geometric figures by increasing 

number of sides, changing size or 

orientation (e.g., polygons with 

more and more sides). 

b) Assemble simple plane shapes to 

construct a given shape. 

b) Apply geometric properties and 

relationships in solving simple 

problems in two and three 

dimensions. 

b) Apply geometric properties and 

relationships in solving multi-step 

problems in two and three 

dimensions (including rigid and 

non-rigid figures). 

c) Recognize two-dimensional faces 

of three-dimensional shapes. 

c) Represent problem situations with 

simple geometric models to solve 

mathematical or real-world 

problems. 

c) Represent problem situations with 

geometric models to solve 

mathematical or real-world 

problems. 

d) Use the Pythagorean theorem to 

solve problems. 

d) Use the Pythagorean theorem to 

solve problems in two- or threedimensional situations. 

e) Describe and analyze properties of 

circles (e.g., perpendicularity of 

tangent and radius, angle inscribed 

in a semicircle). 

f) Describe and compare properties 

of simple and compound figures 

composed of triangles, squares, 

and rectangles. 

f) Describe or analyze simple 

properties of, or relationships 

between, triangles, quadrilaterals, 

and other polygonal plane figures. 

f) Analyze properties or 

relationships of triangles, 

quadrilaterals, and other 

polygonal plane figures. 

g) Describe or analyze properties and 

relationships of parallel or 

intersecting lines. 

g) Describe or analyze properties and 

relationships of parallel, 

perpendicular, or intersecting 

lines, including the angle 

relationships that arise in these 

cases. 

continued on page 25 

24 
Mathematics Framework for the 2007 NAEP 

Geometry (continued) 

4) Position and direction 

GRADE 4 GRADE 8 GRADE 12 

a) Describe relative positions of 

points and lines using the 

geometric ideas of parallelism or 

perpendicularity. 

a) Describe relative positions of 

points and lines using the 

geometric ideas of midpoint, 

points on common line through a 

common point, parallelism, or 

perpendicularity. 

b) Describe the intersection of two or 

more geometric figures in the 

plane (e.g., intersection of a circle 

and a line). 

b) Describe the intersection of two or 

more geometric figures in the 

plane (e.g., intersection of a circle 

and a line). 

c) Visualize or describe the cross 

section of a solid. 

c) Visualize or describe the cross 

section of a solid. 

d) Construct geometric figures with 

vertices at points on a coordinate 

grid. 

d) Represent geometric figures using 

rectangular coordinates on a 

plane. 

d) Represent geometric figures using 

rectangular coordinates on a 

plane. 

e) Use vectors to represent velocity 

and direction. 

5) Mathematical reasoning 

GRADE 4 GRADE 8 GRADE 12 

a) Distinguish which objects in a 

collection satisfy a given 

geometric definition and explain 

choices. 

a) Make and test a geometric 

conjecture about regular polygons. 

a) Make, test, and validate geometric 

conjectures using a variety of 

methods including deductive 

reasoning and counterexamples. 

DATA ANALYSIS AND PROBABILITY 

Data analysis covers the entire process of collecting, organizing, summarizing, and 

interpreting data. This is the heart of the discipline called statistics; it is in evidence 

whenever quantitative information is used to determine a course of action. To 

emphasize the spirit of statistical thinking, data analysis should begin with a question to 

be answered, not with the data. Data should be collected only with a specific question 

(or questions) in mind and only after a plan (usually called a design) for collecting data 

relevant to the question is thought out. Beginning at an early age, students should grasp 

the fundamental principle that looking for questions in an existing data set is far 

different from the scientific method of collecting data to verify or refute a well-posed 

question. A pattern can be found in almost any data set if one looks hard enough, but a 

pattern discovered in this way is often meaningless, especially from the point of view of 

statistical inference. 

25 
Mathematics Framework for the 2007 NAEP 

In the context of data analysis, or statistics, probability can be thought of as the study 

of potential patterns in outcomes that have not yet been observed. We say that the 

probability of a balanced coin coming up heads when flipped is one-half because we 

believe that about half of the flips would turn out to be heads if we flipped the coin 

many times. Under random sampling, patterns for outcomes of designed studies can be 

anticipated and used as the basis for making decisions. If the coin actually turned up 

heads 80 percent of the time, we would suspect that it was not balanced. The whole 

probability distribution of all possible outcomes is important in most statistics problems 

because the key to decisionmaking is to decide whether a particular observed outcome is 

unusual (located in a tail of the probability distribution). For example, 4 as a grade point 

average is unusually high among most groups of students, 4 as the pound weight of a 

baby is unusually low, and 4 as the number of runs scored in a baseball game is not 

unusual in either direction. 

By grade 4, students should be expected to apply their understanding of number and 

quantity to pose questions that can be answered by collecting appropriate data. They 

should be expected to organize data in a table or a plot and summarize the essential 

features of center, spread, and shape both verbally and with simple summary statistics. 

Simple comparisons can be made between two related data sets, but more formal 

inference based on randomness should come later. The basic concept of chance and 

statistical reasoning can be built into meaningful contexts, though, such as, “If I draw 

two names from among those of the students in the room, am I likely to get two girls?” 

Such problems can be addressed through simulation. 

Building on the same definition of data analysis and the same principles of describing 

distributions of data through center, spread, and shape, grade 8 students will be expected 

to use a wider variety of organizing and summarizing techniques. They can also begin to 

analyze statistical claims through designed surveys and experiments that involve 

randomization, with simulation being the main tool for making simple statistical 

inferences. They will begin to use more formal terminology related to probability and 

data analysis. 

Students in grade 12 will be expected to use a wide variety of statistical techniques 

for all phases of the data analysis process, including a more formal understanding of 

statistical inference (but still with simulation as the main inferential analysis tool). In 

addition to comparing univariate data sets, students at this level should be able to 

recognize and describe possible associations between two variables by looking at twoway tables for categorical variables or scatter plots for measurement variables. 

Association between variables is related to the concepts of independence and 

dependence, and an understanding of these ideas requires knowledge of conditional 

probability. These students should be able to use statistical models (linear and nonlinear equations) to describe possible associations between measurement variables and 

should be familiar with techniques for fitting models to data. 

26 
Mathematics Framework for the 2007 NAEP 

Data Analysis and Probability 

1) Data representation 

GRADE 4 GRADE 8 GRADE 12 

The following representations of data are indicated for each grade level. Objectives in which only a subset of these 

representations is applicable are indicated in the parenthesis associated with the objective. 

Pictographs, bar graphs, circle 

graphs, line graphs, line plots, tables, 

and tallies. 

Histograms, line graphs, scatter plots, 

box plots, circle graphs, stem and leaf 

plots, frequency distributions, tables, 

and bar graphs. 

Histograms, line graphs, scatter plots, 

box plots, circle graphs, stem and leaf 

plots, frequency distributions, and 

tables. 

a) Read or interpret a single set of 

data. 

a) Read or interpret data, including 

interpolating or extrapolating from 

data. 

a) Read or interpret data, including 

interpolating or extrapolating from 

data. 

b) For a given set of data, complete a 

graph (limits of time make it 

difficult to construct graphs 

completely). 

b) For a given set of data, complete a 

graph and then solve a problem 

using the data in the graph 

(histograms, line graphs, scatter 

plots, circle graphs, and bar 

graphs). 

b) For a given set of data, complete a 

graph and then solve a problem 

using the data in the graph 

(histograms, scatter plots, line 

graphs). 

c) Solve problems by estimating and 

computing within a single set of 

data. 

c) Solve problems by estimating and 

computing with data from a single 

set or across sets of data. 

c) Solve problems by estimating and 

computing with univariate or 

bivariate data (including scatter 

plots and two-way tables). 

d) Given a graph or a set of data, 

determine whether information is 

represented effectively and 

appropriately (histograms, line 

graphs, scatter plots, circle graphs, 

and bar graphs). 

d) Given a graph or a set of data, 

determine whether information is 

represented effectively and 

appropriately (bar graphs, box 

plots, histograms, scatter plots, 

line graphs). 

e) Compare and contrast the 

effectiveness of different 

representations of the same data. 

