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. CONTRADICTIONS OF A CURRICULIJH BEFORE:
THE LDHTS 0F INSTRUMENTAL IDEOINY OR HATHDIATICS EDUCATION
IN AN URBAN DISTRICT

By
Sandra K. Wilcox

A DISSERTATION

Sub-itted to
Michigan State University
in partial fulfillment of the require-ants
for the degree of

DOCTOR OF PHILOSOPHY

Department of Teacher Education

1989

ABSTRACT
CONTRADICTIONS 0F.A.CURRICUEUH REFORM:

THE LIMITS OF INSTRUMENTAL IDEOLDGY ON MATHEMATICS EDUCATION
IN AN URBAN DISTRICT

By

Sandra K. Wilcox

This study examines an elementary curriculum reform effort being
carried out in a large urban school district. Teachers were charged
'with implementing city-wide mandated curricula in seven content areas
over a three year period. Using the mathematics curriculum as the
case, this study explores contextual factors, in and out of schools,
that influenced the course of this reform. The study aims to show the
relationship between the nature of a district-wide educational reform
and the larger social context in which it was conceived. Three
questions frame the study. 1) In what ways and to what extent had
‘political, economic and organizational factors shaped the design of
this curriculum and mandates for its implementation? 2) What was the
:nature of the curriculum reform and what was its potential to develop
«children's mathematical knowledge and power? 3) In what ways did the
district prepare teachers to implement the newly-mandated curriculum in
their classrooms?

The data collected for this field-based study included:
observations of teacher inservice; interviews with district teachers
and administrators, community people and experts on the city's economy;
district prepared curriculum and instructional materials; documents
from the federal desegregation order and the decentralization, then

recentralization of the district; figures on school financing, student

achievement and drop-out; and economic and demographic statistics for
'the city. The study's results include analysis of the formal
curriculum documents and the inservice opportunities provided for
‘teachers using a framework that poses three orientations to mathematics
education: certification of numeracy; comprehension of mathematical
ideas; and enabling mathematical inquiry.

Using the concept of genealogy, the reform is located within a set
of linked events, ideas, practices and people that shaped the
initiative. The study provides a genealogical account of the reform,
the antecedants that preceeded the reform and out of which it emerged.
The genealogy provides a useful way to investigate what appear to be
contradictions between a rhetoric of empowering youngsters and a

curriculum and a model of instruction embedded in instrumentalism.

Copyright by
SANDRA K. VILCOX
1989

To

Jack and Michael

ACKNOWLEDGUHIHS

This dissertation represents the culmination of a half decade of
study, teaching and research. A number of people have contributed in
multiple ways, as teachers and mentors, as friends and colleagues. As
this thesis awaits transport to the printer and binder, I am honored to
complete these final pages by acknowledging my appreciation for all who
have helped make it possible.

First to my husband, Jack Russell, who for nearly twenty years has
been my mentor, political compatriot, best friend and constant source
of support and love. He saw possibilities for my work before I did and
helped create the conditions under which I have matured politically and
intellectually. His contributions to this dissertation were many;
financial, intellectual, political. His own writing and his powerful
use of language became models for mine. When I was stuck with a
problem of description, he provided just what I needed. He leaves his
mark here in the title and the terms for defining the orientations to
mathematics education. As we both begin new phases in our careers, I
pledge to help us reclaim more time for the pleasures of our daily life
together that this long journey has interrupted.

To my son, Michael Buist, a wonderful young man whose caring for
others and whose sense of himself has made me very proud. If I have
one regret it is that his experiences with school mathematics did not
engender an enthusiasm that we could share. At the same time, 1

vi

pleasure in the unfolding of his life as he pursues his own interests.
His independence, personal responsibility and support allowed me to be
a sometimes absent parent without worry and without much guilt.

I dedicate this work to Jack and Michael as an expression of my
deep love and respect.

To my mother, Jean Wilcox, who has always been a supporter of my
pursuit of learning opportunities and taken pride in my
accomplishments.

