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Mathematics Framework for the 2005 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 toward the National Education Goals.

The National Assessment Governing Board

The National Assessment Governing Board (NAGB) was created by Congress to formulate policy for NAEP. Among the Board’s responsibilities are developing objectives and test specifications, and designing the assessment methodology for NAEP.

Members

Darvin M. Winick, Chair President Winick & Associates Dickinson, Texas

Honorable Jo Ann Pottorff, Vice Chair

State Legislator Topeka, Kansas

Amanda Avallone Eighth-Grade Teacher Boulder, Colorado

Barbara Byrd-Bennett Chief Executive Officer Cleveland Municipal School District Cleveland, Ohio

Carl Cohn Clinical Professor University of Southern California Los Angeles, California

Shirley Dickson Educational Consultant Laguna Niguel, California

Edward Donley Former Chair

Air Products and Chemicals, Inc.

Allentown, Pennsylvania

John Q. Easton Executive Director Consortium on Chicago School

Research University of Chicago Chicago, Illinois

Honorable Dwight Evans State Legislator Philadelphia, Pennsylvania

Sheila Ford Elementary School Principal Washington, D.C.

David Gordon County Superintendent of Schools Sacramento County Office of

Education Sacramento, California

Juanita Haugen Local School Board President Pleasanton, California

Henry L. Johnson Superintendent of Education Mississippi Department of Education Jackson, Mississippi

Kathi M. King Twelfth-Grade Teacher Oakland, Maine

Kim Kozbial-Hess Fourth-Grade Teacher Toledo, Ohio

Honorable Ronnie Musgrove Former Governor of Mississippi Jackson, Mississippi

Diane Ravitch Senior Research Scholar New York, New York

Mark Reckase Professor, Measurement and

Quantitative Methods Michigan State University East Lansing, Michigan

Sister Lourdes M. Sheehan Associate General Secretary United States Catholic Conference Washington, D.C.

John H. Stevens Executive Director Texas Business and Education

Coalition Austin, Texas

Honorable Michael E. Ward State Superintendent of Public

Instruction North Carolina Public Schools Raleigh, North Carolina

Eileen L. Weiser Member, State Board of Education Michigan Department of Education Lansing, Michigan

Dennie Palmer Wolf Director, Opportunity and

Accountability Initiatives Annenberg Institute for School Reform Providence, Rhode Island

Grover (Russ) Whitehurst Ex-Officio Director, Institute of

Education Sciences U.S. Department of EducationWashington, D.C.

Charles Smith Executive Director, NAGB Washington, D.C.

Mathematics Framework for the 2005 National Assessment of Educational Progress

National Assessment Governing Board U.S. Department of Education

National Assessment Governing Board

Darvin M. Winick Chair

Jo Ann Pottorff Vice Chair

Charles E. Smith Executive Director

Sharif M. Shakrani Contracting Officer

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 the National Assessment Governing Board:

800 North Capitol Street, NW.

Suite 825

Washington, DC 20002-4233

www.nagb.org

September 2004

For sale by the U.S. Government Printing Office Superintendent of Documents, Mail Stop: SSOP, Washington, DC 20402–9328

Table of Contents

Chapter One: Overview of Recommendations ..........................................................................1Introduction.................................................................................................................................1Context for Planning the 2005 Assessment ................................................................................1The Consensus Approach ...........................................................................................................4Achievement Levels....................................................................................................................4Recommendations for the 2005 NAEP Mathematics Assessment .............................................5

Chapter Two: Framework for the Assessment...........................................................................7Content Areas..............................................................................................................................7Mathematical Complexity of Items ............................................................................................8Distribution of Items...................................................................................................................9Special Study at Grade 8 ..........................................................................................................10Calculators ................................................................................................................................10

Chapter Three: 2005 NAEP Mathematics Objectives ............................................................13Mathematical Content Areas.....................................................................................................13

Chapter Four: Mathematical Complexity of Items.................................................................37Low Complexity .......................................................................................................................38Moderate Complexity ...............................................................................................................40High Complexity.......................................................................................................................43Balance of Mathematical Complexity ......................................................................................45

Chapter Five: Item Formats ......................................................................................................47Multiple-Choice Items ..............................................................................................................47Short Constructed-Response Items...........................................................................................48Extended Constructed-Response Items ....................................................................................49Scoring Extended Constructed-Response Items.......................................................................50