e) Compare and contrast the 

effectiveness of different 

representations of the same data. 

continued on page 28 

27 
Mathematics Framework for the 2007 NAEP 

Data Analysis and Probability (continued) 

2) Characteristics of data sets 

GRADE 4 GRADE 8 GRADE 12 

a) Calculate, use, or interpret mean, 

median, mode, or range. 

a) Calculate, interpret, or use mean, 

median, mode, range, interquartile 

range, or standard deviation. 

b) Given a set of data or a graph, 

describe the distribution of the 

data using median, range, or 

mode. 

b) Describe how mean, median, 

mode, range, or interquartile 

ranges relate to the shape of the 

distribution. 

b) Recognize how linear transformations of one-variable data affect 

mean, median, mode, and range 

(e.g., effect on the mean by adding 

a constant to each data point). 

c) Identify outliers and determine 

their effect on mean, median, 

mode, or range. 

c) Determine the effect of outliers on 

mean, median, mode, range, 

interquartile range, or standard 

deviation. 

d) Compare two sets of related data. d) Using appropriate statistical 

measures, compare two or more 

data sets describing the same 

characteristic for two different 

populations or subsets of the same 

population. 

d) Compare two or more data sets 

using mean, median, mode, range, 

interquartile range, or standard 

deviation describing the same 

characteristic for two different 

populations or subsets of the same 

population. 

e) Visually choose the line that best 

fits given a scatter plot and 

informally explain the meaning of 

the line. Use the line to make 

predictions. 

e) Given a set of data or a scatter 

plot, visually choose the line of 

best fit and explain the meaning of 

the line. Use the line to make 

predictions. 

f) Use or interpret a normal distribution as a mathematical model 

appropriate for summarizing 

certain sets of data. 

 g) Given a scatter plot, make 

decisions or predictions involving 

a line or curve of best fit. 

h) Given a scatter plot, estimate the 

correlation coefficient (e.g., Given 

a scatter plot, is the correlation 

closer to 0, .5, or 1.0? Is it a 

positive or negative correlation?). 

continued on page 29 

28 
Mathematics Framework for the 2007 NAEP 

Data Analysis and Probability (continued) 

3) Experiments and samples 

GRADE 4 GRADE 8 GRADE 12 

a) Given a sample, identify possible 

sources of bias in sampling. 

a) Identify possible sources of bias in 

data collection methods and 

describe how such bias can be 

controlled and reduced. 

b) Distinguish between a random and 

non-random sample. 

b) Recognize and describe a method 

to select a simple random sample. 

d) Evaluate the design of an 

experiment. 

c) Make inferences from sample 

results. 

d) Identify or evaluate the characteristics of a good survey or of a 

well-designed experiment. 

4) Probability 

GRADE 4 GRADE 8 GRADE 12 

a) Use informal probabilistic 

thinking to describe chance events 

(i.e., likely and unlikely, certain 

and impossible). 

a) Analyze a situation that involves 

probability of an independent 

event. 

a) Analyze a situation that involves 

probability of independent or 

dependent events. 

b) Determine a simple probability 

from a context that includes a 

picture. 

b) Determine the theoretical 

probability of simple and 

compound events in familiar 

contexts. 

b) Determine the theoretical 

probability of simple and 

compound events in familiar or 

unfamiliar contexts. 

c) Estimate the probability of simple 

and compound events through 

experimentation or simulation. 

c) Given the results of an experiment 

or simulation, estimate the 

probability of simple or 

compound events in familiar or 

unfamiliar contexts. 

d) Use theoretical probability to 

evaluate or predict experimental 

outcomes. 

d) Use theoretical probability to 

evaluate or predict experimental 

outcomes. 

e) List all possible outcomes of a 

given situation or event. 

e) Determine the sample space for a 

given situation. 

e) Determine the number of ways an 

event can occur using tree 

diagrams, formulas for 

combinations and permutations, or 

other counting techniques. 

f) Use a sample space to determine 

the probability of the possible 

outcomes of an event. 

f) Determine the probability of the 

possible outcomes of an event. 

g) Represent the probability of a 

given outcome using a picture or 

other graphic. 

g) Represent probability of a given 

outcome using fractions, decimals, 

and percents. 

continued on page 30 

29 
Mathematics Framework for the 2007 NAEP 

Data Analysis and Probability (continued) 

4) Probability (continued) 

GRADE 4 GRADE 8 GRADE 12 

h) Determine the probability of 

independent and dependent 

events. (Dependent events should 

be limited to linear functions with 

a small sample size.) 

h) Determine the probability of 

independent and dependent 

events. 

i) Determine conditional probability 

using two-way tables. 

j) Interpret probabilities within a 

given context. 

j) Interpret probabilities within a 

given context. 

ALGEBRA 

Algebra was pioneered in the Middle Ages by mathematicians in the Middle East and 

Asia as a method of solving equations easily and efficiently by manipulation of 

symbols, rather than by the earlier geometric methods of the Greeks. The two 

approaches were eventually united in the analytic geometry of René Descartes. Modern 

symbolic notation, developed in the Renaissance, greatly enhanced the power of the 

algebraic method; from the 17th century forward, algebra in turn promoted advances in 

all branches of mathematics and science. 

The widening use of algebra led to the study of its formal structure. Out of this were 

gradually distilled the “rules of algebra,” a compact summary of the principles behind 

algebraic manipulation. A parallel line of thought produced a simple but flexible 

concept of function and also led to the development of set theory as a comprehensive 

background for mathematics. When it is taken liberally to include these ideas, algebra 

reaches from the foundations of mathematics to the frontiers of current research. 

These two aspects of algebra, a powerful representational tool and a vehicle for 

comprehensive concepts such as function, form the basis for the expectations throughout 

the grades. By grade 4, students are expected to be able to recognize and extend simple 

numeric patterns as one foundation for a later understanding of function. They can begin 

to understand the meaning of equality and some of its properties, as well as the idea of 

an unknown quantity as a precursor to the concept of variable. 

As students move into middle school, the ideas of function and variable become more 

important. Representation of functions as patterns, via tables, verbal descriptions, 

symbolic descriptions, and graphs, can combine to promote a flexible grasp of the idea 

of function. Linear functions receive special attention. They connect to the ideas of 

proportionality and rate, forming a bridge that will eventually link arithmetic to 

calculus. Symbolic manipulation in the relatively simple context of linear equations is 

reinforced by other means of finding solutions, including graphing by hand or with 

calculators. 

30 
Mathematics Framework for the 2007 NAEP 

In high school, students should become comfortable in manipulating and interpreting 

more complex expressions. The rules of algebra should come to be appreciated as a 

basis for reasoning. Non-linear functions, especially quadratic functions, and also power 

and exponential functions, are introduced to solve real-world problems. Students should 

become accomplished at translating verbal descriptions of problem situations into 

symbolic form. Expressions involving several variables, systems of linear equations, 

and the solutions to inequalities are encountered by grade 12. 