To the members of my doctoral committee, each of whom brought a
unique talent and who together provided a richness in perspectives on
my work. Bill Cole served as my academic advisor throughout. When he
suspected that I might abandon mathematics for a more general course of
study, he encouraged me to keep mathematics as a partial interest. As
this dissertation indicates, mathematics has become my lens for
examining larger issues of educational reform. Doug Campbell urged me
throughout to try to understand the actions of professional educators
in terms of the context in which they worked. His careful reading of
drafts, the copious notes he wrote for my further reflection, the time
he gave for talking about my latest efforts to make sense of the
situation were significant contributions to the quality of this work.
Cleo Cherryholmes supervised my cognate studies. He helped me fashion
a program of study on critical theory that has shaped my teaching and
my research. His suggestion that I consider Foucault's concept of
history as genealogy as a way to bring the context of this reform to

the fore was the piéce de resistance of this dissertation.

vii

To the fourth member of my committee, Susan Melnick, who has been
my mentor, my friend, and my most consistent and principled critic. As
my teacher and critic, she refused to accept work that was
intellectually sloppy, that failed to go beyond the obvious. When I
struggled over "What is this study about,” she helped me to see that
this dissertation was perhaps several studies in one, framed by more
than one question. She understood the political interest that informed
my ways of thinking about schooling and encouraged me to pursue that
interest in every undertaking. And when the politics got lost in the
details of writing, she was the voice of conscience.

To Pam Schram, Glenda Lappan, Perry Lanier and Ruhama Even with
whom I have worked for nearly three years. I returned to the
university dissatisfied with much of my own teaching of mathematics.
Their visions of how mathematics education might be conceived and our
efforts to create classrooms that embody that vision have had an
enormous influence on my own thinking about the teaching and learning
of mathematics. They will see that influence in multiple ways in this
dissertation and to them I am extremely grateful.

To Eliot Singer, my fieldwork instructor and my friend. Our shared
interest in the politics of schooling provided the focus for many
discussions about teaching and research. And the guest room at his and
Suzanne's house became a home away from home when deep fatigue
cautioned against the commute back to Detroit.

To the many who contributed in their own unique ways. Pam, Ruhama

and Deborah Ball, who read analytic memos and offered critique and

viii

suggested further questions. To my teachers who provided the challenge
to reconsider the taken-for-granted. And to fellow graduate students
with whom I have ”shared an ordeal."

' To those who participated in this study, the dedicated educators in
the Detroit Public Schools who face enormous constraints in trying to

make a difference in the lives of the children they teach.

ix

TABLE OF CONTENTS

LIST OF TABLES ....................................................... xv

LIST OF FIGURES ..................................................... xvi

CHAPTER 1 LOCATING THE STUDY IN A BROADER SOCIAL,

POLITICAL AND HISTORICAL CONTEXT ........................... 1
Introduction .................................................... l
The Laboratory of the Study: Detroit Public Schools ............. 3

Desegregation and Decentralization: Politics
and Unintended Consequences ................................ 4

Structural Changes in the Economy:

Problems for the Schools .................................. 10
Educational Crises: Academic Achievement

and Drop-outs ............................................. 17

The District's Response to a Growing Crisis ............... 25

Structure of the Dissertation .................................. 27

CHAPTER 2 WHY THE CASE OF MATHEMATICS ............................... 30

Introduction ................................................... 30

Post-Sputnik Reforms in Mathematics Education .................. 33

"New Math” ................................................ 33

'Back-to-basics" .......................................... 36

The Persistence of Traditional Instructional

Practices ................................................. 40
Reforming Mathematics Education in the 19805 ................... 41
"Conceptual Understanding" and ”Problem Solving” .......... 41
Orientations to Mathematics Education ..................... 43

The Imperative for Mathematical Inquiry ........................ 53

CHAPTER 3 METHODS OF DATA COLLECTION AND ANALYSIS ................... 56
Introduction ................................................... 56
Evolution of the Study ......................................... S7

The Beginning: Acting to Influence Teacher
Knowledge and Beliefs About Conceptually-oriented

Mathematics Through Inservice ............................. 57

What Does ”Teaching for Conceptual Understanding”

