The Final Report of the 

National Mathematics 

Advisory Panel 

2008 

U.S. Department of Education 
National Mathematics Advisory Panel FINAL REPORT 

This report was produced under U.S. Department of Education Contract No. 

ED04CO0082/0001 for Widmeyer Communications and No. ED04CO0015/0006 

for Abt Associates Inc. Ida Eblinger Kelley served as the contracting officer’s 

technical representative. The views expressed herein do not necessarily represent 

the positions or policies of the Department of Education. No official endorsement 

by the U.S. Department of Education of any product, commodity, service or 

enterprise mentioned in this publication is intended or should be inferred. 

National Mathematics Advisory Panel 

Tyrrell Flawn 

Executive Director 

March 2008 

This report is in the public domain. Authorization to reproduce it in whole or in part 

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should be: National Mathematics Advisory Panel. Foundations for Success: The 

Final Report of the National Mathematics Advisory Panel, U.S. Department of 

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National Mathematics Advisory Panel FINAL REPORT iii

CONTENTS 

List of Tables......................................................................................................iv 

List of Abbreviations ...........................................................................................v 

Members of the National Mathematics Advisory Panel ......................................vii 

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

Executive Summary............................................................................................xi 

Background...................................................................................................xi 

Principal Messages......................................................................................xiii 

The National Mathematics Advisory Panel...................................................xv 

Main Findings and Recommendations.........................................................xvi 

Curricular Content ..........................................................................xvi

Learning Processes .......................................................................xviii

Teachers and Teacher Education......................................................xx

Instructional Practices....................................................................xxii

Instructional Materials..................................................................xxiv

Assessment ..................................................................................... xxv

Research Policies and Mechanisms ...............................................xxvi

Chapter 1: Background for the President’s Charge ...............................................1 

Chapter 2: The National Mathematics Advisory Panel .........................................7 

Chapter 3: Principal Messages ...........................................................................11 

First Things First..........................................................................................11 

Learning as We Go Along............................................................................12 

Chapter 4: Curricular Content ............................................................................15 

The Nature of School Algebra......................................................................15 

Critical Foundations of Algebra ...................................................................17 

Benchmarks for the Critical Foundations......................................................19 

A Need for Coherence..................................................................................20 

Integrated versus Single Subject Approach...................................................22 

Universal Availability of Authentic Education in Algebra............................23 

Chapter 5: Learning Processes ...........................................................................25 

Readiness for Learning.................................................................................25 

Whole Number Arithmetic: Computational Proficiency Plus Conceptual 

Understanding ........................................................................................26 

Number Sense..............................................................................................27 

Fractions ......................................................................................................28 

Geometry and Measurement.........................................................................29 

General Principles of Learning.....................................................................30 

Social, Motivational, and Affective Influences .............................................31 

Considerations Specific to Algebra...............................................................32 

Chapter 6: Teachers and Teacher Education.......................................................35 

Teachers’ Mathematical Knowledge.............................................................35 

Teachers’ Education: Preparation, Induction, and 

Professional Development......................................................................39 

Recruitment and Retention Strategies to Attract and Retain Effective 

Teachers of Mathematics........................................................................41 

Elementary Mathematics Specialist Teachers...............................................43 
National Mathematics Advisory Panel FINAL REPORT iv

Chapter 7: Instructional Practices .......................................................................45 

Teacher-Directed and Student-Centered Instruction in Mathematics ................45 

Using Formative Assessment........................................................................46 

Teaching Low-Achieving Students and Students with 

Learning Disabilities...............................................................................48 

Using “Real-World” Problems to Teach Mathematics ..................................49 

Technology and Applications of Technology: Calculators and 

Computer-Based Instruction ...................................................................50 

Teaching Mathematically Gifted Students ....................................................52 

Chapter 8: Instructional Materials.......................................................................55 

Accuracy of Textbooks.................................................................................55 

Length, Coherence, and Sequencing of Topics..............................................55 

Chapter 9: Assessment of Mathematics Learning................................................57 

Content.........................................................................................................58 

Performance Categories................................................................................59 

Item and Test Design....................................................................................60 

Chapter 10: Research Policies and Mechanisms .................................................63 

BIBLIOGRAPHY..............................................................................................67 

APPENDIX A: Presidential Executive Order 13398...........................................71 

APPENDIX B: Roster of Panel Members, Staff, and Consultants.......................75 

APPENDIX C: Organization and Operation of the Panel....................................79 

APPENDIX D: Dates and Locations of Meetings...............................................87 

APPENDIX E: Rosters of Task Groups and Subcommittees...............................89 

Tables 

Table 1: The Major Topics of School Algebra....................................................16 

Table 2: Benchmarks for the Critical Foundations..............................................20 

Table 3: Suggested Reorganization of NAEP Content Strands............................58 
National Mathematics Advisory Panel FINAL REPORT v

Abbreviations 

ABCTE American Board for Certification of Teacher Excellence 

ACT American College Testing 

CAI Computer-Assisted Instruction 

ETS Educational Testing Service 

IDA STPI Institute for Defense Analyses Science and Technology Policy Institute 

LA Low Achieving 

LD Learning Disabilities 

NAEP National Assessment of Educational Progress 

NCTM National Council of Teachers of Mathematics 

NES National Evaluation Systems 

NVS NAEP Validity Study 

SES Socioeconomic Status 

STEM Science, Technology, Engineering, and Mathematics 

TAI Team Assisted Individualization 

TIMSS Trends in International Mathematics and Science Study 

PISA Programme for International Student Assessment 
National Mathematics Advisory Panel FINAL REPORT vi
National Mathematics Advisory Panel FINAL REPORT vii

Members of the National 

Mathematics Advisory Panel 

Larry R. Faulkner, Chair 

Camilla Persson Benbow, Vice Chair 

Deborah Loewenberg Ball 

A. Wade Boykin 

Douglas H. Clements 

Susan Embretson 

Francis “Skip” Fennell 

Bert Fristedt 

David C. Geary 

Russell M. Gersten 

Tom Loveless 

Liping Ma 

Valerie F. Reyna 

Wilfried Schmid 

Robert S. Siegler 

James H. Simons 

Sandra Stotsky 

Vern Williams 

Hung-Hsi Wu 

Ex Officio Members 

Irma Arispe 

Daniel B. Berch 

Joan Ferrini-Mundy 

Raymond Simon 

Grover J. “Russ” Whitehurst 

U.S. Department of Education Staff 

Tyrrell Flawn, Executive Director 

Marian Banfield Jennifer Graban 

Ida Eblinger Kelley 
National Mathematics Advisory Panel FINAL REPORT viii
National Mathematics Advisory Panel FINAL REPORT ix

Acknowledgments 

The members of the National Mathematics Advisory Panel express the 

greatest appreciation to those who facilitated this work, beginning with our own 

employers. This work of the Panel required extraordinary time commitments over 

a two-year period, and the gifts of that time represent substantial, concrete support 

for this project by the institutions that employ the members. They are recognized 

with gratitude below. 

This report was developed and adopted by the panelists listed on page vii, 

who constituted the membership at the end of the project. During the two years of 

the Panel’s work, there were changes in membership caused by shifts in personal 

and professional circumstances and by the Panel’s direct request for augmentation 

in particular areas of expertise. Appendix B is a list of all members ever appointed 

to the Panel. Here, the concluding members wish to express appreciation for the 

contributions of colleagues who were once part of the Panel: Nancy Ichinaga, 

Diane Auer Jones, Thomas W. Luce, III, and Kathie Olsen. 

From time to time, the Panel was fortunate to be able to call on the 

specific expertise of many colleagues. They examined materials as needed, 

offered opinions on specialized topics, and examined drafts of sections. We thank 

them all for their able and generous contributions: Mark Ashcraft, Richard A. 

Askey, Scott K. Baker, Arthur J. Baroody, Hyman Bass, Benjamin Samuel 

Clarke, Carol S. Dweck, Anne Foegen, Karen C. Fuson, Dan Goldhaber, Thomas 

L. Good, Eric A. Hanushek, James Hiebert, Heather C. Hill, Roger Howe, 

Andrew G. Izsak, Nancy C. Jordan, Jeremy Kilpatrick, Kenneth R. Koedinger, 

James W. Lewis, David F. Lohman, R. James Milgram, Anthony Ralston, 

William H. Schmidt, Catherine Sophian, Jon R. Star, Joyce VanTassel-Baska, 

Patrick W. Thompson, Johannes E.H. Van Luit, Linda Dager Wilson, and Bradley 

Witzel. Of course, the Panel itself is responsible for the language, findings, and 

recommendations in this report. 

The budget dedicated to this project by the U.S. Department of Education 

was augmented by funds from donors who are also recognized below. Their 

generosity enabled work that was more thorough and more expertly supported 

than would ever have been otherwise possible. 

Finally, the Panel expresses the deepest appreciation to the U.S. 

Department of Education staff, headed by Executive Director Tyrrell Flawn. They 

carried out essential tasks with skill and dedication. Without them, this report 

would have not been realized. 
National Mathematics Advisory Panel FINAL REPORT 

x

Panel Member Institutions 

Brookings Institution 

Carnegie Mellon University 

Cornell University 

Georgia Institute of Technology 

Harvard University 

Houston Endowment Inc. 

Howard University 

Instructional Research Group 

Longfellow Middle School, Fairfax 

County Public Schools, VA 

McDaniel College 

National Council of Teachers of 

Mathematics 

National Institutes of Health 

National Science Foundation 

Renaissance Technologies 

Corporation 

The Carnegie Foundation for the 

Advancement of Teaching 

U.S. Department of Education 

University at Buffalo, State 

University of New York 

University of Arkansas 

University of California at Berkeley 

University of Michigan 

University of Minnesota, Twin Cities 

University of Missouri 

Vanderbilt University 

White House Office of Science and 

Technology Policy 

Gift Fund Donors 

Exxon Mobil Corporation 

Houston Endowment Inc. 

Math for America 

Northrop Grumman 

Texas Instruments 
National Mathematics Advisory Panel FINAL REPORT xi

Executive Summary 

Background 

The eminence, safety, and well-being of nations have been entwined for 

centuries with the ability of their people to deal with sophisticated quantitative 

ideas. Leading societies have commanded mathematical skills that have brought 

them advantages in medicine and health, in technology and commerce, in 

navigation and exploration, in defense and finance, and in the ability to understand 

past failures and to forecast future developments. History is full of examples. 

During most of the 20th century, the United States possessed peerless 

mathematical prowess—not just as measured by the depth and number of the 

mathematical specialists who practiced here but also by the scale and quality of 

its engineering, science, and financial leadership, and even by the extent of 

mathematical education in its broad population. But without substantial and 

sustained changes to its educational system, the United States will relinquish its 

leadership in the 21st century. This report is about actions that must be taken to 

strengthen the American people in this central area of 

learning. Success matters to the nation at large. It matters, 

too, to individual students and their families, because it 

opens doors and creates opportunities. 

Much of the commentary on mathematics and science in 

the United States focuses on national economic competitiveness 

and the economic well-being of citizens and enterprises. There is 

reason enough for concern about these matters, but it is yet more 

fundamental to recognize that the safety of the nation and the 

quality of life—not just the prosperity of the nation—are at issue. 

In the contemporary world, an educated technical 

workforce undergirds national leadership. Yet the United 

States faces a future in which there will be accelerating 

retirements affecting a large fraction of the current science 

and engineering workforce, even as the growth of job opportunities in this sector 

is expected to outpace job growth in the economy at large. These trends will 

place substantial stress on the nation’s ability to sustain a workforce with 

adequate scale and quality. For many years, our country has imported a great 

volume of technical talent from abroad, but the dramatic success of economies 

overseas in the age of the Internet casts doubt on the viability of such a strategy 

in the future, because attractive employment for technical workers is developing 

in countries that have been supplying invaluable talent for U.S. employers. From 

1990 to 2003, research and development expenditures in Asian countries other 

than Japan grew from an insignificant percentage to almost half of American 

R&D expenditures. There are consequences to a weakening of American 

During most of the 20th 

century, the United States 

possessed peerless 

mathematical prowess—not 

just as measured by the 

depth and number of the 

mathematical specialists 

who practiced here, but 

also by the scale and 

quality of its engineering, 

science, and financial 

leadership.... 
National Mathematics Advisory Panel FINAL REPORT 

xii

independence and leadership in mathematics, the natural sciences, and 

engineering. We risk our ability to adapt to change. We risk technological 

surprise to our economic viability and to the foundations of our country’s 

security. National policy must ensure the healthy development of a domestic 

technical workforce of adequate scale with top-level skills. 

But the concerns of national policy relating to mathematics education go far 

beyond those in our society who will become scientists or engineers. The national 

workforce of future years will surely have to handle quantitative concepts more 

fully and more deftly than at present. So will the citizens and policy leaders who 

deal with the public interest in positions of civic leadership. Sound education in 

mathematics across the population is a national interest. 

Success in mathematics education also is important for individual citizens, 

because it gives them college and career options, and it increases prospects for 

future income. A strong grounding in high school mathematics through Algebra II 

or higher correlates powerfully with access to college, graduation from college, 

and earning in the top quartile of income from employment. The value of such 

preparation promises to be even greater in the future. The 

National Science Board indicates that the growth of jobs in the 

mathematics-intensive science and engineering workforce is 

outpacing overall job growth by 3:1. 

International and domestic comparisons show that 

American students have not been succeeding in the 

mathematical part of their education at anything like a level 

expected of an international leader. Particularly disturbing is the 

consistency of findings that American students achieve in 

mathematics at a mediocre level by comparison to peers 

worldwide. On our own “National Report Card”—the National Assessment of 

Educational Progress (NAEP)—there are positive trends of scores at Grades 4 and 

8, which have just reached historic highs. This is a sign of significant progress. 

Yet other results from NAEP are less positive: 32% of our students are at or 

above the “proficient” level in Grade 8, but only 23% are proficient at Grade 12. 

Consistent with these findings is the vast and growing demand for remedial 

mathematics education among arriving students in four-year colleges and 

community colleges across the nation. 

Moreover, there are large, persistent disparities in mathematics 

achievement related to race and income—disparities that are not only devastating 

for individuals and families but also project poorly for the nation’s future, given 

the youthfulness and high growth rates of the largest minority populations. 

International and domestic 

comparisons show that 

American students have 

not been succeeding in the 

mathematical part of their 

education at anything like a 

level expected of an 

international leader.
National Mathematics Advisory Panel FINAL REPORT xiii

Although our students encounter difficulties with many aspects of 

mathematics, many observers of educational policy see Algebra as a central 

concern.1 The sharp falloff in mathematics achievement in the U.S. begins as 

students reach late middle school, where, for more and more students, algebra 

course work begins. Questions naturally arise about how students can be best 

prepared for entry into Algebra. 

These are questions with consequences, for Algebra is a demonstrable 

gateway to later achievement. Students need it for any form of higher 

mathematics later in high school; moreover, research shows that completion of 

Algebra II correlates significantly with success in college and earnings from 

employment. In fact, students who complete Algebra II are more than twice as 

likely to graduate from college compared to students with less 

mathematical preparation. Among African-American and 

Hispanic students with mathematics preparation at least through 

Algebra II, the differences in college graduation rates versus the 

student population in general are half as large as the differences 

for students who do not complete Algebra II. 

For all of these considerations, the President created the 

National Mathematics Advisory Panel in April 2006, with the 

responsibilities of relying upon the “best available scientific evidence” and 

recommending ways “...to foster greater knowledge of and improved 

performance in mathematics among American students.” 

Principal Messages 

This Panel, diverse in experience, expertise, and philosophy, agrees 

broadly that the delivery system in mathematics education—the system that 

translates mathematical knowledge into value and ability for the next 

generation—is broken and must be fixed. This is not a conclusion about any 

single element of the system. It is about how the many parts do not now work 

together to achieve a result worthy of this country’s values and ambitions. 

On the basis of its deliberation and research, the Panel can report that 

America has genuine opportunities for improvement in mathematics education. 

This report lays them out for action. 

The essence of the Panel’s message is to put first things first. There are six 

elements, expressed compactly here, but in greater detail later. 

• The mathematics curriculum in Grades PreK–8 should be streamlined and 

should emphasize a well-defined set of the most critical topics in the early 

grades. 

 

1

 The word “algebra” is capitalized when referring to a particular course or course sequence, such 

as Algebra I and II. 

Students who complete 

Algebra II are more than 

twice as likely to graduate 

from college compared 

to students with less 

mathematical preparation. 
National Mathematics Advisory Panel FINAL REPORT 

xiv

• Use should be made of what is clearly known from rigorous research 

about how children learn, especially by recognizing a) the advantages for 

children in having a strong start; b) the mutually reinforcing benefits of 

conceptual understanding, procedural fluency, and automatic (i.e., quick 

and effortless) recall of facts; and c) that effort, not just inherent talent, 

counts in mathematical achievement. 

• Our citizens and their educational leadership should recognize 

mathematically knowledgeable classroom teachers as having a central role 

in mathematics education and should encourage rigorously evaluated 

initiatives for attracting and appropriately preparing prospective teachers, 

and for evaluating and retaining effective teachers. 

• Instructional practice should be informed by high-quality research, when 

available, and by the best professional judgment and experience of 

accomplished classroom teachers. High-quality research does not support 

the contention that instruction should be either entirely “student centered” 

or “teacher directed.” Research indicates that some forms of particular 

instructional practices can have a positive impact under specified 

conditions. 

• NAEP and state assessments should be improved in quality and should 

carry increased emphasis on the most critical knowledge and skills leading 

to Algebra. 

• The nation must continue to build capacity for more rigorous research in 

education so that it can inform policy and practice more effectively. 

Positive results can be achieved in a reasonable time at accessible cost, but 

a consistent, wise, community-wide effort will be required. Education in the 

United States has many participants in many locales—teachers, students, and 

parents; state school officers, school board members, superintendents, and 

principals; curriculum developers, textbook writers, and textbook editors; those 

who develop assessment tools; those who prepare teachers and help them to 

continue their development; those who carry out relevant research; association 

leaders and government officials at the federal, state, and local levels. All carry 

responsibilities. All can be important to success. 

The network of these many participants is linked through interacting 

national associations. A coordinated national approach toward improved 

mathematics education will require an annual forum of their leaders for at least a 

decade. The Panel recommends that the U.S. Secretary of Education take the lead 

in convening the forum initially, charge it to organize in a way that will sustain an 

effective effort, and request a brief annual report on the mutual agenda adopted 

for the year ahead. 
National Mathematics Advisory Panel FINAL REPORT xv

The President asked the Panel to use the best available scientific research 

to advise on improvements in the mathematics education of the nation’s children. 

Our consistent respect for sound research has been the main factor enabling the 

Panel’s joint conclusions on so many matters, despite differences of perspective 

and philosophy. At the same time, we found no research or insufficient research 

relating to a great many matters of concern in educational policy and practice. In 

those areas, the Panel has been very limited in what it can report. 

The Panel lays out many concrete steps that can be taken now toward 

significantly improved mathematics education, but it also views them only as a 

best start in a long process. This journey, like that of the post-Sputnik era, will 

require a commitment to “learning as we go along.” The nation should recognize 

that there is much more to discover about how to achieve better results. Models of 

continuous improvement have proven themselves in many other areas, and they 

can work again for America in mathematics education. 

The National Mathematics Advisory Panel 

The President established the Panel via Executive Order 13398 (Appendix 

A), in which he also assigned responsibility to the U.S. Secretary of Education for 

appointment of members and for oversight of the Panel. While the presidential 

charge contains many explicit elements, there is a clear emphasis on the preparation 

of students for entry into, and success in, Algebra. 

Over a period of 20 months, the Panel received public testimony as a 

committee of the whole but worked largely in task groups and subcommittees 

dedicated to major components of the presidential charge. Questions like the 

following illustrate the scope of the Panel’s inquiry: 

• What is the essential content of school algebra and what do children need 

to know before starting to study it? 

• What is known from research about how children learn mathematics? 

• What is known about the effectiveness of instructional practices and 

materials? 

• How can we best recruit, prepare, and retain effective teachers of 

mathematics? 

• How can we make assessments of mathematical knowledge more accurate 

and more useful? 

• What do practicing teachers of algebra say about the preparation of 

students whom they receive into their classrooms and about other relevant 

matters? 

• What are the appropriate standards of evidence for the Panel to use in 

drawing conclusions from the research base? 
National Mathematics Advisory Panel FINAL REPORT 

xvi

Each of five task groups carried out a detailed analysis of the available 

evidence in a major area of the Panel’s responsibility: Conceptual Knowledge and 

Skills, Learning Processes, Instructional Practices, Teachers and Teacher Education, 

and Assessment. Each of three subcommittees was charged with completion of a 

particular advisory function for the Panel: Standards of Evidence, Instructional 

Materials, and the Panel-commissioned National Survey of Algebra Teachers (see 

sidebar, page 9). Each task group and subcommittee produced a report supporting this 

document. All eight reports are separately available. 

The Panel took consistent note of the President’s emphasis on “the best 

available scientific evidence” and set a high bar for admitting research results into 

consideration. In essence, the Panel required the work to have been carried out in 

a way that manifested rigor and could support generalization at the level of 

significance to policy. One of the subcommittee reports covers global 

considerations relating to standards of evidence, while individual task group 

reports amplify the standards in the particular context of each task group’s work. 

In all, the Panel reviewed more than 16,000 research publications and policy 

reports and received public testimony from 110 individuals, 

of whom 69 appeared before the Panel on their own and 41 

others were invited on the basis of expertise to cover 

particular topics. In addition, the Panel reviewed written 

commentary from 160 organizations and individuals, and 

analyzed survey results from 743 active teachers of algebra. 

In late 2007, the Panel synthesized this Final Report 

by drawing together the most important findings and 

recommendations, which are hereby issued with the Panel’s 

full voice. This report connects in many places to the eight reports of the task 

groups and subcommittees, which carry detailed analyses of research literature 

and other relevant materials. These supporting reports cover work carried out as 

part of the Panel’s overall mission, but they are presented by only those members 

who participated in creating them. This Final Report represents findings and 

recommendations of the Panel as a whole. 

Main Findings and Recommendations 

The Panel had a broad scope and reached many individual findings and 

recommendations, all conveyed in the main report under headings corresponding 

to those below. This Executive Summary generally contains only abbreviated 

versions of the most important points. 

Curricular Content 

1) A focused, coherent progression of mathematics learning, with an 

emphasis on proficiency with key topics, should become the norm in 

elementary and middle school mathematics curricula. Any approach that 

continually revisits topics year after year without closure is to be avoided. 

The Panel took consistent

note of the President’s

emphasis on “the best 

available scientific 

evidence” and set a high 

bar for admitting research 

results into consideration. 
National Mathematics Advisory Panel FINAL REPORT xvii

By the term focused, the Panel means that curriculum must include (and 

engage with adequate depth) the most important topics underlying 

success in school algebra. By the term coherent, the Panel means that the 

curriculum is marked by effective, logical progressions from earlier, less 

sophisticated topics into later, more sophisticated ones. Improvements 

like those suggested in this report promise immediate positive results 

with minimal additional cost. 