Bibliography ................................................................................................................................53

Appendix A: NAEP Mathematics Achievement Level Descriptions ......................................55Grade 4......................................................................................................................................57Grade 8......................................................................................................................................57Grade 12....................................................................................................................................58

Appendix B: Mathematical Complexity Items Scoring Guide...............................................61Low Complexity .......................................................................................................................63Moderate Complexity ...............................................................................................................65High Complexity.......................................................................................................................69

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Appendix C: NAEP Mathematics Project Staff and Committees..........................................73NAEP Mathematics Project ......................................................................................................75

Steering Committee ..................................................................................................................75Planning Committee .................................................................................................................77Technical Advisory Panel .........................................................................................................78Committee Representation........................................................................................................78

Acknowledgments .......................................................................................................................81

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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 problemsolving 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 2005 NAEP mathematics assessment. 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 mathematics should be assessed in 2005 on NAEP? The answer to this question must necessarily take into account the constraints of a large-scale assessment such as NAEP, with its limitations on time and resources. Of critical importance is the fact that this document does not attempt to answer the question: What mathematics 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. (An example would be 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 2005 NAEP mathematics assessment.

Note: To comply with the provisions of the No Child Left Behind Act of 2001, the National Assessment Governing Board (NAGB) adopted a revised NAEP assessment schedule in March 2002. This schedule moved the 2004 mathematics assessment to 2005 because NAEP will test mathematics and reading in odd-numbered years.

Context for Planning the 2005 Assessment

The National Assessment Governing Board, created by Congress in 1988, is responsible for formulating policy for NAEP. NAGB 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.

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Congress also authorized the trial state assessment program (more commonly known as the state NAEP program) in 1988. The voluntary trial state NAEP began with the 1990 assessment of mathematics at grade 8; the state NAEP program continued with the 1992 mathematics assessment at grades 4 and 8. The regular national mathematics assessment of students at grades 4, 8, and 12 also occurred in 1990 and 1992.

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. As a part of the work, 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. The design was reported in Mathematics Objectives: 1990 Assessment.

To prepare for the next NAEP mathematics assessment, NAGB awarded a contract in fall 1991 to the College Board to develop the assessment and item specifications for the 1994 mathematics assessment. The College Board work was completed between September 1991 and March 1992. Due to a budget shortfall, the 1994 mathematics assessment and the science assessment were rescheduled for 1996.

The NAEP mathematics project conducted by the College Board had two primary purposes: (1) to recommend a framework for the overall design of the mathematics assessment—a structure for describing what students know and can do in mathematics, and (2) to develop specifications for the assessment items, paying particular attention to the mix of item formats, the item distribution for content areas within mathematics, and the conditions under which items are presented to students (for example, use of manipulatives, use of calculators, and time to complete tasks). The work of the project was also to be considered in light of the three NAEP achievement levels adopted by NAGB—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. The design was reported in Mathematics Framework for the 1996 and 2000 National Assessment of Educational Progress. The 1996/2000 Framework was used in the design of the assessments for those years. Those assessments maintained the trend lines that began in 1990.

Other changes have taken place in NAEP in recent years. A number of policy, legislative, civil rights, and technical considerations caused NAEP to look more closely at its administration and assessment procedures and to consider implementing changes that can increase participation among students with disabilities and limited-English-proficient students. In the early 1990s, participation of these students in NAEP was limited, with schools excluding them from the assessment in accordance with criteria provided by the program at that time. This exclusion raised concerns that some special-needs students who could be meaningfully assessed were being excluded from NAEP.

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In the mid-1990s, based on requirements from the reauthorization of the Individuals with Disabilities Education Act (IDEA), the general trend of state assessments toward more inclusion of special-needs students, and recommendations from various offices of the U.S. Department of Education, NAEP modified its procedures with the aim of increasing the participation of special-needs students. For the 1996 assessment, modifications were made in two areas: inclusion criteria were revised to make them clearer, 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. This shift toward greater inclusiveness allowed NAEP to be more similar to state and district testing policies. NAEP used a split-sample approach to study the changes made and their possible impact on trends, and issued several reports from their findings (see Olson and Goldstein, 1997; Reese et al., 1997; Mazzeo et al., 1999).