Mean in This Setting? ..................................... 64
The Contexts of Educational Policymaking ....................... 71

Historical, Political, and Organizational Contexts ........ 71

Economic Contexts ......................................... 74
A Plan for Presenting the Findings ............................. 75

CHAPTER 4 THE INFLUENCE OF SOCIOPOLITICAL AND

ORGANIZATIONAL CONTEXTS IN SHAPING THE REFORM ............. 77
Introduction ................................................... 77
The Testing Component of the Desegregation Order ............... 77

The Assessment of Basic Curriculum Skills ................. 77
The Mandate for City-wide Implementation ....................... 84

Improving Educational Equity .............................. 84

Increasing Central Administrative Control
Over Curriculum ........................................... 86

The Dilemmas in Increasing Centralization and
Standardization ........................................... 91

Summary ........................................................ 92

xi

CHAPTER 5 THE FORMAL CURRICULUM.DOCUMENTS ........................... 97

Introduction ................................................... 97
The Analytic Framework .................................... 99
The K-5 Mathematics Curriculum ................................ 100
Goals for Students as Learners of Mathematics ............ 100
The Curriculum Strands and Objectives .................... 102
Grade 3 Instructional Sequence ................................ 105

Objectives for Student Learning About Numbers,
Numeration and Operations with Whole Numbers ............. 105

Verbs That Described Student Actions ..................... 113

The Relative Emphasis Placed on Each Strand

and the Development of New Ideas ......................... 117

Summary ....................................................... 124

The Textbook and the Instructional Sequence ................... 126
CHAPTER 6 THE MODEL LESSONS ........................................ 137
Introduction .................................................. 137

The Analytic Framework ................................... 140

Model Lessons to Promote Thinking ............................. 142

The Introduction to the Model Lessons .................... 142

The Content of the Grade 3 Model Lessons ................. 146

The Form of the Model Lessons ............................ 167

Summary ....................................................... 172
CHAPTER 7 INSERVICE FOR TEACHERS CHARGED WITH IMPLEMENTATION ....... 174
Introduction .................................................. 174

The Provider of Inservice ..................................... 175

xii

The Context of Inservice ...................................... 178

The Structure of the Inservice Sessions .................. 178

The Settings of Participating Teachers' Practice ......... 181
The Content of Inservice ...................................... 183

What Did Teaching for Conceptual Understanding Mean? ..... 184

The Curriculum of Teaching for Conceptual

Understanding ............................................ 187

What Did Teaching for Conceptual Understanding

Look Like? ............................................... 191
Inservice as Teacher Education ................................ 204

Assumptions About What Teachers Needed to Know ........... 204

Assumptions About How Teachers Would Learn

What They Needed To Know ................................. 206

The Underlying Model of Changing Teachers' Practice ...... 207

Summary ....................................................... 209
CHAPTER8 THEGENEAIDG’YOFTHECURRICULUMREFORM ................... 212
Introduction .................................................. 212

A Genealogical Account ........................................ 213
Introduction ............................................. 213
Instrumental Ideology .................................... 215

The Inheritance of an Instrumental Ideology
in the Detroit Reform .................................... 222

The Inheritance of the Politics of Race and
School Governance in the Detroit Reform .................. 224

The Inheritance of Reform Agendas for Mathematics

Education and Changing Economic Realities ................ 226
Summary .................................................. 227
A New Challenge to the Domination of Instrumentalism .......... 229

xiii

An Agenda for Enabling Mathematical Inquiry .............. 229
What is Required to Create the Vision in Classrooms? ..... 231

A Necessary but Not Sufficient Condition

for Liberation ........................................... 236
LIST OF REFERENCES .................................................. 238
APPENDIX A Teacher Interview Schedule .......................... 250
APPENDIX B Grade 3 Objectives and Instructional Sequence ....... 251
APPENDIX.C Quiz on Effective Math Teaching ..................... 259

xiv

LIST OF TABLES

Table 1.1
SEV Per Pupil, Operating Millage Rates, Local Tax Yield,
State Formula Yield and Operating Expenditures for