By the term proficiency, the Panel means that students should understand 

key concepts, achieve automaticity as appropriate (e.g., with addition and 

related subtraction facts), develop flexible, accurate, and automatic 

execution of the standard algorithms, and use these competencies to solve 

problems.2

2) To clarify instructional needs in Grades PreK–8 and to sharpen future 

discussion about the role of school algebra in the overall mathematics 

curriculum, the Panel developed a clear concept of school algebra via its 

list of Major Topics of School Algebra (Table 1, page 16). 

School algebra is a term chosen to encompass the full body of algebraic 

material that the Panel expects to be covered through high school, 

regardless of its organization into courses and levels. The Panel expects 

students to be able to proceed successfully at least through the content of 

Algebra II. 

3) The Major Topics of School Algebra in Figure 1 should be the focus for 

school algebra standards in curriculum frameworks, algebra courses, 

textbooks for algebra, and in end-of-course assessments. 

4) A major goal for K–8 mathematics education should be proficiency with 

fractions (including decimals, percents, and negative fractions), for such 

proficiency is foundational for algebra and, at the present time, seems to be 

severely underdeveloped. Proficiency with whole numbers is a necessary 

precursor for the study of fractions, as are aspects of measurement and 

geometry. These three areas—whole numbers, fractions, and particular 

aspects of geometry and measurement—are the Critical Foundations of 

Algebra. Important elements within each of these three categories are 

delineated on page 17 of this report. 

 

2

 This meaning is in keeping with Adding It Up (National Research Council, 2001, p. 116), in 

which five attributes were associated with the concept of proficiency: 1) conceptual understanding 

(comprehension of mathematical concepts, operations, and relations), 2) procedural fluency (skills 

in carrying out procedures flexibly, fluently, and appropriately), 3) strategic competence (ability to 

formulate, represent, and solve mathematical problems), 4) adaptive reasoning (capacity for 

logical thought, reflection, explanation, and justification), and 5) productive disposition (habitual 

inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in 

diligence and one's own efficacy). 
National Mathematics Advisory Panel FINAL REPORT 

xviii

The Critical Foundations are not meant to comprise a complete 

mathematics curriculum leading to algebra; however, they deserve 

primary attention and ample time in any mathematics curriculum. 

5) To encourage the development of students in Grades PreK–8 at an 

effective pace, the Panel recommends a set of Benchmarks for the Critical 

Foundations (Table 2, page 20). They should be used to guide classroom 

curricula, mathematics instruction, textbook development, and state 

assessments. 

6) All school districts should ensure that all prepared students have access to 

an authentic algebra course—and should prepare more students than at 

present to enroll in such a course by Grade 8. The word authentic is used 

here as a descriptor of a course that addresses algebra consistently with the 

Major Topics of School Algebra (Table 1, page 16). Students must be 

prepared with the mathematical prerequisites for this course according to 

the Critical Foundations of Algebra (page 17) and the Benchmarks for the 

Critical Foundations (Table 2, page 20). 

7) Teacher education programs and licensure tests for early childhood 

teachers, including all special education teachers at this level, should fully 

address the topics on whole numbers, fractions, and the appropriate 

geometry and measurement topics in the Critical Foundations of Algebra, 

as well as the concepts and skills leading to them; for elementary 

teachers, including elementary level special education teachers, all topics 

in the Critical Foundations of Algebra and those topics typically covered 

in an introductory Algebra course; and for middle school teachers, 

including middle school special education teachers, the Critical 

Foundations of Algebra and all of the Major Topics of School Algebra.

Learning Processes 

8) Most children acquire considerable knowledge of numbers and other 

aspects of mathematics before they enter kindergarten. This is important, 

because the mathematical knowledge that kindergartners bring to school 

is related to their mathematics learning for years thereafter—in 

elementary school, middle school, and even high school. Unfortunately, 

most children from low-income backgrounds enter school with far less 

knowledge than peers from middle-income backgrounds, and the 

achievement gap in mathematical knowledge progressively widens 

throughout their PreK–12 years. 

9) Fortunately, encouraging results have been obtained for a variety of 

instructional programs developed to improve the mathematical 

knowledge of preschoolers and kindergartners, especially those from lowincome backgrounds. There are effective techniques—derived from 

scientific research on learning—that could be put to work in the 

classroom today to improve children’s mathematical knowledge. 
National Mathematics Advisory Panel FINAL REPORT xix

However, tests of both short-term and long-term effects of these 

interventions with larger populations of children from low-income 

families are urgently needed. 

10) To prepare students for Algebra, the curriculum must simultaneously 

develop conceptual understanding, computational fluency, and problemsolving skills. Debates regarding the relative importance of these aspects 

of mathematical knowledge are misguided. These capabilities are 

mutually supportive, each facilitating learning of the others. Teachers 

should emphasize these interrelations; taken together, conceptual 

understanding of mathematical operations, fluent execution of 

procedures, and fast access to number combinations jointly support 

effective and efficient problem solving. 

11) Computational proficiency with whole number operations is dependent 

on sufficient and appropriate practice to develop automatic recall of 

addition and related subtraction facts, and of multiplication and related 

division facts. It also requires fluency with the standard algorithms for 

addition, subtraction, multiplication, and division. Additionally it requires 

a solid understanding of core concepts, such as the commutative, 

distributive, and associative properties. Although the learning of concepts 

and algorithms reinforce one another, each is also dependent on different 

types of experiences, including practice. 

12) Difficulty with fractions (including decimals and percents) is pervasive 

and is a major obstacle to further progress in mathematics, including 

algebra. A nationally representative sample of teachers of Algebra I who 

were surveyed for the Panel rated students as having very poor preparation 

in “rational numbers and operations involving fractions and decimals.” 

As with learning whole numbers, a conceptual understanding of 

fractions and decimals and the operational procedures for using them 

are mutually reinforcing. One key mechanism linking conceptual and 

procedural knowledge is the ability to represent fractions on a number 

line. The curriculum should afford sufficient time on task to ensure 

acquisition of conceptual and procedural knowledge of fractions and of 

proportional reasoning. Instruction focusing on conceptual knowledge 

of fractions is likely to have the broadest and largest impact on 

problem-solving performance when it is directed toward the accurate 

solution of specific problems. 

13) Mathematics performance and learning of groups that have traditionally 

been underrepresented in mathematics fields can be improved by 

interventions that address social, affective, and motivational factors. 

Recent research documents that social and intellectual support from peers 

and teachers is associated with higher mathematics performance for all 

students, and that such support is especially important for many AfricanAmerican and Hispanic students. There is an urgent need to conduct 

experimental evaluations of the effectiveness of support-focused 
National Mathematics Advisory Panel FINAL REPORT 

xx

interventions both small- and large-scale, because they are promising 

means for reducing the mathematics achievement gaps that are prevalent 

in U.S. society. 

14) Children’s goals and beliefs about learning are related to their 

mathematics performance. Experimental studies have demonstrated that 

changing children’s beliefs from a focus on ability to a focus on effort 

increases their engagement in mathematics learning, which in turn 

improves mathematics outcomes: When children believe that their efforts 

to learn make them “smarter,” they show greater persistence in 

mathematics learning. Related research demonstrates that the engagement 

and sense of efficacy of African-American and Hispanic students in 

mathematical learning contexts not only tends to be lower than that of 

white and Asian students but also that it can be significantly increased. 

Teachers and other educational leaders should consistently help students 

and parents to understand that an increased emphasis on the importance 

of effort is related to improved mathematics performance. This is a 

critical point because much of the public’s self-evident resignation about 

mathematics education (together with the common tendencies to dismiss 

weak achievement and to give up early) seems rooted in the erroneous 

idea that success is largely a matter of inherent talent or ability, not effort. 

15) Teachers and developers of instructional materials sometimes assume that 

students need to be a certain age to learn certain mathematical ideas. 

However, a major research finding is that what is developmentally 

appropriate is largely contingent on prior opportunities to learn. Claims 

based on theories that children of particular ages cannot learn certain 

content because they are “too young,” “not in the appropriate stage,” or “not 

ready” have consistently been shown to be wrong. Nor are claims justified 

that children cannot learn particular ideas because their brains are 

insufficiently developed, even if they possess the prerequisite knowledge for 

learning the ideas. 

Teachers and Teacher Education 

16) Teachers who consistently produce significant gains in students’ 

mathematics achievement can be identified using value-added analyses 

(analyses that examine individual students’ achievement gains as a 

function of the teacher). The impact on students’ mathematics learning is 

compounded if students have a series of these more effective teachers. 
National Mathematics Advisory Panel FINAL REPORT xxi

Unfortunately, little is known from existing high-quality research about 

what effective teachers do to generate greater gains in student learning. 

Further research is needed to identify and more carefully define the skills 

and practices underlying these differences in teachers’ effectiveness, and 

how to develop them in teacher preparation programs. 

17) Research on the relationship between teachers’ mathematical knowledge 

and students’ achievement confirms the importance of teachers’ content 

knowledge. It is self-evident that teachers cannot teach what they do not 

know. However, because most studies have relied on proxies for teachers’ 

mathematical knowledge (such as teacher certification or courses taken), 

existing research does not reveal the specific mathematical knowledge and 

instructional skill needed for effective teaching, especially at the 

elementary and middle school level. Direct assessments of teachers’ actual 

mathematical knowledge provide the strongest indication of a relation 

between teachers’ content knowledge and their students’ achievement. 

More precise measures are needed to specify in greater detail the 

relationship among elementary and middle school teachers’ mathematical 

knowledge, their instructional skill, and students’ learning. 

18) Teaching well requires substantial knowledge and skill. However, 

existing research on aspects of teacher education, including standard 

teacher preparation programs, alternative pathways into teaching, support 

programs for new teachers (e.g., mentoring), and professional 

development, is not of sufficient rigor or quality to permit the Panel to 

draw conclusions about the features of professional training that have 

effects on teachers’ knowledge, their instructional practice, or their 

students’ achievement. 

Currently there are multiple pathways into teaching. Research indicates 

that differences in teachers’ knowledge and effectiveness between these 

pathways are small or nonsignificant compared to very large differences 

among the performance of teachers within each pathway. 

19) The mathematics preparation of elementary and middle school teachers 

must be strengthened as one means for improving teachers’ effectiveness in 

the classroom. This includes preservice teacher education, early career 

support, and professional development programs. A critical component of 

this recommendation is that teachers be given ample opportunities to learn 

mathematics for teaching. That is, teachers must know in detail and from a 

more advanced perspective the mathematical content they are responsible 

for teaching and the connections of that content to other important 

mathematics, both prior to and beyond the level they are assigned to teach. 
National Mathematics Advisory Panel FINAL REPORT 

xxii

High-quality research must be undertaken to create a sound basis for the 

mathematics preparation of elementary and middle school teachers within 

preservice teacher education, early-career support, and ongoing 

professional development programs. Outcomes of different approaches 

should be evaluated by using reliable and valid measures of their effects on 

prospective and current teachers’ instructional techniques and, most 

importantly, their effects on student achievement. 

20) In an attempt to improve mathematics learning at the elementary level, a 

number of school districts around the country are using “math specialist 

teachers” of three different types—math coaches (lead teachers), full-time 

elementary mathematics teachers, and pull-out teachers. However, the Panel 

found no high-quality research showing that the use of any of these types of 

math specialist teachers improves students’ learning. 

The Panel recommends that research be conducted on the use of full-time 

mathematics teachers in elementary schools. These would be teachers with 

strong knowledge of mathematics who would teach mathematics full-time 

to several classrooms of students, rather than teaching many subjects to one 

class, as is typical in most elementary classrooms. This recommendation for 

research is based on the Panel’s findings about the importance of teachers’ 

mathematical knowledge. The use of teachers who have specialized in 

elementary mathematics teaching could be a practical alternative to 

increasing all elementary teachers’ content knowledge (a problem of huge 

scale) by focusing the need for expertise on fewer teachers. 

21) Schools and teacher education programs should develop or draw on a 

variety of carefully evaluated methods to attract and prepare teacher 

candidates who are mathematically knowledgeable and to equip them 

with the skills to help students learn mathematics. 

22) Research on teacher incentives generally supports their effectiveness, 

although the quality of the studies is mixed. Given the substantial number 

of unknowns, policy initiatives involving teacher incentives should be 

carefully evaluated. 

Instructional Practices 

23) All-encompassing recommendations that instruction should be entirely 

“student centered” or “teacher directed” are not supported by research. If 

such recommendations exist, they should be rescinded. If they are being 

considered, they should be avoided. High-quality research does not 

support the exclusive use of either approach. 

24) Research has been conducted on a variety of cooperative learning 

approaches. One such approach, Team Assisted Individualization (TAI), has 

been shown to improve students’ computation skills. This highly structured 

pedagogical strategy involves heterogeneous groups of students helping 
National Mathematics Advisory Panel FINAL REPORT xxiii

each other, individualized problems based on student performance on a 

diagnostic test, specific teacher guidance, and rewards based on both group 

and individual performance. Effects of TAI on conceptual understanding 

and problem solving were not significant. 

25) Teachers’ regular use of formative assessment improves their students’ 

learning, especially if teachers have additional guidance on using the 

assessment to design and to individualize instruction. Although research 

to date has only involved one type of formative assessment (that based on 

items sampled from the major curriculum objectives for the year, based 

on state standards), the results are sufficiently promising that the Panel 

recommends regular use of formative assessment for students in the 

elementary grades. 

26) The use of “real-world” contexts to introduce mathematical ideas has been 

advocated, with the term “real world” being used in varied ways. A 

synthesis of findings from a small number of high-quality studies indicates 

that if mathematical ideas are taught using “real-world” contexts, then 

students’ performance on assessments involving similar “real-world” 

problems is improved. However, performance on assessments more 

focused on other aspects of mathematics learning, such as computation, 

simple word problems, and equation solving, is not improved. 

27) Explicit instruction with students who have mathematical difficulties has 

shown consistently positive effects on performance with word problems and 

computation. Results are consistent for students with learning disabilities, as 

well as other students who perform in the lowest third of a typical class. By 

the term explicit instruction, the Panel means that teachers provide clear 

models for solving a problem type using an array of examples, that students 

receive extensive practice in use of newly learned strategies and skills, that 

students are provided with opportunities to think aloud (i.e., talk through the 

decisions they make and the steps they take), and that students are provided 

with extensive feedback. 

This finding does not mean that all of a student’s mathematics instruction 

should be delivered in an explicit fashion. However, the Panel 

recommends that struggling students receive some explicit mathematics 

instruction regularly. Some of this time should be dedicated to ensuring 

that these students possess the foundational skills and conceptual 

knowledge necessary for understanding the mathematics they are learning 

at their grade level. 

28) Research on instructional software has generally shown positive effects 

on students’ achievement in mathematics as compared with instruction 

that does not incorporate such technologies. These studies show that 

technology-based drill and practice and tutorials can improve student 

performance in specific areas of mathematics. Other studies show that 

teaching computer programming to students can support the development 

of particular mathematical concepts, applications, and problem solving. 
National Mathematics Advisory Panel FINAL REPORT 

xxiv

However, the nature and strength of the results vary widely across these 

studies. In particular, one recent large, multisite national study found no 

significant effects of instructional tutorial (or tutorial and practice) 

software when implemented under typical conditions of use. Taken 

together, the available research is insufficient for identifying the factors 

that influence the effectiveness of instructional software under 

conventional circumstances. 

29) A review of 11 studies that met the Panel’s rigorous criteria (only one 

study less than 20 years old) found limited or no impact of calculators on 

calculation skills, problem solving, or conceptual development over 

periods of up to one year. This finding is limited to the effect of 

calculators as used in the 11 studies. However, the Panel’s survey of the 

nation’s algebra teachers indicated that the use of calculators in prior 

grades was one of their concerns. The Panel cautions that to the degree 

that calculators impede the development of automaticity, fluency in 

computation will be adversely affected. 

The Panel recommends that high-quality research on particular uses of 

calculators be pursued, including both their short- and long-term effects on 

computation, problem solving, and conceptual understanding. 

30) Mathematically gifted students with sufficient motivation appear to be 

able to learn mathematics much faster than students proceeding through 

the curriculum at a normal pace, with no harm to their learning, and 

should be allowed to do so. 

Instructional Materials 

31) U.S. mathematics textbooks are extremely long—often 700–1,000 pages. 

Excessive length makes books more expensive and can contribute to a lack 

of coherence. Mathematics textbooks are much smaller in many nations 

with higher mathematics achievement than the U.S., thus demonstrating that 

the great length of our textbooks is not necessary for high achievement. 

Representatives of several publishing companies who testified before the 

Panel indicated that one substantial contributor to the length of the books 

was the demand of meeting varying state standards for what should be 

taught in each grade. Other major causes of the extreme length of U.S. 

mathematics textbooks include the many photographs, motivational stories, 

and other nonmathematical content that the books include. Publishers 

should make every effort to produce much shorter and more focused 

mathematics textbooks. 

32) States and districts should strive for greater agreement regarding which 

topics will be emphasized and covered at particular grades. Textbook 

publishers should publish editions that include a clear emphasis on the 

material that these states and districts agree to teach in specific grades. 
National Mathematics Advisory Panel FINAL REPORT xxv

33) Publishers must ensure the mathematical accuracy of their materials. 

Those involved with developing mathematics textbooks and related 

instructional materials need to engage mathematicians, as well as 

mathematics educators, at all stages of writing, editing, and reviewing 

these materials. 

Assessment 

34) NAEP and state tests for students through Grade 8 should focus on and 

adequately represent the Panel’s Critical Foundations of Algebra. Student 

achievement on this critical mathematics content should be reported and 

tracked over time. 

35) The Panel suggests that the NAEP strand on “Number Properties and 

Operations” be expanded and divided into two parts. The former should 

include a focus on whole numbers, including whole number operations 

(i.e., addition, subtraction, multiplication, division), at Grade 4, and on all 

integers (negative and positive) at Grade 8. The second content area 

involving number should focus on fractions. At Grade 4, it should involve 

beginning work with fractions and decimals, including recognition, 

representation, and comparing and ordering. The coverage should be 

expanded to include operations with fractions, decimals, and percents at 

Grade 8. Similarly, the content of work with whole numbers and fractions 

on state tests should expand and cover these concepts and operations as 

they develop from year to year, particularly at Grades 5, 6, and 7, which are 

grade levels when the NAEP test is not offered. 

36) The Panel recommends a more appropriate balance in how algebra is 

defined and assessed at both the Grade 4 and Grade 8 levels of the NAEP. 

The Panel strongly recommends that “algebra” problems involving 

patterns should be greatly reduced in these tests. The same consideration 

applies to state tests. 

37) State tests and NAEP must be of the highest mathematical and technical 

quality. To this end, states and NAEP should develop procedures for item 

development, quality control, and oversight to ensure that test items reflect 

the best item-design features, are of the highest mathematical and 

psychometric quality, and measure what is intended, with non-constructrelevant sources of variance in performance minimized (i.e., with 

nonmathematical sources of influence on student performance minimized). 

38) Calculators should not be used on test items designed to assess 

computational facility. 
National Mathematics Advisory Panel FINAL REPORT 

xxvi

Research Policies and Mechanisms 

39) It is essential to produce methodologically rigorous scientific research in 

crucial areas of national need, such as the teaching and learning of 

mathematics. Researchers, educators, state and federal policymakers, 

private foundations, and research agencies have made and can continue to 

make important contributions toward this goal. Specifically, more research 

is needed that identifies: 1) effective instructional practices, materials, and 

principles of instructional design, 2) mechanisms of learning, 3) ways to 

enhance teachers’ effectiveness, including teacher education, that are 

directly tied to objective measures of student achievement, and 4) item and 

test features that improve the assessment of mathematical knowledge. 

Although the number of such studies has grown in recent years due to 

changes in policies and priorities at federal agencies, these studies are only 

beginning to yield findings, and their number remains comparatively 

small. 

40) As in all fields of education, the large quantity of studies gathered in 

literature searches on important topics in mathematics education is 

reduced appreciably once contemporary criteria for rigor and 

generalizability are applied. Therefore, the Panel recommends that 

governmental agencies that fund research give priority not only to 

increasing the supply of research that addresses mathematics education but 

also to ensuring that such projects meet stringent methodological criteria, 

with an emphasis on the support of studies that incorporate randomized 

controlled designs (i.e., designs where students, classrooms, or schools are 

randomly assigned to conditions and studied under carefully controlled 

circumstances) or methodologically rigorous quasi-experimental designs. 

These studies must possess adequate statistical power, which will require 

substantial funding. 

Both smaller-scale experiments on the basic science of learning and 

larger-scale randomized experiments examining effective classroom 

practices are needed to ensure the coherent growth of research addressing 

important questions in mathematics education. Basic research on causal 

mechanisms of learning, as well as randomized trials, are essential, and, 

depending on their methodologies, both can be rigorous and relevant to 

educational practice. Basic research, in particular, is necessary to 

develop explicit predictions and to test hypotheses, which are 

underemphasized in current research on mathematics education. 

41) Leaders of graduate programs in education and related fields should ensure 

attention to research design, analysis, and interpretation for teachers and 

those entering academic and educational leadership positions in order to 

increase the national capacity to conduct and utilize rigorous research. 

42) New funding should be provided to establish support mechanisms for 

career shifts (K, or career development, awards from the National 
National Mathematics Advisory Panel FINAL REPORT xxvii

Institutes of Health represent one example). Many accomplished 

researchers who study the basic components of mathematics learning are 

not directly engaged in relevant educational research. While this more 

basic kind of research is important both in its own right and as a crucial 

foundation for designing classroom-level learning projects, at least some 

of these investigators have the potential to make more directly relevant 

contributions to educational research. Consequently, providing incentives 

for them to change the emphasis of their research programs could 

enhance research capacity in the field. 

43) Support should be provided to encourage the creation of crossdisciplinary research teams, including expertise in educational 

psychology, sociology, economics, cognitive development, mathematics, 

and mathematics education. 

44) PreK–12 schools should be provided with incentives and resources to 

provide venues for, and encourage collaboration in, educational research. 

45) Unnecessary barriers to research should be lowered. Although existing 

guidelines for the protection of human subjects must be fully respected, 

Institutional Review Board procedures should be streamlined for 

educational research that qualifies as being of low or minimal risk. The 

resolutions of the National Board for Education Sciences concerning 

making individual student data available to researchers with appropriate 

safeguards for confidentiality should be supported. 

National Mathematics Advisory Panel FINAL REPORT 1

Chapter 1: 

Background for the President’s Charge 

The eminence, safety, and well-being of nations have been entwined for 

centuries with the ability of their people to deal with sophisticated quantitative 

ideas. Leading societies have commanded mathematical skills that have brought 

them advantages in medicine and health, in technology and commerce, in 

navigation and exploration, in defense and finance, and in the ability to understand 

past failures and to forecast future developments. History is full of examples. 