Algebra 

1) Patterns, relations, and functions 

GRADE 4 GRADE 8 GRADE 12 

a) Recognize, describe, or extend 

numerical patterns. 

a) Recognize, describe, or extend 

numerical and geometric patterns 

using tables, graphs, words, or 

symbols. 

a) Recognize, describe, or extend 

arithmetic, geometric 

progressions, or patterns using 

words or symbols. 

b) Given a pattern or sequence, 

construct or explain a rule that can 

generate the terms of the pattern 

or sequence. 

b) Generalize a pattern appearing in 

a numerical sequence or table or 

graph using words or symbols. 

b) Express the function in general 

terms (either recursively or 

explicitly), given a table, verbal 

description, or some terms of a 

sequence. 

c) Given a description, extend or 

find a missing term in a pattern or 

sequence. 

c) Analyze or create patterns, 

sequences, or linear functions 

given a rule. 

d) Create a different representation 

of a pattern or sequence given a 

verbal description. 

e) Recognize or describe a 

relationship in which quantities 

change proportionally. 

e) Identify functions as linear or nonlinear or contrast distinguishing 

properties of functions from 

tables, graphs, or equations. 

e) Identify or analyze distinguishing 

properties of linear, quadratic, 

inverse (y = k/x) or exponential 

functions from tables, graphs, or 

equations. 

f) Interpret the meaning of slope or 

intercepts in linear functions. 

g) Determine the domain and range 

of functions given various 

contexts. 

 h) Recognize and analyze the general 

forms of linear, quadratic, inverse, 

or exponential functions (e.g., in y 

= ax + b, recognize the roles of a 

and b). 

i) Express linear and exponential 

functions in recursive and explicit 

form given a table or verbal 

description. 

continued on page 32 

31 
Mathematics Framework for the 2007 NAEP 

Algebra (continued) 

2) Algebraic representations 

GRADE 4 GRADE 8 GRADE 12 

a) Translate between the different 

forms of representations (symbolic, numerical, verbal, or 

pictorial) of whole number relationships (such as from a written 

description to an equation or from 

a function table to a written 

description). 

a) Translate between different representations of linear expressions 

using symbols, graphs, tables, 

diagrams, or written descriptions. 

a) Translate between different representations of algebraic 

expressions using symbols, 

graphs, tables, diagrams, or 

written descriptions. 

b) Analyze or interpret linear 

relationships expressed in 

symbols, graphs, tables, diagrams, 

or written descriptions. 

b) Analyze or interpret relationships 

expressed in symbols, graphs, 

tables, diagrams, or written 

descriptions. 

c) Graph or interpret points with 

whole number or letter 

coordinates on grids or in the first 

quadrant of the coordinate plane. 

c) Graph or interpret points that are 

represented by ordered pairs of 

numbers on a rectangular 

coordinate system. 

c) Graph or interpret points that are 

represented by one or more 

ordered pairs of numbers on a 

rectangular coordinate system. 

d) Solve problems involving coordinate pairs on the rectangular 

coordinate system. 

d) Perform or interpret 

transformations on the graphs of 

linear and quadratic functions. 

e) Verify a conclusion using 

algebraic properties. 

e) Make, validate, and justify 

conclusions and generalizations 

about linear relationships. 

e) Use algebraic properties to 

develop a valid mathematical 

argument. 

f) Use an algebraic model of a 

situation to make inferences or 

predictions. 

g) Identify or represent functional 

relationships in meaningful 

contexts including proportional, 

linear, and common non-linear 

(e.g., compound interest, bacterial 

growth) in tables, graphs, words, 

or symbols. 

g) Given a real-world situation, 

determine if a linear, quadratic, 

inverse, or exponential function 

fits the situation (e.g., half-life 

bacterial growth). 

 h) Solve problems involving 

exponential growth and decay. 

continued on page 33 

32 
Mathematics Framework for the 2007 NAEP 

Algebra (continued) 

3) Variables, expressions, and operations 

GRADE 4 GRADE 8 GRADE 12 

a) Use letters and symbols to 

represent an unknown quantity 

in a simple mathematical 

expression. 

b) Express simple mathematical 

relationships using number 

sentences. 

b) Write algebraic expressions, 

equations, or inequalities to 

represent a situation. 

b) Write algebraic expressions, equations, 

or inequalities to represent a situation. 

c) Perform basic operations, 

using appropriate tools, on 

linear algebraic expressions 

(including grouping and order 

of multiple operations 

involving basic operations, 

exponents, roots, simplifying, 

and expanding). 

c) Perform basic operations, using 

appropriate tools, on algebraic 

expressions (including grouping and 

order of multiple operations involving 

basic operations, exponents, roots, 

simplifying, and expanding). 

d) Write equivalent forms of algebraic 

expressions, equations, or inequalities to 

represent and explain mathematical 

relationships. 

continued on page 34 

33 
Mathematics Framework for the 2007 NAEP 

Algebra (continued) 

4) Equations and inequalities 

GRADE 4 GRADE 8 GRADE 12 

a) Find the value of the unknown in 

a whole number sentence. 

a) Solve linear equations or 

inequalities (e.g., ax + b = c or ax 

+ b = cx + d or ax + b > c). 

a) Solve linear, rational, or quadratic 

equations or inequalities. 

b) Interpret "=" as an equivalence between two expressions and use 

this interpretation to solve 

problems. 

c) Analyze situations or solve 

problems using linear equations 

and inequalities with rational 

coefficients symbolically or 

graphically (e.g., ax + b = c or ax 

+ b = cx + d). 

c) Analyze situations or solve 

problems using linear or quadratic 

equations or inequalities symbolically or graphically. 

d) Interpret relationships between 

symbolic linear expressions and 

graphs of lines by identifying and 

computing slope and intercepts 

(e.g., know in y = ax + b, that a is 

the rate of change and b is the 

vertical intercept of the graph). 

d) Recognize the relationship 

between the solution of a system 

of linear equations and its graph. 

e) Use and evaluate common 

formulas [e.g., relationship 

between a circle’s circumference 

and diameter (C = πd), distance 

and time under constant speed]. 

e) Solve problems involving more 

advanced formulas [e.g., the 

volumes and surface areas of three 

dimensional solids; or such 

formulas as: A = P(1 + r)t, A = 

Pert]. 

f) Given a familiar formula, solve 

for one of the variables. 

g) Solve or interpret systems of 

equations or inequalities. 

34 
Chapter Four: Mathematical Complexity 

of Items 

Each NAEP item assesses an objective that can be associated with a single content 

area of mathematics, such as number or geometry. The item also makes certain 

demands on students’ thinking. These demands constitute the mathematical 

complexity of the item, which is the second dimension of the mathematics framework. 

The demands on thinking that an item makes (what it asks the student to recall, 

understand, reason about, and do) are determined on the assumption that the student is 

familiar with the mathematics of the task. If a student has not studied the mathematics, 

the task is likely to make different and heavier demands, and the student may well not 

be successful. Items are chosen for administration at a given grade level in part on the 

basis of their appropriateness for typical curricula, but the complexity of those items is 

always independent of the particular curriculum a student has experienced. 

The categories (low complexity, moderate complexity, and high complexity) form an 

ordered description of the demands an item may make on a student. Items at the low 

level of complexity, for example, may ask a student to recall a property. At the moderate 

level, an item may ask the student to make a connection between two properties; at the 

high level, an item may ask a student to analyze the assumptions made in a 

mathematical model. This is an example of the distinctions made in item complexity to 

provide balance in the item pool. The ordering is not intended to imply that mathematics 

is learned or should be taught in such an ordered way. 

The complexity dimension is both similar to and different from the levels of 

mathematical ability (conceptual understanding, procedural knowledge, and problem 

solving) that were used in the NAEP Mathematics Framework for the 1996 and 2000 

assessments. The dimensions are similar in that both attempt to address the kind of 

thinking that the student is doing when working on an item. They are also similar in that 

although neither dimension is used to define specific percentages of items in each 

content area, both are used to help define item descriptors and achieve a balance across 

the tasks administered at each grade level. Level of complexity is different from level of 

mathematical ability, however, in that complexity describes the mathematical 

expectations of an item, whereas mathematical ability (along with the associated 

construct of mathematical power) requires an inference about the skill, knowledge, and 

background of the students taking the item. 

The mathematical complexity of an item is not directly related to its format (multiple 

choice, short constructed response, or extended constructed response). Items requiring 

that the student generate a response tend to make somewhat heavier demands on 

students than items requiring a choice among alternatives, but that is not always the 

case. Any type of item can deal with mathematics of greater or lesser depth and 

sophistication. There are multiple-choice items that assess complex mathematics, and 

constructed-response items can be crafted to assess routine mathematical ideas. 

Moreover, the mathematical complexity of an item is constant; it does not vary 

depending on the score given for a certain kind or level of response. 

35 
Mathematics Framework for the 2007 NAEP 

Low Complexity 

This category relies heavily on the recall and recognition of previously learned 

concepts and principles. Items typically specify what the student is to do, which is often 

to carry out some procedure that can be performed mechanically. It is not left to the 

student to come up with an original method or solution. The following are some, but not 

all, of the demands that items in the low-complexity category might make: 

• Recall or recognize a fact, term, or property. 

• Recognize an example of a concept. 

• Compute a sum, difference, product, or quotient. 