Selected Michigan School Districts, 1986-1987 ........................ 16
Table 1.2

Percentage of 4th Grade Students City-Wide in Each

Achievement Category in Mathematics, MEAP, 1978-1987 ................. 18
Table 1.3

Percentage of 4th Grade Students from Two Schools in

Each Achievement Category in Mathematics, MEAP, 1978-1987 ............ 20
Table 3 . 1

Percent of Students at Each Grade Achieving At or Above

601 on Each Mathematics Competency, ABCS, May 1983 ................... 82

LIST OF FIGURES

Figure 1.1

Student Progression Through the Detroit Public High Schools,

Regular Day School Program, 1981-82 9th Grade Class .................. 23
Figure 5.1

The Strands and Objectives Devoted to Number, Numeration

and Operations With Whole Numbers ................................... 108
Figure 5.2

Number of Objectives by Category of Level of Objectives ............. 115
Figure 5.3

The Distribution of Grade 3 Objectives Among the

Curriculum Strands .................................................. 118
Figure 6.1

Distribution of Model Lessons Among Curriculum Strands and

Features of the Lessons ............................................. 147
Figure 7.1

Rectangles With Area 12 ............................................. 200
Figure 8.1

Genealogy of Reform ................................................. 214

CHAPTER 1

LOCATING THE STUDY IN A.BROADER SOCIAL, POLITICAL
AND HISTORICAL.CONTEXT

INTRODUCTION

While the educational reform agenda of the eighties has been aimed
at improving schooling for all students, those who have benefitted most
are college bound youth. It is only recently that the educational
community has begun to explicitly address the needs of ”at-risk” and
non-college bound youth. In a recent report entitled The Forgotten
Half; Non-College Youth in America, the W. T. Grant Foundation
Commission on Work, Family and Citizenship called attention to the
bleak economic future facing many of our young people who finish their
education when they leave high school. Although many young people
without post-secondary education do succeed, an increasingly larger
number are finding it harder than ever to find work that is rewarding,
provides steady employment, decent wages and benefits, and offers
opportunities for further training and advancement. This is
particularly true for minorities in impoverished inner cities. The
picture is even more bleak for those who drop out of high school.

The worsening economic situation for many young adults is due in
large measure to structural changes in the U.S. economy (Wilson, 1987),
a situation over which schools have no control. Yet schools continue
to be the targets of criticism for failing to prepare young people with
the appropriate skills and attitudes to find and hold steady, decent-
paying jobs.

School boards have responded to this growing criticism by
increasing the graduation requirements in English, mathematics, social

1

2
studies and science. Urban districts with large working-class, poor
and minority populations have tended to focus their efforts on
increasing proficiency in basic communication, reading and mathematics
skills. However, minimum competency initiatives alone will not provide
the educational experiences to help all young people become critical,
flexible, interactive, self-confident problem solvers, arguably the
single most important skill for work and citizenship in the let
century (Giroux, 1984; Jacobs, 1987). Literacy and numeracy must take
on new meaning, not in just a limited economic and technical sense, but
also in a political and social sense. Classrooms must become
communities of critical dialogue, where multiple forms of intellectual
inquiry are promoted, where young people collaborate to solve problems
they have helped to define, where they share understandings and build
upon each others' insights, and where they learn to make reasoned
judgments about their own thinking and that of others. Education must
provide the knowledge, skills and dispositions for work in an
increasingly complex, technological economy. But it must also
contribute to a citizenry educated in civic responsibility by
developing a commitment and a set of analytic tools to engage in the
sociopolitical life of a democratic society (Giroux, 1984; Katz, 1982;
Reich, 1988). As Young and Melnick (1988) point out, schools "have a
dismal record for having bettered the lives of most of the youthful
poor entrusted to their care” (p. 387). Districts with significant
poor and minority populations can no longer support policies, programs
and practices that continue to ill-serve children and youth who are
”'truly' educationally at-risk” (p. 387). These districts must

conceive an educational program that empowers young people to become

#:-:l.a__z-—-

3

problem posers and problem solvers, able to use the power of
disciplinary knowledge to understand, participate in, and shape the
world around them.

