the algebra solution to the mathematics reform completing the equation pdf

Understanding the Variables in Mathematics Education 15 How Did We Get Here from There?: Search for the Identity of the High School in the United States 19 Who Should Study Algebra?: Ch

The Algebra Solution

to Mathematics Reform

Completing the Equation

The Algebra Solution

Copyright © 2011 by Teachers College, Columbia University

All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, or any information storage and retrieval system, without permission from the publisher.

Library of Congress Cataloging-in-Publication Data

Spielhagen, Frances R.,

1946-The algebra solution to mathematics reform : completing the equation / Frances

R Spielhagen.

p cm.

Includes bibliographical references and index.

ISBN 978-0-8077-5231-9 (pbk : alk paper)

1 Algebra—Study and teaching (Middle school)—United States 2 Algebra— Study and teaching (Middle school) —Social aspects I Title

QA159.S66 2011

512.9071’2—dc22 2011008527

ISBN 978-0-8077-5231-9 (paper)

Printed on acid-free paper

Manufactured in the United States of America

18 17 16 15 14 13 12 11 8 7 6 5 4 3 2 1

daily to create opportunities for the students in their care Especially among

them is Gerard Spielhagen, my husband and partner, who has empowered

so many to achieve excellence and who has truly been the wind beneath

my wings.

as if it were for yourself.

—Confucius

Acknowledgments ix

1 Mathematics Success: Solving the Equation! 1

Dissension and Confl ict About When to Teach Algebra 4

A Template for Change: Looking at a Model of Success 5 How Did They Do It? The Organization of the Book 10

2 Understanding the Variables in Mathematics Education 15

How Did We Get Here from There?:

Search for the Identity of the High School in the United States 19 Who Should Study Algebra?:

Challenging the Status Quo in the Mid-20th Century 21

Forward Movement and

3 Keeping Track: Setting the Course in Mathematics 27

Case Study: Taking Charge of Change in Chesterfi eld, Virginia 31

4 Student Voices for Change: “Would Have, Should Have,

Could Have Studied Algebra in 8th Grade” 39

Students’ Views on Social Life in Middle School 41

The Relationship of Academics to the Students’ Lives 42 How Were They Selected (or Not) for 8th-Grade Algebra? 42

5 Following the Rules: Teachers as Gatekeepers to Advanced Math 51

Ready or Not, Here They Come!: Meeting Diverse Algebra Students 56

6 Changing the Rules: Long-Term Effects of 8th-Grade Algebra 63

Benefi ts Derived from Studying Algebra in 8th Grade 65 Effects of 8th-Grade Algebra on High School Math Performamce 66

The Effect of 8th-Grade Algebra on College Attendance 67

Mandate for Change in Local Mathematics Policies 70

7 The Solution:

How Educational Policy Changes Mathematics Outcomes 71

Changing the Game Plan for Mathematics Curriculum Delivery 72

References 85 Index 92

At the turn of the 21st century, the words equity and excellence resounded at all

levels of the educational establishment As a career educator with over 30 years’ experience in secondary classrooms, I was as concerned as any teacher who had spent her life trying to connect students to learning and to pave the way for their future success At the same time, I had just begun my doctoral study in educational leadership at Fordham University’s Graduate School of Education Therefore, I resolved that I would explore the ways in which school organization infl uences student learning and how policy affects the opportunities that are afforded to students In the face of all the rhetoric surrounding the “achievement gap” in education, I was more concerned about opportunity gaps that perpetuated low achievement, especially among underserved populations of students

I would like to thank Dr Bruce Cooper, professor of education at Fordham University, for encouraging me to wade boldly into the waters of policy analysis to explore how schools affect student achievement by the structures they create and maintain As my dissertation mentor, Bruce demanded rigor and encouraged me to explore my research questions into hypotheses that produced defensible fi ndings

He has continued to offer support throughout all the subsequent stages of this work

My initial study resulted in more questions than answers, which I then sued from 2003 to 2006 as a postdoctoral research fellow, funded by the American Educational Research Association and the Institute for Educational Sciences I am grateful to Felice Levine, executive director of AERA, and members of the Post-doctoral Fellows Advisory Committee, especially Dr Robert Floden and Dr Wil-liam Trent, for their support and guidance I particularly want to thank Dr Joyce Van Tassel-Baska, professor emerita at the College of William and Mary, who was

pur-my mentor during this fellowship Her wisdom and guidance, coupled with her unfl agging energy, provided a model of scholarship that continues to inform my work My colleagues and the administration at my home institution, Mount Saint Mary College, in Newburgh, New York, have provided collegial support in count-less tacit and overt ways

Indeed, this book would not have been possible without the cooperation and vision of the school leaders of Chesterfi eld County, Virginia, especially Dr Glenn

Miller, assistant superintendent (now retired), and Mr Michael Bolling, ematics specialist, who is currently a curriculum specialist in the State of Virginia Department of Education The teachers and students who shared their stories with

math-me helped math-me understand the issues in depth I am grateful to them for their ness and willingness to explore the many important questions I posed to them.Finally, I would like to thank my husband, Jerry Spielhagen, for his support

open-as a mathematician and statistician, open-as well open-as being my personal cheerleader I

am grateful to my daughter, Amy, and her husband, Briggs, and my son, Jeremy, for their patience over the years in which I have delved deeply into this work My daughter is fond of saying “If you ask my mom about her research, she will tell you.” I am happy to share the success story that will unfold in the chapters that follow in the hopes that it can serve as a source of conversation and a catalyst for change

Mathematics Success

Solving the Equation!

All school districts should ensure that all prepared students should have access to an authentic algebra course—and should prepare more students than at present to enroll in such a course by Grade 8

—National Mathematics Advisory Panel, 2008, p 23One of the most diffi cult challenges facing teachers of mathematics, and other subjects, is the wide range of students they teach Mathematics classes often include students with low motivation and weak knowledge alongside others with advanced understanding and high motivation

—Boaler, 2006, p 40The dilemma that confronts educational policymakers is clear How can we edu-cate all students in a democracy? The United States was founded on the unalien-able principle of equality for all, but what does that really mean for how and what we teach, especially in math? If mathematics study is a key to success for both the individual and the nation, how can schools guarantee that success is at least a possible outcome of their endeavors? The task of providing appropriate mathematics curriculum for students across the diverse school situations in the United States has plagued school leaders for decades Algebra remains at the core

of the dilemma

The National Mathematics Advisory Panel soundly endorsed the study of algebra as a cornerstone of mathematics literacy, but the key words in their pre-

scription are all prepared students The Panel further advised that school districts

prepare more students for “authentic” algebra instruction Boaler’s (2006) knowledgment of the range of students in mathematics classes focuses the chal-lenge on the recipients of this instruction What constitutes suitable preparation for the study of algebra? How can teachers deliver curriculum effectively to meet the varying needs of their students?

ac-ALGEBRA AS A CIVIL RIGHT

The debate about algebra rages in the national arena, in state departments of cation, and in local school districts, which ultimately implement policies enact-

edu-ed at the state and local level Should algebra be requiredu-ed of all students? Where should it reside in the traditional mathematics curriculum? Is the study of algebra,

in fact, a civil right?