• Recognize an equivalent representation. 

• Perform a specified procedure. 

• Evaluate an expression in an equation or formula for a given variable. 

• Solve a one-step word problem. 

• Draw or measure simple geometric figures. 

• Retrieve information from a graph, table, or figure. 

Examples 

See appendix B for solutions and scoring guides. 

• Recall or recognize a fact, term, or property. 

How many fourths make a whole?

Answer: ___________________ 

Source: 1996 NAEP (grade 4) Percent correct: 50 

• Compute a sum, difference, product, or quotient. 

+6 + -12 = 

A. -6 

B. +6 

C. -18 

D. +18 

Source: 1990 NAEP (grade 8) Percent correct: 68 

• Provide or recognize equivalent representations. 

N stands for the number of stamps John had. He gave 12 stamps to his sister. 

Which expression tells how many stamps John has now? 

A. N + 12 

36 
Mathematics Framework for the 2007 NAEP 

B. N – 12 

C. 12 – N 

D. 12 x N 

Source: 1996 NAEP (grade 4) Percent correct: 67 

• Perform a specified procedure. 

How many hours are equal to 150 minutes? 

A. 11⁄2 

B. 21⁄4 

C. 21/3 

D. 21⁄2 

E. 25/6 

Source: 1990 NAEP (grade 12) Percent correct: 74 

• Retrieve information from a graph, table, or figure. 

What is the weight shown on the scale? 

A. 6 pounds 

B. 7 pounds 

C. 51 pounds 

D. 60 pounds 

Source: 1992 NAEP (grade 4) Percent correct: 44 

Moderate Complexity 

Items in the moderate-complexity category involve more flexibility of thinking and 

choice among alternatives than do those in the low-complexity category. They require a 

37
Mathematics Framework for the 2007 NAEP 

response that goes beyond the habitual, is not specified, and ordinarily has more than a 

single step. The student is expected to decide what to do, using informal methods of 

reasoning and problem-solving strategies, and to bring together skill and knowledge 

from various domains. The following illustrate some of the demands that items of 

moderate complexity might make: 

• Represent a situation mathematically in more than one way. 

• Select and use different representations, depending on situation and purpose. 

• Solve a word problem requiring multiple steps. 

• Compare figures or statements. 

• Provide a justification for steps in a solution process. 

• Interpret a visual representation. 

• Extend a pattern. 

• Retrieve information from a graph, table, or figure and use it to solve a problem 

requiring multiple steps. 

• Formulate a routine problem, given data and conditions. 

• Interpret a simple argument. 

Examples 

See appendix B for solutions and scoring guides. 

The following shapes were provided to students. (Shapes were larger than shown.) 

(Grade 8 version) 

Bob, Carmen, and Tyler were comparing the areas of N and P. Bob said that N and 

P have the same area. Carmen said that the area of N is larger. Tyler said that the 

area of P is larger. 

Who was correct? _______________________________ 

Use words or pictures (or both) to explain why. 

Source: 1996 NAEP (grades 4 and 8) Percent correct: 27 

38 
Mathematics Framework for the 2007 NAEP 

• Interpret a visual representation. 

In this figure, how many small cubes were put together to form the large cube? 

A. 7 

B. 8 

C. 12 

D. 24 

Source: 1996 NAEP (grade 4) Percent correct: 33 

• Extend a pattern. 

From any vertex of a 4-sided polygon, 1 diagonal can be drawn. 

From any vertex of a 5-sided polygon, 2 diagonals can be drawn. 

From any vertex of a 6-sided polygon, 3 diagonals can be drawn. 

From any vertex of a 7-sided polygon, 4 diagonals can be drawn. 

How many diagonals can be drawn from any vertex of a 20-sided polygon? 

Answer: ___________________________ 

Source: 1996 NAEP (grade 8) Percent correct: 54 

• Interpret a simple argument. 

Tracy said, “I can multiply 6 by another number and get an answer that is smaller 

than 6.” 

Pat said, “No, you can’t. Multiplying 6 by another number always makes the 

answer 6 or larger.” 

Who is correct? Give a reason for your answer. 

Source: 1992 NAEP (grades 8 and 12) Percent correct: 49 (grade 8); 63 (grade 12) 

39 
Mathematics Framework for the 2007 NAEP 

High Complexity 

High-complexity items make heavy demands on students, who must engage in more 

abstract reasoning, planning, analysis, judgment, and creative thought. A satisfactory 

response to the item requires that the student think in abstract and sophisticated ways. 

Items at the level of high complexity may ask the student to do any of the following: 

• Describe how different representations can be used for different purposes. 

• Perform a procedure having multiple steps and multiple decision points. 

• Analyze similarities and differences between procedures and concepts. 

• Generalize a pattern. 

• Formulate an original problem, given a situation. 

• Solve a novel problem. 

• Solve a problem in more than one way. 

• Explain and justify a solution to a problem. 

• Describe, compare, and contrast solution methods. 

• Formulate a mathematical model for a complex situation. 

• Analyze the assumptions made in a mathematical model. 

• Analyze or produce a deductive argument. 

• Provide a mathematical justification. 

Examples 

See appendix B for solutions and scoring guides. 

• Generalize a pattern. 

A pattern of dots is shown below. At each step, more dots are added to the 

pattern. The number of dots added at each step is more than the number added in 

the previous step. The pattern continues infinitely. 

Marcy has to determine the number of dots in the 20th step, but she does not 

want to draw all 20 pictures and then count the dots. Explain or show how she 

could do this and give the answer that Marcy should get for the number of dots. 

Source: 1992 NAEP (grade 8) Percent correct: 6 

40 
Mathematics Framework for the 2007 NAEP 

• Analyze or produce a deductive argument. 

Jaime knows the following facts about points A, B, and C. 

• Points A, B, and C are on the same line, but might not be in that order. 

• Point C is twice as far from point A as it is from point B. 

Jaime concluded that point C is always between points A and B. 

Is Jaime’s conclusion correct?

Yes No

In the space provided, use a diagram to explain your answer. 

Source: 1996 NAEP (grade 8) Percent correct: 23 

• Provide a mathematical justification 

In Mr. Bell’s classes, the students voted for their favorite shape for a symbol. 

Here are the results. 

Class 1 Class 2 Class 3 

Shape N 9 14 11 

Shape P 1 9 17 

Shape Q 22 7 2 

Using the information in the chart, Mr. Bell must select one of the shapes to be 

the symbol. Which one should he select and why? 

The shape Mr. Bell should select: _________________ 

Explain. 

Source: 1996 NAEP (grades 4 and 8) Percent correct: 31 (grade 4); 58 (grade 8) 

41 
Mathematics Framework for the 2007 NAEP 

Balance of Mathematical Complexity 

The ideal balance sought for NAEP is not necessarily the balance one would wish for 

curriculum or instruction in mathematics education. Balance here must be considered in 

the context of the constraints of an assessment such as NAEP. These constraints include 

the timed nature of the test and its paper-and-pencil format. Items of high complexity, 

for example, often take more time to complete. At the same time, some items of all three 

types are essential to assess the full range of students’ mathematical achievement. 

Within that context, the ideal balance would be that half of the score on the assessment 

is based on items of moderate complexity, with the remainder of the score based equally 

on items of low and high complexity. 

42 
Chapter Five: Item Formats 

Central to the development of the NAEP assessment in mathematics is the careful 

selection of items/tasks. Since the 1992 assessment, items have consisted of 

three formats: multiple choice, short constructed response, and extended 

constructed response. Testing time on NAEP is divided evenly between multiple-choice 

items and both types of constructed-response items, as shown below: 

Multiple-Choice Items 

Multiple-choice items require students to read, reflect, or compute, and then to select 

the alternative that best expresses what they believe the answer to be. This format is 

appropriate for quickly determining whether students have achieved certain knowledge 

and skills. A carefully constructed multiple-choice item can assess any of the levels of 

mathematical complexity (described in chapter four), from simple procedures to more 

sophisticated concepts. Such items are limited in the extent to which they can provide 

evidence of the depth of students’ thinking. 