In the late 1990s, states had implemented assessments that included most, if not all, of their students and were beginning to report assessment results that included the performance of special-needs students. An array of assessment accommodations was being offered by states routinely, and by NAEP as well. 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. In 2000, NAEP reported the mathematics assessment results for the first time on both a non-accommodated sample of students as well as a sample of regular and special-needs students who received accommodations in The Nation’s Report Card: Mathematics 2000 (Braswell et al., 2001).

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 2005 mathematics framework enables the continuation of this best practices approach, and the assessment and item specifications document (Assessment and Item Specifications for the 2005 NAEP Mathematics Assessment) provides clear and detailed guidance on how to develop a more inclusive assessment.

The long-range plan established by NAGB 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 (at grades 4, 8, and 12) and the states (at grade 8 and later grade 4) that began with the 1990 mathematics assessment. The short-term trend lines were being used by many states 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.

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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

The 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. The members of all three committees and project staff are presented in appendix C.

The steering committee and planning committee both began their work with a review of the framework used for the 1996 and 2000 assessments. 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 the recent report issued by the National Research Council of the National Academy of Sciences, Adding It Up (Findell et al., 2001). 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 the framework for the 2005 mathematics assessments 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 clearer and more specific objectives for each grade level.

Achievement Levels

The updated 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,

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application of such knowledge to real-world 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 NAGB, the three levels per grade are a major means of reporting NAEP data. The updated mathematics framework was developed with these levels in mind to ensure congruence between the levels and the test content. See appendix A for the NAEP Mathematics Achievement Level Descriptions. Note: Since the grade 12 NAEP mathematics assessment will start a new trend line, the achievement level descriptions will change for this grade. New achievement level descriptions will be finalized in late 2004.

Recommendations for the 2005 NAEP Mathematics Assessment

As a result of analysis and review, the steering committee and planning committee endorsed the following recommendations for the 2005 NAEP mathematics assessment:

1. Content Areas

The 2005 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 3.

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 2, and the levels are described at length in chapter 4.

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 and 2000 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 specific recommended percentages are discussed in chapter 2.

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,

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with the remaining half devoted to constructed-response items of both types. Further description of the item formats can be found in chapter 5. 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. Special Study

A special study should be designed to probe grade 8 students’ understanding of mathematical topics that are critical for further study of algebra and more advanced mathematics. The study would answer questions about the depth of students’ knowledge of such algebraic ideas as rate of change and variable, and their understanding of proportionality. Further discussion of this special study is provided in chapter 2.

6. Manipulatives

The assessment should continue to utilize 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.

7. Calculators

It is appropriate for some portions (two-thirds) of the NAEP at all grade levels to assess students’ mathematical knowledge and skills without access to a calculator, but for other portions (onethird) of the test to allow the use of a calculator. At grade 4, a four-function calculator should be supplied by NAEP with appropriate training at the time of administration. Eighth- and twelfth-grade students should be allowed to bring whatever calculator, graphing or otherwise, they are accustomed to using in the classroom. More details and discussion about the calculator recommendations can be found in chapter 2. Note: NAGB determined that eighth-grade students will be provided with scientific calculators for the 2005 assessment based on a survey of state testing programs.

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Chapter Two: Framework for the Assessment

This chapter further discusses the rationale for recommendations, especially those thatreflect a change from current policy.

Content Areas

Since its first mathematics assessments in the late 1970s and early 1980s, 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 2005 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. Separating these two areas of mathematics at grade 12 becomes forced and unnecessary.

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

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and Operations, which encompasses a wide range of concepts about our numeration system (see chapter 3 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. While the main NAEP assessment is not designed to report a separate score for computation, results from the long-term NAEP assessment can provide insight into students’ computational abilities.

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 making sure 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.

There were many laudable features of that framework. Notions of conceptual understanding, procedural knowledge, and problemsolving 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 the 2005 framework is to build on the former framework, retaining its strengths while addressing some of its weaknesses. The purpose remains the same: to make sure 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 nonroutine problems. Ordering of the levels is not intended to imply a developmental sequence or the sequencing in which teaching or learning occur. Rather, it is a description of the different demands made on students by particular test items. See chapter 5 for further discussion of the levels of mathematical complexity.

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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 the 2005 assessment. Table 1 provides the recommended balance of items in the assessment by content area for each grade (4, 8, and 12) in the 2005 assessment. Note that the percentages refer to numbers of items, not the amount of student testing time (see chapter 5 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 framework for the 1996 and 2000 assessments. 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 the 2005 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 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.