Deep within the collective understanding of some American educators resides the opinion that students must have a “rare, innate, ability the math gene” (NACME, 1996, p 4) to learn algebra, an ability possessed only by a select few Taylor (1990) calls this a self-fulfi lling prophecy (p 45) Nevertheless, for every curriculum specialist who recommends the study of algebra, another insists that universal attainment of algebra is unnecessary and even impossible Nor is this a new phenomenon

In 1990, the National Research Council advised that formerly exclusive ematics courses must become inclusive “To achieve an equitable society, we must change the algebra course from a fi lter that screens out segments of our popula-tion to a pump that propels all students toward opportunity” (Taylor, 1990, p 45)

math-In 1995, the College Board weighed in on the side of increased access to algebra as

a means to increased equity and to increased excellence Quite simply, it is sible for a student who does not study algebra by at least 9th grade to attain the mathematics courses needed to attend a 4-year college

impos-The vested interest of the College Board in promoting the study of algebra may well lie in the justifi cation of its elaborate testing system Nevertheless, the re-sults of its research (Pelavin & Kane, 1990) suggested that minority students who master algebra and geometry and have aspirations to go to college actually enroll

in college at the same rate as their nonminority peers who take those same courses Therefore, the College Board’s Equity 2000 strategy advocated that all students complete Algebra I by the 9th grade

At the turn of the 21st century, the National Council of Teachers of ematics (NCTM) published a series of dialogues on the role of algebra in Ameri-can schools In this volume, one experienced mathematics teacher protested, “To think that all students are ready to study algebra in high school is nạve and cer-tainly contrasts with the reality of our society Although everyone needs skills

Math-to solve problems, not everyone needs Math-to use algebra Math-to solve them” (Bagwell,

2000, p 9) In the same journal, noted educational philosopher Nel Noddings (2000) concurred that the study of algebra is not necessary for all students In-stead, arguing from a social justice perspective, she proposed a course on “how students contribute to their own lower economic status by refl ective resistance to mathematics courses ” (p 2)

Despite these voices to the contrary, many policymakers have long regarded gebra as critical to personal, professional, and economic advancement Taylor (1990; see also Lawson, 1990; McKnight et al., 1987; Oakes, 1985; Ravitch, 2000; and Sells, 1978) considered algebra to be “the fork in the road where one direction leads to opportunity and the other to limited options for further education and promising careers None of the other disciplines has a similar decision point” (p 45) Moses and Cobb (2001) further asserted that algebra is more than the key to advanced study and understanding of technology, because “in today’s world, economic access and full citizenship depend crucially on math and science literacy” (p 5)

al-In May 1996, the National Action Council for Minorities in Engineering ported that “A majority of children (about 60%) and the bulk of minority children (about 85%) are channeled out of mathematics and science when they are only 13 years old” (p 1) In 1995, NACME had reported that students were opting out of mathematics study without real understanding of the implications of their deci-sions in terms of further career options “Fifty-three percent of young minority students and 52 percent of young non-minority students say no one has spoken

re-to them about their academic options regarding math courses or about the pact this will have on their future” (NACME, 1995, p 3) Moses and Cobb (2001) concurred that in 1995, Blacks comprised only 2.1% of the PhDs in engineering, 1.8% in computer science, and 0.6% in mathematics They maintain that “because

im-of how access to the learning im-of algebra was organized in the industrial era, its place in society under the old jurisdiction, it has become, not a barrier to college entrance, but a barrier to citizenship” (p 14) While making its deliberations about the status of algebra in the curriculum, the National Mathematics Advisory Panel was informed that the study of algebra is foundational for all mathematics and science (Bass, 2006)

IS 9TH GRADE EARLY ENOUGH TO STUDY ALGEBRA?

In 2008, Skip Fennell, then president of the National Council of Teachers of ematics, reported that at that time about 40% of students in the United States were enrolled in algebra or a higher-level math course He agreed that “the path to

Math-a good job often begins with Math-algebrMath-a” (Fennell, 2008), but he questioned whether mandating algebra for 8th-grade students (or even earlier) was a good idea He ultimately concluded that early access to algebra might be a good idea for many students, but he emphasized the importance of preparing students for algebra be-fore they actually begin its study

Preparation for algebra begins long before 8th or 9th grade School districts that fail to ramp up the curriculum leading to the study of algebra are doomed

for failure In 1997 Chicago began requiring algebra for all students in 9th grade Twenty years later, researchers reported that algebra enrollment had increased, but that failure rates in algebra had also increased (Allensworth, 2009) Without sup-port for both teachers and students, the Chicago initiative failed to achieve the goal of signifi cant test-score gains, a universal measure of the success of school reform This measure of success, however, begs the question of the possible posi-tive effects of algebra study for individual students and the potential to open up advanced mathematics to a wider range of students, notably those who have been traditionally underrepresented in advanced and college preparatory courses.

Despite this cautionary tale, school districts have begun to implement algebra

for all policies for 8th-grade students In 2008, California’s State Board of

Educa-tion ruled that all students must pass the state algebra test in 2011, giving local districts 3 years to ramp up to the mandate This prescription was quickly struck down by the California State Supreme Court on the grounds that the state board

of education had overstepped its jurisdiction Vigorous, and as yet unresolved, litigation and counter litigation followed the Supreme Court decision

Meanwhile, a report by the Noyce Foundation indicated that nationally explainably large numbers of 8th-grade algebra students are being assigned to re-peat algebra in high school, to their detriment At least half of these students end

“un-up doing worse in the course the second time around A high proportion of the repeaters are non-Asian minority students, the data indicate” (Fensterwald, 2010) However, the report also explains that from 2003 to 2008 enrollment of 8th-graders

in algebra increased 63%, and the number of students who actually passed the state Algebra I test increased 76% Despite the unexplained repetition of the al-gebra course among non-Asian minority students, overall the rate of profi ciency actually rose from 39 to 42%

DISSENSION AND CONFLICT ABOUT WHEN TO TEACH ALGEBRA

Soon after the publication of the report from the National Panel in March 2008,

Education Week reported that some of the members of the National Panel were

reported to consider the California 8th-grade algebra initiative to be ill-advised (Cavanaugh, 2008) Indeed, some of these panelists were known to have dis-agreed with one another in the writing of the report Among them, Tom Loveless (2008) of the Brookings Institute has been the most vocal about those he consid-ers “misplaced math students,” that is, students who are unprepared for the study

of algebra He notes that these misplaced students are more likely to come from low-income families qualifying for federal free or reduced-price lunch and are overwhelmingly Black and Hispanic

Backlash to the National Panel’s report itself yielded interesting insights into this debate Scholars associated with the American Educational Research Asso-ciation (December 2008) produced a concerted response to the report, basically assailing it for its research methodology and reliance on quasi-experimental re-search studies Boaler (2008b) chided the panel for ignoring an important study

by Burris, Heubert, and Levin (2006), which found that participation of minority students and low-income students increased in advanced math courses when they were grouped heterogeneously Boaler’s objections centered not on the National Panel’s recommendation that greater numbers of students should be prepared to study algebra, but rather on the failure to include the Burris study because its methodology did not meet the Panel’s narrow guidelines for acceptable studies.Confusion seems to pervade all discussion of the place of algebra in the na-tional mathematics curriculum Indeed, the debate over the delivery of algebra rages primarily because there is no national curriculum in place Even as rhetoric about national standards permeates discussions among education policymakers,

the states’ rights debate continues to sabotage algebra for all initiatives The lack of

consensus about the appropriate place of algebra in the mathematics curriculum

is mired in the outdated concept of local control of educational standards.Meanwhile, generations of students in the United States continue to be at the mercy of a disorganized mathematics curriculum and fl ounder on national and global measures of achievement The most recent publication of international assessments by the Program in International Student Assessment (PISA) in 2006 revealed a drop in the mathematics scores of 15-year-olds in the United States in the 3 years since the last report (Organisation for Economic Cooperation and De-velopment, 2007) Moreover, Darling-Hammond (2010) maintains that this drop

is indicative of unequal access of underserved student populations to higher-order thinking and advanced math courses She attributes this decline to the “political forces [that] have repeatedly pushed most mathematics teaching in the United States back to drill-and-practice methods at odds with what research shows are the most effective strategies for developing high levels of mathematical competence” (p 12) The most recent PISA results (Organisation for Economic Cooperation and Development, 2010) indicated that the 2009 scores resulted in an even lower score for students in the United States