This study examines a curriculum reform effort being carried out in
a large urban school district. In 1987, the Superintendent and the
Board approved and mandated city-wide curricula in seven content
areas--1anguage arts, social studies, mathematics, science, music, art,
and health/physical education. Teachers were charged with implementing
the newly mandated curricula over a three year period beginning with
the 1987-88 school year. Using the mathematics curriculum as the case,
this study explores contextual factors, in and out of schools, that
influenced the course of this reform. Three questions frame the study.
1) In what ways and to what extent had political, economic and
organizational factors shaped the design of this curriculum and
mandates for its implementation? 2) What was the nature of the
curriculum reform and what was its potential to develop children's
mathematical knowledge and power? 3) In what ways did the district

prepare teachers to implement the newly-mandated curriculum in their

classrooms?

THE LABORATORY OF THE STUDY: DETROIT PUBLIC SCHOOLS
This study was conducted in the Detroit Public Schools during the
1987-88 school year. The city and its schools provided an
extraordinary laboratory in which to examine the curriculum reform
efforts of a large urban district faced with a number of educational
dilemmas. Some of the dilemmas arose in the context of the schools

themselves; others grew out of unequal social conditions in the larger

a
community. To begin, it is necessary to situate the story in a
sociopolitical and historical context, to take a look back at events
that changed the Detroit schools in fundamental ways and created the

conditions for a new curriculum reform initiative.

Desegregation and Decentralization: Politics and Unintended
Consequences

In 1971, the National Association for the Advancement of Colored
People (NAACP) filed suit in Federal Court charging the Detroit School
Board with intentional racial segregation in the district. This event
set the stage for five years of litigation challenging various plans to
desegregate the schools including altering attendance boundaries of
high schools and creating voluntary magnet schools. In 1972, Federal
District Court Judge Stephen Roth, who heard the case until his death
in 1974, selected a metropolitan remedy involving Detroit and 52
suburban school districts in Wayne, Oakland and Macomb counties,
involving about 780,000 children. However, in 1974, the United States
Supreme Court ruled that, because only Detroit had been found guilty of
de jure segregation, the remedy must be confined to the city. In 1975,
Judge Robert DeMascio was assigned to the case and in the next year
issued a series of orders on student and teacher reassignment and a set

of educational components.1

 

1The desegregation orders also created a 55-member monitoring
commission to act as an arm of the Court to oversee the order. It was
to function as a citizen's committee, auditing the District's efforts
and assisting the Court with reports and recommendations regarding
implementation of the components of the desegregation plan. For a
history of the desegregation of the Detroit schools, see Eleanor P.

Wolf's, W.

5

Stndgnt reassignneng. In January, 1976, bussing for the purposes
of student reassignment began. The district had been decentralized by
act of the state legislature in 1969, dividing the district into eight
regions. The school board was determined not to bus students across
regional boundaries and to limit the number of white pupils reassigned
to predominantly black schools. In 1961 when data on enrollment by
race was first released, 54% of the pupils in the system were white.
By 1975, the schools were 75% black.2 The desegregation order
specified that schools 301 or more black would be considered
integrated. Those schools that were less than 302 black could be
involved in bussing to achieve integration. In January, 1976, nearly
28,000 students (111 of the student enrollment) were reassigned.

The reassignment plan changed the racial balance of 105 of the 300
schools. Sixty-seven of the 80 schools with white majorities received
black students. Thirty-eight schools that were at least 802 black

received white students. But half the schools in the district remained

 