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Special Study at Grade 8

The past decade has brought major changes to the middle school mathematics curriculum in the form of more students taking courses called algebra and more algebraic concepts being taught in other mathematics courses at grades 7 and 8. From the 2000 NAEP assessment, we know that 31 percent of eighth graders reported taking prealgebra and 26 percent reported taking either first-or second-year algebra. We have also learned from research and from international studies such as Trends in International Mathematics and Science Study that the topic of proportionality is particularly challenging for students around the world. Proportionality (such as “a is to b as c is to d”) is foundational for understanding many algebraic concepts, such as rate of change, variation, linear relationships, slope, and functions, as well as concepts in other advanced mathematics courses and in other disciplines. Taken together, these two findings form the basis for the purpose of a special study of grade 8 students: to examine the depth and breadth of eighth-grade students’ understanding of proportionality and other fundamental topics in algebra.

The special study would be designed to answer two research questions:

1. What is the depth of understanding by grade 8 students of fundamental topics in algebra?

2. What is the depth of understanding by grade 8 students of proportionality?

Items for the special study should be administered to a random sample of eighth-grade students who also participate in regular NAEP. The items would be administered along with the main NAEP items, but in separate blocks, so that only a subset of students is taking special study items.

Because the main NAEP assessment is designed to assess all five content areas of mathematics at grade 8, it is not possible to obtain indepth data about students’ understanding of fundamental topics in algebra or proportionality. This special study would provide an opportunity to more closely study students’ mathematical development in proportionality and related topics in algebra during the critical middle school years. It would also yield a clearer description of the depth and breadth of grade 8 students’ knowledge in these areas, which would be helpful in the design of future NAEP frameworks.

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.

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● At grade 12, students should be allowed to bring whatever calculator, graphing or otherwise, they are accustomed to using in the classroom with some restrictions for test security purposes (see below). For students who do not bring a calculator to use on the assessment, NAEP will provide a scientific calculator.

No items on the 2005 NAEP at either grade 8 or grade 12 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 noncalculator block, the developer should take into consideration 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 noncalculator 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 = –x2).

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Chapter Three: 2005 NAEP Mathematics Objectives

In 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 utilizing the five content areas, as described in chapter 2. 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, grade 8, and grade 12. 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 2005 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 the 2005 NAEP 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. Numbers let us talk in a precise way about anything that can be counted, measured, or located in space.

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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 also 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 2005 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. In middle school also, 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.

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Number Properties and Operations

1) Number sense

GRADE 4 GRADE 8 GRADE 12

a) Identify the place value and a) Use place value to model and actual value of digits in whole describe integers and decimals. numbers.

b) Represent numbers using b) Model or describe rational nummodels such as base 10 rep bers or numerical relationships resentations, number lines, using number lines and diagrams. and two-dimensional models.

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 d) Write or rename rational d) Write, rename, represent, or (e.g., 10: 5 + 5, 12 – 2, 2 x 5). numbers. compare real numbers (e.g., �,

�2, numerical relationships using number lines, models, or diagrams).

e) Connect model, number word, e) Recognize, translate between, or number using various models or apply multiple representations and representations for whole of rational numbers (fractions, numbers, fractions, and decimals. decimals, and percents) in

meaningful contexts.

f) Express or interpret numbers f) Represent very large or very using scientific notation from small numbers using scientific real-life contexts. notation in meaningful contexts.

g) Find or model absolute value g) Find or model absolute value or apply to problem situations. 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 num j) Order or compare rational num j) Order or compare real numbers, bers, decimals, or fractions. bers including very large and including very large or small

small integers, and decimals real numbers. and fractions close to zero.

continued on page 16

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Number Properties and Operations (continued)

2) Estimation


a) Use benchmarks (well-known a) Establish or apply benchmarks a) Establish or apply benchmarks numbers used as meaningful for rational numbers and com for real numbers in contexts. points for comparison) for whole mon irrational numbers (e.g., �) numbers, decimals, or fractions in contexts. in contexts (e.g., 1⁄2 and .5 may be used as benchmarks for fractions and decimals between 0 and 1.00).

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 c) Verify solutions or determine c) Verify solutions or determine the reasonableness of results the reasonableness of results in the reasonableness of results in in meaningful contexts. a variety of situations including a variety of situations including

calculator and computer results. scientific notation, calculator, and computer results.

d) Estimate square or cube roots d) Estimate square or cube roots of numbers less than 1,000 be- of numbers less than 1,000 between two whole numbers. tween two whole numbers.