A TEMPLATE FOR CHANGE: LOOKING AT A MODEL OF SUCCESS

In the face of the rhetoric surrounding mathematics literacy, in spite of (or cause of) the advice and recommendations of NCTM and the National Panel, and

be-in the absence of cohesive national be-initiatives, local school districts have struggled

to defi ne and redefi ne their math curricula One large southern district,

Chester-fi eld, Virginia, took active measures to revamp its mathematics curriculum and provide strong algebra instruction for all of its students in the 8th grade To begin this initiative, the district sought the input of an independent and objective evalu-ator, who would analyze existing data for possible evidence of inequitable access

to its advanced mathematics classes

As that evaluator, I examined entrance and performance data of students

in all the 8th-grade mathematics classes in the district starting in 2000 and lowed their performance over the next 8 years The initial project began as pro bono analysis that developed into my doctoral dissertation From 2003 to 2006, I conducted follow-up analysis as a post-doctoral research fellow sponsored by the American Educational Research Association and the Institute for Educational Sci-ence From 2006 to 2008, I continued my analyses pro bono I reported annually

fol-to district administrafol-tors and received no payment for my evaluation In this role,

I was able to observe fi rsthand the efforts made by the district to address pockets

of inequity and the success that followed This book chronicles the story of the success of this district (Please note that the names of all teachers and students discussed in this book have been changed.)

Understanding the Context of Change

Located in the southeastern United States, Chesterfi eld County, Virginia, serves 60,000 students and was similar to other school districts across the nation, where long-standing tracking policies institutionalize inequity The district covers ap-proximately 450 square miles Two-thirds of the population of 300,000 is White or Caucasian/European, approximately one third is Black or African American, and a small percentage (about 3%) is “other,” mixed Asian, Hispanic, or Native American

Previous District Policy

In 2000, district-level administrators became actively concerned about both the quality and the equity of the established mathematics curriculum and launched a systematic effort to examine the placement and tracking policies that had long defi ned placement in 8th-grade algebra Long-standing district policy involved a complex nomination system that provided enriched mathematics in-struction to some students starting as early as 3rd grade In the 6th grade, some of these students took a local testing instrument that had been designed by teachers

to predict readiness for algebraic concepts Scores on this readiness test as well

as teacher nomination provided entrance into honors mathematics in 7th grade Students who successfully completed 7th-grade honors-level mathematics usually

advanced to Algebra I in 8th grade However, standardized test scores (at that time, SAT 9) were also used to select students for 8th-grade algebra

Reform Efforts

This complex, somewhat Byzantine, selection process proved to be inherently

fl awed, because it resulted in underrepresentation of Black and minority students

in the 8th-grade algebra classes Based on the analysis of their existing test scores and curriculum protocols, district-level administrators, notably the mathemat-ics coordinator, began to revamp both the curriculum and the protocols for en-trance into to 8th-grade algebra Gradually but consistently, each year the schools opened access 8th-grade algebra to larger numbers of students, steadily increasing the number of students in the 8th-grade algebra classes After 8 years, in 2009, the district provided standard and rigorous algebra instruction for all students by 8th grade, with the exception of the most challenged special education students Some students completed algebra in 7th grade Others took 2 years to fi nish the standard course Most completed the course in their 8th-grade year Of these students, 92% passed the state algebra test, ranking among other students across the state, stu-dents that were in 9th grade!

This success was not easily attained, nor was it simple This curriculum sion required rigorous refl ection at all levels in the district, from the classroom teacher to the superintendent Consistent policy changes were implemented grad-ually, with supports as needed for all personnel, as well as for the students It took

revi-8 years to achieve this goal, and each step of the transformation involved thought, forethought, and refl ection

In every possible way, this success story meets the stringent prescriptions of the most vocal critics of the 8th-grade algebra program Loveless (2008) lists four key “Elements of a Realistic Algebra Program” (pp 12–13)

1 Get the goal right: Focus on learning, not completing a course

2 Teach and assess prerequisite skills: Focus on elementary mathematics leading to algebra

3 Early intervention: Build an infrastructure for all students leading to 8th grade

4 Collect data, conduct research: Examine the results of what you are doing

As the researcher of record for this transformation, I can attest to the oughness with which this district approached this policy reform Every step of the way, district administrators requested specifi c data analysis from me, their external and objective evaluator They asked good questions and took my fi ndings into

thor-consideration in their planning during the 8 years that I worked with them over, they also conducted their own research and shared their results with me so that my conclusions about their efforts could be cohesive and accurate This was clearly a situation where policymakers “got it right” about implementing research-based reform.

More-Reform Outcomes

As a result of these efforts, this district turned around mathematics ment for its students over a period of 5 years Whereas only 30.7% of the dis-trict’s 8th-grade students were enrolled in algebra in 2004–2005, effectively 90% (89.9%) of the 8th-graders were studying algebra in the 2008–2009 school year (see Figure 1.1)

achieve-Moreover, in this increased pool of algebra course-takers, 92% passed the state’s Algebra I examination, scoring a bit lower than the state average (94%) that includes students studying algebra in 9th grade Even more powerful are the results for various subgroups within the 8th-grade population that was studying Algebra I In every group, students scored comparably with the other test-takers

in the state, even though the larger sample included students who were studying algebra in the 9th grade (see Figure 1.2) The extra year of mathematics prepara-tion did not seem to make an appreciable difference in the test scores Admittedly, the high-passing scores on the state algebra test may indicate a larger problem of the rigor of the test Nevertheless, the fact remains that on the state measure, the 8th-grade students in this district score comparably well when compared to others

in the larger population across the state As long as state tests are a universally cepted a measure of student success, these results are worth consideration

ac-Figure 1.1 Changes in Algebra I Enrollment, 2005–2009

2004–2005

District administrators were not content with simple passing scores They pointed out that the “advanced pass rate” among their students taking the state Algebra I test has also increased somewhat steadily since they have implemented the 8th-grade algebra policy (see Figure 1.3)

Figure 1.2 Passing Rates by Demographics, 2009

Male Female

Figure 1.3 Advanced Pass Rates on State Algebra 1 Test, 2005–2009

2006–2007 2005–2006

HOW DID THEY DO IT? THE ORGANIZATION OF THE BOOK

The case study in this book chronicles the intricacies of this policy reform because

it was both well-planned and effi ciently implemented The experiences of those involved in this reform also reveal the dynamics of real policy change My role has been simply to help them implement change by providing feedback on what their policy involved and how the changes they made over time were affecting student performance Ultimately, the efforts of these school leaders resulted in both excel-lence and equity that extends beyond the rhetoric usually implied by those loaded terms This book outlines how all stakeholders were brought into the process of changing policy for the good of the students, the stakeholders most affected by the change This recipe for success can be replicated by other districts seeking to revamp curriculum in realistic ways and achieve the much-debated goal of 8th-grade algebra for all students The chapters that follow provide a comprehensive view of the problem presented by this case study and how this district successfully solved the algebra equation