During most of the 20th century, the United States possessed peerless 

mathematical prowess—not just as measured by the depth and number of the 

mathematical specialists who practiced here but also by the scale and quality of its 

engineering, science, and financial leadership, and even by the extent of 

mathematical education in its broad population. But without substantial and 

sustained changes to its educational system, the United States will relinquish its 

leadership in the 21st century. This report is about actions that 

must be taken to strengthen the American people in this central 

area of learning. Success matters to the nation at large. It matters, 

too, to individual students and their families, because it opens 

doors and creates opportunities. 

Much of the commentary on mathematics and science in 

the United States focuses on national economic competitiveness 

and the economic well-being of citizens and enterprises. There is 

reason enough for concern about these matters, but it is yet more 

fundamental to recognize that the safety of the nation and the 

quality of life—not just the prosperity of the nation—are at issue. 

In the contemporary world, an educated technical 

workforce undergirds national leadership. Yet the United States 

faces a future in which a large fraction of the current science and 

engineering workforce will be retiring. In the latest analysis, based on data from 

2003, 26% of the science and engineering degree holders in the workforce (40% of 

doctoral degree holders) were age 50 or older (National Science Board, 2008). At 

the same time, the demand for employees in this sector is expected to outpace job 

growth in the economy at large. In the last decade for which Census data are 

available (1990 to 2000), growth in employment in science and engineering 

occupations tripled that in other occupations (National Science Board, 2008). The 

combination of retirements and increasing demand for technologically 

knowledgeable workers will stress the nation’s ability to sustain a workforce of 

adequate scale and quality. We are not the first to note this danger (National 

Science Board, 2003). 

During most of the 20th 

century, the United States 

possessed peerless 

mathematical prowess—not 

just as measured by the 

depth and number of the 

mathematical specialists 

who practiced here, but 

also by the scale and 

quality of its engineering, 

science, and financial 

leadership.... 
National Mathematics Advisory Panel FINAL REPORT 2

For many years, the United States has imported a great volume of technical 

talent from abroad. Census data show that our domestic reliance on scientists and 

engineers from abroad significantly increased from 1990 to 2000—from 14% to 

22% across the whole technical workforce and from 24% to 38% at the doctoral 

level. The dramatic success of economies overseas in the age of the Internet casts 

doubt on the viability of such a strategy in the future, because attractive 

employment for technical workers is developing in countries that have been 

supplying invaluable talent for U.S. employers. This point is underscored by the 

rapid growth of research and development (R&D) expenditures in China, 

Singapore, South Korea, and Taiwan. From 1990 to 2004, the volume of R&D in 

these four countries increased from an insignificant percentage to almost half of 

American R&D expenditures (National Science Foundation, 2007). By 2004, 

China’s expenditures alone nearly reached parity with Japan’s, and each country 

was funding R&D at about a third of the commitment in the U.S. (National Science 

Foundation, 2007).3

There are consequences to a weakening of U.S. independence and 

leadership in mathematics, the natural sciences, and engineering. Looking at the 

fast pace of technological advancement in the United States, Schacht (2005) 

commented, “[I]t is widely accepted that technological progress is responsible for 

up to one-half the growth of the U.S. economy, and is one principal driving force 

in long-term growth and increases in living standards.” Ignoring threats to the 

nation’s ability to advance in the science, technology, engineering, and 

mathematics (STEM) fields will put our economic viability and our basis for 

security at risk. 

For decades, the education pipeline has not produced the necessary number 

of U.S. students for jobs in the STEM fields—jobs that the National Science Board 

indicates are outpacing overall job growth by 3:1 (National Science Board, 2008). 

As a result of this shortfall of citizens going into these fields, the United States has 

relied increasingly on immigrant and temporary nonimmigrant scientists and 

engineers (National Science Board, 2008). The fraction of U.S. students pursuing 

STEM-related degrees, according to recent numbers from the General 

Accountability Office (Ashby, 2006), has declined from 32% in academic year 

1994–95 to 27% in academic year 2003–04. In addition, a report by the 

Commission on Professionals in Science and Technology stated, “[O]ver the past 

40 years, there has been a significant decrease in the proportion of doctorates 

earned by U.S. citizens and permanent residents in STEM fields. In 1966, they 

earned 83.5% of all STEM doctorates awarded, but in 2004, they earned just 

59.8%” (Babco, 2006). This strategy may not work in the future, however, because 

the supply of immigrant and temporary nonimmigrant STEM professionals may 

become more uncertain for reasons addressed above. It is therefore in the national 

 

3

 These data are Gross Expenditures on Research and Development, as defined by the OECD. This 

quantity comprises the total expenditure on R&D by all domestic enterprises, including 

businesses, institutes, universities, and government laboratories. R&D expenditures performed 

abroad by domestic enterprises are not included. 
National Mathematics Advisory Panel FINAL REPORT 3

interest to increase the number of domestic students studying and receiving degrees 

in STEM areas. 

National policy must ensure the healthy development of a domestic technical 

workforce with adequate scale and top-level skill. But the concerns of national 

policy relating to mathematics education go well beyond those in our society who 

will become scientists or engineers. The national workforce of future years will 

surely have to handle quantitative concepts more fully and more deftly than at 

present. So will the citizens and policy leaders who deal with the 

public interest in positions of civic leadership. Sound education in 

mathematics across the population is a national interest. 

Mathematics literacy is a serious problem in the United 

States. According to Philips (2007), 78% of adults cannot explain 

how to compute the interest paid on a loan, 71% cannot calculate 

miles per gallon on a trip, and 58% cannot calculate a 10% tip for a 

lunch bill. Further, it is clear from the research that a broad range of students and 

adults also have difficulties with fractions (e.g., Hecht, Vagi, & Torgeson, 2007; 

Mazzocco & Devlin, in press), a foundational skill essential to success in algebra. 

The recent National Assessment of Educational Progress (NAEP, “the Nation’s 

Report Card”) shows that 27% of eighth-graders could not correctly shade 1/3 of a 

rectangle and 45% could not solve a word problem that required dividing fractions 

(U.S. Department of Education, 2004). 

Labor economists Richard J. Murnane and Frank Levy have spoken to the 

vital importance of mathematical skill (Murnane & Levy, 1996): 

Close to half of all seventeen year olds cannot read or do math at 

the level needed to get a job at a modern automobile plant. Barring 

some other special knowledge or talent that would allow them to 

earn a living as, say, a plumber or artist, they lack the skills to earn 

a middle-class paycheck in today’s economy. 

Algebra has emerged as a central concern, for it is a demonstrable gateway 

to later achievement. Students need Algebra4

 for more advanced mathematics 

course work in high school (Evan, Gray, & Olchefske, 2006). Yet, problems in 

mathematics learning in the U.S. increase in late middle school before students 

move into algebra. We see this in the scores on the NAEP. Results at Grade 4 

have improved considerably over the past 15 years and have just reached historic 

highs; scores at Grade 8 have also increased somewhat; but no progress is evident 

at Grade 12. In addition, NAEP results show that only 39% of our students are at 

or above the “proficient” level in Grade 8 (U.S. Department of Education, 2007), 

and even fewer, 23%, are at that level by Grade 12 (U.S. Department of 

Education, 2005). 

 

4

 The word “algebra” is capitalized when referring to a particular course or course sequence, such 

as Algebra I and II. 

National policy must 

ensure the healthy 

development of a 

domestic technical 

workforce with adequate 

scale and top-level skill. 
National Mathematics Advisory Panel FINAL REPORT 4

International comparisons also show that American students have not been 

succeeding in the mathematical part of their education at anything like a level 

expected of an international leader. In the Trends in Mathematics and Science 

Study (TIMSS), an international assessment, U.S. students do less well in Grade 8 

than Grade 4. The performance is still poorer in Grade 12, although the data for 

Grade 12, dating from 1995, are now quite old (Evan et al., 2006). Similarly, in 

the 2007 Programme for International Student Assessment (PISA), U.S. 15-yearolds ranked 25th among 30 developed nations in math literacy and problem 

solving (Baldi, Jin, Shemer, Green, Hergert, & Xie, 2007). Even in elementary 

school, the U.S. is not among the world leaders; only 7% of U.S. fourth-graders 

scored at the advanced level in TIMSS, compared to 38% of fourth-graders in 

Singapore, a world leader in mathematics achievement. 

From all of these results and analyses, questions naturally arise about how 

American students can be generally better prepared in mathematics and, in 

particular, how they can make a good start in secondary education by being well 

prepared for entry into Algebra. 

Given the importance of mathematics education, we must also take a hard 

look at who will be teaching this subject in school. All the efforts to ensure that 

mathematics is given the attention it deserves in the nation’s schools will be for 

naught without an adequate supply of mathematically knowledgeable and 

properly trained mathematics teachers. 

Success in mathematics education matters at the level of individual 

citizens because it opens options for college and career and increases prospects 

for future income. The probability that a student will enroll in a four-year college 

correlates substantially with completion of high school mathematics programs 

beyond the level of Algebra II (Horn & Nuñez, 2000; Horowitz, 2005). In fact, 

students who complete Algebra II are more than twice as likely to graduate from 

college as students who lack such preparation (Adelman, 1999; Evan et al., 2006). 

Although such correlations do not establish cause-and-effect linkages, they are 

clear and notable, because they connect with policy concerns of leaders and 

practical choices that students and parents must make. College participation and 

graduation rates are critical for our nation, because college graduates offer many 

benefits to civic life and to the economy rooted in their additional education. 

College graduates are more likely to vote, use new technology, and become civic 

leaders, and are less likely to be involved in criminal activity (Pascarella & 

Terenzini, 1991). 

Consistent with the NAEP findings is the vast and growing demand for 

remedial mathematics education among arriving students in four-year colleges 

and community colleges across the nation. Data from the year 2000 showed that 

71% of America’s degree-granting institutions offered an average of 2.5 remedial 
National Mathematics Advisory Panel FINAL REPORT 5

courses5 in mathematics (Business Higher Education Forum, 2005). This need 

for remediation reveals weakness in the preparation of students for college and 

may limit a student’s ability to advance toward a degree in a timely manner. 

Moreover, there are large, persistent disparities in mathematics achievement 

related to race and income—disparities that are not only devastating for 

individuals and families but also project poorly for the nation’s future, given the 

youthfulness and high growth rates of the largest minority populations. 

Attending college is a social escalator. It levels opportunities for success 

across all socioeconomic groups (Pascarella & Terenzini, 1991). Among students 

from the principal ethnic and racial groups in the U.S. who have completed 

mathematics courses at least through Algebra II, the differences in college 

graduation rates versus the student population in general are half as large as the 

differences for students who do not complete Algebra II (Achieve, Inc., 2006). 

According to research, “The achievement gap between students of differing ethnic 

and socioeconomic groups can be significantly reduced or even eliminated if lowincome and minority students increase their success in high school mathematics and 

science courses” (Evan et al., 2006, p. 11). 

Once out of college, an individual’s past participation in mathematics 

courses and higher education continues to be correlated with benefits. Individuals 

who receive college degrees earn more and have better career mobility 

(McGregor, 1994). The majority of workers who earn more than $40,000 annually 

have two or more high school credits at the Algebra II level or higher (Achieve, 

Inc., 2006). A national poll found that more than two-thirds of students who took 

Algebra II in high school reported that they were well-prepared for demands of 

the workplace (Carnevale & Desrochers, 2003). 

No longer can we accept that a rigorous mathematics education is reserved 

for the few who will go on to be engineers or scientists. Mathematics may indeed 

be “the new literacy” (Schoenfeld, 1995); at the least, it is essential for any citizen 

who is to be prepared for the future. 

 

5

 Remedial courses cover precollegiate mathematics and normally do not bear credit that can be 

counted toward graduation from college. Some institutions do not offer remedial course work. 
National Mathematics Advisory Panel FINAL REPORT 6
National Mathematics Advisory Panel FINAL REPORT 7

Chapter 2: 

The National Mathematics Advisory Panel 

Because mathematics education bears on the policy concerns delineated in 

the preceding section, the President created the National Mathematics Advisory 

Panel in April 2006 via Executive Order 13398 (Appendix A). He assigned 

responsibility to the U.S. Secretary of Education for appointment of members and 

for oversight. 

The Panel’s precise charge, given in the Executive Order, is to advise the 

President and the Secretary on ways “...to foster greater knowledge of and 

improved performance in mathematics among American students...with respect 

to the conduct, evaluation, and effective use of the results of research relating to 

proven-effective and evidence-based mathematics instruction.” The Executive 

Order further calls for recommendations “...based on the best available scientific 

evidence.” Moreover, the Executive Order also defines a particular set of topics 

for the Panel to examine: 

a) the critical skills and skill progressions for students to acquire 

competence in algebra and readiness for higher levels of 

mathematics; b) the role and appropriate design of standards and 

assessment in promoting mathematical competence; c) the 

processes by which students of various abilities and backgrounds 

learn mathematics; d) instructional practices, programs, and 

materials that are effective for improving mathematics learning; 

e) the training, selection, placement, and professional development 

of teachers of mathematics in order to enhance students’ learning 

of mathematics; f) the role and appropriate design of systems for 

delivering instruction in mathematics that combine the different 

elements of learning processes, curricula, instruction, teacher 

training and support, and standards, assessments, and accountability; 

g) needs for research in support of mathematics education; h) ideas 

for strengthening capabilities to teach children and youth basic 

mathematics, geometry, algebra, and calculus and other 

mathematical disciplines; i) such other matters relating to 

mathematics education as the Panel deems appropriate; and j) such 

other matters relating to mathematics education as the Secretary 

may require. 
National Mathematics Advisory Panel FINAL REPORT 8

The first item in the President’s list indicates that the Panel’s focus should 

be on the preparation of students for entry into and success in algebra, which itself 

is a foundation for higher mathematics. Thus, the Panel has seen its role 

as addressing the teaching and learning of mathematics from preschool 

(PreK) through Grade 8 or so, with a particular emphasis on the 

concepts and skills most relevant to the learning of algebra. 

Over a period of 20 months, the Panel received public testimony 

as a committee of the whole but worked largely in task groups and 

subcommittees. Each of five task groups carried out a detailed analysis 

of the available evidence in a major area of the Panel’s responsibility: 

Conceptual Knowledge and Skills, Learning Processes, Instructional 

Practices, Teachers and Teacher Education, and Assessment. Each of 

three subcommittees was charged with completion of a particular 

advisory function for the Panel: Standards of Evidence, Instructional 

Materials, and the Panel-commissioned National Survey of Algebra Teachers 

(National Mathematics Advisory Panel, 2008; see sidebar, page 9). 

The task groups and subcommittees produced reports supporting this 

document. Those reports cover work carried out as part of the Panel’s overall 

mission, but they are presented by only those members who participated in 

creating them. This Final Report represents findings and recommendations 

formally adopted by the Panel as a whole. All eight reports6

 are separately 

available in printed form and via the Web site that houses the Panel’s work.7

 

Details of the Panel’s work can be found in Appendixes B–E, which cover 

the membership and processes of the Panel. 

 

6

 National Mathematics Advisory Panel (2008). 

7

 http://www.ed.gov/mathpanel. 

The first item in the

President’s list

indicates that the 

Panel’s focus should be 

on the preparation of 

students for entry into 

and success in algebra, 

which itself is a 

foundation for higher 

mathematics. 
National Mathematics Advisory Panel FINAL REPORT 9

Voices from the Field: What Algebra Teachers Say 

To understand the experiences of Algebra teachers in the classroom, the National Math

Panel commissioned a national survey of randomly chosen Algebra I teachers designed to elicit their

views on student preparation, work-related attitudes and challenges, and use of instructional

materials. The National Opinion Research Center at the University of Chicago conducted the survey

in the spring and summer of 2007. Of the 310 public schools identified, 258 agreed to participate, and

743 teachers—a 72% response rate—completed the questionnaire. 

The survey revealed that teachers rate their students’ background preparation for Algebra I as

weak. The three areas in which teachers report their students to have the poorest preparation are

rational numbers, word problems, and study habits. When asked to provide a brief description of any

changes they would like to see in the curriculum leading up to Algebra I, teachers most often cited the

need for a greater focus at the elementary school level on proficiency with basic mathematical

concepts and skills. 

Sample responses representing this predominant view include: 

• “Students need to be better prepared in basic math skills and not be quite so 

calculator dependent. Also, more training in thinking skills.” 

• “Make sure the 1st–8th grade teachers teach the foundations of math and that the 

students know their basic skills.” 

• “More focus on basics—students should already know order of operations, positive 

vs. negative numbers, fractions and decimals.” 

With regard to instructional materials, teachers, for the most part, do not regularly use

technological tools. On average, teachers said they use these tools less than once a week. Low

levels of computer use do not appear to be a reflection of insufficient access. About one-third of

teachers never use the graphing calculator, and manipulative materials are used only occasionally. 

In response to 10 options describing the challenges they face, a majority of the teachers

(62%) rated “working with unmotivated students” as the “single most challenging aspect of teaching

Algebra I successfully.” Their second highest-rated challenge—11%—was making mathematics

accessible and comprehensible. However, the written-in responses most frequently mentioned

handling different skill levels in a single classroom. A substantial number of teachers consider mixedability groupings to be a “moderate” (30%) or “serious” (23%) problem, an item with a combined rating

of 53% for “moderate” and “serious,” second only to the combined rating of 64% for “too little

parent/family support.” 

The survey results reinforce the research findings presented in this report, particularly the

need to strengthen students’ proficiency with rational numbers. Further, the Panel suggests that

greater attention be focused on ways in which negative attitudes toward mathematics develop and

how to overcome students’ lack of motivation. 

A full report on the survey is available (National Mathematics Advisory Panel, 2008). 

The Panel wishes to express its appreciation to the teachers who participated in this survey.

Their voices and experience proved valuable to the Panel’s work. 

The Panel took consistent note of the President’s emphasis on “the best 

available scientific evidence” and set a high bar for admitting research results into 

consideration. In essence, the Panel required the work to have been carried out in 

a way that manifested rigor and could support generalization at the level of 

significance to policy. One of the subcommittee reports covers global 

considerations relating to standards of evidence, while individual task group 

reports amplify the standards in the particular context of each task group’s work. 
National Mathematics Advisory Panel FINAL REPORT 10

In all, the Panel reviewed more than 16,000 research publications and policy 

reports and received public testimony from 110 individuals, of whom 69 appeared 

before the Panel on their own and 41 others were invited on the basis of expertise 

to cover particular topics. Those who testified included parents, teachers, school 

administrators, members of boards of education, educational researchers, textbook 

publishers, and other individuals interested in improving mathematics education. 

In addition, the Panel reviewed written commentary from 160 organizations and 

individuals, and analyzed survey results from 743 active teachers of algebra. 

In late 2007, the Panel synthesized this Final Report by drawing together 

the most important findings and recommendations. They are hereby issued with 

the Panel’s full voice. This report connects in many places to the eight reports of 

the task groups and subcommittees, which carry detailed analyses of research 

literature and other relevant materials. The sections below are not extensively 

referenced, because the goal of this report is to communicate the Panel’s main 

conclusions without distractions from detail. Readers interested in a particular 

topic should examine the relevant task group and subcommittee reports. 
National Mathematics Advisory Panel FINAL REPORT 11

Chapter 3: 

Principal Messages 

This Panel, diverse in experience, expertise, and philosophy, agrees 

broadly that the delivery system in mathematics education—the system that 

translates mathematical knowledge into value and ability for the next 

generation—is broken and must be fixed. This is not a conclusion about teachers 

or school administrators, or textbooks or universities or any other single element 

of the system. It is about how the many parts do not now work together to achieve 

a result worthy of this country’s values and ambitions. 

On the basis of its deliberation and research, the Panel can report that 

America has genuine opportunities for improvement in mathematics education. 

This report lays them out for action. 

First Things First 

The essence of the Panel’s message is to put first things first. There are six 

elements, expressed compactly here, but in greater detail later. 

• The mathematics curriculum in Grades PreK–8 should be streamlined 

and should emphasize a well-defined set of the most critical topics in the 

early grades. 

• Use should be made of what is clearly known from rigorous research about 

how children learn, especially by recognizing a) the advantages for children 

in having a strong start; b) the mutually reinforcing benefits of conceptual 

understanding, procedural fluency, and automatic (i.e., quick and effortless) 

recall of facts; and c) that effort, not just inherent talent, counts in 

mathematical achievement. 

• Our citizens and their educational leadership should recognize 

mathematically knowledgeable classroom teachers as having a central role 

in mathematics education and should encourage rigorously evaluated 

initiatives for attracting and appropriately preparing prospective teachers, 

and for evaluating and retaining effective teachers. 

• Instructional practice should be informed by high-quality research, when 

available, and by the best professional judgment and experience of 

accomplished classroom teachers. High-quality research does not support the 

contention that instruction should be either entirely “student-centered” or 

“teacher-directed.” Research indicates that some forms of particular 

instructional practices can have a positive impact under specified conditions. 
National Mathematics Advisory Panel FINAL REPORT 12

• NAEP and state assessments should be improved in quality and should 

carry increased emphasis on the most critical knowledge and skills leading 

to Algebra. 

• The nation must continue to build capacity for more rigorous research in 

education so that it can inform policy and practice more effectively. 

Positive results can be achieved in a reasonable time at accessible cost, but 

a consistent, wise, community-wide effort will be required. Education in the 

United States has many participants in many locales—teachers, students, and 

parents; state school officers, school board members, superintendents, and 

principals; curriculum developers, textbook writers, and textbook editors; those 

who develop assessment tools; those who prepare teachers and help them to 

continue their development; those who carry out relevant research; association 

leaders and government officials at the federal, state, and local levels. All carry 

responsibilities. All are important to success. 

The network of these participants is linked through interacting national 

associations. A coordinated national approach toward improved mathematics 

education will require an annual forum of their leaders for at least a decade. The 

Panel recommends that the U.S. Secretary of Education take the lead in convening 

the forum initially, charge it to organize in a way that will sustain an effective effort, 

and request a brief annual report on the mutual agenda adopted for the year ahead. 

Learning as We Go Along 

The President asked the Panel to use the best available scientific research to 

advise on improvements in the mathematics education of the nation’s children, and 

we have delivered here on his request. Our consistent respect for sound research has 

been the main factor enabling the Panel’s joint conclusions on so many matters, 

despite differences of perspective and philosophy. 

However, we also found no research or insufficient research relating to a 

great many matters of concern in education policy and practice related to 

mathematics. In those areas, the Panel has been very limited in what it can report 

to the President, to the Secretary, and to the public. 

A small number of questions have been deemed to have such currency as 

to require comment from the Panel, even if the scientific evidence was not 

sufficient to justify research-based findings. In those instances, the Panel has 

spoken on the basis of collective professional judgment, but it has also 

endeavored to minimize both the number and the scope of such comments. 