2Although the departure of white families from the city had begun
in earnest in the late 19505, two events hastened the exodus. Detroit
was one of the nation's urban centers where black communities erupted
in the summer of 1967. Sparked by a police raid on a "blind pig," the
rebellion symbolized the growing resentment of the black community
toward a nearly all-white, openly racist police department. The
violence on the streets was met by organized state violence (President
Johnson ordered Federal troops into the city) leaving hundreds of homes
and stores in ruin, and thousands of people homeless. The turmoil in
the neighborhoods spilled over into the schools. Militant groups of
students, black radicals and white right-wingers, precipitated major
confrontations at some of the high schools. On Malcolm X Day in 1969,
hundreds of high school students across the city staged black power
demonstrations. The growing polarization among black and white
students and the inability of school administrators to find a common
ground for community dialogue led to the white population's abandonment
of the city's schools. By 1970, fewer than 402 of the students were
white (see Conot, 1974; Georgakas & Surkin, 1975). The second event
that further accelerated the flight of whites from the city was the
series of court decisions from 1970 until 1974 that ultimately led to
the Detroit school desegregation order.

6

more than 902 black. Regions 1, 5, and 8, those with the greatest
proportion of black pupils (90.32, 96.71, 95.21, respectively) were
excluded from the student reassignment plan (United States District
Court, Eastern District of Michigan, Southern Division, May 11, 1976).
There was considerable fear that the reassignment of white pupils to
schools in Regions 1 and 8, inner-city regions with areas of extreme
poverty, would exacerbate the problem of "white flight.“ There seemed
to be a further explanation for the exclusion of Regions 1 and 5 from
the reassignment p1an--political clout of residents in those Regions.
Region 1 contained a part of the city -- in an earlier time called
”Black Bottom" -- where black residents had a long and distinguished
history of community activism. Region 5 contained one of the middle-
class enclaves where black and white professionals, city government
officials, managers and administrators of social service agencies and
educational institutions made their homes.3 Several sources suggested
that organized political pressure from influential members of these
communities put sufficient pressure on the school board to exclude
these Regions from the student reassignment plan.

In 1978, the NAACP sought to have bussing expanded and the Sixth
Court of Appeals ordered such an expansion, even though by then fewer
than 151 of the students in the district were white. By 1987, when

this study was conducted, less than 102 of the students were white.

 

3City residency for workers was required by a number of employers
in the city. Workers employed by the City of Detroit were required to
live within the city. Also, members of OSAS, the Organization of
School Administrators in the Detroit Public Schools, were required to
be city residents. There was no such condition of employment for
teachers in the system. The Detroit school board had tried in several
contract negotiations to win a residency requirement for teachers in
the Detroit Public Schools, a concession they were never able to win
from the Detroit Federation of Teachers.

7
The district reported the following racial/ethnic composition of the
student population in 1987: Black, 159,275 (88.45%); White, 15,713
(9.01%); Hispanic, 3,515 (1.65%); Asian, 1,106 (.621); and American
Indian, 497 (.272). Most white families had abandoned the Detroit
schools, either leaving the city altogether, or -- for those who
remained and could afford several thousand dollars a year in tuition --
enrolling their children in private or parochial schools.

Edncggignal ggnnonents. The Court's overview of the district was
not limited to assuring racial integration of the schools. A second
stipulation required the district to include educational components to
”eradicate the effects of past discrimination” (United States District
Court, Eastern District of Michigan, Southern Division, August 15,
1975). In the first year of the Court's oversight, Judge Robert
DeMascio ordered the implementation of eight educational components--
reading, testing, bilingual education, inservice training, school-
community relations, vocational education, counseling and guidance, and
a student code of conduct. The specific set of proposals that created
the educational components originated with the central administration
of the District and not the Court (with the exception of the Code of
Conduct).

At the time of the desegregation order, central administration had
limited authority over educational programs. The Michigan legislature
had enacted a law in 1969 decentralizing administrative authority in

the Detroit schools.“ The legislation divided the district into eight

 

4The key architect of the bill was then Senator Coleman A. Young
from Detroit. The legislation was intended to give the black
community some measure of input to and control over its schools.
Young was elected mayor of Detroit in 1973, the first black to serve
as the city's chief executive, a position he continues to hold.

8
Regions, each with its locally elected school board. Each board
appointed a regional superintendent. In addition, the district
maintained a central administration, with a general superintendent and
a central board, some of whose members came from the regional boards,
others who were elected in city-wide elections. Regional
superintendents and boards had control over curriculum and staffing
decisions in their schools. The central administration maintained
control over financial matters for the district. However, in many
other areas of the day-to-day operations of the schools, there was
considerable ambiguity about lines of authority among schools, regions,
and the central administration.