Example 

In the graph above, each dot shows the number of sit-ups and the corresponding age for 

1 of 13 people. According to this graph, what is the median number of sit-ups for these 

13 people? 

43 
Mathematics Framework for the 2007 NAEP 

A. 15 

B. 20 

C. 45 

D. 50 

E. 55 

This item assesses students’ understanding of two concepts: median, and how data are 

displayed on a graph. For students who are able to combine these concepts, little or no 

computation is required to determine the correct answer (D). 

Short Constructed-Response Items 

To provide more reliable and valid opportunities for extrapolating about students’ 

approaches to problems, NAEP assessments have included items that are often referred 

to as short constructed-response items. These are short-answer items that require 

students to give either a numerical result or the correct name or classification for a 

group of mathematical objects, draw an example of a given concept, or perhaps write a 

brief explanation for a given result. 

Example 

Product 

2 x 2 = 4 

2 x 2 x 2 = 8 

2 x 2 x 2 x 2 = 16 

2 x 2 x 2 x 2 x 2 = 32 

If the pattern shown continues, could 375 be one of the products in this pattern? 

Explain why or why not. 

44 
Mathematics Framework for the 2007 NAEP 

Scoring Guide 

Score and Description 

1 - Correct 

 Correct reason: 

“Because 375 is not divisible by 2 (or is not even)” 

OR 

“Because 375 is between two of the numbers in the pattern.” 

Acceptable: “Because 375 is uneven.” 

Not acceptable: 

“Because you’re counting by 2s.” 

OR 

“Because you’re multiplying by 2.” 

0 - Incorrect 

Any incorrect or incomplete reason. 

This item assesses students’ understanding of number patterns and of odd and even 

numbers. Students are given the opportunity to explain why 375 could not be one of the 

products in the pattern. Scoring for this item was done on a correct/incorrect basis. Items 

such as this are valuable in providing greater evidence of the depth of students’ 

understanding than might be accomplished in other formats. Scoring for some short 

constructed-response items uses more than two points in the scale to award partial credit 

for work that is partially correct. Full credit is reserved for work that is mathematically 

correct and complete. 

Extended Constructed-Response Items 

Extended constructed-response items require students to consider a situation that 

demands more than a numerical response or a short verbal communication. If it is a 

problem to solve, the student is asked to carefully consider a situation within or across 

content areas, understand what is required to “solve” the situation, choose a plan of 

attack, carry out the attack, and interpret the solution derived in terms of the original 

situation. In the example that follows, the student is asked to analyze a situation and 

provide a mathematical explanation. 

45 
Mathematics Framework for the 2007 NAEP 

Example 

In a game, Carla and Maria are making subtraction problems using tiles numbered 1 to 

5. The player whose subtraction problem gives the largest answer wins the game. 

Look at where each girl placed two of her tiles. 

In this item, students are given the opportunity to display their understanding of place 

value and subtraction. They must examine the numbers left on each player’s tiles, 

consider all the possible placements of those tiles in the remaining spaces, then compare 

the results of the subtraction and construct an explanation of the situation. 

Scoring Extended Constructed-Response Items 

Extended constructed-response items in mathematics should be evaluated according 

to an established grading scale developed from a sample of actual student responses. 

The scale used should follow a multiple-point format similar to the following (this is the 

scoring guide for the previous example item): 

46 
Mathematics Framework for the 2007 NAEP 

Scoring Guide 

Solution: 

Maria will win the game. 

The following reasons may be given: 

a. The largest possible difference for Carla is less than 100 and the smallest 

possible difference for Maria is 194. 

b. Carla will only get a difference of 91 or less but Maria will get several larger 

differences. 

c. Carla can have only up to 143 as her top number, but Maria can have 435 as 

her largest number. 

d. Carla has only 1 hundred but Maria can have 2, 3, or 4 hundreds. 

e. Maria can never take away as much as Carla. 

f. Any combination of problems to show that Maria’s difference is greater. 

47 
Mathematics Framework for the 2007 NAEP 

Score and Description 

4 - Extended 

Student answers Maria and gives explanation such as (a) or (b), or an appropriate 

combination of the other explanations. 

3 - Satisfactory 

Student answers Maria and gives explanation such as (c), (d), or (e). 

2 - Partial 

Student answers Maria with partially correct, or incomplete but relevant, explanation. 

1 - Minimal 

Student answers Maria and gives example such as in (f) but no explanation OR answers 

Maria with an incorrect explanation. 

0 - Incorrect 

Incorrect response 

A scoring scale adapted to a particular problem and applied by experienced scorers 

provides rich information about students’ understanding of concepts and procedures, 

and about their ability to solve a problem and communicate understanding of the 

process. 

48 
Bibliography 

Achieve, Inc. (2000). Foundations for Success: Mathematics for the Middle Grades 

(consultation draft). Washington, DC: Mathematics Achievement Partnership. 

Anderson, N.E., F.F. Jenkins, and K.E. Miller (1995). NAEP Inclusion Criteria and 

Testing Accommodations: Findings From the NAEP 1995 Field Test in Mathematics. 

Princeton, NJ: Educational Testing Service. 

Blank, R., and L.D. Wilson (1999). Improving Mathematics Education Using Results 

From NAEP and TIMSS. Washington, DC: Council of Chief State School Officers. 

Bourque, M.L., and S. Byrd (2000). Student Performance Standards on the National 

Assessment of Educational Progress: Affirmation and Improvements. Washington, DC: 

National Assessment Governing Board. 

Braswell, J., A. Lutkus, W. Grigg, S. Santapau, B. Tay-Lim, and M. Johnson (2001). 

The Nation’s Report Card: Mathematics 2000. Washington, DC: National Center for 

Education Statistics. 

Carlson, J.E., A.D. Lutkus, J. Mazzeo, and K.E. Voelkl (2000). Increasing the 

Participation of Special Needs Students in NAEP. Jessup, MD: U.S. Department of 

Education, Office of Educational Research and Improvement. 

Council for Basic Education (2000). Issues in Mathematics Education Assessment: 

NAEP Mathematics 2005. Washington, DC: Council of Chief State School Officers. 

Diperna, P., and T. Loveless (2000). The Brown Center Report on American Education: 

How Well Are American Students Learning? Washington, DC: The Brookings 

Institution. 

Dossey, J.A., J. Mazzeo, K.E. Miller, and C.M. Reese (1997). NAEP 1996 Mathematics 

Report Card for the Nation and the States. Washington, DC: Educational Testing 

Service and National Center for Education Statistics. 

Educational Testing Service (1988). Mathematics Objectives 1990 Assessment. 

Princeton, NJ: Educational Testing Service. 

Findell, B., J. Kilpatrick, and J. Swafford (2001). Adding It Up: Helping Children Learn 

Mathematics. Washington, DC: National Research Council. 

Kenney, P.A. (1999). An Analysis of the 1996 NAEP Mathematics Framework and 

Items for Grade 8. Pittsburgh, PA: University of Pittsburgh, Learning Research and 

Development Center. 

Kenney, P.A., E.A. Silver, C. Alacaci, and J.S. Zawojewski (1998). Content Analysis 

Project—State and NAEP Mathematics Assessment (final report). Pittsburgh, PA: 

University of Pittsburgh, Learning Research and Development Center. 

49 
Mathematics Framework for the 2007 NAEP 

Mazzeo, J., J. Carlson, K. Voelkl, and A. Lutkus (1999). Increasing the Participation of 

Special Needs Students in NAEP: A Report on 1996 Research Activities (NCES 

Publication No. 2000-473). Washington, DC: National Center for Education Statistics. 

Mitchell, J., E.F. Hawkins, F.B. Stancavage, and J. Dossey (1999). Estimation Skills, 

Mathematics-in-Context, and Advanced Skills in Mathematics. Washington, DC: 

National Center for Education Statistics. 

Mullis, I.V.S., M.O. Martin, T.A. Smith, R.A. Garden, K.D. Gregory, E.J. Gonzalez, 

S.J. Chrostowski, and K.M. O’Connor (2001). TIMSS Assessment Frameworks and 

Specifications 2003. Boston, MA: Boston College, The International Study Center. 