The United States has been in a similar situation with respect to education 

concerns at least once before. When the country was jarred by the challenge of 

Sputnik, its people responded, in essence, “We see clearly what is broken (mathscience education and research), and we are going to fix it by taking the best first 
National Mathematics Advisory Panel FINAL REPORT 13

steps we can, and then by learning as we go along.” And America did. The nation 

moved rapidly from the doubt of October 1957 into an extended era of 

achievement and leadership in science and engineering. 

The Panel lays out many concrete steps that can be taken now toward 

significantly improved mathematics education, but it also views them only as a 

best start in a long process. This journey, like that of the post-Sputnik era, will 

require a commitment to “learning as we go along.” The nation should recognize 

that there is much more to discover about how to achieve better results. Models of 

continuous improvement have proven themselves in many other areas, and they 

can work again for America in mathematics education. 
National Mathematics Advisory Panel FINAL REPORT 14
National Mathematics Advisory Panel FINAL REPORT 15

Chapter 4: 

Curricular Content 

The Nature of School Algebra 

To clarify instructional needs in Grades K–8 and to sharpen future 

discussion about the role of school algebra in the overall mathematics 

curriculum,8 the Panel made a specific effort to delineate the content and demands 

of school algebra, which is the term used here to encompass the full body of 

algebraic material that the Panel expects to be covered through high school, 

regardless of its organization into courses and levels. Most commonly, school 

algebra is organized into two courses, Algebra I and II. Less commonly, the 

content of school algebra is interwoven with that of geometry, trigonometry, 

statistics, and other mathematical subjects in an integrated curriculum covering 

several courses in high school. Even when the traditional pattern of Algebra I and 

II is used, the course called Algebra II may include elements of statistics or 

trigonometry in place of some of the more advanced elements of school algebra, 

which may be offered in a subsequent precalculus course. When the Panel 

addressed the effective preparation of students for the study of Algebra, its 

expectation was that students should be able to proceed successfully at least 

through the content of Algebra II, wherever the elements might be taught in the 

high school curriculum. 

Consequently, the Panel reviewed the algebra topics addressed 1) in 

current state standards for Algebra I and Algebra II courses and for integrated 

curricula, 2) in current textbooks for school algebra and integrated mathematics, 

3) in the algebra objectives in NAEP’s 2005 Grade 12 mathematics assessment, 4) 

in the American Diploma Project’s benchmarks for a high school exit test and its 

forthcoming Algebra II end-of-course test, and 5) in the algebra standards in 

Singapore’s mathematics curriculum for Grades 7–10. With professional 

judgment advised by these comparisons, the Panel sets out the Major Topics of 

School Algebra, shown in Table 1, as central for the teaching of algebra.9

Recommendation: The Panel recommends that school algebra be 

consistently understood in terms of the Major Topics of 

School Algebra given in Table 1. 

Recommendation: The Major Topics of School Algebra, accompanied by a 

thorough elucidation of the mathematical connections 

among these topics,10 should be the focus of Algebra I and 

 

8

 The detailed work underlying this section was carried out by the Task Group on Conceptual 

Knowledge and Skills, whose report carries relevant references and more elaborate discussion 

(National Mathematics Advisory Panel, 2008). 

9

 The list of Major Topics of School Algebra is meant as a catalog for coverage, not as a template 

for how courses should be sequenced or texts should be written. 

10 As presented, for example, in National Mathematics Advisory Panel (2008). 
National Mathematics Advisory Panel FINAL REPORT 16

Algebra II standards in state curriculum frameworks, in 

Algebra I and Algebra II courses, in textbooks for these 

two levels of Algebra whether for integrated curricula or 

otherwise, and in end-of-course assessments of these two 

levels of Algebra. The Panel also recommends use of the 

Major Topics of School Algebra in revisions of 

mathematics standards at the high school level in state 

curriculum frameworks, in high school textbooks 

organized by an integrated approach, and in grade-level 

state assessments using an integrated approach at the 

high school, by Grade 11 at the latest. 

Table 1: The Major Topics of School Algebra 

Symbols and Expressions 

• Polynomial expressions 

• Rational expressions 

• Arithmetic and finite geometric series 

Linear Equations 

• Real numbers as points on the number line 

• Linear equations and their graphs 

• Solving problems with linear equations 

• Linear inequalities and their graphs 

• Graphing and solving systems of simultaneous linear equations 

Quadratic Equations 

• Factors and factoring of quadratic polynomials with integer coefficients 

• Completing the square in quadratic expressions 

• Quadratic formula and factoring of general quadratic polynomials 

• Using the quadratic formula to solve equations 

Functions 

• Linear functions 

• Quadratic functions—word problems involving quadratic functions 

• Graphs of quadratic functions and completing the square 

• Polynomial functions (including graphs of basic functions) 

• Simple nonlinear functions (e.g., square and cube root functions; absolute value; 

rational functions; step functions) 

• Rational exponents, radical expressions, and exponential functions 

• Logarithmic functions 

• Trigonometric functions 

• Fitting simple mathematical models to data 

Algebra of Polynomials 

• Roots and factorization of polynomials 

• Complex numbers and operations 

• Fundamental theorem of algebra 

• Binomial coefficients (and Pascal’s Triangle) 

• Mathematical induction and the binomial theorem 

Combinatorics and Finite Probability 

• Combinations and permutations, as applications of the binomial theorem and 

Pascal’s Triangle

Source: National Mathematics Advisory Panel, 2008. 
National Mathematics Advisory Panel FINAL REPORT 17

Critical Foundations of Algebra 

The mathematics that children learn from preschool through the middle 

grades provides the basic foundation for Algebra. What is taught at particular 

grades is determined at local and state levels, and reflects the interests of a variety 

of national, state, and local agencies and organizations, as well as parents and the 

general public. In the past, there has been no research base to guide them. 

However, the results of TIMSS and other international tests showing student 

achievement across the participating countries have led to international 

comparisons of curricula and provided much information on what high-achieving 

countries teach their students in elementary and middle school. 

To suggest what essential concepts and skills should be learned as 

preparation for algebra course work, the Panel reviewed the skills and concepts 

listed in 1) the Grades 1–8 curricula of the highest-performing countries on TIMSS 

(Singapore, Japan, Korea, Hong Kong, Flemish Belgium, and the Czech 

Republic), sometimes called the “A+ countries,” 2) National Council of Teachers 

of Mathematics Curriculum Focal Points for Prekindergarten through Grade 8 

Mathematics: A Quest for Coherence (hereinafter Focal Points), 3) Grades K–8 in 

the six highest-rated state curriculum frameworks in mathematics, 4) a 2007 

American College Testing (ACT) survey, and 5) a Panel-sponsored survey of 743 

teachers of introductory Algebra across the country who were asked what students 

need to learn to be prepared for success in Algebra. 

The Panel also took into consideration the structure of mathematics itself, 

which requires teaching a sequence of major topics (from whole numbers to 

fractions, from positive numbers to negative numbers, and from the arithmetic of 

rational numbers to algebra) and an increasingly complex progression from 

specific number computations to symbolic computations. The structural reasons 

for this sequence and its increasing complexity dictate what must be taught and 

learned before students take course work in Algebra. 

Based on all these considerations, the Panel proposes three clusters of 

concepts and skills—called the Critical Foundation of Algebra—reflecting their 

judgment about the most essential mathematics for students to learn thoroughly 

prior to algebra course work. 

1. Fluency with Whole Numbers. By the end of Grade 5 or 6, children 

should have a robust sense of number. This sense of number must include an 

understanding of place value and the ability to compose and decompose whole 

numbers. It must clearly include a grasp of the meaning of the basic operations of 

addition, subtraction, multiplication, and division. It must also include use of the 

commutative, associative, and distributive properties; computational facility; and 

the knowledge of how to apply the operations to problem solving. Computational 

facility requires the automatic recall of addition and related subtraction facts, and 

of multiplication and related division facts. It also requires fluency with the 

standard algorithms for addition, subtraction, multiplication, and division. Fluent 
National Mathematics Advisory Panel FINAL REPORT 18

use of the algorithms not only depends on the automatic recall of number facts 

but also reinforces it. A strong sense of number also includes the ability to 

estimate the results of computations and thereby to estimate orders of magnitude, 

e.g., how many people fit into a stadium or how many gallons of water are 

needed to fill a pool. 

2. Fluency with Fractions. Before they begin algebra course work, middle 

school students should have a thorough understanding of positive as well as 

negative fractions. They should be able to locate positive and negative fractions 

on a number line; represent and compare fractions, decimals, and related percents; 

and estimate their size. They need to know that sums, differences, products, and 

quotients (with nonzero denominators) of fractions are fractions, and they need to 

be able to carry out these operations confidently and efficiently. They should 

understand why and how (finite) decimal numbers are fractions and know the 

meaning of percentages. They should encounter fractions in problems in the many 

contexts in which they arise naturally, for example, to describe rates, 

proportionality, and probability. Beyond computational facility with specific 

numbers, the subject of fractions, when properly taught, introduces students to the 

use of symbolic notation and the concept of generality, both being integral parts 

of algebra.

3. Particular Aspects of Geometry and Measurement. Middle grade 

experience with similar triangles is most directly relevant for the study of Algebra: 

Sound treatments of the slope of a straight line and of linear functions depend 

logically on the properties of similar triangles. Furthermore, students should be able 

to analyze the properties of two- and three-dimensional shapes using formulas to 

determine perimeter, area, volume, and surface area. They should also be able to 

find unknown lengths, angles, and areas.

Recommendation: Proficiency with whole numbers, fractions, and 

particular aspects of geometry and measurement should 

be understood as the Critical Foundations of Algebra. 

Emphasis on these essential concepts and skills must be 

provided at the elementary and middle grade levels. 

Recommendation: The coherence and sequential nature of mathematics 

dictate the foundational skills that are necessary for the 

learning of algebra. The most important foundational 

skill not presently developed appears to be proficiency 

with fractions (including decimals, percents, and 

negative fractions). The teaching of fractions must be 

acknowledged as critically important and improved 

before an increase in student achievement in algebra 

can be expected. 
National Mathematics Advisory Panel FINAL REPORT 19

To prepare students for Algebra, the curriculum must simultaneously 

develop conceptual understanding, computational fluency, and problem-solving 

skills. These three aspects of learning are mutually reinforcing and should not be 

seen as competing for class time. 

The Critical Foundations of Algebra identified and discussed here are not 

meant to comprise a complete preschool-to-algebra curriculum. However, the 

Panel aims to recognize the Critical Foundations of Algebra, whether as part of a 

dedicated algebra course in Grade 7, 8, or 9, or within an integrated mathematics 

sequence in the middle and high school grades. These Critical Foundations of 

Algebra deserve ample time in any mathematics curriculum.

Recommendation: Teacher education programs and licensure tests for 

early childhood teachers, including all special education 

teachers at this level, should fully address the topics on 

whole numbers, fractions, and the appropriate 

geometry and measurement topics in the Critical 

Foundations of Algebra, as well as the concepts and 

skills leading to them; for elementary teachers, 

including elementary level special education teachers, 

all topics in the Critical Foundations of Algebra and 

those topics typically covered in an introductory 

Algebra course; and for middle school teachers, 

including middle school special education teachers, the 

Critical Foundations of Algebra and all of the Major 

Topics of School Algebra. 

Benchmarks for the Critical Foundations 

In view of the sequential nature of mathematics, the Critical Foundations 

of Algebra described in the preceding section require judicious placement in the 

grades leading up to Algebra. To encourage the development of students in 

Grades PreK–8 at an effective pace, the Panel suggests the Benchmarks for the 

Critical Foundations in Table 2 as guideposts for state frameworks and school 

districts. There is no empirical research on the placement of these benchmarks, 

but they find justification in a comparison of national and international curricula. 
National Mathematics Advisory Panel FINAL REPORT 20

Table 2: Benchmarks for the Critical Foundations 

Fluency With Whole Numbers 

1) By the end of Grade 3, students should be proficient with the addition and subtraction of 

whole numbers. 

2) By the end of Grade 5, students should be proficient with multiplication and division of 

whole numbers. 

Fluency With Fractions 

1) By the end of Grade 4, students should be able to identify and represent fractions and 

decimals, and compare them on a number line or with other common representations of 

fractions and decimals. 

2) By the end of Grade 5, students should be proficient with comparing fractions and decimals 

and common percents, and with the addition and subtraction of fractions and decimals. 

3) By the end of Grade 6, students should be proficient with multiplication and division of 

fractions and decimals. 

4) By the end of Grade 6, students should be proficient with all operations involving positive 

and negative integers. 

5) By the end of Grade 7, students should be proficient with all operations involving positive 

and negative fractions. 

6) By the end of Grade 7, students should be able to solve problems involving percent, ratio, 

and rate and extend this work to proportionality. 

Geometry and Measurement 

1) By the end of Grade 5, students should be able to solve problems involving perimeter and 

area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., 

trapezoids). 

2) By the end of Grade 6, students should be able to analyze the properties of two-dimensional 

shapes and solve problems involving perimeter and area, and analyze the properties of threedimensional shapes and solve problems involving surface area and volume. 

3) By the end of Grade 7, students should be familiar with the relationship between similar 

triangles and the concept of the slope of a line. 

Source: National Mathematics Advisory Panel, 2008. 

Recommendation: The Benchmarks for the Critical Foundations in Table 

2 should be used to guide classroom curricula, 

mathematics instruction, and state assessments. They 

should be interpreted flexibly, to allow for the needs of 

students and teachers. 

A Need for Coherence 

There seem to be two major differences between the curricula in topperforming countries and those in the U.S.—in the number of mathematical 

concepts or topics presented at each grade level and in the expectations for learning. 

U.S. curricula typically include many topics at each grade level, with each receiving 

relatively limited development, while top-performing countries present fewer topics 

at each grade level but in greater depth. In addition, U.S. curricula generally review 

and extend at successive grade levels many (if not most) topics already presented at 

earlier grade levels, while the top-performing countries are more likely to expect 

closure after exposure, development, and refinement of a particular topic. These 

critical differences distinguish a spiral curriculum (common in many subjects in 
National Mathematics Advisory Panel FINAL REPORT 21

U.S. curricula) from one built on developing proficiency—a curriculum that 

expects proficiency in the topics that are presented before more complex or difficult 

topics are introduced. 

The Singapore standards (Singapore Ministry of Education, 2006) provide 

an established example of curriculum standards designed to develop proficiency 

in a relatively small number of important mathematics topics, as validated by a 

recent analysis (Ginsburg et al., 2005). The desirability of emphasizing fewer 

important mathematics topics in greater depth has also been recognized by some 

U.S. educators. 

In 2005, the Fordham Foundation report on state mathematics standards 

(Klein et al., 2005) ranked state mathematics curriculum standards based on 

mathematics content, clarity, and reasoning, as well as negative qualities, 

assigning different weights to each criterion for the overall assessment. The 

standards of California, Indiana, Massachusetts, Alabama, New Mexico, and 

Georgia achieved the highest ranking. The curricular profiles of the standards of 

these six states do, on the whole, provide an emphasis on fewer important topics 

per year than most states; but compared with the “A+ countries” (Singapore, 

Japan, Korea, Hong Kong, Flemish Belgium, and the Czech Republic), they all 

spend a great deal of time in the primary grades on topics other than arithmetic. 

A more recent development in the national discussion is the publication of 

Focal Points (National Council of Teachers of Mathematics, 2006), which offers 

curricular direction to teachers and administrators by suggesting areas of 

emphasis for the concepts, skills, and procedures that connect important 

mathematics topics from grade to grade, and form the foundation for more 

advanced mathematics, beginning with Algebra. The message of Focal Points is 

also one of curriculum coherence with an emphasis on fewer important topics per 

year. Focal Points does not represent a set of standards but calls for a curriculum 

which reduces the number of important topics per year. In effect, Focal Points

asks for greater emphasis on key topics, particularly with whole numbers and 

fractions and particular aspects of geometry and measurement. Yet Focal Points

still implies more time on non-number topics, especially in the primary grades, 

than is the case in the A+ countries but less than the intended mathematics 

curriculum as represented in the frameworks of the six states. 

The Panel also notes that a state’s (or a country’s) mathematics standards, 

however highly their quality may be judged, cannot ensure high student 

achievement. For example, the six leading states in the Fordham study exhibit a 

wide range of student achievement on the 2007 NAEP mathematics tests for Grades 

4 and 8. The quality of a state’s assessments and the extent to which its standards 

drive sound school curricula, as well as appropriate programs for teacher 

preparation and professional development, are intervening variables that strongly 

influence achievement. They may well override the quality of the standards. 
National Mathematics Advisory Panel FINAL REPORT 22

Recommendation: A focused, coherent progression of mathematics 

learning, with an emphasis on proficiency with key 

topics, should become the norm in elementary and 

middle school mathematics curricula. Any approach 

that continually revisits topics year after year without 

closure is to be avoided. 

By the term focused, the Panel means that the curriculum must include 

(and engage with adequate depth in) the most important topics underlying 

success in school algebra, particularly the Critical Foundations of Algebra. By 

the term coherent, the Panel means that the curriculum is marked by effective, 

logical progressions from earlier, less sophisticated topics into later, more 

sophisticated ones. 

By the term proficiency, the Panel means that students should understand 

key concepts, achieve automaticity as appropriate (e.g., with addition and related 

subtraction facts), develop flexible, accurate, and automatic execution of the 

standard algorithms, and use these competencies to solve problems.11

Integrated versus Single-Subject Approach 

An integrated approach is defined as one in which the topics of high 

school mathematics are presented in some order other than the customary 

sequence in the United States of yearlong courses in Algebra I, Geometry, 

Algebra II, and Precalculus. 

The curricula of most high-achieving nations in the TIMSS study do not 

follow the single-subject sequence of Algebra I, Geometry, and Algebra II, but they 

also differ from the approach used in most U.S. integrated curricula. Instead, 

Algebra, Geometry, and Trigonometry are divided into blocks. The teaching of 

each block typically extends over several months and aims for mathematical 

closure. As a result, these curricula avoid the need to revisit essentially the same 

material over several years, often referred to as “spiraling.” 

A search of the literature did not produce studies that clearly examined 

whether an integrated approach or a single-subject sequence is more effective for 

algebra and more advanced mathematics course work. The Panel finds no basis in 

research for preferring one or the other. 

 

11 This meaning is in keeping with Adding It Up (National Research Council, p. 116), in which 

five attributes were associated with the concept of proficiency: 1) conceptual understanding 

(comprehension of mathematical concepts, operations, and relations), 2) procedural fluency (skills 

in carrying out procedures flexibly, fluently, and appropriately), 3) strategic competence (ability to 

formulate, represent, and solve mathematical problems), 4) adaptive reasoning (capacity for 

logical thought, reflection, explanation, and justification), and 5) productive disposition (habitual 

inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in 

diligence and one's own efficacy). 
National Mathematics Advisory Panel FINAL REPORT 23

An analysis of high school mathematics standards, and one state’s 

standards in particular, suggests that high school students enrolled in mathematics 

courses using an integrated approach to mathematics may find it more difficult to 

take advanced mathematics course work (e.g., calculus or precalculus) in their 

senior year than high school students who are able to enroll in an Algebra II 

course in their sophomore or junior year. 

Universal Availability of Authentic Education in Algebra 

Recommendation: All school districts should ensure that all prepared 

students have access to an authentic algebra course—

and should prepare more students than at present to 

enroll in such a course by Grade 8. The word 

“authentic” is used here as a descriptor of a course that 

addresses algebra consistently with the Major Topics of 

School Algebra (Table 1, page 16). Students must be 

prepared with the mathematical prerequisites for this 

course according to the Critical Foundations of Algebra 

(page 17) and the Benchmarks for the Critical 

Foundations (Table 2, page 20). 
National Mathematics Advisory Panel FINAL REPORT 24
National Mathematics Advisory Panel FINAL REPORT 25

Chapter 5: 

Learning Processes 

Readiness for Learning 

The mathematics that children learn from preschool through the middle 

grades provides the basic foundation for Algebra and more advanced mathematics 

course work.12 Even before they enter kindergarten, most children develop 

considerable knowledge of numbers and other aspects of mathematics. The 

mathematical knowledge that children bring to school influences their math 

learning for many years thereafter, and probably throughout their education. 

When they enter kindergarten, most children from families with the 

combination of low-parental education levels, low incomes, and single parents 

bring less foundational knowledge for learning school mathematics than does the 

average child from more advantaged backgrounds. Fortunately, a variety of 

promising instructional programs have been developed to improve the 

mathematical knowledge of preschoolers and kindergartners, especially those 

from at-risk backgrounds, and have yielded encouraging results. 

Recommendation: Research that scales up early interventions capable of 

strengthening mathematical knowledge, evaluates their 

utility in prekindergarten and kindergarten settings, 

and examines long-term effects are urgently needed, 

with a particular focus on at-risk learners. 

Recommendation: Teachers in Head Start and other programs serving 

preschoolers from low-income backgrounds should be 

made aware of the importance of early mathematical 

knowledge for long-term educational success and of 

the availability of effective techniques for improving 

that knowledge. Training in how to implement these 

teaching techniques must be included in the 

intervention studies carried out pursuant to the above 

recommendation and should be made available to 

interested teachers and preschools. 

 

12 The detailed work underlying this section was carried out by the Task Group on Learning 

Processes, whose report carries relevant references and more elaborate discussion (National 

Mathematics Advisory Panel, 2008). 
National Mathematics Advisory Panel FINAL REPORT 26

Whole Number Arithmetic: Computational Proficiency 

Plus Conceptual Understanding 

Debates regarding the relative importance of conceptual knowledge, 

procedural skills (e.g., the standard algorithms), and the commitment of addition, 

subtraction, multiplication, and division facts to long-term memory are 

misguided. These capabilities are mutually supportive, each facilitating learning 

of the others. Conceptual understanding of mathematical operations, fluent 

execution of procedures, and fast access to number combinations together support 

effective and efficient problem solving. 

Computational facility with whole number operations rests on the 

automatic recall of addition and related subtraction facts, and of multiplication 

and related division facts. It requires fluency with the standard algorithms for 

addition, subtraction, multiplication, and division. Fluent use of the algorithms not 

only depends on the automatic recall of number facts but also reinforces it. 

Studies of children in the United States, comparisons of these children 

with children from other nations with higher mathematics achievement, and even 

cross-generational changes within the United States indicate that many 

contemporary U.S. children do not reach the point of fast and efficient solving of 

single-digit addition, subtraction, multiplication, and division with whole 

numbers, much less fluent execution of more complex algorithms as early as 

children in many other countries. Surprisingly, many never gain such proficiency. 

The reasons for differences in the computational fluency of children in the 

United States and peers in countries with higher mathematics achievement are 

multifaceted. They include quantity and quality of practice, emphases within 

curricula, and parental involvement in mathematics learning. As an example, in 

elementary school textbooks in the United States, easier arithmetic problems are 

presented far more frequently than harder problems. The opposite is the case in 

countries with higher mathematics achievement, such as Singapore. 