Chapter 2—Understanding the Variables in Mathematics Education

The historical foundations of the national algebra debate reveal the “givens,” that is, the factors, in the mathematics equation that have defi ned the mathemat-ics curriculum in the United States for over the past 150 years More than a his-tory lesson, this chapter provides a substantial account of the forces at play when discussing mathematics in a democratic society These forces, fi rst evidenced in the 19th and 20th centuries, continue to control the debate about where algebra belongs in the curriculum and who should study it Understanding these forces can further inform our efforts in the 21st century to benefi t greater numbers of students and increase mathematics literacy in this country

Chapter 3—Keeping Track: Setting the Course in Mathematics

One of the inherent problems with promoting mathematics literacy has been the reality that mathematics learning has long been regarded as largely unattain-able by the general population After all, performance records seem to indicate that some people simply don’t get math! As a result, tracking of students has been the predominant delivery model for offering algebra to students and has affected the ways in which math is perceived by the general population This chapter presents a clear explanation of tracking and detracking in math classes in the United States and the implications of tracking policies for equity and overall student performance It examines tracking policies that restrict the study of advanced math concepts and

destroy the infrastructure for algebra readiness Building on the work of Julia Smith and Adam Gamoran, recognized critics of mathematics tracking policies, this chap-ter shows how mathematics tracking in this district impeded student success in algebra in both 8th and 9th grades This chapter summarizes the original analysis

of the case study district’s mathematics program in 2002 that served as the impetus for the reform of the mathematics curriculum Despite commonly held confi dence

in the tracking system by the teachers and some administrators, this chapter plains how the system of tracking students was fl awed and inequitable

ex-Chapter 4—Student Voices for Change:

“Would Have, Should Have, Could Have Studied Algebra in 8th Grade”

Interviews with students who were tracked in math reveal the dynamics of having algebra in 8th grade and the effects of tracking on the students’ lives in terms of the types of instruction they received, the opportunities deriving from their studies, and their future aspirations related to their high school experiences This chapter presents strong research-based evidence of the importance of early mathematics instruction and the ways in which algebra instruction transforms students’ personal goals and overall achievement The students reported that be-ing in higher-level math and science courses broadened their horizons beyond high school and provided a peer group with similarly shared aspirations These students universally attributed their success in high school in both math and sci-ence courses to their early algebra experience Signifi cantly, they credited the cur-riculum they explored, the expectations of well-qualifi ed teachers, the rigorous activities of their courses, and co-curricular benefi ts they accrued because of their mathematics placement

The voices of these students provide on-the-ground insight into the cations of tracking policies on the lives of students who stand to benefi t from instruction that is provided for them The students in this chapter offer a view

impli-of what happens in the various levels impli-of math classes and how their mathematics experiences have affected their high school careers and college application and acceptance In contrast with the perceived success of students who were in higher- level math classes, those who were not in those classes expressed regret about their perceived lack of academic opportunities because they had not taken advanced math courses and reported that their college selection and career plans were af-fected by their early mathematics experiences Moreover, they reported that place-ment in the lower track in mathematics led to interactions with mathematics and science teachers they considered to be less qualifi ed than teachers in the advanced classes They cited rapid turnover among those teachers and a lack of engagement among the teachers and the students

Chapter 5—Following the Rules: Teachers as Gatekeepers for Learning

Teachers are the foot soldiers of mathematics instruction, especially in tary and intermediate grades, when the infrastructure for understanding mathe-matics must be built However, in these grades, teachers often are unprepared for teaching the basic math concepts required in the early grades They may even be math phobic! Through interviews with teachers from grades 3 through 12, this chapter addresses their dispositions toward teaching mathematics and their re-sponses to the policy change to provide algebra for greater numbers of students Mathematics involves a continuum of skills and concepts that build incrementally Articulation across grade levels allows teachers in the intermediate grades to see how the concepts they are teaching provide foundation for skills and concepts their students will learn in middle school Changes in access to advanced mathematics must address the ways in which teachers perceive that curriculum across all levels.This chapter further chronicles the teachers’ reactions to the challenges of teaching higher-level concepts in the intermediate grades and providing sup-ports in the middle-school grades, especially for students who could not afford

elemen-to hire private tuelemen-tors The concepts of “readiness” dominated the teachers’ cerns Preparation, professional development, and collegial support of the teach-ers emerge as key components of the changing mathematics policy

con-Chapter 6—Changing the Rules:

The Long-Term Effects of 8th-Grade Algebra

Schools across the nation are changing their policies regarding math tion This chapter describes efforts in one state to open access to the study of algebra in 8th grade to greater numbers of students Contrary to conventional wisdom, students who were not traditionally expected to succeed in 8th-grade al-gebra not only succeeded but exceeded expectations regarding their performance Given the opportunity to study algebra in 8th grade and the supports they needed, like after-school tutoring and differentiated instruction, students across the entire population achieved at higher levels than usually anticipated Access to 8th-grade algebra resulted in higher test scores and long-term benefi ts in college attendance (Spielhagen, 2007)

instruc-Chapter 7—The Solution:

How Educational Policy Changes Mathematics Outcomes

This chapter summarizes the ways in which policymakers can change the livery and support of mathematics instruction It helps the reader make sense out

de-of current developments in mathematics education by explaining the most recent recommendations of the National Council of Teachers of Mathematics It also ex-plores the implications of the report from the National Mathematics Advisory Panel New directions in mathematics instruction promise improvement in over-all mathematics performance—and those directions all revolve around incorpo-rating early instruction of algebra This chapter summarizes how policy changes can bring about success across student populations

Following the template of successful districts like the one chronicled in this book, successful change results from several specifi c strategies First, districts can require algebra for most students by the end of 9th grade This may mean that some students take 2 years to study algebra It may also mean that some students may study it before 9th grade Success also lies in opening access to 8th-grade algebra to greater numbers of students and providing support for that study As greater numbers of students study algebra in 8th grade, there will be those who are able to study it earlier, in 7th grade Algebra is a milepost, not a goalpost Most important, however, is the provision for algebraic reasoning throughout the in-termediate grades to provide the infrastructure and scaffold for students when-ever they engage in formal algebra study This type of instructional change, with

a policy based in the commitment to increasing mathematics literacy for all dents by increasing opportunities to interact at higher levels across the spectrum

stu-of mathematics instruction, will foster success for the majority stu-of students across the total population

Understanding the Variables

in Mathematics Education

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 fi nancial 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

—Executive Summary, The National Mathematics Advisory Panel, Final Report, 2008The infl ammatory language of the National Mathematics Advisory Panel is eerily

reminiscent of the rhetoric of A Nation of Risk (1983), published a quarter century

earlier, when the mathematics curriculum in schools in the United States came under fi re for providing too little substance to too many students to the detriment

of our national identity and position in the world community At the core of that problem has been the question of appropriate mathematics education for a diverse American population in order to maintain world leadership This is not a new problem Since the early days of the American republic, educators and politicians have sought to defi ne the purpose of education for students in a democracy that

by defi nition seeks to provide opportunities for all citizens, regardless of creed, ethnicity, race, or gender