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3) Number operations


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:

d) Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a real number by:

• zero, or • zero, or

• a number less than zero, or • a number less than zero, or

• a number between zero and • a number between zero and one, one, or

• one, or • one, or

• a number greater than one. • a number greater than one.

e) Provide a mathematical argument to explain operations with two or more fractions.

f) Interpret whole number opera- f) Interpret rational number operations and the relationships be tions and the relationships between them. tween them.

g) Solve application problems involving numbers and operations.

g) Solve application problems involving rational numbers and operations using exact answers

g) Solve application problems involving numbers, including rational and common irrationals,

or estimates as appropriate. using exact answers or estimates as appropriate.


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4) Ratios and proportional reasoning


a) Use simple ratios to describe a) Use ratios to describe problem problem situations. situations.

b) Use fractions to represent and b) Use proportions to model express ratios and proportions. problems.

c) Use proportional reasoning to c) Use proportional reasoning to model and solve problems (in- solve problems (including rates). cluding rates and scaling).

d) Solve problems involving d) Solve problems involving percentages (including percent percentages (including percent increase and decrease, interest increase and decrease, interest rates, tax, discount, tips, or rates, tax, discount, tips, or part/whole relationships). part/whole relationships).

5) Properties of number and operations


a) Identify odd and even numbers. a) Describe odd and even integers and how they behave under different operations.

b) Identify factors of whole b) Recognize, find, or use factors, b) Solve problems involving numbers. multiples, or prime factorization. factors, multiples, or prime

factorization.

c) Recognize or use prime and c) Use prime or composite numbers composite numbers to solve to solve problems. problems.

d) Use divisibility or remainders in d) Use divisibility or remainders in problem settings. problem settings.

e) Apply basic properties of e) Apply basic properties of e) Apply basic properties of operations. operations. operations.

f) Explain or justify a mathematical f) Explain or justify a mathematical f) Provide a mathematical arguconcept or relationship (e.g., ex- concept or relationship (e.g., ment about a numerical property plain why 15 is an odd number explain why 17 is prime). or relationship. or why 7–3 is not the same as 3–7).

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

titatively. 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 nonstandard, 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 part of geometry; this is reflected in the proportion of items recommended for these two areas.

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Measurement

1) Measuring physical attributes


a) Identify the attribute that is appropriate to measure in a given situation.

b) Compare objects with respect to b) Compare objects with respect a given attribute, such as length, to length, area, volume, angle area, volume, time, or temperature. measurement, weight, or mass.

c) Estimate the size of an object c) Estimate the size of an object c) Estimate or compare perimeters with respect to a given measure- with respect to a given measure- or areas of two-dimensional ment attribute (e.g., length, ment attribute (e.g., area). geometric figures. perimeter, or area using a grid).

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 measure- g) Select or use appropriate measment instruments such as ruler, urement instrument to determine meter stick, clock, thermometer, or create a given length, area, or other scaled instruments. 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

h) Solve mathematical or real-world problems involving perimeter or area of plane fig-

triangles, rectangles, circles, ures such as polygons, circles, or composite figures. or composite figures.

i) Solve problems involving area of squares and rectangles.

j) Solve problems involving volume j) Solve problems involving volor surface area of rectangular ume or surface area of rectansolids, cylinders, prisms, or gular solids, cylinders, cones, composite shapes. pyramids, prisms, spheres, or

composite shapes.


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Measurement (continued)

1) Measuring physical attributes (continued)


k) Solve problems involving indi k) Solve problems involving indirect measurement such as finding rect measurement such as find-the height of a building by com ing the height of a building by paring its shadow with the height finding the distance to the base and shadow of a known object. of the building and the angle of

elevation to the top.

l) Solve problems involving rates l) Solve problems involving rates such as speed or population such as speed, density, populadensity. tion density, or flow rates.

m)Use trigonometric relations in right triangles to solve problems.

2) Systems of measurement


a) Select or use appropriate type a) Select or use appropriate type of a) Select or use appropriate type of unit for the attribute being unit for the attribute being meas of unit for the attribute being measured such as length, time, ured such as length, area, angle, measured such as volume or or temperature. time, or volume. surface area.

b) Solve problems involving conver b) Solve problems involving conver b) Solve problems involving consions within the same measure sions within the same measure- versions within or between ment system such as conversions ment system such as conversions measurement systems, given the involving inches and feet or hours involving square inches and relationship between the units. and minutes. square feet.