Few curricula in the United States provide sufficient practice to ensure 

fast and efficient solving of basic fact combinations and execution of the 

standard algorithms. 

Too many American students also have a poor grasp of many core 

arithmetical concepts. For example, many U.S. middle school students do not 

understand the concept of mathematical equality. Understanding core concepts is 

a necessary component of proficiency with arithmetic and is needed to transfer 

previously learned procedures to solve novel problems. U.S. students’ poor 

knowledge of the core arithmetical concepts impedes their learning of algebra and 

is an unacceptable indication of a substantive gap in the mathematics curricula 

that must be addressed. 
National Mathematics Advisory Panel FINAL REPORT 27

Number Sense 

In its most fundamental form, number sense entails an ability to 

immediately identify the numerical value associated with small quantities (e.g., 

3 pennies), a facility with basic counting skills, and a proficiency in approximating 

the magnitudes of small numbers of objects and simple numerical operations. An 

intuitive sense of the magnitudes of small whole numbers is evident even among 

most 5-year-olds who can, for example, accurately judge which of two single digits 

is larger, estimate the number of dots on a page, and determine the approximate 

location of single digit numerals on a number line that provides only the numerical 

endpoints. These competencies comprise the core number sense that children often 

acquire informally prior to starting school. 

A more advanced type of number sense that children must acquire through 

formal instruction requires a principled understanding of place value, of how whole 

numbers can be composed and decomposed, and of the meaning of the basic 

arithmetic operations of addition, subtraction, multiplication, and division. It also 

requires understanding the commutative, associative, and distributive properties and 

knowing how to apply these principles to solve problems. This more highly 

developed form of number sense should extend to numbers written in fraction, 

decimal, percentage, and exponential forms. Far too many middle and high school 

students lack the ability to accurately compare the magnitudes of such numbers. 

This is a serious problem, because poor number sense interferes with learning 

algorithms and number facts and prevents use of strategies to verify if solutions to 

problems are reasonable. Analysis of the literature on number sense suggests two 

specific recommendations: 

Recommendation: Teachers should broaden instruction in computational 

estimation beyond rounding. They should ensure that 

students understand that the purpose of estimation is to 

approximate the exact value and that rounding is only 

one estimation strategy. 

Recommendation: Textbooks need to explicitly explain that the purpose of 

estimation is to produce an appropriate approximation. 

Illustrating multiple useful estimation strategies for a 

single problem, and explaining how each procedure 

achieves the goal of an appropriate estimate, is a useful 

means for achieving this goal. Contrasting these 

procedures with others that produce less appropriate 

estimates is also likely to be helpful. 
National Mathematics Advisory Panel FINAL REPORT 28

Fractions 

Difficulty with the learning of fractions is pervasive and is an obstacle to 

further progress in mathematics and other domains dependent on mathematics, 

including algebra. It also has been linked to difficulties in adulthood, such as 

failure to understand medication regimens. Algebra I teachers who were surveyed 

for the Panel as part of a large, nationally representative sample rated students as 

having very poor preparation in “rational numbers and operations involving 

fractions and decimals” (see Panel-commissioned National Survey of Algebra 

Teachers, National Mathematics Advisory Panel, 2008). 

Preschool and early elementary school children have a rudimentary 

understanding of simple fractional relations. The relation between this informal 

knowledge and the learning of formal mathematical fractional concepts and 

procedures is not well understood, and is an area in critical need of further study. 

Elementary and middle school children should begin fraction instruction 

with the prerequisite ability to quickly and easily retrieve basic arithmetic facts, 

execute arithmetic procedures involving whole numbers, and deeply understand 

core concepts involving whole numbers. Teachers should not assume that children 

understand the magnitudes represented by fractions, even if the children can 

perform arithmetic operations with them, or that children understand what the 

operations mean (e.g., understand what it means to multiply or divide one fraction 

by another). Instruction focusing on conceptual knowledge of fractions is likely to 

have the broadest and largest impact on problem-solving performance, provided it 

is aimed at accurately solving problems that tap conceptual knowledge. 

Procedural knowledge is also essential and is likely to enhance conceptual 

knowledge and vice versa. 

Studies in the scientific literature reveal features of children’s 

understanding of fractions that should be transferable to their learning in 

classrooms. These potential interventions include using fraction names that 

demarcate parts and wholes and linking common fraction representations to 

locations on number lines. Conceptual and procedural knowledge about fractions 

with magnitudes less than 1 do not necessarily transfer to fractions with 

magnitudes greater than 1. Therefore, understanding of fractions with magnitudes 

in each range needs to be taught directly, and the relation between them needs to 

be discussed. 

As with learning whole numbers, conceptual and procedural knowledge of 

fractions reinforce one another and influence such varied tasks as estimation, 

computation, and the solution of word problems. One key mechanism linking 

conceptual and procedural knowledge is the ability to represent fractions on a 

number line. 
National Mathematics Advisory Panel FINAL REPORT 29

Recommendation: The curriculum should allow for sufficient time to 

ensure acquisition of conceptual and procedural 

knowledge of fractions (including decimals and 

percents) and of proportional reasoning. The 

curriculum should include representational supports 

that have been shown to be effective, such as number 

line representations, and should encompass instruction 

in tasks that tap the full gamut of conceptual and 

procedural knowledge, including ordering fractions on 

a number line, judging equivalence and relative 

magnitudes of fractions with unlike numerators and 

denominators, and solving problems involving ratios 

and proportion. The curriculum also should make 

explicit connections between intuitive understanding 

and formal problem solving involving fractions. 

Recommendation: Research is needed on how children can be taught to 

appropriately estimate the magnitudes of fractions and 

on how learning to estimate those magnitudes influences 

acquisition of other skills involving fractions, such as 

arithmetic and algebra. 

Geometry and Measurement 

Although early exposure to basic geometric shapes, names, and other 

concepts may be helpful in developing children’s formal geometric knowledge 

and skills, this is not sufficient. Despite the widespread use of mathematical 

manipulatives such as geoboards and dynamic software, evidence regarding their 

usefulness in helping children learn geometry is tenuous at best. Students must 

eventually transition from concrete (hands-on) or visual representations to 

internalized abstract representations. The crucial steps in making such transitions 

are not clearly understood at present and need to be a focus of learning and 

curriculum research. 

Recommendation: Teachers should recognize that from early childhood 

through the elementary school years, the spatial 

visualization skills needed for learning geometry have 

already begun to develop. In contrast to the claims of 

Piagetian theory, young children appear to possess at 

least an implicit understanding of basic facets of 

Euclidean concepts. However, formal instruction is 

necessary to ensure that children build upon this 

knowledge to learn geometry. 
National Mathematics Advisory Panel FINAL REPORT 30

General Principles of Learning 

Students learn by building on prior knowledge, extending as far back as 

early childhood. Learning and development are incremental processes that occur 

gradually and continuously over many years. Even during the preschool period, 

children have considerably greater reasoning and problem-solving ability than 

was suspected until recently. 

For all content areas, conceptual understanding, computational fluency, and 

problem-solving skills are each essential and mutually reinforcing, influencing 

performance on such varied tasks as estimation, word problems, and computation. 

For all content areas, practice allows students to achieve automaticity of 

basic skills—the fast, accurate, and effortless processing of content information—

which frees up working memory for more complex aspects of problem solving. 

The issue of transfer, that is, the ability to use skills learned to solve one class of 

problems, such as similar triangles, to solve another class of problems, such as 

linear algebra, is a vital part of mathematics learning. Of particular importance is 

determining the variables that impede or facilitate transfer. Studies of transfer 

suggest that people’s ability to make links between related domains is limited; 

studies on how to foster transfer in key mathematical domains are needed. 

Teachers and developers of instructional materials sometimes assume that 

students need to be a certain age to learn certain mathematical ideas. However, a 

major finding, documented in a National Research Council synthesis of studies 

about science learning and reaffirmed in the review of learning studies in 

mathematics conducted by the Task Group on Learning Processes, is, “What is 

developmentally appropriate is not a simple function of age or grade, but rather is 

largely contingent on prior opportunities to learn” (Duschl et al., 2007, p. 2). 

Claims based on Piaget’s highly influential theory, and related theories of 

“developmental appropriateness” that children of particular ages cannot learn 

certain content because they are “too young,” “not in the appropriate stage,” or 

“not ready” have consistently been shown to be wrong. Nor are claims justified 

that children cannot learn particular ideas because their brains are insufficiently 

developed, even if they possess the prerequisite knowledge for learning the ideas. 

The sociocultural perspective of Vygotsky has also been influential in 

education. It characterizes learning as a social induction process through which 

learners become increasingly independent through the tutelage of more 

knowledgeable peers and adults. However, its utility in mathematics classrooms 

and mathematics curricula remains to be scientifically tested. 
National Mathematics Advisory Panel FINAL REPORT 31

Social, Motivational, and Affective Influences 

Understanding how children gain proficiency in mathematics requires 

more than knowledge about how they learn in content areas. Children’s goals and 

beliefs about learning are also critical. 

Children who seek to master an academic topic are said to have masteryoriented goals. These children show better long-term academic development in 

mathematics than do their peers whose main goals are to get good grades or 

outperform others. Students who believe learning mathematics is strongly related 

to innate ability show less persistence on complex tasks than peers who believe 

that effort is more important. 

Experimental studies have demonstrated that children’s beliefs about the 

relative importance of effort and ability or inherent talent can be changed, and that 

increased emphasis on the importance of effort is related to greater engagement in 

mathematics learning and, through this engagement, improved mathematics 

grades and achievement. 

Research demonstrating that beliefs about effort matter and that these 

beliefs can be changed is critical. Much of the public’s resignation about 

mathematics education (together with the common tendencies to dismiss weak 

achievement and to give up early) seems rooted in the idea that success in 

mathematics is largely a matter of inherent talent, not effort. 

Recommendation: The Panel recommends that teachers and other 

educational leaders use research-based interventions to 

help students and parents understand the vital 

importance of effort in learning mathematics. 

Anxiety about mathematics performance is related to low mathematics 

grades, failure to enroll in advanced mathematics courses, and poor scores on 

standardized tests of mathematics achievement. It also may be related to failure to 

graduate from high school. At present, however, little is known about its onset or the 

factors responsible for it. Potential risk factors for mathematics anxiety include low 

mathematics aptitude, low working memory capacity, vulnerability to public 

embarrassment, and negative teacher and parent attitudes. 

Recommendation: The Panel recommends research that assesses potential 

risk factors for mathematics anxiety; it also recommends 

development of promising interventions for reducing 

serious mathematics anxiety. 
National Mathematics Advisory Panel FINAL REPORT 32

Mathematics performance and learning of groups that have traditionally been 

underrepresented in mathematics fields can be considerably improved by 

interventions that address social, affective, and motivational factors. Recent research 

documents that social and intellectual support from peers and teachers is associated 

with higher mathematics performance for all students, and that such support is 

especially important for many African-American and Hispanic students. 

Recommendation: The Panel recommends the scaling-up and experimental 

evaluation of support-focused interventions that have 

been shown to improve the mathematics outcomes of 

African-American and Hispanic students. These and 

related studies focused on improving task engagement 

and self-efficacy of such students hold promise for 

helping to close the mathematics achievement gaps that 

are prevalent in U.S. society. 

Average gender differences are small or nonexistent, and our society’s 

focus on them has diverted attention from the essential task of raising the scores 

of both boys and girls. 

Progress has been made in understanding the difficulties that children with 

learning disabilities have with the learning of concepts, procedures, and facts in 

some areas of arithmetic. However, little is known about the source of their 

difficulties in other core areas, including fractions and algebra. Preliminary 

research has identified some of the mechanisms that contribute to exceptional 

mathematics learning, but much remains to be discovered. 

Recommendation: Research on the cognitive mechanisms that contribute 

to learning disabilities and precocious learning in 

mathematical domains beyond whole number 

arithmetic is needed to better understand the sources of 

individual differences in children’s mathematical 

learning. 

Considerations Specific to Algebra 

There are many gaps in the current understanding of how students learn 

algebra and the preparation that is needed before they enter Algebra. What is 

known indicates that too many students in middle or high school algebra classes 

are woefully unprepared for learning even the basics of algebra. The types of 

errors these students make when attempting to solve algebraic equations reveal 

they do not have a firm understanding of many basic principles of arithmetic. 

Many students also have difficulty grasping the syntax or structure of algebraic 

expressions and do not understand procedures for transforming equations or why 

transformations are done the way they are. These and other difficulties are 

compounded as equations become more complex and when students attempt to 

solve word problems. 
National Mathematics Advisory Panel FINAL REPORT 33

Algebra teachers should not assume that all students understand even basic 

concepts, such as mathematical equality. Many students will not have a sufficient 

understanding of the commutative and distributive properties, for example, to take 

full advantage of instruction in algebra. Many students will need extensive 

practice at solving algebraic equations and explanation as to why the equations 

are solved in a particular way—for instance, to maintain equality across the two 

sides of an equation. Examining common errors with students may provide an 

opportunity to discuss and remediate overgeneralizations or misconceptions. 

Recommendation: Longitudinal research is needed to identify early 

predictors of success or failure in algebra. The 

identification of these predictors will help to guide the 

design of interventions that will build the foundational 

skills needed for success in algebra. 
National Mathematics Advisory Panel FINAL REPORT 34
National Mathematics Advisory Panel FINAL REPORT 35

Chapter 6: 

Teachers and Teacher Education 

Substantial differences in mathematics achievement of students are 

attributable to differences in teachers.13 Teachers are crucial to students’ 

opportunities to learn and to their learning of mathematics. 

There are large, measurable differences in the effectiveness of 

mathematics teachers in generating achievement gains: 

• Differences in teachers account for 12% to 14% of total variability 

in students’ mathematics achievement gains during an elementary school 

year. 

• When teachers are ranked according to their ability to produce student 

achievement gains, there is a 10 percentile point difference across the 

course of a school year between achievement gains of students of topquartile teachers versus bottom-quartile teachers. 

• The effects of teachers on student achievement compound dramatically if 

students receive a series of effective or ineffective teachers. 

Vital, therefore, to the Panel’s inquiry and recommendations is the best 

available evidence on how teachers’ own knowledge matters for students’ 

achievement and how effective teachers can be best recruited, prepared, 

supported, and rewarded. The Panel found an uneven research base to address 

these questions. 

Teachers’ Mathematical Knowledge 

Teachers’ mathematical knowledge was estimated in three different ways 

across the research we reviewed: certification, courses completed, and direct tests 

of teachers’ knowledge of mathematics. The Panel appraised what is known 

about the relationships between teachers’ content knowledge, as measured in each 

of these ways, and students’ achievement. 

Teacher Certification as a Measure of Mathematical Knowledge 

Overall, findings about the relationship between teacher certification (i.e., 

licensure) and student achievement in mathematics have been mixed, even among 

the most rigorous and highest-quality studies. Research in this area has not provided 

consistent or convincing evidence that students of teachers who are certified to teach 

 

13 The detailed work underlying this section was carried out by the Task Group on Teachers and 

Teacher Education, whose report carries relevant references and more elaborate discussion 

(National Mathematics Advisory Panel, 2008). 
National Mathematics Advisory Panel FINAL REPORT 36

mathematics gain more than those whose teachers are not. The relationship between 

teacher certification status, the most inexact proxy for teachers’ content knowledge, 

and students’ mathematics achievement remains ambiguous. 

Content Course Work and Degrees as Measures of 

Mathematical Knowledge 

Studies that used the mathematics courses that teachers have taken as a 

proxy for their mathematical knowledge showed mixed results regarding the 

relationship of teachers’ content knowledge to their students’ achievement at the 

elementary and middle school level. At the secondary school level, there appears 

to be some effect of teachers’ content knowledge when it is measured in terms of 

teachers’ course-taking. However, the available evidence does not support this 

relationship below ninth grade. 

Tests and Ad Hoc Assessments as Measures of 

Mathematical Knowledge 

Some studies of practicing teachers at the elementary and middle school 

level that used tests of specific mathematical knowledge for teaching and other ad 

hoc measures also yielded mixed results. Overall, however, the evidence suggests 

a positive relationship between teachers’ mathematical knowledge and gains in 

student achievement. The one study that used test items specifically designed to 

directly measure the mathematical content knowledge used in teaching did 

produce findings whose magnitude is substantially larger than the others. 

Because these studies were focused at the elementary level, comparisons with 

other findings are difficult. 

The inability to draw solid conclusions from this literature is in part due to a 

historical lack of high-quality measures of mathematics content knowledge, as well 

as a paucity of high-quality studies using these types of measures. 

The Mathematical Content and Nature of Teacher 

Licensure Exams 

Recent research treating teacher licensure as a proxy for teachers’ 

mathematical knowledge has not shown consistently or convincingly that students 

of teachers who are licensed to teach mathematics gain more academically than 

those whose teachers are not. However, since the 1998 reauthorization of the 

Higher Education Act, teacher licensure exams have come to play an important 

role in determining the quality and quantity of teachers available to teach 

mathematics. Therefore the Panel also attempted to assess the mathematical 

content covered on teachers’ licensure exams and the rigor and relevance of those 

exams. There are three major developers of teacher licensure tests: The 

Educational Testing Service (ETS), National Evaluation Systems (NES), and the 

American Board for Certification of Teacher Excellence (ABCTE). For those 

states that contract with ETS, prospective teachers take the Praxis series, which 

comprises two separate exams, the Praxis I and II. The Praxis I exams are 
National Mathematics Advisory Panel FINAL REPORT 37

designed to measure basic skills in reading, writing, and mathematics. Most ETS 

states currently require the Praxis I tests for licensure, and often for admission 

into their teacher education programs. The Praxis II exams for those who will 

teach mathematics as content specialists or as generalists vary in the amount and 

level of mathematical knowledge assessed. Some of these tests do not assess 

mathematics content. To analyze the effectiveness of these exams in assessing 

teachers’ content knowledge, the Panel sought access to exams together with data 

on teachers’ performance on each item. Due to issues of confidentiality, however, 

the Panel was not able to gather sufficiently complete information to assess the 

mathematical quality of these exams. 

Conclusions About the Relationship of Teachers’ Mathematical 

Knowledge to Students’ Achievement Gains 

Overall, across the studies reviewed by the Panel, it is clear that teachers’ 

knowledge of mathematics is positively related to student achievement. However, 

evidence about the relationship of elementary and middle school teachers’ 

mathematical knowledge to students’ mathematics achievement remains uneven 

and has been surprisingly difficult to produce. One important reason has been the 

lack of valid and reliable measures of teachers’ mathematical knowledge. The 

literature has been dominated by the use of proxies for such knowledge, such as 

certification status and mathematics course work completed. A second reason for 

the inconsistent findings has been weak study designs. Too few studies exist that 

set up proper comparisons or use sufficient sample sizes or appropriate analytic 

methods. Selection bias and failure to isolate potentially important variables from 

confounding variables have further plagued these studies, as have inadequate 

measures of students’ mathematics achievement. Finally, with the exception of 

one study that directly measured the mathematical knowledge used in teaching, no 

studies identified by the Panel probed the dynamic that would examine how 

elementary and middle school teachers’ mathematical knowledge affects 

instructional quality, students’ opportunities to learn, and gains in achievement 

over time. 

In the context of a body of literature as inexact as this one, the positive 

trends we identified do support the importance of teachers’ knowledge of 

mathematics as a factor in students’ achievement. 

Recommendation: Teachers must know in detail the mathematical content 

they are responsible for teaching and its connections to 

other important mathematics, both prior to and beyond 

the level they are assigned to teach. However, because 

most studies have relied on proxies for teachers’ 

mathematical knowledge (e.g., course work as part of a 

certification program), existing research does not 

provide definitive evidence for the specific mathematical 

knowledge and skill that are needed for teaching. 
National Mathematics Advisory Panel FINAL REPORT 38

Recommendation: More precise measures should be developed to uncover 

in detail the relationships among teachers’ knowledge, 

their instructional skill, and students’ learning, and to 

identify the mathematical and pedagogical knowledge 

needed for teaching. 

Recommendation: The mathematics preparation of elementary and middle 

school teachers must be strengthened as one means for 

improving teachers’ effectiveness in the classroom. This 

includes preservice teacher education, early career 

support, and professional development programs. A 

critical component of this recommendation is that 

teachers be given ample opportunities to learn 

mathematics for teaching. That is, teachers must know 

in detail and from a more advanced perspective the 

mathematical content they are responsible for teaching 

and the connections of that content to other important 

mathematics, both prior to and beyond the level they 

are assigned to teach. 

High-quality research must be undertaken to create a 

sound basis for the mathematics preparation of 

elementary and middle school teachers within 

preservice teacher education, early-career support, 

and ongoing professional development programs. 

Outcomes of different approaches should be evaluated 

by using reliable and valid measures of their effects on 

prospective and current teachers’ instructional 

techniques and, most important, their effects on 

student achievement. 
National Mathematics Advisory Panel FINAL REPORT 39

Teachers’ Education: Preparation, Induction, and 

Professional Development 

The Panel investigated evidence on the impact of: 

• Preservice teacher preparation: Initial teacher training, conventionally 

offered in institutions of higher education;

• Alternative pathways: Initial teacher preparation offered outside of 

conventional teacher education programs; 

• Induction programs: Programs of professional support and additional 

training within the first years of teachers’ practice;

• Professional development: Ongoing programmatic professional education 

of practicing teachers. 

The focus was on what is known about the relationship between different 

forms of teacher education and the learning of teachers and their students: 

Preservice Teacher Preparation 

Very few empirical studies were found that addressed the impacts of 

preparation programs on student achievement or teachers’ mathematical 

knowledge. Unfortunately, none of these studies was of sufficient rigor or quality 

to allow the Panel to draw conclusions about the relationship of particular features 

of teacher preparation programs and their effects. 

Alternative Pathways Into Teaching 

With respect to alternative pathways into teaching, there were a few studies 

that compared the effectiveness of standard and alternative preparation programs on 

student achievement, suggesting that there is no basis in research for preferring one 

pathway to another. The extant evidence suggests that there are not significant 

differences among current pathways. Moreover, the variation within programs 

appears to be greater than that found across programs. 

Early-Career Teacher Support Programs 

No peer-reviewed studies could be found that focused on the effects of 

programs for first- and second-year mathematics teachers (i.e., induction 

programs) on student achievement or on teachers’ mathematics knowledge. The 

key outcome for much of the extant induction literature is teacher retention. There 

also is a wealth of literature examining the qualitative effects of induction 

programs on teacher beliefs, satisfaction, and practices. Induction programs 

continue to expand, some with and some without mandated funding. Given the 

expansion, it is important to assess the effectiveness of induction programs on 

student achievement as well as on teacher retention. 
National Mathematics Advisory Panel FINAL REPORT 40

Professional Development 

The Panel searched for peer-reviewed research and national reports that 

would offer high-quality evidence regarding the impact of professional 

development programs for teachers, but found that many studies were 

descriptive. Most of the studies that were intended to be empirical tests of 

hypotheses did not include a comparison group, but used a “one-group 

pretest/posttest design.” Moreover, many studies relied on teachers’ self-reports 

about their knowledge before and after the professional development rather 

than on objective measures of teacher knowledge. Consequently, the studies 

that the Panel was able to include were only ones that examined the 

relationship between teacher professional development programs and students’ 

mathematics achievement. 