National Assessment of Educational Progress (1988). Mathematics Objectives: 1990 

Assessment. Princeton, NJ: Educational Testing Service. 

National Assessment Governing Board (1994). Mathematics Framework for the 1996 

and 2000 National Assessment of Educational Progress. Washington, DC: National 

Assessment Governing Board. 

National Assessment Governing Board (1992). 1994 National Assessment of 

Educational Progress: Mathematics Assessment Framework and Specifications. 

Washington, DC: The College Board and The National Assessment Governing Board. 

Olson, J.F., and A.A. Goldstein (1997). The Inclusion of Students With Disabilities and 

Limited English Proficient Students in Large-Scale Assessments: A Summary of Recent 

Progress (NCES Publication No. 97-482). Washington, DC: National Center for 

Education Statistics. 

Reese, C.M., K.E. Miller, J. Mazzeo, and J.A. Dossey (1997). 1996 Mathematics Report 

Card for the Nation and the States. Washington, DC: U.S. Department of Education, 

National Center for Education Statistics. 

Silver, E., and P. Kenney (eds.) (2000). Results From the Seventh Mathematics 

Assessment of the National Assessment of Educational Progress. Reston, VA: National 

Council of Teachers of Mathematics. 

U.S. Department of Education (2000). Before It’s Too Late. Washington, DC: U.S. 

Department of Education, National Commission of Mathematics and Science Teaching 

for the 21st Century. 

U.S. Department of Education (1999). The NAEP Guide: A Description of the Content 

and Methods of the 1999 and 2000 Assessments. Washington, DC: National Center for 

Education Statistics and U.S. Department of Education, Office of Educational Research 

and Improvement. 

50 
Mathematics Framework for the 2007 NAEP 

Appendix A 

NAEP Mathematics Achievement Level Descriptions 

51 
Mathematics Framework for the 2007 NAEP 

Grade 4 

Basic Fourth-grade students performing at the Basic level should show 

some evidence of understanding the mathematical concepts and 

procedures in the five NAEP content areas. 

Fourth graders performing at the Basic level should be able to estimate 

and use basic facts to perform simple computations with whole numbers; 

show some understanding of fractions and decimals; and solve some 

simple real-world problems in all NAEP content areas. Students at this 

level should be able to use—though not always accurately—four-function 

calculators, rulers, and geometric shapes. Their written responses are often 

minimal and presented without supporting information. 

Proficient Fourth-grade students performing at the Proficient level should 

consistently apply integrated procedural knowledge and conceptual 

understanding to problem solving in the five NAEP content areas. 

Fourth graders performing at the Proficient level should be able to use 

whole numbers to estimate, compute, and determine whether results are 

reasonable. They should have a conceptual understanding of fractions and 

decimals; be able to solve real-world problems in all NAEP content areas; 

and use four-function calculators, rulers, and geometric shapes 

appropriately. Students performing at the Proficient level should employ 

problem-solving strategies such as identifying and using appropriate 

information. Their written solutions should be organized and presented 

both with supporting information and explanations of how they were 

achieved. 

Advanced Fourth-grade students performing at the Advanced level should apply 

integrated procedural knowledge and conceptual understanding to 

complex and non-routine real-world problem solving in the five 

NAEP content areas. 

Fourth graders performing at the Advanced level should be able to solve 

complex non-routine real-world problems in all NAEP content areas. They 

should display mastery in the use of four-function calculators, rulers, and 

geometric shapes. These students are expected to draw logical conclusions 

and justify answers and solution processes by explaining why, as well as 

how, they were achieved. They should go beyond the obvious in their 

interpretations and be able to communicate their thoughts clearly and 

concisely. 

53 
Mathematics Framework for the 2007 NAEP 

Grade 8 

Basic Eighth-grade students performing at the Basic level should exhibit 

evidence of conceptual and procedural understanding in the five 

NAEP content areas. This level of performance signifies an 

understanding of arithmetic operations—including estimation—on 

whole numbers, decimals, fractions, and percents. 

Eighth graders performing at the Basic level should complete problems 

correctly with the help of structural prompts such as diagrams, charts, and 

graphs. They should be able to solve problems in all NAEP content areas 

through the appropriate selection and use of strategies and technological 

tools, including calculators, computers, and geometric shapes. Students at 

this level also should be able to use fundamental algebraic and informal 

geometric concepts in problem solving. As they approach the Proficient 

level, students at the Basic level should be able to determine which of the 

available data are necessary and sufficient for correct solutions and use 

them in problem solving. However, these eighth graders show limited skill 

in communicating mathematically. 

Proficient Eighth-grade students performing at the Proficient level should apply 

mathematical concepts and procedures consistently to complex 

problems in the five NAEP content areas. 

Eighth graders performing at the Proficient level should be able to 

conjecture, defend their ideas, and give supporting examples. They should 

understand the connections among fractions, percents, decimals, and other 

mathematical topics such as algebra and functions. Students at this level 

are expected to have a thorough understanding of Basic level arithmetic 

operations—an understanding sufficient for problem solving in practical 

situations. Quantity and spatial relationships in problem solving and 

reasoning should be familiar to them, and they should be able to convey 

underlying reasoning skills beyond the level of arithmetic. They should be 

able to compare and contrast mathematical ideas and generate their own 

examples. These students should make inferences from data and graphs, 

apply properties of informal geometry, and accurately use the tools of 

technology. Students at this level should understand the process of 

gathering and organizing data and be able to calculate, evaluate, and 

communicate results within the domain of statistics and probability. 

Advanced Eighth-grade students performing at the Advanced level should be 

able to reach beyond the recognition, identification, and application of 

mathematical rules in order to generalize and synthesize concepts and 

principles in the five NAEP content areas. 

Eighth graders performing at the Advanced level should be able to probe 

examples and counterexamples in order to shape generalizations from 

which they can develop models. Eighth graders performing at the 

54 
Mathematics Framework for the 2007 NAEP 

Advanced level should use number sense and geometric awareness to 

consider the reasonableness of an answer. They are expected to use 

abstract thinking to create unique problem-solving techniques and explain 

the reasoning processes underlying their conclusions. 

55 
Mathematics Framework for the 2007 NAEP 

Appendix B 

Mathematical Complexity Items Scoring Guide 

57 
Mathematics Framework for the 2007 NAEP 

Low Complexity 

How many fourths make a whole? 

Answer: ___________________ 

Source: 1996 NAEP (grade 4) Percent correct: 50 

Scoring Guide 

 Scoring Rubric 

1 - Correct 

4, 4 fourths, etc. 

0 - Incorrect 

Any incorrect response 

+6 + –12 = 

*A. –6 

B. +6 

C. –18 

D. +18 

Source: 1990 NAEP (grade 8) Percent correct: 68 

N stands for the number of stamps John had. He gave 12 stamps to his sister. 

Which expression tells how many stamps John has now?

 A. N + 12 

*B. N – 12 

C. 12 – N

 D. 12 x N 

Source: 1996 NAEP (grade 4) Percent correct: 67 

How many hours are equal to 150 minutes? 

A. 11⁄2 

B. 21⁄4 

C. 21/3 

*D. 21⁄2 

E. 25/6 

Source: 1990 NAEP (grade 12) Percent correct: 74 

59 
Mathematics Framework for the 2007 NAEP 

What is the weight shown on the scale?

 A. 6 pounds 

B. 7 pounds 

C. 51 pounds 

*D. 60 pounds 

Source: 1992 NAEP (grade 4) Percent correct: 44 

Moderate Complexity 

The following shapes were provided to students. (Shapes were larger than shown.) 

(Grade 8 version) 

Bob, Carmen, and Tyler were comparing the areas of N and P. Bob said that N and P

have the same area. Carmen said that the area of N is larger. Tyler said that the area of

P is larger. 

Who was correct? _______________________________ 

Use words or pictures (or both) to explain why. 

Source: 1996 NAEP (grade 8) Percent correct: 27 

60 
Mathematics Framework for the 2007 NAEP 

2X

area = 1⁄2 x (2X) 

= X 2 

X P 

Scoring Rubic 

1 – Correct 

An adequate explanation with or without Bob. 

May say “neither” or “both.” 