As demographics in the United States have evolved over the last 2 centuries, educational policy has also evolved, starting with the earliest philosophical debates

in the 19th century, about the essential purpose of school in a democratic society Redefi nition of educational philosophy during the early 20th century ultimately led to intense scrutiny of the issues of equity and access, particularly in math-ematics, that began during the civil rights movement in the mid-20th century and continues into the 2nd decade of the 21st century History shines a light on the

variables that have infl uenced conventional ideas about what constitutes priate mathematics instruction and that have brought us to the impasse today that stymies education policymakers at local, state, and national levels

appro-Recent national and international studies reveal that mathematics literacy among students in the United States has declined over the past 60 years When compared with their peers internationally on the Program in International Stu-dent Assessment (PISA), students in the United States have lost ground in math and science, falling below the international average in both subjects In 2010, the United States ranked 29th out of 30 countries among the Organization for Economic Cooperation and Development Domestic comparisons using NAEP scores indicate a slight uptick in performance among 8th-grade math scores from the previous NAEP test in 2007, but persistent gaps in scores between White and Black students, and White and Hispanic students remained unchanged (Darling-Hammond, 2010)

The key to mathematics literacy is the study of algebra Earlier NAEP reports indicated that effective middle schools offered algebra to 8th-grade students (Na-tional Center for Education Statistics [NCES], 1994a) Subsequent NAEP (NCES, 2001) data showed that students who studied algebra in the 8th or even 7th grade did as well as or even better on state standardized tests than their peers who had not studied algebra So, why and how did educational policymakers in the United States arrive at a plan that restricts the study of algebra to high school students or

to those deemed more capable to study it in 8th grade?

HOW DID WE GET HERE FROM THERE?:

THE HISTORY OF MATHEMATICS CURRICULUM DELIVERY

Solving the mathematics reform equation requires an understanding of the ables that interact when solving the problem of providing appropriate mathemat-ics curriculum for all students The fi rst variable is, quite simply, convention All education, but especially mathematics, relies on established protocols and routines that have been in place and have become institutionalized For example, “everyone knows” that algebra has always been studied in high school The social, political, and philosophical bases for this conventional knowledge are open to discussion but, for some, this conventional knowledge is absolute The second variable is the concept of readiness Related to the fi rst variable but more specifi c in focus, stu-dents’ readiness for algebra involves a complex combination of the requirements

vari-of curriculum and attitudes and beliefs about students in general, including ments about their developmental and socioeconomic attributes Finally, the com-pound variable of excellence and equity, frequently a slogan-like catchword for

judg-policymakers, plagues curriculum designers and practitioners alike in the struggle

to “reach” all students

Each variable drives the solution of the algebra problem For example, since the consolidation of the comprehensive high school in the early part of the 20th century, algebra has enjoyed the status in the traditional high school as gatekeeper

to advanced courses in math and science As a status course, algebra was reserved for those students deemed capable of doing the coursework and for those who would benefi t from algebra as a basis for further studies, that is, those who plan

to go to college However, this was not the case prior to the modern high school.During the fi rst century of America’s existence as an independent nation, the number of schools in both Europe and North America increased drastically These schools, characterized by a variety of styles and an equal variety of funding sources, could be divided into three basic types: the English school or one-room schoolhouse, in which children between the ages of 2 and 14 learned basic skills; the two-part system, with the primary school, for children ages 5 to 9; the Latin grammar school, for boys, ages 9 to 14, in which they learned Latin, Greek, algebra, geometry, and trigonometry; and fi nally, the academy, a private catchall school that followed the Latin grammar school curriculum Learning specifi c facts was

a mainstay of education, and students were socialized to gather as many of these facts as possible and to strengthen their memories

Popular education in the fi rst century of the American Republic helped

“de-fi ne an American paideia and teach it to a polyglot population spread across a

continent” (Cremin, 1980, p 507) Soon, the concept of the American Common School led to thousands of small, tax-supported, local schools that were open to all students living in a particular district State systems, rather than a large na-tional system of governance, became the norm, but the states delegated much of their responsibility to the local districts The diffusion of power that characterized popular democracy in the Jacksonian era also resulted in the diffusion of authority and responsibility for education to local communities

In the 19th century, unlike their contemporary English counterparts, who feared that education would allow people to rise above their station, the mem-bers of the new American democracy promoted universal education, albeit primarily for free White males As a result, schools evolved in the form of the academies, which fell into three funding categories The fi rst were private, en-trepreneurial “for-profi t” schools, reserved for those with resources to fi nance

an elite education The second group included parochial or religious schools that had the additional mandate of providing a specifi c philosophical base in a pluralistic nation The fi nal group comprised those with limited public fi nancial support and control These schools struggled to maintain both curriculum and integrity

As a reform, the academy was not without controversy As early as 1795, uel Adams had feared that academies would lead to class distinctions based on wealth (Curti, 1959) However, as an outgrowth of Jacksonian democracy, acad-emies “replaced the Latin grammar schools and made secondary education avail-able and accessible to larger numbers of people” (Gutek, 1986, p 113) In addition

Sam-to promoting democracy, through these academies and the grammar schools ing up to them, mass education became a means of maintaining social stability and training citizens for citizenship (Dewey, 1899/1997)

lead-Throughout this evolution of the concept of school, the study of mathematics was intertwined with school success However, in the early days of American edu-cation, algebra was not necessarily a high school subject but was taught to students

as they progressed individually through the arithmetic curriculum In the 19th to early 20th century, educators attempted to provide a school curriculum that met the varying needs of a vast range of students At that time, the study of algebra came to be reserved for those who were capable or deserving of studying mathematics in high school, and the standardization of curriculum resulted in the movement of algebra to the high school

mid-As result of this shift, the convention of algebra in high school was lished At the same time, the comprehensive high school became an important link between public elementary school and the state-supported college or university However, this path was not open to all students Promotion from the grammar school to the high school was solely related to the presumed interests of the stu-dents In fact, more affl uent students were more likely to advance to high school because they were, in turn, more likely to attend college Less affl uent students left the educational pipeline after grammar school and, therefore, did not “need”

estab-to study algebra This judgment was further institutionalized as standards were imposed on the high school admissions process

By 1855, high school admissions tests were established as one of the fi rst temic attempts to measure academic progress and ultimately became a sorting mechanism that set the stage for entrenched inequities that would be challenged

sys-a century lsys-ater Prior to these sys-admissions tests, the prevsys-ailing system of one-room schools relied on holistic appraisal rubrics that became suspect and antiquated, as policymakers began a quest for predictability in education High school admis-sions tests were designed as a means of control as well as measurement Before the 1870s, only students of “good moral character,” as determined by the teacher, were allowed to take admissions tests

This initial standards-based reform movement employed high school missions tests as a tool for raising standards in public schools (Cremin, 1980) Ultimately, the content of the tests prescribed the curriculum of the grammar schools As greater numbers of students were allowed to take the admissions

ad-tests, the tests forced those prospective secondary students to master level subjects As a result of the larger pool of potential high school candidates, the introduction of the high school admissions tests resulted in the removal of advanced subjects like algebra from the grammar school curriculum and made those subjects the domain of the high school What had happened? Items pertain-ing to the study of algebra had been removed from the high school admissions test Since students were not required to answer test items related to algebra on the high school admissions tests, it became unnecessary to study algebra before entering high school Even though non-college-bound students had potentially studied algebra in the old grammar school curriculum, under the new plan your station in life determined your course of study Cremin (1980) highlighted the discrimination that resulted from the removal of the study of algebra from the grammar school curriculum.

grammar-The elimination of advanced subjects below the high school meant lost opportunities for pupils unable to advance Most high school pupils were from the middle and upper classes; thus, removing advanced subjects from grammar schools restricted chances for the talented but less advantaged youth to taste the higher learning (p 154)