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 5⁄8 of a mile.

• Money conversion: US dollar is approximately 1.5 Canadian dollars.

• Temperature conversion: Fahrenheit to Celsius

d) Determine appropriate size of d) Determine appropriate size of unit of measurement in problem unit of measurement in problem situation involving such attributes situation involving such attributes as length, time, capacity, or as length, area, or volume. weight.


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Measurement (continued)

2) Systems of measurement (continued)


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

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 meas

measurement of the dimensions to obtain a specified accuracy of area) and find the measure to

ure to that degree of accuracy. that degree of accuracy.

f) Construct or solve problems f) Construct or solve problems (e.g., floor area of a room) in- (e.g., number of rolls needed volving scale drawings. 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).

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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).

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Geometry

1) Dimension and shape


a) Explore properties of paths be a) Draw or describe a path of tween points. shortest length between points

to solve problems in context.

b) Identify or describe (informally) b) Identify a geometric object b) Use two-dimensional represen-real-world objects using simple given a written description of its tations of three-dimensional plane figures (e.g., triangles, rec- properties. objects to visualize and solve tangles, squares, and circles) and problems involving surface area simple solid figures (e.g., cubes, and volume. spheres, and cylinders).

c) Identify or draw angles and other c) Identify, define, or describe geo c) Give precise mathematical de-geometric figures in the plane. metric shapes in the plane and in scriptions or definitions of geo-

three-dimensional space given a metric shapes in the plane and visual representation. in three-dimensional space.

d) Draw or sketch from a written d) Draw or sketch from a written description polygons, circles, or semicircles.

description plane figures (e.g., isosceles triangles, regular polygons, curved figures) and planar images of three-dimensional figures (e.g., polyhedra, spheres, and hemispheres).

e) Represent or describe a three- e) Describe or analyze properties dimensional situation in a two- of spheres and hemispheres. dimensional drawing from different views.

f) Describe attributes of two- and three-dimensional shapes.

f) Demonstrate an understanding about the two- and three-dimensional shapes in our world through identifying, drawing, modeling, building, or taking apart.


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Geometry (continued)

2) Transformation of shapes and preservation of properties


a) Identify whether a figure is a) Identify lines of symmetry in a) Recognize or identify types of symmetrical, or draw lines of plane figures or recognize and symmetries (e.g., point, line, rota-symmetry. classify types of symmetries of tional, self-congruences) of two-

plane figures. 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

c) Recognize or informally describe the effect of a transformation on

c) Perform or describe the effect of a single transformation on two-

(translations), or turns (rotations). two-dimensional geometric and three-dimensional geometric shapes (reflections across lines of shapes (reflections across lines symmetry, rotations, translations, of symmetry, rotations, translamagnifications, and contractions). tions, and dilations).

d) Recognize which attributes (such d) Predict results of combining, sub- d) Describe the final outcome of as shape and area) change or dividing, and changing shapes successive transformations. don’t change when plane figures of plane figures and solids (e.g., are cut up or rearranged. paper folding, tiling, and cutting

up and rearranging pieces).

e) Match or draw congruent e) Justify relationships of congru e) Justify relationships of congrufigures in a given collection. ence and similarity, and apply ence and similarity, and apply

these relationships using scaling these relationships using scaling and proportional reasoning. and proportional reasoning.

f) For similar figures, identify and use the relationships of conservation of angle and of proportionality of side length and perimeter.


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3) Relationships between geometric figures


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

b) Apply geometric properties and relationships in solving multi-

problems in two and three step problems in two and three dimensions. dimensions (including rigid and

nonrigid figures).

c) Recognize two-dimensional c) Represent problem situations c) Represent problem situations faces of three-dimensional with simple geometric models with geometric models to solve shapes. to solve mathematical or real- mathematical or real-world

world problems. problems.

d) Use the Pythagorean theorem d) Use the Pythagorean theorem to to solve problems. solve problems in two- or three-

dimensional 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 f) Describe or analyze simple prop- f) Analyze properties or relation-of simple and compound figures erties of, or relationships between, ships of triangles, quadrilaterals, composed of triangles, squares, triangles, quadrilaterals, and and other polygonal plane and rectangles. other polygonal plane figures. figures.

g) Describe or analyze properties g) Describe or analyze properties and relationships of parallel or and relationships of parallel, per-intersecting lines. pendicular, or intersecting lines,

including the angle relationships that arise in these cases.