Although the Panel did find some positive effects of professional 

development on students’ achievement gains, research does not yield sufficient 

evidence on the features of any particular approach to permit detailed conclusions 

about the forms of or approaches to effective professional development. 

Conclusions About the Impact of Teachers’ Education 

on Teachers’ Mathematical Knowledge or Students’ 

Achievement Gains 

Despite widespread beliefs about the qualities that make teacher education 

effective, the Panel did not find strong evidence for the impact of any specific form 

of, or approach to, teacher education on either teachers’ knowledge or students’ 

learning. Even for the few studies that did produce significant effects, little detail 

was provided about the features of the training that might account for the impact of 

the program. Such deficiencies of the research impeded the Panel’s ability to identify 

crucial components of teacher education. 

Much more needs to be known about features of professional development 

programs that are able to equip teachers with the knowledge and skills they need 

to facilitate student learning. 

Recommendation: The Panel recommends that a sharp focus be placed on 

systematically strengthening teacher preparation, earlycareer mentoring and support, and ongoing professional 

development for teachers of mathematics at every level, 

with special emphasis on ways to ensure appropriate 

content knowledge for teaching. 

Recommendation: A well-designed program of research and evaluation, 

meeting standards permitting the generalization of 

results, should be undertaken to create a sound basis for 

the education of teachers of mathematics. 
National Mathematics Advisory Panel FINAL REPORT 41

Key questions on which robust evidence is needed include: 

• Does teacher education (including preservice training of different kinds, 

professional development, and early career induction programs) have an 

impact on teachers’ capacity and on students’ achievement? 

• What are the key features of teacher education (e.g., duration, structure, 

quantity, content, pedagogy, structure, relationship to practice) that have 

effects on teachers’ capacity and on students’ achievement? 

• How do contexts (e.g., school, students, teachers, policy) affect the 

outcomes of professional development? 

• How do different amounts of teacher education affect its outcomes 

and effects? 

Given the vast investment made in teacher education, knowledge about the 

effectiveness of different approaches is vitally needed. Well-conceived efforts to 

improve the outcomes of teacher education, to improve measures of those outcomes, 

and to implement better research strategies should be supported. 

Recruitment and Retention Strategies to Attract and Retain 

Effective Teachers of Mathematics 

Because compensation is often cited as a key factor in improving teacher 

quality, the Panel investigated evidence on how different salary schemes work to 

recruit, reward, and retain skillful teachers. 

In the business sector, pay is typically contingent on performance and area 

of specialization as well as on years of experience and level of education. In 

universities, for example, economists typically receive higher salaries than 

historians, reflecting the greater demand for economists outside academe. Parallels 

in K–12 education would take the form of paying more to teachers who have 

technical skills that are in demand in other sectors of the economy, such as teachers 

with degrees in mathematics (skills-based pay), and paying more to mathematics 

teachers who are more productive in raising student achievement (performancebased pay). Another type of incentive has the purpose of compensating teachers for 

working in conditions they view as unfavorable (location-based pay), such as those 

associated with high-poverty, low-achieving schools. The Panel examined research 

on each of these approaches to teacher compensation. 

Skills-Based Pay 

College students’ decisions to prepare for and enter into teaching depend on 

how the salary structure for teachers compares with those in competing 

occupations. The magnitude of the salary differential between the private sector and 

the teaching profession for those who enter teaching with technical training is large, 

with a negligible difference on entry but a rapidly increasing gap over the first 10 
National Mathematics Advisory Panel FINAL REPORT 42

years of employment. Teachers of mathematics and science are significantly more 

likely to leave their teaching jobs because of job dissatisfaction than are other 

teachers (40% of math and science teachers and 29% of all teachers). Of those 

mathematics and science teachers who depart because of job dissatisfaction, the 

most common reason given is low salaries (57% of respondents). 

Location-Based Pay 

Research on the effects of location-based pay, intended to attract or retain 

skilled teachers in schools that serve under-resourced communities, yields mixed 

results. The effectiveness of such salary schemes is affected by the amount of 

differential in pay, the gender and experience of the teacher, and whether the bonus 

is a one-time signing bonus or a permanently higher salary, as well as other factors. 

Performance-Based Pay 

The Panel identified four different aspects of “merit” pay: whether salary 

differentials are tied to schools’ performance or that of individual teachers, how 

significant the pay differential is, the degree to which the scheme is focused on 

student performance, and whether the plan seems continuous or is a short-term 

experiment. Across the studies reviewed, each found some positive effects on 

student achievement, but none was sufficient to reach strong conclusions about the 

effectiveness of performance-based pay schemes. 

The results from research on teacher incentives generally support the 

effectiveness of incentives, although the methodological quality of the studies in 

terms of causal conclusions is mixed. The substantial body of economic research 

in other fields indicating that salary affects the number of workers entering a field 

and their job performance is relevant. In the context of the totality of the evidence, 

and acknowledging the substantial number of unknowns, the NMP recommends 

policy initiatives that put in place and carefully evaluate the effects of: 

• Raising base salaries for teachers of mathematics to attract more 

mathematically qualified teachers into the workforce; 

• Salary incentives for teachers of mathematics for working in locations that 

are difficult to staff, and; 

• Opportunities for teachers of mathematics to increase their base salaries 

substantially by demonstrable effectiveness in raising student achievement. 

Considerable work remains to be done before enough will be known to put 

particular pay-for-performance systems in place and to confidently predict their 

outcomes. Knowing more about how various incentive systems affect teachers 

would enable the design of more effective and efficient incentives. Additional 

evidence also shows that teachers’ decisions to remain in teaching and to continue 

teaching in particular schools are affected by factors in addition to salary, including 

work conditions, the proximity of teachers’ residences to the school, support from 

school administrators, teaching assignments, and characteristics of students. 
National Mathematics Advisory Panel FINAL REPORT 43

Elementary Mathematics Specialist Teachers 

There have been many calls in recent years for the use of “math 

specialists” at the K–5 level, but what is meant by “math specialists” can take 

different forms. The Panel sought to learn what is known about such specialists at 

the elementary level. 

Models of Math Specialists 

The Panel identified at least three types of “math specialist teachers”: the 

math coach (lead teacher), the full-time elementary mathematics teacher, and the 

pull-out program teacher. Math coaches are more common than the other two 

types, but there is considerable blurring across types and roles. Math coaches

(sometimes called lead teachers) tend to act as resources for their colleagues and 

do not directly instruct students. They work at the state, district, and school levels, 

providing leadership and information to teachers and staff and often coordinating 

mathematics programs within a school, a district, or across districts. Full-time 

mathematics teachers are responsible for the direct instruction of students. They 

work at the school and district levels, but most frequently take responsibilities in 

one school. Pull-out program teachers represent a variation of the specialized 

teacher model. In this model, math specialists directly instruct individuals or 

small groups of students within or outside a regular classroom. 

Effects on Student Achievement of Using Math Specialists 

Very few studies were identified that probed the effectiveness of 

elementary mathematics specialists of any of the three types. Out of 114 

potentially relevant pieces of literature, only 1 explored the effects of mathematics 

specialists on student achievement in elementary schools. These authors found no 

difference in the mathematics gain scores of students in an elementary school with 

a departmentalized structure compared to students in a school with a selfcontained structure. 

Costs Associated with Using Math Specialists 

One cost has to do with the funding of the personnel involved and depends 

on the model used. The use of a full-time mathematics teacher involves only a 

redistribution of responsibilities among the existing staff, whereas the use of math 

coaches and pull-out teachers requires the hiring of additional staff. A second cost 

is that of the additional training needed for teachers to gain the specialized 

knowledge needed to fill these roles. 
National Mathematics Advisory Panel FINAL REPORT 44

The Use of Math Specialists in Other Countries 

Full-time elementary mathematics teachers are not widely used in most of 

the countries that produce high levels of student achievement in mathematics. 

Only three (China, Singapore, and Sweden) deploy such teachers at the 

elementary level. That elementary teachers in those countries may enter teaching 

with stronger backgrounds in mathematics may be a factor in the success of those 

countries with mathematics education. 

Recommendation: The Panel recommends that research be conducted on 

the use of full-time mathematics teachers in elementary 

schools. These would be teachers with strong knowledge 

of mathematics who would teach mathematics full-time 

to several classrooms of students, rather than teaching 

many subjects to one class, as is typical in most 

elementary classrooms. This recommendation for 

research is based on the Panel’s findings about the 

importance of teachers’ mathematical knowledge. The 

use of teachers who have specialized in elementary 

mathematics teaching could be a practical alternative to 

increasing all elementary teachers’ content knowledge 

(a problem of huge scale) by focusing the need for 

expertise on fewer teachers. 
National Mathematics Advisory Panel FINAL REPORT 45

Chapter 7: 

Instructional Practices 

Teacher-Directed and Student-Centered Instruction 

in Mathematics 

A controversial issue in the field of mathematics education is whether 

classroom instruction should be more teacher directed or more student centered.14

These terms encompass a wide array of meanings, with teacher-directed 

instruction ranging from highly scripted direct instruction approaches to 

interactive lecture styles, and with student-centered instruction ranging from 

students having primary responsibility for their own mathematics learning to 

highly structured cooperative groups. Schools and districts must make choices 

about curricular materials and instructional approaches that often seem more 

aligned with one instructional orientation than another. This leaves teachers 

wondering about when to organize their instruction one way or the other, whether 

certain topics are taught more effectively with one approach or another, and 

whether certain students benefit more from one approach than another. 

In the Panel’s review, the search was limited to studies that directly 

compared these two extreme positions. We defined teacher-directed instruction as 

instruction in which it is the teacher who is primarily communicating the 

mathematics to the students directly and student-centered instruction as 

instruction in which students are primarily doing the teaching. 

We found only eight studies that met our standards for quality and that 

compared versions of teacher-directed and student-centered instruction consistent 

with our definition. The studies presented a mixed and inconclusive picture of the 

relative effect of these two approaches to instruction. Although there were some 

significant effect sizes in some studies in both groups, all had limitations and no 

generalizations can be made. Additional high-quality research is needed, using clear 

definitions of “teacher directed” and “student centered.” 

Recommendation: All-encompassing recommendations that instruction 

should be entirely “student centered” or “teacher 

directed” are not supported by research. If such 

recommendations exist, they should be rescinded. If 

they are being considered, they should be avoided. 

High-quality research does not support the exclusive 

use of either approach. 

 

14 The detailed work underlying this section was carried out by the Task Group on Instructional 

Practices, whose report carries relevant references and more elaborate discussion (National 

Mathematics Advisory Panel, 2008). 
National Mathematics Advisory Panel FINAL REPORT 46

One of the major shifts in education over the past 25–30 years has been 

advocacy for the increased use of cooperative learning groups and peer-to-peer 

learning (e.g., structured activities for students working in pairs) in the teaching and 

learning of mathematics. Use of cooperative or collaborative learning has been 

advocated in various mathematics education reports and in state curricular 

frameworks, policies, and instructional guidelines. Cooperative learning is used for 

multiple purposes: for tutoring, for enrichment and for remediation, as an 

occasional substitute for independent seatwork, for intricate extension activities, for 

initial brainstorming, and for numerous other purposes. 

The Panel located high-quality studies in the following areas of cooperative 

and collaborative learning: Team Assisted Individualization (four studies), Student 

Teams-Achievement Division (six studies), peer-to-peer learning strategies (five 

studies), other cooperative learning strategies (five studies), studies combining 

cooperative learning with other instructional practices (three studies), and studies 

investigating cooperative learning in the context of computers (eight studies). 

Research has been conducted on a variety of cooperative learning 

approaches. One such approach, Team Assisted Individualization (TAI), has been 

shown to improve students’ computation skills. This highly structured 

instructional approach involves heterogeneous groups of students helping each 

other, individualized problems based on student performance on a diagnostic test, 

specific teacher guidance, and rewards based on both group and individual 

performance. Effects of TAI on conceptual understanding and problem solving 

were not significant. 

There is suggestive evidence that peer tutoring improves computation 

skills in the elementary grades. However, additional research is needed. 

Using Formative Assessment 

Formative assessment—the ongoing monitoring of student learning to 

inform instruction—is generally considered a hallmark of effective instruction in 

any discipline. 

Our key findings from a review of the high-quality studies of this topic are: 

• The average gain in learning provided by teachers’ use of formative 

assessments is marginally significant. Results suggest that use of formative 

assessments benefited students at all ability levels. When teachers are 

provided with additional “enhancements” (i.e., specific suggestions on how 

to use the assessment data to provide differentiated instruction), the pooled 

effect is significant. 

• The studies describe a set of tools and procedures (what are called 

“enhancements”) that can accompany formative assessment. Given the 

nature of the evidence, the Panel would more cautiously call these 

practices promising as opposed to evidence-based. 
National Mathematics Advisory Panel FINAL REPORT 47

• Only one type of formative assessment has been studied with rigorous 

experimentation, viz. assessment that includes a random sampling of items 

that address state standards. The assessments tend to take between two and 

eight minutes to administer and, thus, are feasible for regular use. 

Teachers’ regular use of formative assessments improves their students’ 

learning, especially if teachers have additional guidance on using the assessment 

results to design and individualize instruction. The research to date has only 

involved formative assessment based on items sampled from the major curriculum 

objectives for the year as specified by state standards. Findings regarding use of 

this type of formative assessment were consistently positive and significant. 

Recommendation: Based on its review of research, the Panel recommends 

regular use of formative assessment, particularly for 

students in the elementary grades. These assessments 

need to provide information not only on their content 

validity but also on their reliability and their criterionrelated validity (i.e., correlation of these measures with 

other measures of mathematics proficiency). For 

struggling students, frequent (e.g., weekly or biweekly) 

use of these assessments appears optimal, so that 

instruction can be adapted based on student progress. 

Although the research base is smaller, and less consistent than that on the 

general effectiveness of formative assessment, the research does suggest that several 

specific tools and strategies can help teachers use formative assessment information 

more effectively. The first promising strategy is providing formative assessment 

information to teachers (via technology) on content and concepts that require 

additional work with the whole class. The second promising strategy involves using 

technology to specify activities needed by individual students. Both of these aids can 

be implemented via tutoring, computer-assisted instruction, or help provided by a 

professional (teacher, mathematics specialist, trained paraprofessional). 

Recommendation: The Panel recommends that professional 

organizations, school districts, and state agencies 

provide tools that inform teachers about specific ways 

of using formative assessment information to provide 

differentiated instruction. 

The Panel also recommends that research be conducted 

regarding the content and criterion-related validity and 

reliability of other types of formative assessments (such 

as unit mastery tests included with many published 

mathematics programs, performance assessments, and 

dynamic assessments involving “think alouds”). This 

research should include studies of consequential validity 

(i.e., the impact they have on helping teachers improve 

their effectiveness). 
National Mathematics Advisory Panel FINAL REPORT 48

Use of formative assessments in mathematics can lead to increased 

precision in how instructional time is used in class and can assist teachers in 

identifying specific instructional needs. Formative measures provide guidance as 

to the specific topics needed for assistance. Formative assessment should be an 

integral component of instructional practice in mathematics. 

Teaching Low-Achieving Students and Students with 

Learning Disabilities 

The Panel conducted a review of 26 high-quality studies, mostly using 

randomized control designs. These studies provide a great deal of guidance 

concerning some defining features of effective instructional approaches for 

students with learning disabilities (LD) as well as low-achieving (LA) students. 

The review indicated that explicit methods of instruction are effective with LD 

and LA students. 

Some key findings: 

• Explicit systematic instruction was found to improve the performance of 

students with learning disabilities in computation, solving word problems, 

and solving problems that require the application of mathematics to novel 

situations. Explicit systematic instruction typically entails teachers 

explaining and demonstrating specific strategies and allowing students 

many opportunities to ask and answer questions and to think aloud about 

the decisions they make while solving problems. It also entails careful 

sequencing of problems by the teacher or through instructional materials 

to highlight critical features. Significant positive effects were also found 

for Direct Instruction (a specific type of explicit instruction that provides 

teachers with scripts and that calls for frequent interactions between 

students and teachers, clear feedback to students on the accuracy of their 

work, and sequencing of problems so that critical differences are 

highlighted). Other forms of explicit systematic instruction have been 

developed with applications for students with learning disabilities. These 

developments reflect the infusion of research findings from cognitive 

psychology, with particular emphasis on automaticity and enhanced 

problem representation. 

• Most of the small number of studies that investigated the use of visual 

representations yielded nonsignificant effects. However, studies that 

included visual representations along with the other components of 

explicit instruction tended to produce significant positive effects. 

Recommendation: The Panel recommends that students with learning 

disabilities and other students with learning problems 

receive, on a regular basis, some explicit systematic 

instruction that includes opportunities for students to 

ask and answer questions and think aloud about the 
National Mathematics Advisory Panel FINAL REPORT 49

decisions they make while solving problems. This kind 

of instruction should not comprise all the mathematics 

instruction these students receive. However, it does 

seem essential for building proficiency in both 

computation and the translation of word problems into 

appropriate mathematical equations and solutions. 

Some of this time should be dedicated to ensuring that 

students possess the foundational skills and conceptual 

knowledge necessary for understanding the 

mathematics they are learning at their grade level. 

Recommendation: The Panel identified surprisingly few methodologically 

rigorous studies (given a literature base that spanned 

the past 30 years) that examined instructional practices 

designed to improve the performance of low-achieving 

students and students with learning disabilities. 

Although the actual quantity of such studies was small, 

their quality was high. There is a critical need for 

stimulating and supporting through federal funding of 

additional high-quality research to address this major 

national challenge. 

Using ‘Real-World’ Problems to Teach Mathematics 

The Panel’s review of the literature addressed the question of whether 

using “real-world” contexts to introduce and teach mathematical topics and 

procedures is preferable to using more typical instructional approaches. The 

meaning of the term “real-world” problem varies by mathematician, researcher, 

developer, and teacher. Doing research in this area is complex; fidelity of the 

teachers’ implementation of the instructional materials or instructional strategy is 

difficult to assess; contextual features, such as socioeconomic status or the 

school’s orientation toward reform, matter and, most likely, although not 

addressed in the studies examined by the Panel, teachers’ knowledge and capacity 

to use such problems effectively varies greatly. 

The body of high-quality studies on this topic is small. We located 10 

studies that met our criteria for quality. Five of these addressed the question of 

whether the use of “real-world” problems as the instructional approach led to 

improved performance on outcome measures of ability to solve “real-world” 

problems, as well as on more traditional assessments. Four of these were similar 

enough to combine in a meta-analysis. They involved upper elementary and middle 

school students, as well as ninth-grade remedial students; the mathematical topics 

included fraction computation and beginning equation solving. The analysis 

revealed that if mathematical ideas are taught using “real-world” contexts, then 

students’ performance on assessments involving similar problems is improved. 

However, performance on assessments of other aspects of mathematics learning, 

such as computation, simple word problems, and equation solving, is not improved. 
National Mathematics Advisory Panel FINAL REPORT 50

For certain populations (upper elementary and middle grade students, and 

remedial ninth-graders) and for specific domains of mathematics (fraction 

computation, basic equation solving, and function representation), instruction that 

features the use of “real-world” contexts has a positive impact on certain types of 

problem solving. However, these results are not sufficient as a basis for 

widespread policy recommendations. Additional research is needed to explore the 

use of “real-world” problems in other mathematical domains, at other grade 

levels, and with varied definitions of “real-world” problems. 

Technology and Applications of Technology: Calculators 

and Computer-Based Instruction

Although young in historic terms, computer technology has a strong 

presence in people’s lives and in the research literature. The Panel reviewed 

research on the role of technology, including computer software and calculators, 

in mathematics instruction and learning. 

A review of 11 studies that met the Panel’s rigorous criteria (only one 

study less than 20 years old) found limited or no impact of calculators on 

calculation skills, problem solving, or conceptual development over periods of up 

to 1 year. This finding is limited to the effect of calculators as used in the 11 

studies. Unfortunately, these studies cannot be used to judge the advantages or 

disadvantages of multiyear calculator use beginning in the early years because 

such long-term use has not been adequately investigated. 

The Panel’s survey of the nation’s algebra teachers indicated that the use 

of calculators in prior grades was one of their concerns (National Mathematics 

Advisory Panel, 2008). The Panel cautions that to the degree that calculators 

impede the development of automaticity, fluency in computation will be 

adversely affected. 

Recommendation: The Panel recommends that high-quality research on 

particular uses of calculators be pursued, including 

both their short- and long-term effects on computation, 

problem solving, and conceptual understanding. 

Research on instructional software has generally shown positive effects on 

students’ achievement in mathematics as compared with instruction that does not 

incorporate such technologies. These studies show that technology-based drill and 

practice and tutorials can improve student performance in specific areas of 

mathematics. Other studies show that teaching computer programming to students 

can support the development of particular mathematical concepts, applications, 

and problem solving. 

However, the nature and strength of the results vary widely across these 

studies. In particular, one recent large, multisite national study found no 

significant effects of instructional tutorial (or tutorial and practice) software when 
National Mathematics Advisory Panel FINAL REPORT 51

implemented under typical conditions of use. Taken together, the available 

research is insufficient for identifying the factors that influence the effectiveness 

of instructional software under conventional circumstances. 

Recommendation: The Panel recommends that high-quality computerassisted instruction (CAI) drill and practice, 

implemented with fidelity, be considered as a useful tool 

in developing students’ automaticity (i.e., fast, accurate, 

and effortless performance on computation), freeing 

working memory so that attention can be directed to the 

more complicated aspects of complex tasks. 

Research has demonstrated that tutorials (i.e., CAI programs, often 

combined with drill and practice) that are well designed and implemented can have 

a positive impact on mathematics performance, particularly at the middle and high 

school levels. CAI tutorials have been used effectively to introduce and teach new 

subject-matter content. Research suggests that tutorials that are designed to help 

specific populations meet specific educational goals have a positive impact. 

However, these studies also suggest several important caveats. Care must be taken 

to ensure that there is evidence that the software to be used has been shown to 

increase learning in the specific domain and with students who are similar to those 

who will use the software. Educators should critically inspect individual software 

packages and the studies that evaluate them. Furthermore, the requisite support 

conditions to use the software effectively (sufficient hardware and software; 

technical support; adequate professional development, planning, and curriculum 

integration) should be in place, especially in large-scale implementations, to 

achieve optimal results. 