SEARCH FOR THE IDENTITY OF THE HIGH SCHOOL IN THE UNITED STATES

Since only select students even considered attending high school, those confi ned

to simply a grammar school education were deprived of exposure to algebra and, conversely, came to be regarded as not ready to study algebra The second variable, readiness, is tied, at least in part, to the school evolution that took place between

1880 and 1920, when the American high school struggled to defi ne its identity Was the high school a college preparatory institution, or was it simply an exten-sion of the elementary school? Prior to World War I, school attendance beyond the grammar school was voluntary, but by 1918, new child labor laws forced nearly all children and youth to remain in school Waves of immigrants changed the demo-graphic composition of the schools Education reformers combined skepticism about the capabilities of the new student population and optimism about the ca-pacity of the schools to meet their needs (Powell, Farrar, & Cohen, 1985)

As a result, a critical shift occurred in the mission of the high school in the United States Cremin (1956) compared the emphases of two major policymaking bodies in the history of American education at this time period Whereas in 1893, the Committee of Ten focused on providing rigorous curriculum that students were to strive to meet, in 1918 the Commission on Reorganization of Secondary

Education mandated that the schools offer something of value for the various needs of their diverse students In 1893, the students had to measure up to the schools’ standards By 1918, however, the schools had to measure up to the stu-dents’ needs

Diverse, often non-academic courses were added to the curriculum and demic courses were watered down to meet the needs of these new constituencies (French, 1964) More importantly, the study of mathematics in general and alge-bra in particular came under fi re as not being “socially effi cient, that is, based on the future social destination of the students.” In 1913, David Snedden, the Com-missioner of Education in Massachusetts, stated that the study of algebra was use-less for all but very small numbers who would use it as a professional tool In New York City, algebra was attacked for high failure rates and pressures on pupils Algebra was charged with injuring the mind, destroying health, and wrecking the lives of children It was especially dangerous for girls The Superintendent of the Los Angeles schools maintained that algebra “caused many a girl to lose her soul” (Krug, 1964, p 347)

aca-As a result of this outcry, in 1920 the Commission on the Reorganization

of Secondary Education (CRSE) recommended a common course in basic ematics in grades 7, 8, and 9, and reserved algebra for differentiated groupings

math-in high school—that is, the “prospective engmath-ineers” (Krug, 1964, p 351) This scenario was a far cry from the 21st-century emphasis on scaffolding learning

in math and science as early as kindergarten, with Congress considering tion, endorsed by businesses such as I.B.M and Lockheed Martin, to promote engineering education starting in kindergarten In contrast, by the early 1930s, “a new system of secondary studies had been installed, in which a small minority of college-bound students were expected to pursue intellectually serious work while everyone else was taking courses explicitly designed for those less able, less willing,

legisla-or less interested” (Powell, Farrar, & Cohen, 1985, p 245) In a mere 70 years, the study of an advanced subject like algebra was no longer available to most students before high school Basing their policies on a presumed social class system, educa-tion reformers reserved the study of algebra for students who were older and/or presumably more capable, or “ready.”

WHO SHOULD STUDY ALGEBRA?:

PHILOSOPHICAL AND PSYCHOLOGICAL FOUNDATIONS

The reorganization of the curriculum not only evolved from the politics of the previous century It also derived from the cognitive theory and educational philos-ophy that emerged during that same time period In the 20th century, two names

dominated the theoretical landscape of educational structure and reform in the United States: John Dewey, a political theorist and philosopher, and Jean Piaget,

a cognitive theorist and psychologist Indeed, if you ask veteran teachers to name important educational theorists, either Dewey or Piaget will top their lists These two men have shaped the form and function of education in the United States Dewey (1899/1997) maintained that schools were the laboratories in which children gained the skills of citizenship By engaging in genuine exploration and problem solving, children acquire insight into how society functions and appre-ciation for the work of all members of society As a result, education promotes democracy How to achieve this lofty goal can be found in the cognitive theories

of French educator Jean Piaget (1937, 1964), who infl uenced educational thought

in the United States throughout most of the 20th century Piaget’s concepts of developmental stages and cognitive readiness validated reserving algebra for high school or, at most, for those students who exhibit advanced readiness or ability before high school At the same time, Piaget’s defi nition of the formal period as beginning at age 12 opens the door to providing algebra instruction before high school, a trend that reformers, researchers, and curriculum specialists began to advocate toward the end of the 20th century

According to Thompson (1976), the Great Depression and World War II scured the educational inadequacies of the Progressive movement In the 1930s and 1940s, poor children left school to go to work or to war, whereas wealthy children went to private school or elite public high schools Public schools became the domain of the middle class, who were largely satisfi ed or had no means to complain During this time, NCTM (1941) reorganized the mathematics curricu-lum, and outlined strategies for teachers to pave the way for algebra instruction However, this NCTM report also created the two-track math curriculum in high school, reserving algebra for “interested” students, removing the danger of course failure for those who might not need to take the course This set the scene for the questions of equity and access When post–World War II peace and prosperity inevitably led to the civil rights movement of the 1950s and 1960s, mathematics emerged at the core of the debate about the third variable, equity and appropriate education for all citizens

ob-CHALLENGING THE STATUS QUO IN THE MID-20TH CENTURY

In the midst of this unrest, the Soviet Union’s launch of Sputnik in 1958 was a wake-up call for American educators The National Science Foundation expanded its interaction with the high school curriculum, promoting new curricula, espe-cially in math and science, for college-bound students but with little emphasis on

advanced study for underserved populations (Ravitch, 1983) However, equity of access to advanced math and science study was not an issue, even though during the next quarter century in the United States civil unrest in the schools intensifi ed,

as it did in all of society Education policymakers and practitioners focused instead

on the concept of excellence, translated as dominance in the global arena Despite post-Sputnik rhetoric, reformers retained their traditional concept of advanced mathematics study for specifi c, specialized populations, and not for the general population

Much of the impetus for change came from outside the educational arena, from educational entrepreneurs and business leaders termed “delta forces” by researchers Reyes, Wagstaff, and Fusarelli (1999) Consequently, in 1983, in its

landmark report, A Nation at Risk, the United States Commission on Educational

Excellence, under the auspices of the U.S Department of Education, issued a ion call to reform and recommended 3 years of mathematics in high school and

clar-a decreclar-ase in the number of students in the “generclar-al trclar-ack.” With these reforms in place, the United States would be able to restore its leadership role among the na-tions of the world as well as its economic supremacy

Educators scrambled to respond to this stirring body of rhetoric that placed the word “risk” on the lips of teachers, parents, and lawmakers and created a sense

of urgency unmatched since the Sputnik scare of 25 years earlier The aftermath of the Progressive movement in the early 20th century was decried for distorting and diluting the true focus of schools Theodore Sizer (1984) stepped into the lime-light as the “leading voice of contemporary American Progressivism” (Ravitch,

2000, p 418), but with the essential difference that he prescribed rigor and basic skills where Dewey had prescribed hands-on experiential learning Sizer regarded high school attendance as “an opportunity, not an obligation” (1984, p 88) and advised that compulsory attendance should end when the minimum require-ments are mastered—that is, before senior high school