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4) Position and direction


a) Describe relative positions of a) Describe relative positions of points and lines using the geo points and lines using the geometric ideas of parallelism or metric ideas of midpoint, points perpendicularity. on common line through a com

mon point, parallelism, or perpendicularity.

b) Describe the intersection of b) Describe the intersections of two or more geometric figures lines in the plane and in space, in the plane (e.g., intersection intersections of a line and a of a circle and a line). plane, or of two planes in space.

c) Visualize or describe the cross c) Describe or identify conic sec-section of a solid. tions and other cross sections

of solids.

d) Construct geometric figures d) Represent geometric figures d) Represent two-dimensional with vertices at points on a using rectangular coordinates figures algebraically using coordinate grid. on a plane. coordinates and/or equations.

e) Use vectors to represent velocity and direction.

5) Mathematical reasoning


a) Distinguish which objects in a) Make and test a geometric a) Make, test, and validate geoa collection satisfy a given conjecture about regular metric conjectures using a geometric definition and polygons. variety of methods including explain choices. deductive reasoning and

counterexamples.

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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.

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 two-way tables for categorical variables or scatterplots for

28

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.

Data Analysis and Probability

1) Data representation


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, scatterplots, 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, scatterplots, 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 scatterplots and two-way tables).

d) Given a graph or a set of data, determine whether information is represented effectively and appropriately (histograms, line graphs, scatterplots, 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, scatterplots, 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.


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Data Analysis and Probability (continued)

2) Characteristics of data sets


a) Calculate, use, or interpret mean, a) Calculate, interpret, or use mean, median, mode, or range. median, mode, range, interquar

tile 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 c) Determine the effect of outliers their effect on mean, median, on mean, median, mode, range, mode, or range. interquartile range, or standard

deviation.

d) Compare two sets of related data.

d) Using appropriate statistical measures, compare two or more

d) Compare two or more data sets using mean, median, mode,

data sets describing the same range, interquartile range, or characteristic for two different standard deviation describing populations or subsets of the the same characteristic for two same population. different populations or subsets

of the same population.

e) Visually choose the line that e) Given a set of data or a scatter-best fits given a scatterplot and plot, visually choose the line of informally explain the meaning best fit and explain the meaning of the line. Use the line to make of the line. Use the line to make predictions. predictions.

f) Use or interpret a normal distribution as a mathematical model appropriate for summarizing certain sets of data.

g) Given a scatterplot, make decisions or predictions involving a line or curve of best fit.

h) Given a scatterplot, estimate the correlation coefficient (e.g., Given a scatterplot, is the correlation closer to 0, .5, or 1.0? Is it a positive or negative correlation?).


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3) Experiments and samples


a) Given a sample, identify possible a) Identify possible sources of sources of bias in sampling. bias in data collection methods

and describe how such bias can be controlled and reduced.

b) Distinguish between a random b) Recognize and describe a and nonrandom sample. method to select a simple

random sample.

c) Make inferences from sample results.

d) Evaluate the design of an d) Identify or evaluate the characexperiment. teristics of a good survey or of

a well-designed experiment.

4) Probability


a) Use informal probabilistic think- a) Analyze a situation that involves a) Analyze a situation that involves ing to describe chance events probability of an independent probability of independent or (i.e., likely and unlikely, certain event. dependent events. and impossible).

b) Determine a simple probability b) Determine the theoretical proba b) Determine the theoretical probafrom a context that includes a bility of simple and compound bility of simple and compound picture. events in familiar contexts. events in familiar or unfamiliar

contexts.

c) Estimate the probability of c) Given the results of an experisimple and compound events ment or simulation, estimate through experimentation or the probability of simple or simulation. compound events in familiar or

unfamiliar contexts.

d) Use theoretical probability to d) Use theoretical probability to evaluate or predict experimental evaluate or predict experimental outcomes. 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 f) Determine the probability of the the probability of the possible possible outcomes of an event. outcomes of an event.

g) Represent the probability of a g) Represent probability of a given given outcome using a picture outcome using fractions, decior other graphic. mals, and percents.


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4) Probability (continued)


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 o