Recommendation: The Panel recommends that high-quality computerassisted instruction (CAI) tutorials, implemented with 

fidelity, be considered as a potentially useful tool in 

introducing and teaching specific subject-matter 

content to specific populations. The Panel also 

recommends that additional high-quality research be 

pursued to identify which goals and which populations 

are served well by tutorials, as well as the particular 

features of effective tutorials and of their 

implementation in the classroom. 

Research indicates that learning to write computer programs improves 

students’ performance compared to conventional instruction, with the greatest 

effects on understanding of concepts and applications, especially geometric 

concepts, and weaker effects on computation. However, computer programming 

by students can be employed in a wide variety of situations using distinct 

pedagogies, not all of which may be effective. Therefore, the findings are limited 

to the careful, targeted application of computer programming for learning used in 

the studies reviewed. 
National Mathematics Advisory Panel FINAL REPORT 52

Recommendation: The Panel recommends that computer programming be 

considered as an effective tool, especially for elementary 

school students, for developing specific mathematics 

concepts and applications, and mathematical problemsolving abilities. Effects are larger if the computer 

programming language is designed for learning (e.g., 

Logo) and if students’ programming is carefully guided 

by teachers so as to explicitly teach students to achieve 

specific mathematical goals. 

There are insufficient rigorous studies of other categories of software to 

make recommendations about their use. Problem-solving software may have 

potential, but more research is needed on this category of software, as well as on 

the effects of simulations, games, and Internet applications.

Finally, research is needed on specific features of software that 

theoretically should contribute to learning. Information regarding critical features 

of software is important, because decisions about whether to use existing software 

and how to develop new software could be guided by the software’s inclusion or 

omission of these critical features. More research is also needed on issues relevant 

to software use, such as fidelity of implementation, curriculum integration, and 

use software as a replacement or supplement to other instruction.

Teaching Mathematically Gifted Students 

The Panel’s review of the literature about what kind of mathematics 

instruction would be most effective for gifted students focused on the impact of 

programs involving acceleration, enrichment, and the use of homogeneous 

grouping. Although many syntheses and summaries of research in these areas 

have been conducted, our searches yielded surprisingly few studies that met the 

Panel’s methodologically rigorous criteria for inclusion; thus for this section we 

relaxed these criteria to fulfill the charge of evaluating the “best available 

scientific evidence.” The Panel could formulate its recommendations only on the 

basis of one randomized control trial study and seven quasi-experimental studies. 

These studies have limitations. For instance, motivation is a confounding variable, 

just as it is a selection criterion for being considered a candidate for acceleration. 

The Panel’s key findings are the following: 

• The studies reviewed provided some support for the value of 

differentiating the mathematics curriculum for students with sufficient 

motivation, especially when acceleration is a component (i.e., pace and 

level of instruction are adjusted). 

• A small number of studies indicated that individualized instruction, in 

which pace of learning is increased and often managed via computer 

instruction, produces gains in learning. 
National Mathematics Advisory Panel FINAL REPORT 53

• Gifted students who are accelerated by other means not only gained time 

and reached educational milestones earlier (e.g., college entrance) but also 

appear to achieve at levels at least comparable to those of their equally 

able same-age peers on a variety of indicators even though they were 

younger when demonstrating their performance on the various 

achievement benchmarks. 

• Gifted students appeared to become more strongly engaged in science, 

technology, engineering, or mathematical areas of study. There is no 

evidence in the research literature that gaps and holes in knowledge have 

occurred as a result of student acceleration. 

In the case of gifted (or academically advanced) students who are 

advanced in their skill and concept attainment and can learn new material at a 

much more rapid rate than their same-age peers, it is the professional judgment of 

those in gifted education that they need a curriculum that is differentiated (by 

level, complexity, breadth, and depth), developmentally appropriate, and 

conducted at a more rapid rate. 

Support also was found for supplemental enrichment programs. Of the 

two programs analyzed, one explicitly utilized acceleration as a program 

component and the other did not. Self-paced instruction supplemented with 

enrichment yielded the greater benefits. This supports the widely held view in the 

field of gifted education that combined acceleration and enrichment should be the 

intervention of choice. 

Recommendation: Mathematically gifted students with sufficient 

motivation appear to be able to learn mathematics 

much faster than students proceeding through the 

curriculum at a normal pace, with no harm to their 

learning, and should be allowed to do so. 

There is a need for more high-quality experimental and quasi-experimental 

research to study the effectiveness of interventions designed to meet the learning 

needs of gifted students. Especially vital are evaluations of academically rigorous 

enrichment programs. 

It is important for school policies to support appropriately challenging 

work in mathematics for gifted and talented students. Acceleration, combined 

with enrichment, is a promising practice that is moderately well supported by the 

research literature, especially when the full range of available literature 

is considered. 
National Mathematics Advisory Panel FINAL REPORT 54
National Mathematics Advisory Panel FINAL REPORT 55

Chapter 8: 

Instructional Materials 

Accuracy of Textbooks 

One would like to assume that textbooks for middle school and high 

school mathematics are free of errors.15 But when mathematicians have reviewed 

recently published middle and high school textbooks, they have identified many 

errors and a large number of ambiguous and confusing statements and problems. 

One such review of widely used Algebra I textbooks was conducted on behalf of 

the Panel. Many of the detected errors and ambiguities arose in word problems 

that were intended to elicit use of the mathematical concepts and procedures in 

“real-world” contexts. 

Recommendation: Publishers must ensure the mathematical accuracy of 

their materials. Those involved with developing 

mathematics textbooks and related instructional 

materials need to engage mathematicians, as well as 

mathematics educators, at all stages of writing, editing, 

and reviewing these materials. 

Length, Coherence, and Sequencing of Topics 

U.S. mathematics textbooks are extremely long. Not counting study guides 

and answers at the end of the books, middle and high school textbooks typically 

range from 600 to more than 900 pages. With the study guides and answers, they 

sometimes exceed 1,000 pages. Even elementary school textbooks sometimes 

exceed 700 pages. Mathematics textbooks were much shorter in previous decades 

and continue to be much shorter in many nations with higher mathematics 

achievement than in the United States. Thus, the great length is not needed for 

effective instruction. The excessive length also makes books unnecessarily 

expensive and difficult to transport between school and home, consequences that 

may undermine their effectiveness as tools for learning. 

Recommendation: All parties involved in the publication and adoption of 

textbooks should strive for more compact and more 

coherent mathematics texts for use by students in 

Grades K–8 and beyond. 

 

15 The detailed work underlying this section was carried out by the Subcommittee on Instructional 

Materials, whose report carries relevant references and more elaborate discussion (National 

Mathematics Advisory Panel, 2008). 
National Mathematics Advisory Panel FINAL REPORT 56

Textbook publishers emphasize that a major source of the textbooks’ 

length is the need to cover all of the curricular expectations encompassed in any 

state’s mathematics standards, as a topic covered in sixth grade in one state may 

be covered in seventh grade in another state and in eighth grade in a third state. 

This situation leads to the topic being included in all three grades’ math 

textbooks. The large influence of this factor is illustrated by the fact that the statespecific editions of Algebra I textbooks published for California, Texas, and 

Florida are roughly 25% (more than 200 pages) shorter than the national edition 

published for the other 47 states. Coverage of all 50 states’ benchmarks for a 

given grade increases length and decreases coherence—this despite the fact that 

mathematics is especially amenable to a coherent treatment. Integrating new 

concepts with previous ones is impossible when textbook writers cannot 

anticipate the topics students already have encountered. 

Recommendation: States and districts should strive for greater agreement 

regarding the topics to be emphasized and covered at 

particular grades. Textbook publishers should publish 

editions that include a clear emphasis on the material 

that these states and districts agree to teach in specific 

grades. 

Another source of lack of coherence and potential confusion in some 

textbooks is the table of contents. Tables of contents should provide students, 

teachers, and textbook adoption teams with a sense of the organization of the 

mathematical topics in the book. In some textbooks, however, tables of contents 

emphasize not the mathematics but rather specific applications (e.g., Ferris 

wheels, penny jars). Tables of contents that emphasize the mathematical content 

seem more likely to help teachers and students appreciate the coherence inherent 

in mathematics. 

Other potentially useful ways of decreasing length and increasing 

coherence are: 1) reducing the number of photographs that are not essential to the 

mathematical content; 2) placing content aimed at providing extended review, 

enrichment activities, or motivation in supplements rather than in the main 

textbook; and 3) reducing applications in which the primary challenge is posed by 

the social studies or science content. 
National Mathematics Advisory Panel FINAL REPORT 57

Chapter 9: 

Assessment of Mathematics Learning 

Achievement tests are widely used to estimate what students know and 

can do in specific subject areas.16 Tests make visible to teachers, parents, and 

policymakers some of the outcomes of student learning. They also can drive 

instruction. Due to their important role in education today, the Panel examined 

released items from the mathematics portions of the NAEP and six state tests and 

reviewed the relevant scientific literature on the appropriate content of such tests, 

the setting of performance categories (e.g., by determining cut scores), and factors 

affecting the quality of measurement, accuracy, and appropriate test design. 

On the basis of the work of the Task Group on Assessment, the Panel 

developed two broad recommendations that lead to several specific 

recommendations: 

Recommendation: NAEP and state tests for students through Grade 8 

should focus on and adequately represent the Panel’s 

Critical Foundations of Algebra. Student achievement 

on this critical mathematics content should be reported 

and tracked over time. 

Recommendation: State tests and NAEP must be of the highest 

mathematical and technical quality. To this end, states 

and NAEP should develop procedures for item 

development, quality control, and oversight to ensure that 

test items reflect the best item-design features, are of the 

highest mathematical and psychometric quality, and 

measure what is intended, with non-construct-relevant 

sources of variance in performance minimized (i.e., with 

nonmathematical sources of influence on student 

performance minimized). 

These recommendations are not independent of each other. What one tests 

and how one chooses to test are intertwined. The background for these 

recommendations is discussed in this section, and additional specific 

recommendations are presented. 

 

16 The detailed work underlying this section was carried out by the Task Group on Assessment 

whose report carries relevant references and more elaborate discussion (National Mathematics 

Advisory Panel, 2008). 
National Mathematics Advisory Panel FINAL REPORT 58

Content 

The mathematical content strands in many state tests are highly similar to 

those in the NAEP tests, although there are striking differences in the weights 

attached to these strands. Thus, the Panel focused its investigation on the NAEP 

content strands, knowing that any suggestions for the NAEP would have 

implications for state mathematics tests as well. 

Table 3 shows the Panel’s recommended content strands for NAEP’s 

mathematics assessments. This new structure is intended to ensure that the content 

strands address what students should be learning. In the Panel’s view, this begins 

with the Critical Foundations of Algebra. 

Table 3: Suggested Reorganization of NAEP Content Strands 

Grade 4 Grade 8 

Number: Whole Numbers Number: Integers 

Number: Fractions and Decimals Number: Fractions, Decimals, and Percents 

Geometry and Measurement Geometry and Measurement 

Algebra Algebra 

Data Display Data Analysis and Probability 

Source: National Mathematics Advisory Panel, 2008. 

Because the most critical skills leading to Algebra concern whole 

numbers, whole-number operations, and facility with fractions, we make the 

following recommendation: 

Recommendation: The Panel suggests that the NAEP strand on “Number 

Properties and Operations” be expanded and divided 

into two parts. The first part should include a focus on 

whole numbers, including whole number operations (i.e., 

addition, subtraction, multiplication, division), at Grade 

4, and on all integers (negative and positive) at Grade 8. 

The second content area involving number should focus 

on fractions. At Grade 4, it should involve beginning 

work with fractions and decimals, including recognition, 

representation, and comparing and ordering. The 

coverage should be expanded to include operations with 

fractions, decimals and percents at Grade 8. Similarly, 

the content of work with whole numbers and fractions 

on state tests should expand and also should cover these 

concepts and operations as they develop from year to 

year, particularly at Grades 5, 6, and 7, which are grade 

levels when the NAEP test is not offered. 

One of the Panel’s greatest concerns is that fractions are underrepresented 

on NAEP. The NAEP Validity Study (NVS; Daro et al., 2007), as well as others, 

have noted the relative paucity of items assessing fractions, particularly within the 

Grade 8 NAEP. (And, teachers have noted the importance of ensuring 

proficiency with fractions before beginning the study of algebra; see the Panel-
National Mathematics Advisory Panel FINAL REPORT 59

commissioned National Survey of Algebra Teachers, National Mathematics 

Advisory Panel, 2008.) Moreover, Daro et al. indicate that half of the Data 

Analysis and Probability section in the Grade 4 NAEP test is probability-related. 

Given the importance of fractions for the conceptual understanding of probability, 

the Panel questions the appropriateness of items related to probability within 

NAEP at Grade 4. Thus, the Panel recommends that this strand at Grade 4 

emphasize well-organized representations of data pictorially and numerically and 

be re-titled as “Data Display.” 

The Panel also recommends a more appropriate balance in how algebra is 

defined and assessed in both the Grade 4 and Grade 8 NAEP. At Grade 4, most of 

the NAEP algebra items relate to patterns or sequences (Daro et al., 2007). 

Although states’ inclusion of patterns in textbooks or as curriculum expectations 

may reflect their views of what constitutes algebra, patterns are not emphasized in 

high-achieving countries (Schmidt, 2007). In the Major Topics of School Algebra 

set forth in this report, patterns are not a topic of major importance. The prominence 

given to patterns in PreK–8 is not supported by comparative analyses of curricula or 

mathematical considerations (Wu, 2007). Thus, the Panel strongly recommends that 

“algebra” problems involving patterns should be greatly reduced in the NAEP. 

It should be noted that the TIMSS content domains were recently changed 

(Mullis et al., 2007), independent of the Panel’s work. If the above 

recommendation was to be adopted by the National Assessment Governing 

Board, NAEP would be brought into greater alignment with TIMSS. 

Performance Categories 

Once content is selected, decisions must be made as to what constitutes 

acceptable performance. The Panel did not investigate what the cut scores or 

standards ought to be, but rather looked at how they should be determined. 

Although the states and NAEP vary in both process and method for such standard 

setting, all six studied states and NAEP employ acceptable educational practices 

to quantify judgments of the standard-setting panelists and to map their judgments 

on to test scores. 

The Panel examined the background of the panelists in NAEP and the six 

states and found that classroom teachers predominate, many of whom are not 

mathematics specialists. The panels used to set performance categories should 

draw on the expertise of mathematicians, mathematics educators, and curriculum 

specialists in education and academia, as well as of teachers and the general 

public. The Panel also found that the standard-setting panelists often do not take 

the complete test as examinees before attempting to set the performance 

categories, and that their judgments are not consistently informed by international 

performance data. Thus, the Panel also suggests that these deficiencies be 

addressed. On the basis of international performance data, there are indications 

that the NAEP cut scores for the two highest performance categories are set too 
National Mathematics Advisory Panel FINAL REPORT 60

high. This does not mean, however, that the mathematical content of the test is too 

hard; it is simply a statement about the location of cut scores for qualitative 

categories such as “proficient” and “beyond proficient.” 

Recommendation: Mathematicians should be included in greater numbers, 

along with mathematics educators, mathematics 

education researchers, curriculum specialists, classroom 

teachers, and the general public, in the standard-setting 

process and in the review and design of mathematical 

test items for state, NAEP, and commercial tests. 

Item and Test Design 

It is important not only that appropriate content is measured and cut scores 

for student proficiency are set appropriately, but also that test scores are valid and 

reliable, and reflect the competencies that are intended to be measured. That is, the 

measurement itself must be carried out in a high-quality and appropriate manner. 

The Panel first examined whether constructed-response formats measure 

different aspects of mathematics competency in comparison with the multiplechoice format. Many educators believe that constructed-response items (e.g., short 

answers) are superior to multiple-choice items in measuring mathematical 

competencies and that they represent a more authentic measure of mathematical 

skill. The Panel examined the literature on the psychometric properties of 

constructed-response items as compared to multiple-choice items. The evidence in 

the scientific literature does not support the assumption that a constructedresponse format, particularly the short-answer type, measures different aspects of 

mathematics competency in comparison with the multiple-choice format. 

The Panel then examined test items for flaws. The NVS reported many 

examples of flawed and marginal items on NAEP and state assessments that could 

affect performance of all or some students (Daro et al., 2007). The Panel probed 

that issue in depth and also concluded that there are too many flawed items on the 

NAEP and state tests, often related to the wording of an item. The Panel classified 

the many flaws discovered in the individual test items into seven general types 

that could introduce non-construct-relevant variance (i.e., unwarranted 

nonmathematical sources of influence) and affect the meaning and accuracy of 

scores. The Panel recommends that test developers be especially sensitive to the 

presence of these types of flaws in the test development process. To further 

ameliorate concerns, significant attention should be devoted to the actual design 

of individual mathematics items and to the evaluation of items for inclusion in an 

assessment. Careful attention must be paid to exactly what mathematical 

knowledge is being assessed by a particular item and the extent to which the item 

is, in fact, focused on that mathematics. To that end, more mathematicians and 

mathematics educators should be involved in the test development process, as 

well as curriculum specialists, linguistics experts, and cognitive psychologists. 
National Mathematics Advisory Panel FINAL REPORT 61

The frequency of flawed items on NAEP and state tests points to another 

possible gap in test development procedures that needs to be addressed. The 

developers of NAEP and state tests use sophisticated psychometric models and 

methods for this highly complex and technical process. Yet, it is the professional 

opinion of the Panel that problems in communication may be an additional 

contributing cause of the number of flawed items found in the NVS and by this 

Panel. Psychometricians are trained to use highly sophisticated statistical models 

and data analysis methods for measurement but are not as familiar with issues of 

designing items to measure specified constructs. In contrast, typical item writers 

and item evaluators have a college degree, but not always in the appropriate 

subject, and, typically, have little or no training in task and item design. 

Moreover, they often receive limited feedback from psychometricians on how the 

items they develop end up functioning. A more interactive feedback mechanism 

with more diagnostic information about item responses would help item writers 

pinpoint the sources of item flaws. 

Use of calculators on assessments is another oft-discussed design issue. 

While findings from the literature indicated that using calculators on assessments 

has no significant short-term impact on performance overall or in problem 

solving, it does affect performance on computation-related items and could also 

change the nature of the competencies assessed. 

Recommendation: Much more attention should be paid to the 

mathematical knowledge being assessed by a particular 

item and to the extent to which the item addresses that 

knowledge. 

Recommendation: Calculators should not be used on test items designed to 

assess computational facility. 

Research Needs 

Recommendation: More research is needed on test item design features 

and how they influence the measurement of the 

knowledge, skills, and abilities that students use when 

solving mathematics problems on achievement tests. 

These design features might have differential impacts 

across various groups (e.g., gender, race, English 

language learners). 
National Mathematics Advisory Panel FINAL REPORT 62
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Chapter 10: 

Research Policies and Mechanisms 

Systematic reviews of research on mathematics education by the task 

groups and subcommittees of the Panel yielded thousands of studies on important 

topics, but only a small proportion met standards for rigor for the causal questions 

the Panel was attempting to answer. The dearth of relevant rigorous research in 

the field is a concern. First, the number of experimental studies in education that 

can provide answers to questions of cause and effect is currently small. Although 

the number of such studies has grown in recent years due to changes in policies 

and priorities at federal agencies, these studies are only beginning to yield 

findings that can inform educational policy and practice. Second, in educational 

research over the past two decades, the pendulum has swung sharply away from 

quantitative analyses that permit inferences from samples to populations. Third, 

there is a need for a stronger emphasis on such aspects of scientific rigor as 

operational definitions of constructs, basic research to clarify phenomena and 

constructs, and disconfirmation of hypotheses. Therefore, debates about issues of 

national importance, which mainly concern cause and effect, have devolved into 

matters of personal opinion rather than scientific evidence. 

Causal knowledge is essential to produce and to evaluate scientific 

research in crucial areas of national need, including mathematics education. 

Specifically, research is needed that identifies: 1) effective instructional practices 

and materials, 2) mechanisms of learning, 3) ways to enhance teachers’ 

effectiveness, including teacher education that focus on learning processes and 

outcomes, and 4) item and test features that improve the assessment of 

mathematical knowledge. 

To achieve these goals, the rigor and scale of the federal government’s 

infrastructure for educational research must be dramatically increased. In particular, 

the nation’s research portfolio should be better diversified, increasing experimental 

research at multiple points along a continuum from smaller-scale (less costly but 

highly informative) experiments to large field trials that address problems of major 

national importance. And, to be ready for even small-scale experiments, basic 

research and intervention development studies are needed to bring interventions and 

models to a point such that studying their efficacy is viable. 

Both smaller-scale experiments on the basic science of learning and largerscale randomized experiments examining effective classroom practices are needed to 

ensure the coherent growth of research addressing important questions in 

mathematics education. Basic research on causal mechanisms of learning, as well as 

randomized trials, are essential, and, depending on their methodologies, both can be 

rigorous and relevant to educational practice. Basic research, in particular, is 

necessary to develop explicit predictions and to test hypotheses, which are 

underemphasized in current research on mathematics education. 
National Mathematics Advisory Panel FINAL REPORT 64

There are three elements that are essential to produce the needed quality 

and quantity of research: 1) a sufficient supply of competent researchers dedicated 

to areas of critical national need; 2) a sufficient supply of willing schools and 

practitioners who have the time, resources, and motivation to be partners in 

research and to use the findings of research in decision making; and 3) a sufficient 

and stable source of funding for quality research and training with appropriate 

peer review. Streamlining human subjects’ protection procedures for qualified, 

low-risk research would be a major factor in encouraging researchers to conduct 

educationally relevant research. In addition, the supply of researchers can be 

increased by improving the training of researchers in education, by encouraging 

qualified researchers from closely related fields to retrain in education, and by 

fostering collaborative, interdisciplinary research teams (such as those developed 

by the Social Science Research Council and others during the post-Sputnik period 

(Brown, 1970; Morrissett & Vinsonhaler, 1965)). 

Recommendation: Leaders of graduate programs in education and related 

fields should ensure attention to research design, 

analysis, and interpretation for teachers and those 

entering academic and educational leadership positions 

in order to increase the national capacity to conduct 

and utilize rigorous research. 

Recommendation: New funding should be provided to establish support 

mechanisms for career shifts (K, or career, awards from 

the National Institutes of Health represent one example). 

Many accomplished researchers who study the basic 

components of mathematics learning are not directly 

engaged in relevant educational research. While this 

more basic kind of research is important both in its own 

right and as a crucial foundation for designing classroomlevel learning projects, at least some of these investigators 

have the potential to make more directly relevant 

contributions to educational research. Consequently, 

providing incentives for them to change the emphasis of 

their research programs could enhance research capacity 

in the field. 

Recommendation: Support should be provided to encourage the creation of 

cross-disciplinary research teams, including expertise in 

educational psychology, sociology, economics, cognitive 

development, mathematics, and mathematics education. 

Recommendation: PreK–12 schools should be provided with incentives and 

resources to provide venues for, and encourage 

collaboration in, educational research. 
National Mathematics Advisory Panel FINAL REPORT 65

Recommendation: Unnecessary barriers to research should be lowered. 