Parts of P overlap N, and part sticks out. The sticking out part is equal to the left out part of 

N. 

OR 

Two P’s match two N’s therefore they have the same area. (Therefore, one N has the same 

area as one P.) 

OR 

Areas are equal because height of P is the same as the height of N, and the base of P is 

twice the base of N. 

OR 

Either of these two figures alone are acceptable 

N N 

N N 

These are 

equal 

These are 

equal 

N 

X

area = X2 

X

61 
Mathematics Framework for the 2007 NAEP 

Scoring Rubric (continued) 

1 - Correct 

Correct response (see page 61). 

0 - Incorrect 

Bob was correct, but explanation not given or inadequate Any response that answers 

Carmen or Tyler to “Who was correct?” or omits the name and gives no satisfactory 

explanation. 

In this figure, how many small cubes were put together to form the large cube?

 A. 7 

*B. 8 

C. 12 

D. 24 

Source: 1996 NAEP (grade 4) Percent correct: 33 

From any vertex of a 4-sided polygon, 1 diagonal can be drawn. 

From any vertex of a 5-sided polygon, 2 diagonals can be drawn. 

From any vertex of a 6-sided polygon, 3 diagonals can be drawn. 

From any vertex of a 7-sided polygon, 4 diagonals can be drawn. 

How many diagonals can be drawn from any vertex of a 20-sided polygon? 

Answer: ___________________________ 

Source: 1996 NAEP (grade 8) Percent correct: 54 

62 
Mathematics Framework for the 2007 NAEP 

Scoring Guide 

Scoring Rubric 

1 - Correct 

17 

Number of diagonals is always 3 less than the number of sides. 

0 - Incorrect 

Any incorrect response. 

Tracy said, “I can multiply 6 by another number and get an answer that is smaller than

6.” 

Pat said, “No, you can’t. Multiplying 6 by another number always makes the answer 6

or larger.” 

Who is correct? Give a reason for your answer. 

Source: 1992 NAEP (grades 8 and 12) Percent correct: 49 (grade 8); 63 (grade 12) 

Scoring Guide 

Scoring Rubric 

1 - Correct 

Tracy, with correct answer given. 

OR 

No name stated but reason given is correct. 

Examples of correct reasons: 

• If you multiply by a number smaller than 1 the result is less than 6. 

• 6 x 0 = 0 

• 6 x 1/2 = 3 

• 6 x –1 = –6 

0 - Incorrect 

Tracy with no reason. 

OR 

Any response that states Pat is correct. 

OR 

No name stated and reason given is incorrect. 

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Mathematics Framework for the 2007 NAEP 

High Complexity 

A pattern of dots is shown below. At each step, more dots are added to the pattern. 

The number of dots added at each step is more than the number added in the previous 

step. The pattern continues infinitely. 

Marcy has to determine the number of dots in the 20th step, but she does not want to 

draw all 20 pictures and then count the dots. Explain or show how she could do this 

and give the answer that Marcy should get for the number of dots. 

Source: 1992 NAEP (grade 8) Percent correct: 6 

Scoring Guide 

Explanation should include one of the following ideas with no false statements. 

a. For each successive step, the number of rows and the number of columns is increasing 

by 1, forming a pattern. For example, the first step forms 1 by 2 rows and columns, the 

next step 2 by 3, the third step 3 by 4, and so on. Continuing this pattern would mean 

that the 20th step has 20 by 21 or 420 dots. 

b. Look at the successive differences between the consecutive steps. The differences 4, 6, 

8, 10,...form a pattern. There are 19 differences forming the pattern 4, 6, 8, 10,...38, 40 

and the sum is (9 x 44) + 22 or 418. However, 2 must be added for the first step, yielding 

a response of 420. 

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Mathematics Framework for the 2007 NAEP 

Scoring Rubric 

4 - Extended 

Correct answer. (Must state 420; must tie step 20 back to beginning of pattern in some 

specific form of generalization.) 

3 - Satisfactory 

Correct explanation of pattern but does not include or omits the correct number of dots. 

2 - Partial 

A partial (incomplete) correct explanation, i.e., does not tie together. 

1 - Minimal 

Any attempt to generalize OR to draw all 20 pictures in the pattern (with a clear 

understanding of the pattern). 

0 - Incorrect 

The work is completely incorrect, irrelevant, or off. 

Jaime knows the following facts about points A, B, and C. 

• Points A, B, and C are on the same line, but might not be in that order. 

• Point C is twice as far from point A as it is from point B. 

Jaime concluded that point C is always between points A and B. 

Is Jaime’s conclusion correct?

Yes No 

In the space provided, use a diagram to explain your answer. 

Source: 1996 NAEP (grade 8) Percent correct: 23 

65 
Mathematics Framework for the 2007 NAEP 

Scoring Guide 

The explanation must include either one of the following diagrams: 

Diagram should include or illustrate the idea that B is halfway between A and C. 

Scoring Rubric 

1 - Correct 

Correct response. Both correct diagrams are given 

OR 

If correct diagram is clearly indicated either by circling and/or the incorrect ones are 

crossed out. 

0 - Incorrect 

Any incorrect reason OR if there is no indication of which diagram is correct (by circling 

or crossing out incorrect one). 

NOTE: Points need to be indicated by some mark, such as a dot or tick mark. A response in which points 

are indicated by letters only is an automatic incorrect. 

In Mr. Bell’s classes, the students voted for their favorite shape for a symbol. 

Here are the results. 

Class 1 Class 2 Class 3 

Shape N 9 14 11 

Shape P 1 9 17 

Shape Q 22 7 2 

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Mathematics Framework for the 2007 NAEP 

Using the information in the chart, Mr. Bell must select one of the shapes to be the 

symbol. Which one should he select and why?

The shape Mr. Bell should select: _________________ 

Explain: 

Source: 1996 NAEP (grades 4 and 8) Percent correct: 31 (grade 4); 58 (grade 8) 

Scoring Guide 

Scoring Rubric 

1 - Correct 

N, because more students chose it. 

OR 

N, because it was first choice in one class and second choice in the other classes. 

“Majority” is acceptable (taken to mean most). If student says most classes, do not accept. 

0 - Incorrect 

Shape Q chosen, with an explanation that refers to a number of votes 

OR 

Shape N chosen, but explanation not given or is inadequate with incorrect computation. 

OR 

Any other incorrect response. 

67 
Mathematics Framework for the 2007 NAEP 

Appendix C 

NAEP Mathematics Project Staff and Committees 

69 
Mathematics Framework for the 2007 NAEP 

NAEP Mathematics Project 

CCSSO Staff 

Rolf Blank 

Director of Indicators Project 

State Education Assessment Center 

Council of Chief State School Officers 

Washington, D.C. 

Wayne Martin 

Director 

State Education Assessment Center 

Council of Chief State School Officers 

Washington, D.C. 

John Olson 

Director of Assessment 

State Education Assessment Center 

Council of Chief State School Officers 

Washington, D.C. 

Frank Philip 

Senior Project Associate 

State Education Assessment Center 

Council of Chief State School Officers 

Washington, D.C. 

Linda D. Wilson 

Consensus Coordinator Consultant 

Council of Chief State School Officers 

Washington, D.C. 

Phoebe Winter 

Project Director 

State Education Assessment Center 

Council of Chief State School Officers 

Richmond, Virginia 

Subcontractors and Consultants 

Christopher Cross 

President (Former) 

Council for Basic Education 

Washington, D.C. 

Kim Gattis 

President, Association of State Supervisors 

of Mathematics 

Education Program Consultant, Kansas 

Department of Education 

Topeka, Kansas 

Patricia Kenney 

Senior Research Associate 

University of Michigan 

Ann Arbor, Michigan 

Rebecca Kopriva 

Director 

Center for the Study of Assessment Validity 

in Education 

Department of Measurement & Statistics 

University of Maryland 

College Park, Maryland 

Linda Plattner 

Director of Policy, Standards & Instruction 

Council for Basic Education 

Washington, D.C. 

NAGB Staff 

Mary Crovo 

Project Officer 

National Assessment Governing Board

Washington, D.C. 