Unfortunately, the incendiary blue-ribbon report essentially restated the guments set forth at the beginning of the 20th century and split educators into two factions On the one hand, many believed that schools had little impact on chil-dren’s capacity to learn, given their heredity and socioeconomic background, fac-tors that overburdened the schools and prevented them from accomplishing their true mission On the other hand, others believed that schools had the responsibil-ity to educate all students regardless of their socioeconomic status (Ravitch, 2000)

ar-As the 20th century drew to a close, educators again faced the questions of which students should study which subjects—and when they should study them The compound emphasis of equity and excellence became the clarion cry of sup-porters of No Child Left Behind, the federal mandate that garnered bipartisan

support in its incontrovertible mantra that all children in the United States are

entitled to education that is “world-class” and equitable After A Nation at Risk

(1983), the debate over excellence versus equity became heated Some theorists advised that striving for excellence would necessarily produce equity (Green, 1983; Murphy, 1989) On the other hand, some policymakers held that increased empha-sis on academic requirements would result in higher drop-out rates, particularly for minorities (Jones-Wilson, 1986; Metz, 1988) At the heart of this debate stand long-standing tracking policies that have reserved algebra for select students.Oakes (1985) decried tracking of students as the major impediment to equity

in the United States She maintained that equity could not exist in American ety as long as some groups continued to retain access to “high-status knowledge” and control its distribution Algebra involves “high-status knowledge” because ac-cess to the study of algebra is the fi rst step on the road to university study Once more, mathematics and science rose to prominence as being essential

soci-to America’s position of dominance in the global community This lofty goal soon became mired in the political muck of high-stakes testing and sanctions on schools that did not perform up to standard More importantly, in the years since the implementation of No Child Left Behind, performance of students from the United States has fallen drastically, compared to peers in the international arena (Darling-Hammond, 2010) and students in schools with failing scores face fewer resources and more sanctions, rather than enhanced opportunity

REFORMING THE REFORM MOVEMENT

The much maligned No Child Left Behind Act might potentially have addressed the opportunity gap that had been created over the 20th century Indeed, the pol-

icy mandates of A Nation at Risk (1983) failed to address the shortfall of

math-ematics course-taking among students in American high schools In response to

the prescriptions of A Nation at Risk, students in the United States did take more

courses in mathematics, but the level of the courses did not increase their level

of profi ciency Schools continued to provide traditional pre-college courses like algebra and geometry for selected, college-bound students However, for the larg-

er student body, schools created low-level courses, like general math, consumer math, and “math for modern-day living,” which defeated the spirit of the commis-sion’s recommendations (Wilson & Rossman, 1993) The aborted effort of Clin-

ton’s Goals 2000 in 1994 to create national standards paved the way for No Child Left Behind in 2002, requiring states to expand educational opportunities for all

students, or at least assess those opportunities

Education practitioners also entered the arena with prescriptions for cess In 1989, the National Council of Teachers of Mathematics (1989) attempted

suc-to revitalize mathematics instruction by issuing new guidelines for the teaching

of mathematics in the United States This group set forth new standards that were student-centered, constructivist, and egalitarian with high standards for all students and the assumption that all students were expected to master enough mathematics to enable them to succeed in college These prescriptions promptly became subject to backlash from powerful groups of mathematics professionals, engineers, parents, and even the students themselves, all favoring a return to the more traditional former curriculum

In 1990, very few students took a full course in algebra before entering high school (Braddock, 1990; Smith, 1996) One popular policy in the mid- to late 1990s involved transition courses in mathematics, which were intended to bridge the gap from arithmetic to pre-algebra to algebra by reinforcing basic informa-tion in arithmetic facts that provide the infrastructure for understanding algebra Transition classes are one policy strategy for ending tracking (Gamoran, Porter, Smithson, & White, 1997) Transitional mathematics courses might be the mini-mal replacement option for the general track, but traditional algebra remains the optimal policy in terms of further mathematics attainment Over a decade ago, Gamoran and Hannigan (2000) confi rmed the effi cacy of the study of algebra regardless of students’ prior achievement level Even “students with average grades who take challenging courses would be better prepared to do college level work than students who achieve high grades through taking undemanding courses” (Roth, Crans, Carter, Ariet, & Resnick, 2001, p 77)

This then leads to the question of whether and when all students should study algebra Policymakers and researchers subsequently suggested that greater access

to algebra would result in greater mathematics “attainment” in high school sey, Mullis, Linquist, & Chambers, 1988; National Center for Educational Statistics [NCES], 1994b; Oakes, 1990; Useem, 1993) In 2000, NCTM disseminated revised standards that were not received negatively and remain in place in curricula and textbooks across the United States

(Dos-Despite these attempts to raise the bar of achievement in mathematics and education in general, Ravitch sounded the death knell for public education in the United States at the end of the 20th century when she declared that, despite the fact that the American dream of universal education had been realized, the quality

of that education was questionable That so many 1st-year college students had not mastered skills that had been traditionally taught in the secondary schools

“made a mockery of the ‘educational ladder’ that idealistic educators had cated at the beginning of the twentieth century” (Ravitch, 2000, p 454)

advo-FORWARD MOVEMENT AND BACKLASH

TO CHANGE IN MATHEMATICS CURRICULUM

While access to algebra remains in the forefront of the debate over the opportunity gap, the readiness of students to study algebra continues to be the prevailing vari-

able that drives the backlash to algebra for all movements In 2008, Skip Fennell,

then president of the National Council of Teachers of Mathematics, questioned whether mandating algebra for all 8th-grade students is a good idea, because

“Teachers of algebra frequently tell me that far too many of their students are not ready for algebra .” (Fennell, 2008) Fennell noted that this readiness is related to knowledge of multiplication, fractions, decimals, percents, ratio, and proportion

He conceded that algebra for all might be possible, but that the content of algebra

courses must be scrutinized for utility and that preparation for algebra must occur

in elementary school

California was in the forefront of requiring algebra for all 8th-grade students

In 2003, the Hoover Institution at Stanford University reported that while learning algebra by the end of 8th grade is a laudable goal, schools must strengthen aca-demics starting in kindergarten The report cited the “overplacement” of students

in 8th-grade algebra In July 2008, the California Board of Education approved a motion requiring all 8th-grade students to take the state Algebra I end-of-course exam In a few months the motion was overturned by the state Superior Court on the grounds that due process had not been followed by the Board of Education The motion itself was met with strong backlash from members of the educational community, with the perennial problem of student readiness as the chief cause

of the unrest and the inevitable cost of ramping up instruction for all students to meet the mandate in time for the 2011–2012 state algebra test Currently, algebra remains a requirement for graduation from high school, with some schools, no-tably the more affl uent districts, offering the subject to their 8th-grade students.One might imagine that taking algebra in 9th grade might be free of contro-versy, but even that has been subject to backlash In Chicago, early efforts in 1997

to increase the rate of 9th-grade algebra course-taking among larger populations

of students foundered on the lack of support and enhanced instruction in the grades leading to algebra A longitudinal study (Allensworth, 2009) of the Chicago mandate found that few benefi ts did result from mandating the algebra course and drop-out rates did not increase Nevertheless, the researchers questioned the effi -cacy of requiring this college-preparatory course, since test scores did not increase

A 2008 Brookings Institute report (Loveless, 2008) underscored the large numbers of students “misplaced” in 8th-grade algebra programs because of their

lack of “readiness” for the coursework The report criticized algebra for all policies

that targeted 8th-grade students, without providing the necessary supports before the students actually entered 8th grade The author maintained that no real gains were realized by requiring larger numbers of students to study algebra in terms of

scores on state tests Relying on the rhetoric of No Child Left Behind, this report

maintained that test scores did not indicate success of the mandates in the states where a high proportion of students were taking algebra in 8th grade, because stu-dents in those states did not outperform their peers on national math assessments, like NAEP Focus on standardized test scores minimizes the potential benefi ts of algebra study for individual students, and NAEP scores can not predict the long-term implications of studying algebra in the lives of those students over the next