Although existing guidelines for the protection of 

human subjects must be fully respected, Institutional 

Review Board procedures should be streamlined for 

educational research that qualifies as being of low or 

minimal risk. The resolutions of the National Board for 

Education Sciences concerning making individual 

student data available to researchers with appropriate 

safeguards for confidentiality should be supported. 

In summary, to produce a steady supply of high-quality research that is 

relevant to classroom instruction, national capacity must be increased: More 

researchers in the field of mathematics education must be prepared, venues for 

research must be made accessible, and a pipeline of research must be funded that 

extends from the basic science of learning, to the rigorous development of 

materials and interventions to help improve learning, to field studies in 

classrooms. The most important criterion for this research is scientific rigor, 

ensuring trustworthy knowledge in areas of national need. 
National Mathematics Advisory Panel FINAL REPORT 66
National Mathematics Advisory Panel FINAL REPORT 67

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APPENDIX A: Presidential Executive Order 13398 
National Mathematics Advisory Panel FINAL REPORT 72
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APPENDIX B: Rosters of Panel Members, Staff, 

and Consultants 

Panelists 

• Larry R. Faulkner (Chair), President, Houston Endowment Inc.; President 

Emeritus, University of Texas at Austin 

• Camilla Persson Benbow (Vice Chair), Patricia and Rodes Hart Dean of 

Education and Human Development, Peabody College, Vanderbilt 

University 

• Deborah Loewenberg Ball, Dean, School of Education and William H. 

Payne Collegiate Professor, University of Michigan 

• A. Wade Boykin, Professor and Director of the Graduate Program, 

Department of Psychology, Howard University 

• Douglas H. Clements, Professor, Graduate School of Education, 

University at Buffalo, State University of New York (Began with the 

Panel March 19, 2007)

• Susan Embretson, Professor, School of Psychology, Georgia Institute of 

Technology (Began with the Panel March 19, 2007) 

• Francis “Skip” Fennell, Professor of Education, McDaniel College 

• Bert Fristedt, Morse-Alumni Distinguished Teaching Professor of 

Mathematics, University of Minnesota, Twin Cities (Began with the Panel 

March 19, 2007) 

• David C. Geary, Curators’ Professor, Department of Psychological 

Sciences, University of Missouri

• Russell M. Gersten, Executive Director, Instructional Research Group; 

Professor Emeritus, College of Education, University of Oregon 

• Nancy Ichinaga, Former Principal, Bennett-Kew Elementary School, 

Inglewood, California (Served with the Panel through May 29, 2007) 

• Tom Loveless, The Herman and George R. Brown Chair, Senior Fellow, 

Governance Studies, The Brookings Institution 

• Liping Ma, Senior Scholar, The Carnegie Foundation for the 

Advancement of Teaching
National Mathematics Advisory Panel FINAL REPORT 76

• Valerie F. Reyna, Professor of Human Development, Professor of 

Psychology, and Co-Director, Center for Behavioral Economics and 

Decision Research, Cornell University 

• Wilfried Schmid, Dwight Parker Robinson Professor of Mathematics, 

Harvard University 

• Robert S. Siegler, Teresa Heinz Professor of Cognitive Psychology, 

Carnegie Mellon University 

• James H. Simons, President, Renaissance Technologies Corporation; 

Former Chairman, Mathematics Department, State University of New 

York at Stony Brook 

• Sandra Stotsky, Twenty-First Century Chair in Teacher Quality, 

University of Arkansas; Member, Massachusetts State Board of Education 

• Vern Williams, Mathematics Teacher, Longfellow Middle School, Fairfax 

County Public Schools, Virginia 

• Hung-Hsi Wu, Professor of Mathematics, University of California 

at Berkeley 

Ex Officio Members 

• Irma Arispe, Assistant Director for Life Sciences and Acting Assistant 

Director for Social and Behavioral Sciences, Office of Science and 

Technology Policy, Executive Office of the President (Began with the 

Panel May 30, 2007) 

• Daniel B. Berch, Associate Chief, Child Development and Behavior 

Branch and Director, Mathematics and Science Cognition and Learning 

Program, National Institute of Child Health and Human Development, 

National Institutes of Health 

• Joan Ferrini-Mundy, Division Director, Division of Research on Learning 

in Formal and Informal Settings, National Science Foundation (On an 

Intergovernmental Personnel Act Assignment from Michigan State 

University. Began with the Panel January 16, 2007) 

• Diane Auer Jones, Deputy to the Associate Director for Science, White 

House Office of Science and Technology Policy (Served with the Panel 

through May 23, 2007) 

• Thomas W. Luce, III, Assistant Secretary for Planning, Evaluation, and 

Policy Development, U.S. Department of Education (Served with the 

Panel through November 1, 2006) 
National Mathematics Advisory Panel FINAL REPORT 77

• Kathie L. Olsen, Deputy Director, National Science Foundation, (Served 

with the Panel through January 11, 2007) 

• Raymond Simon, Deputy Secretary, U.S. Department of Education 

• Grover J. “Russ” Whitehurst, Director, Institute of Education Sciences, 

U.S. Department of Education 

U.S. Department of Education Staff 

• Tyrrell Flawn, Executive Director, National Mathematics Advisory Panel, 

U.S. Department of Education 

• Ida Eblinger Kelley, Special Assistant, National Mathematics Advisory 

Panel, U.S. Department of Education 

• Jennifer Graban, Deputy Director for Research and External Affairs, 

National Mathematics Advisory Panel, U.S. Department of Education 

• Marian Banfield, Deputy Director of Programs and Special Projects, 

National Mathematics Advisory Panel, U.S. Department of Education 

Additional support was provided by the following: Anya Smith, Director 

of Special Events and the Events Team, Office of Communications and Outreach, 

U.S. Department of Education; Holly Clark, Management and Program Analyst, 

Office of Innovation and Improvement, U.S. Department of Education; Mike 

Kestner, Math and Science Partnership Program, Office of Elementary and 

Secondary Education, U.S. Department of Education; Kenneth Thomson, 

Presidential Management Fellow, Office of Planning, Evaluation, and Policy 

Development, U.S. Department of Education; and Jim Yun, Math and Science 

Partnership Program, Office of Elementary and Secondary Education, U.S. 

Department of Education. 
National Mathematics Advisory Panel FINAL REPORT 78

Consultants 

• Alina Martinez, Abt Associates, Inc., Project Director 

• Ellen Bobronnikov, Abt Associates, Inc. 

• Fran E. O’Reilly, Abt Associates, Inc. 

• Mark Lipsey, Vanderbilt University 

• Pamela Flattau, Institute for Defense Analyses Science and Technology 

Policy Institute, Project Director 

• Nyema Mitchell, Institute for Defense Analyses Science and Technology 

Policy Institute 

• Kay Sullivan, Institute for Defense Analyses Science and Technology 

Policy Institute 

• Jason Smith, Widmeyer Communications, Project Director 

• Sara Appleyard, Widmeyer Communications 

• Phyllis Blaunstein, Widmeyer Communications 

• Alix Clyburn, Widmeyer Communications 

• Jessica Love, Widmeyer Communications 
National Mathematics Advisory Panel FINAL REPORT 79

APPENDIX C: Organization and Operation 

of the Panel 

The National Mathematics Advisory Panel (often called the “National 

Math Panel,” NMP, or Panel) comprises 24 members designated by the Secretary 

of Education. Nineteen of the members are experts not employed by the federal 

government and five are ex officio designees from federal agencies. The 

members were sworn into service and the Panel began its work on May 22, 2006. 

Some key dates in the Panel’s work are as follows: 

• April 18, 2006—Establishment of the National Mathematics Advisory 

Panel through Executive Order 13398 

• May 15, 2006—Secretary Spellings announces Panel members 

• May 22, 2006—Panel members sworn into service 

• January 11, 2007—Preliminary report filed and accepted by the Panel 

• March 13, 2008—Official release of the Final Report 

Panel Meetings 

Twelve meetings were held around the country as detailed in Appendix D. 

Please refer to the U.S. Department of Education Web site for more information 

about the meetings: http://www.ed.gov/about/bdscomm/list/mathpanel/ 

meetings.html. 

At most meetings, the Panel used a portion of its time working in task 

groups with the balance in public sessions, receiving testimony and holding 

preliminary public discussions about progress in the task groups. Much of the 

testimony was organized by the Panel to cover particular topics, such as 

textbooks, TIMSS, NAEP, and the use of technology, but a portion was allocated 

to open testimony on a first-come, first-served basis by individual members of the 

public or interested organizations. Seventy-one people provided public testimony 

through the meeting of October 2007. The meetings in November and December 

2007 were entirely dedicated to reports from task groups and to the synthesis of 

this Final Report. All work at these later meetings was carried out in public 

sessions. The proceedings of all meetings have been recorded and documented 

through extensive minutes. Please refer to U.S. Department of Education Web 

site for more information on the public testimony received: http://www.ed.gov/ 

about/bdscomm/list/mathpanel/index.html. 
National Mathematics Advisory Panel FINAL REPORT 80

Organizations likely to have an interest in the Panel’s work were contacted 

by mail to inform them of the work plan, and to solicit their advice and comments on 

matters of particular concern. In December 2006 and October 2007, the Department 

invited these stakeholders to briefings in Washington, D.C., at which the Chair 

discussed the Panel’s process and progress and answered questions from attendees. 

Task Groups and Subcommittees 

The Panel chose to divide into task groups focused on detailed 

examination of topics set forth in the Executive Order. The full range of issues 

was covered in a phased process, with new efforts undertaken as earlier issues 

were completed. Subcommittees were charged with completion of a particular 

advisory function for the Panel. The task groups and subcommittees reported 

periodically to the entire Panel. Their work products were reviewed in progress by 

the Panel as a whole and were formally received by the Panel when completed; 

however, the reports of task groups and subcommittees are presented by only 

those members who participated in creating them. As described below, the Panel 

later incorporated some elements of these reports into this Final Report. These 

reporting documents are all available on the U.S. Department of Education Web 

site at http://www.ed.gov/about/bdscomm/list/ mathpanel/meetings.html. 

The task groups and subcommittees were established as follows: 

• Task Groups 

— Conceptual Knowledge and Skills 

— Learning Processes 

— Instructional Practices 

— Teachers and Teacher Education 

— Assessment 

• Subcommittees 

— Standards of Evidence 

— Survey of Algebra I Teachers 

— Instructional Materials 

Please refer to Appendix E for a roster of task group and subcommittee 

members. 

Synthesis and Submission of This Report 

The Panel as a whole synthesized this report, largely from the reports of the 

task groups and subcommittees. Three synthesis teams were appointed to develop 

parallel concepts for the Final Report, using the reports of the task groups and 

subcommittees as a basis. The team leaders then worked with the Chair and Vice 

Chair to set out a “common concept” for subsequent development by the Panel. The 

synthesis process began prior to the Phoenix, Arizona, public meeting and continued 

until the Final Report was adopted by the Panel as a whole. 
National Mathematics Advisory Panel FINAL REPORT 81

The Panel submitted the Final Report to the Secretary of Education and the 

President of the United States on March 13, 2008. The Final Report was officially 

released to the public on that date, as well. 

Standards of Evidence 

The President’s Executive Order called for the Panel to marshal the best 

available scientific evidence and offer advice on the effective use of the results of 

research related to proven, effective, and evidence-based mathematics instruction. 

The Panel’s assertions and recommendations, therefore, are grounded in the highestquality evidence available from scientific studies. 

So that the Panel could be systematic in identifying the quality of evidence 

on which its assertions and recommendations were based, criteria for classifying 

evidence were developed through a two-level process. The Subcommittee on 

Standards of Evidence formulated general principles applicable to the Panel as a 

whole and to all of its task groups and subcommittees. In general, these principles 

call for strongest confidence to be placed in studies that 

• Test hypotheses 

• Meet the highest methodological standards (internal validity) 

• Have been replicated with diverse samples of students under conditions 

that warrant generalization (external validity) 

These principles are amplified in the excerpt from the subcommittee report 

at the end of this appendix. In addition, the Panel relied on expert professional 

judgment to address questions about the structure and content of mathematics as a 

subject and discipline. 

The Report of the Subcommittee on Standards of Evidence is located with 

the reports of all task groups and subcommittees, and can be found on the U.S. 

Department of Education Web site at http://www.ed.gov/MathPanel. 

Standards of evidence were developed and expressed in a more particular 

way at the task-group level, because the character and form of relevant evidence 

differ across the wide range of concerns addressed by the task groups. 

Accordingly, each task group report includes a detailed description of how the 

task group handled evidence in its particular substantive area. In effect, these 

sections show how the Panel-wide standards of evidence were manifested in the 

work of the individual groups.17

The task groups received support in their survey of the research literature 

and other relevant materials through contracts with Abt Associates and the 

Institute for Defense Analyses Science and Technology Policy Institute (IDA 

 

17 The Report of the Subcommittee on Standards of Evidence also contains brief summaries of 

these sections from the task group reports. 
National Mathematics Advisory Panel FINAL REPORT 82

STPI). Abt carried out searches to capture high-quality, relevant research using 

criteria defined by each task group for its own needs. The results were examined 

directly by the task groups. The criteria set for searches were meant to exclude 

only clearly irrelevant sources. All final decisions about the rigor, adequacy, and 

inclusion of sources in the research literature were made exclusively by Panel 

members working in task groups. IDA STPI performed some original research 

and analysis using a variety of resources such as national reports and state 

education Web sites. 

The Panel as a whole reviewed more than 16,000 research studies and 

related documents. Yet, only a small percentage of available research met the 

standards of evidence and could support conclusions. 

Excerpt from the Report of the Subcommittee on 

Standards of Evidence 

Background: Categories of Internal and External Validity 

There are three broad categories into which one can categorize research and 

the corresponding claims based on that research. First, there is the highest quality 

scientific evidence, based on considerations such as the excellence of the design, 

the validity and reliability of measures, the size and diversity of student samples, 

and similar considerations of internal (scientific rigor and soundness) and external

validity (generalizability to different circumstances and students). The importance 

of scientific theory and hypothesis testing, especially the active search for 

disconfirmation, cannot be overstated in judging the quality of research (see Platt, 

1964). Furthermore, scientifically supported theory provides the surest path to 

generalization (Lewin, 1951). Hence, the Panel’s strongest confidence will be 

reserved for studies that test hypotheses, that meet the highest methodological 

standards (internal validity), and that have been replicated with diverse samples of 

students under conditions that warrant generalization (external validity). 

In addition to reviewing the best scientific evidence, the Panel is also 

charged with considering promising or suggestive findings that should be the 

subject of future research. Promising or suggestive studies do not meet the highest 

standards of scientific evidence, but they represent sound, scientific research that 

needs to be further investigated or extended. For example, laboratory studies 

showing significant effects of “desirable difficulties” (i.e., difficulties produced 

by challenging to-be-learned material) or of repeated testing on long-term 

retention could be extended to actual classrooms or existing curricula (e.g., Bjork, 

1994; Roediger & Karpicke, 2004). The final category corresponds to statements 

based on values or weak evidence; these are essentially unfounded claims and will 

be designated as opinions as opposed to scientifically justified conclusions. 
National Mathematics Advisory Panel FINAL REPORT 83

Quantity, Quality, and Balance of Evidence 

Strong Evidence 

All of the applicable high quality studies support a conclusion (statistically 

significant individual effects, significant positive mean effect size, or equivalent 

consistent positive findings) and they include at least three independent studies 

with different relevant samples and settings or one large high quality multisite 

study. Any applicable studies of less than high quality show either a 

preponderance of evidence consistent with the high quality studies (e.g., mean 

positive effect size) or such methodological weaknesses that they do not provide 

credible contrary evidence. Factors such as error variance and measurement 

sensitivity clearly influence the number of studies needed to support a conclusion 

(reflected in such statistics as p-rep, the probability of replicating an effect; 

Killeen, 2005); the number and balance of studies that are indicated above are, 

therefore, merely illustrative. 

Moderately Strong Evidence 

As above but there are fewer than three high quality studies (but at least 

one), or the effects have not been independently replicated by different 

researchers, or they do not involve different samples (i.e., diversity of 

characteristics) and settings. 

Suggestive Evidence 

One of the following: 

• There are some high quality studies that support the conclusion 

(statistically significant effects, significant mean effects) but others that do 

not (nonsignificant), but those that do not are null, not negative 

(nonsignificant effect or mean effects but not significant negative effect). 

Any applicable moderate quality studies show a comparable pattern or 

better. 

• There are no high quality studies, but all the applicable moderate quality 

studies support the conclusion (statistically significant individual effects, 

significant positive mean effect size, or equivalent consistent positive 

findings) and there are at least three such studies. 

Inconsistent Evidence 

The evaluation of mixed evidence depends crucially on the evaluation of 

the quality of the designs and methods of each study. The results of high-quality 

designs trump inconsistent or null results of low-quality designs. Mixed results 

of high and/or moderate quality studies that are not consistent enough to fall into 

any of the above categories, and cannot be adjudicated by methodological 

criteria, are inconclusive. 
National Mathematics Advisory Panel FINAL REPORT 84

Weak Evidence 

Evidence is considered weak when there are low quality studies but no 

applicable high or moderate quality studies. 

To apply such criteria, the studies on which an assertion or 

recommendation is based must each be characterized as “high quality,” “moderate 

quality,” or “low quality.” The standards for those designations will necessarily 

differ for the different kinds of research that are applicable to different issues and 

inferences (Shavelson & Towne, 2002). The primary interest of the Panel is 

experimental and quasi-experimental research designed to investigate the effects 

of programs, practices, and approaches on students’ mathematics achievement. 

On some matters, however, the relevant studies are surveys (e.g., of students’ 

mathematical knowledge). On other matters, the relevant sources represent 

compilations of practice and informed opinion (e.g., regarding the mathematical 

concepts essential to algebra). The methodological quality of individual studies 

will be categorized as part of the documentation for the database for the panel’s 

work using definitions such as the following. 

For studies of the effects of interventions: 

High quality. Random assignment to conditions; low attrition (<20%); 

valid and reliable measures. 

Moderate quality. Nonrandom assignment to conditions with matching, 

statistical controls, or a demonstration of baseline equivalence on important 

variables; low attrition or evidence that attrition effects are small; valid and reliable 

measures. Correlational modeling with instrumental variables and strong statistical 

controls. Random assignment studies with high attrition. 

Low quality. Nonrandom assignment without matching or statistical 

controls. Pre-post studies. Correlational modeling without strong statistical 

controls. Quasi-experimental studies with high attrition. 

For descriptive surveys of population characteristics: 

High quality. Probability sampling of a defined population; low 

nonresponse rate (< 20%) or evidence that nonresponse is not biasing; large 

sample (achieved sample size gives adequate error of estimate for the study 

purposes); valid and reliable measures. 

Moderate quality. Purposive sampling from a defined population, face 

valid for representativeness; low nonresponse rate; moderate to large sample size; 

valid and reliable measures. Probability sample with high nonresponse rate but 

evidence that nonresponse is not biasing. 
National Mathematics Advisory Panel FINAL REPORT 85

Low quality. Convenience sample; high nonresponse rate or evidence that 

it is biasing; small sample size; invalid or unreliable measures. 

For studies of tests and assessments: 

Psychometric standards such as measures of validity, reliability, and 

sensitivity will be used to evaluate tests and assessments (e.g., Anastasi, 1968; 

Cronbach & Meehl, 1955). 
National Mathematics Advisory Panel FINAL REPORT 86
National Mathematics Advisory Panel FINAL REPORT 87

APPENDIX D: Dates and Locations of Meetings 

May 22, 2006—Washington, D.C. 

Hosted by the National Academy of Sciences 

June 28–29, 2006—Chapel Hill, North Carolina 

Hosted by the University of North Carolina at Chapel Hill 

September 13–14, 2006—Cambridge, Massachusetts 

Hosted by Massachusetts Institute of Technology 

November 5–7, 2006—Palo Alto, California 

Hosted by Stanford University 

January 10–11, 2007—New Orleans, Louisiana 

Hosted by Xavier University of Louisiana 

April 19–20, 2007—Chicago, Illinois 

Hosted by Fermi National Accelerator Laboratory 

June 5–6, 2007—Miami, Florida 

Hosted by Miami Dade College 

September 5–7, 2007—St. Louis, Missouri 

Hosted by Washington University in St. Louis 

October 23–24, 2007—Phoenix, Arizona 

Hosted by Arizona State University 

November 28, 2007—Baltimore, Maryland 

December 14–15, 2007—Baltimore, Maryland 

March 13, 2008—Falls Church, Virginia 

Hosted by Longfellow Middle School 
National Mathematics Advisory Panel FINAL REPORT 88
National Mathematics Advisory Panel FINAL REPORT 89

APPENDIX E: Rosters of Task Groups 

and Subcommittees 

Task Groups 

• Conceptual Knowledge and Skills 

— Francis “Skip” Fennell, Chair 

— Larry R. Faulkner 

— Liping Ma 

— Wilfried Schmid 

— Sandra Stotsky 

— Hung-Hsi Wu 

• Learning Processes 

— David C. Geary, Chair 

— A. Wade Boykin 

— Susan E. Embretson (Beginning March 19, 2007) 

— Valerie F. Reyna 

— Robert S. Siegler 

— Daniel B. “Dan” Berch 

• Instructional Practices 

— Russell M. Gersten, Co-Chair 

— Joan Ferrini-Mundy, Co-Chair (Beginning January 16, 2007) 

— Camilla Benbow 

— Douglas H. Clements (Beginning March 19, 2007) 

— Tom Loveless 

— Vern Williams 

— Irma Arispe (Beginning May 30, 2007) 

— Diane Auer Jones (Through May 23, 2007) 

• Teachers and Teacher Education 

— Deborah Loewenberg Ball, Chair 

— James H. Simons 

— Hung-Hsi Wu 

— Raymond Simon 

— Grover J. “Russ” Whitehurst 

• Assessment 

— Camilla Persson Benbow, Chair 

— Susan E. Embretson (Beginning March 19, 2007) 

— Francis “Skip” Fennell 

— Bert Fristedt (Beginning March 19, 2007) 

— Tom Loveless 

— Wilfried Schmid 

— Sandra Stotsky 

— Irma Arispe (Beginning May 30, 2007) 
National Mathematics Advisory Panel FINAL REPORT 90

Subcommittees 

• Standards of Evidence 

— Valerie F. Reyna, Chair 

— Camilla Persson Benbow 

— A. Wade Boykin 

— Grover J. “Russ” Whitehurst 

• Survey of Algebra I Teachers 

— Tom Loveless, Chair 

— Deborah Loewenberg Ball 

— Francis “Skip” Fennell 

— Vern Williams 

• Instructional Materials 

— Robert S. Siegler, Chair 

— Bert Fristedt (Beginning March 19, 2007) 

— Vern Williams 

— Irma Arispe (Beginning May 30, 2007) 

— Daniel B. Berch