Sharif Shakrani 

Contracting Officer 

National Assessment Governing Board

Washington, D.C. 

Roy Truby 

Executive Director 

National Assessment Governing Board

Washington, D.C. 

Steering Committee 

Eileen Ahearn 

Project Director 

National Association of State Directors 

of Special Education 

Alexandria, Virginia 

Charles Allan 

Mathematics Education Consultant 

Michigan Department of Education 

Lansing, Michigan 

B. Marie Byers 

National School Boards Association 

Hagerstown, Maryland 

71 
Mathematics Framework for the 2007 NAEP 

Randy DeHoff 

Colorado State Board of Education 

6th Congressional District–Littleton 

Denver, Colorado 

M.B. “Sonny” Donaldson 

Superintendent 

Aldine ISD 

Houston, Texas 

Janice Earle 

Senior Program Director 

National Science Foundation 

Arlington, Virginia 

Lou Fabrizio 

Director 

Division of Accountability Services 

North Carolina Department of Public 

Instruction 

Raleigh, North Carolina 

Bettye Forte 

Mathematics Consultant 

Arlington, Texas 

Matt Gandal 

Vice President 

Achieve, Inc. 

Washington, D.C. 

Alice Gill 

Associate Director 

Educational Issues 

American Federation of Teachers 

Washington, D.C. 

M. Kathleen Heid 

The Pennsylvania State University

University Park, Pennsylvania 

Audrey Jackson 

Assistant Principal 

Claymont Elementary School 

Parkway City Schools 

Fenton, Missouri 

James M. Landwehr 

Director 

Data Analysis Research Department 

Avaya Labs 

Basking Ridge, New Jersey 

Sharon Lewis 

Research Director 

Council of the Great City Schools 

Washington, D.C. 

Dane Linn 

Policy Studies Director 

National Governors’ Association 

Washington, D.C. 

Eddie Lucero 

Principal 

Griegos Elementary School

Albuquerque, New Mexico

Lee McCaskill 

Principal 

Brooklyn Technical High School 

Brooklyn, New York 

Barbara Montalto 

Assistant Director of Mathematics 

Texas Education Agency

Austin, Texas 

Judy Rohde 

Mathematics Instructor 

John Glenn Middle School 

Maplewood, Minnesota 

Wilfried Schmid 

Professor of Mathematics

Harvard Department of Mathematics 

Cambridge, Massachusetts

Sr. Mary Frances Taymans 

Executive Director 

Secondary Schools Department 

National Catholic Education Association 

Washington, D.C. 

Zalman Usiskin 

Professor of Education 

Director, University of Chicago School 

Mathematics Project 

Chicago, Illinois 

Judy Walter 

Association for Supervision & Curriculum 

Development 

Alexandria, Virginia 

72 
Mathematics Framework for the 2007 NAEP 

Diana Wearne 

Associate Professor 

School of Education 

University of Delaware 

Newark, Delaware 

Hung-Hsi Wu 

Professor of Mathematics

Department of Mathematics 

University of California–Berkeley

Berkeley, California 

Planning Committee 

Dayo Akinsheye 

Mathematics Resource Teacher 

Seaton Elementary School 

Washington, D.C. 

Geri Anderson-Nielsen 

Mathematics Specialist 

Georgetown Day School 

Washington, D.C. 

Cindy Chapman 

Elementary Teacher 

Albuquerque Public Schools 

Albuquerque, New Mexico

Herbert Clemens 

Professor of Mathematics

Department of Mathematics 

University of Utah 

Salt Lake City, Utah 

Carl Cowen 

Professor of Mathematics

Purdue University

West Lafayette, Indiana 

Jim Ellingson 

Assistant Professor 

Concordia College 

Moorhead, Minnesota 

Joan Ferrini-Mundy 

Associate Dean/Director of Science and 

Mathematics 

College of Natural Science 

Michigan State University 

East Lansing, Michigan 

Kim Gattis 

Education Program Consultant 

Kansas Department of Education 

Association of State Supervisors of 

Mathematics 

Topeka, Kansas 

Anne Gonzales 

Middle School Mathematics Teacher 

South Gate Middle School 

South Gate, California 

Jeremy Kilpatrick 

Professor of Mathematics Education 

University of Georgia 

Athens, Georgia 

Gerald Kulm 

Curtis D. Robert Professor 

of Mathematics Education 

Texas A & M University 

College Station, Texas 

Mary Lindquist 

Fuller E. Callaway Professor 

of Mathematics Education 

Columbus State University 

Columbus, Georgia 

Mary Jo Messenger 

High School Mathematics Teacher 

River Hill High School 

Clarksville, Maryland 

Marjorie Petit 

Senior Associate 

National Center for the Improvement of 

Educational Assessment 

(The Center for Assessment) 

Portsmouth, New Hampshire 

Edward Silver 

Professor 

School of Education 

University of Michigan 

Ann Arbor, Michigan 

Debra Vitale 

Mathematics Specialist 

Arlington Public Schools 

Fairfax, Virginia 

73 
Mathematics Framework for the 2007 NAEP 

Frank Wang 

President/CEO 

Saxon Publishing, Inc. 

Norman, Oklahoma 

Norman Webb 

Senior Research Scientist 

Wisconsin Center for Education Research 

Madison, Wisconsin 

John Wisthoff 

Member, Maryland State Board of 

Education and Mathematics Professor 

Anne Arundel Community College 

Pasadena, Maryland 

Technical Advisory Panel 

Fen Chou 

Psychometrician 

Louisiana Department of Education 

Baton Rouge, Louisiana 

Eugene Johnson 

Chief Psychometrician 

American Institutes for Research 

Washington, D.C. 

Edward Kifer 

Professor and Chairperson 

Department of Educational Policy Studies 

and Evaluation 

College of Education 

University of Kentucky

Lexington, Kentucky

Ina Mullis 

Co-Director 

International Study Center

Boston College 

Chestnut Hill, Massachusetts 

Barbara Plake 

Director 

Buros Center for Testing 

University of Nebraska–Lincoln 

Lincoln, Nebraska 

Roger Trent 

Ohio State Assessment Director (Emeritus) 

Ohio Department of Education 

Columbus, Ohio 

Committee Representation 

Policy Organizations 

Achieve, Inc. 

American Association of School 

Administrators (AASA) 

American Federation of Teachers (AFT) 

American Mathematical Society (AMA) 

American Statistical Association (ASA) 

Association for Supervision and Curriculum 

Development (ASCD) 

Association of State Assessment Programs 

(ASAP) 

Association of State Supervisors of 

Mathematics (ASSM) 

Business Roundtable/National Alliance of 

Business 

Council of the Great City Schools 

Education Leaders Council (ELC) 

National Association of Elementary School 

Principals (NAESP) 

National Association of Secondary School 

Principals (NASSP) 

National Association of State Boards of 

Education (NASBE) 

National Association of State Directors of 

Special Education (NASDE) 

National Catholic Education Association 

(NCEA) 

National Education Association (NEA) 

National Governors’ Association (NGA)

National School Boards Association 

(NSBA) 

National Science Foundation (NSF) 

Representative from national textbook 

publisher 

74 
Mathematics Framework for the 2007 NAEP 

Mathematics Associations and Groups Technical Experts 

Mathematically Correct State testing specialists 

Mathematics Association of America Representatives from private research 

(MAA) organizations 

National Council of Teachers of University professors 

Mathematics (NCTM) 

Third International Mathematics and Science 

Study (TIMSS) 

Educators 

Classroom mathematics teachers from public 

and non-public schools 

District and state mathematics specialists 

Mathematics and mathematics education 

professors from public and private 

universities, colleges, and community 

colleges 

Principals 

75 
Acknowledgments 

The following people were the primary authors of the introductions to the content 

areas: 

Roger Howe, Yale University (Number Properties and Operations, Geometry, and 

Algebra) 

Richard Scheaffer, University of Florida (Data Analysis and Probability) 

Mary Lindquist, Columbus State University (Measurement) 

Frank Philip and Arthur Halbrook, Council of Chief State School Officers, 

designed the layout and edited the document. Natalie Dubois provided word 

processing support. 

77