10 to 20 years

REDEFINING THE VARIABLES

Algebra remains at the core of the debate over appropriate curriculum for all students in the United States The variables of convention, readiness, and equity/excellence continue to confound attempts to solve the complex problem of ensur-

ing mathematics literacy across diverse populations Poorly implemented algebra

for all policies ultimately reinforce the convention of parceling out mathematics

instruction to the worthy few However, once convention is removed as a able, properly implemented algebra programs can address the second variable of readiness, a concept that plagues efforts to attain mathematics literacy across the general population Educators must redefi ne how students attain readiness Readi-ness must not be viewed as an intangible quality that some students possess while others don’t, often because of their ethnicity or socioeconomic status Fostering readiness should be the business of schools, which must create an infrastructure of support that assures success for all students in algebra Schools must also address conventional tracking policies that institutionalize low-level instruction as early

vari-as 3rd grade

Throughout the history of the United States, the explicit and implicit goal

of education has been to create an informed citizenry, capable of understanding complex issues and basic problem solving The current global economic situation has reinforced mathematics as the key to understanding personal and national budgets, investments, and defi cits Eighth-grade algebra is the key to understand-ing mathematics at a mature level and, as such, should be the goal for all students This realization effectively reorganizes the equation It removes convention, rede-

fi nes readiness, and focuses on the most important, and compound, variable of equity and excellence

Keeping Track

Setting the Course in Mathematics

Tracking carries with it class-based damage that can neither be avoided nor compensated for

—Oakes, 1994, p 91Understanding that students have differences in learning rates for different subject areas in different kinds of materials at different stages of development is crucial to school patterns of curriculum and instruction

—Van Tassel-Baska, 2000, p 18Promoting mathematics literacy has been hobbled by a prevailing belief that the study of mathematics is largely unattainable by the general population After all, anecdotal performance records seem to indicate that some people simply don’t get math, while others seem to thrive on it Indeed, in math, perhaps more than other intellectual pursuits, some people simply progress faster than others and should

be able to engage in mathematics activities that provide for continued growth and understanding (Assouline & Shoplik-Lukowski, 2005) Those who “get it” should not be held back, nor should the study of algebra be held hostage However, what about those who need more time or support?

The National Council of Teachers of Mathematics (NCTM, 2000a) strongly recommends appropriately challenging curriculum as the basis for the fi rst princi-

ple for school mathematics, equity In most districts over the last century, tracking

has been the predominant delivery model for offering advanced mathematics to some students before others Students are selected on the basis of “ability,” which is determined by test scores, course grades, or teacher nomination, or a combination

of all three The problem results when the tracks are set and then prevent ment between them

move-The tracking model has affected the ways that math is perceived by the general population, in a self-perpetuating cycle that reinforces the notion that reserves

algebra for the select and deserving few Moreover, historical precedent or vention, as seen in Chapter 2, has traditionally reserved algebra for the 9th grade, boxing out students who are not assigned to the higher track and preventing them from engaging in higher-level mathematics in high school If you don’t study al-gebra in 8th grade, you won’t take high school calculus, the key to college study in math, science, and engineering

con-WHEN SHOULD MOST STUDENTS TAKE ALGEBRA?

Should all students study algebra in 8th grade? Could most students succeed in

8th-grade algebra? Would providing early access to algebra to all students help

close the achievement gap among minority and underserved populations? In

an examination of ability grouping in middle-school mathematics classes, Bode (1996) characterized the controversy over ability grouping as the struggle be-tween equity and excellence, a polarity that pervades any discussion of effective organizational patterns, particularly regarding middle schools She concluded that within-class ability grouping enhances the performance of high achievers without having negative effects on the low achievers Moreover, Smith (1996) maintained that “the question of whether schools should provide advanced coursework to only a select few students remains at the center of this policy debate” (p 149) Gamoran and Hannigan (2000) found that “whether cognitive differences among students lead to variation in learning rates, or not, taking algebra is still a good idea for everyone” (p 250)

The debate about tracking in mathematics permeates research on access to gebra In 1998, Gamoran and Weinstein tackled resistance to de-tracking in 24 re-structured schools and confi rmed that “high-quality instruction in mixed-ability classes is possible, even in secondary school mathematics, which is probably the most diffi cult place to carry out de-tracking” (p 410) At issue in the tracking de-bate is the type of instruction students receive in different tracks

al-In 1999, Mallery and Mallery reviewed arguments by proponents and ponents of tracking and concluded, based on their review of diverse tracking ini-tiatives across curricula, that “tracking as practiced today is detrimental to the U.S educational system” (p 15) Conversely, in a qualitative study of a detracked high school, Rosenbaum (2000) concluded that detracking efforts, while initially greeted with enthusiasm, lost their appeal because the detracked classes provided neither the challenge sought by more capable students nor the mastery needed by lower-performing students

op-Meanwhile, supporters of ability grouping for identifi ed gifted students phasize that grouping and tracking are not identical practices and must not be

em-equated with each other Tieso (2003) maintains that gifted students deserve and require the stimulation of studying with intellectual peers in an advanced en-vironment in order to reach their full potential She warns that the majority of negative research on ability grouping is approximately 25 years old and based on anecdotal rather than statistical evidence However, opportunities for gifted stu-dents and rigorous curriculum for all students should not be mutually exclusive concepts

Dossey, Mullis, Lindquist, and Chambers (1988) captured the one aspect of the problem, maintaining that “too many students leave high school without the mathematical understanding that will allow them to participate fully as workers and citizens in contemporary society and without highly trained personnel, the United States risks forfeiting its competitive edge in world and domestic markets” (p 9) On the other hand, Apple (2001), a leading critical educational theorist, of-fered the controversial opinion that most employment opportunities in the next century will be in the service sector and will not require high levels of education Whether this is a valid opinion is moot, because it is not the role of a public school

in a democracy to predetermine the paths of students by limiting the foundational capacity of students to pursue jobs that require more advanced expertise

The study of algebra in 8th grade by all students could potentially address the issue of mathematics literacy in the United States, especially in terms of global competitiveness For the last decade, scholars have issued warnings that predicted Darling-Hammond’s (2010) recent clarion call for rigorous curriculum One early examination of the Third International Mathematics and Science Study (TIMSS)and the Third International Mathematics and Science Study—Repeat (TIMSS-R) attacked traditional mathematics curricula in the United States and concluded that the 8th-grade mathematics curriculum in the United States was comparable

to the average 7th-grade curriculum in other participating countries (Greene, Herman, & Haury, 2000) Subsequently, Cogan, Schmidt, and Wiley (2001) fo-cused policymakers on strengthening mathematics literacy initiatives, particularly

in the nation’s middle schools, as a means of providing the basis for more rigorous work in high school Schmidt (2003) further decried the U.S pattern of little or

no gain from 8th to 12th grade as the result of a “middle-school curriculum ing coherence, with little rigor or extreme variability in learning opportunities as

lack-a consequence of trlack-acking policies” (p 278) Most recently, Dlack-arling-Hlack-ammond (2010) decried our standing in the world arena when she discovered that the per-formance of U.S 10th-grade students on the PISA was ranked in the lowest quar-tile in 2006, scoring above only Croatia, Portugal, Italy, Greece, and Turkey That ranking was even lower in 2010, when U.S 10th-grade students ranked above only one country, Ireland, in the top 30 countries listed on the 2009 PISA (Organisation for Economic Cooperation and Development, 2010)