The Court considered the decentralized administrative structure a
serious impediment to its desegregation efforts. Judge DeMascio's
order to the District to include educational components in its remedy
opened the door for the general superintendent and his lieutenants to
consolidate their power, with the Court's support. As the district
proposed programs they sought to initiate, the Judge looked for a
rationale to support the implementation, consistent with his order to
remedy past discriminatory practices.

The history of implementation of the educational components was a
mix of consensus and controversy. Those programs that did not affect
local and regional administrative prerogatives and where Region
cooperation was solicited were supported. Those components that
encroached on local administrative authority were met with resistance
or outright opposition by regional and building administrators. Many
school principals actively opposed the establishment of school-

community relations councils and the new standards imposed by the

9

student code of conduct and bilingual education. They also resisted
attempts to redefine the role of guidance counselors. The court-
ordered increase in the delivery of guidance services meant that
counselors no longer had time to handle administrative tasks --
membership accounting, attendance record-keeping, student discipline --
for which they had been previously responsible. The central
administration in its reports to the Court indicated wide support for
the inservice training, testing and reading components. While it
appeared there was substantial support from regional and local school
administrators, teachers were extremely unhappy with the reading
component. The Detroit Federation of Teachers filed a number of
grievances over the increased paper work that resulted from the reading
program.

ec t a i a tho t . By the late seventies, there was
considerable public displeasure with the inefficiency, administrative
chaos and regional board patronage that had come to characterize the
decentralized structure. In addition, Detroit had become a majority
black city. Many of its leaders were black, including the mayor and
the general superintendent. The perceived need for a decentralized
system to provide control to the black community had decreased. In
1981, city voters approved a ballot proposal to recentralize the school
district under a central administration.

The recentralization of the district did not fully restore complete
administrative authority to the central administration. The eight
Regions were reconfigured into six geographical Areas. Regional boards
were eliminated and a central board was constituted by members elected

in both Area and city-wide contests. Area superintendents were

10
appointed by and accountable to the general superintendent and the
central board. Textbook adoptions were centrally approved but
decisions about which among the approved texts would be used resided at
the Area and individual school level. However, the changes in
administrative structuring had done little to clarify lines of
authority over curriculum and instruction. Teachers received
conflicting advisories from different levels in the administrative
hierarchy about the learning goals for children and the knowledge and
skills they should be teaching in the classroom. There remained
considerable confusion among school personnel, particularly teachers,

as to who was in control and to whose directives they were bound.

Structural Changes in the Economy: Problems for the Schools

The Detroit schools curriculum reform occurred in one of the
nation's five largest cities. Structural changes in the economy over
two decades had resulted in increased unemployment, massive
concentrations of poverty, homelessness, crime and welfare dependency.
Beginning in the early 19705, Detroit had lost over half its
manufacturing base through economic disinvestment and industrial
relocation. The automobile industry left the city as it sought new
facilities in the suburbs and sites of cheaper labor in the South and
the Third World. The market for automotive supplier firms
deteriorated. A city that once boasted of 280,000 manufacturing jobs
in 3300 firms by early-1980 employed less than 100,000 in manufacturing
in fewer than 1700 firms (Luria & Russell, 1981). The city had lost

close to 701 of the high-wage, national market industrial jobs.

11

Economic disinvestment from the city had an enormous impact on the
schools. The district was under tremendous pressure to provide an
educational program that responded to the needs and interests of
children and youth in a community "where joblessness and the
relationship between schooling and postschool employment take on
different and defeatest meanings" (Young & Melnick, 1988, p. 387). At
the same time, the loss of manufacturing facilities eroded the property
tax base on which school millage was levied. The Detroit School Board
and its financial officers found it increasingly difficult to raise
sufficient revenues to provide a quality education program for its
young people.

Economic gisinvestment and social dislocation. Between the peak to
peak cycle in the economy from 1979 and 1986, the number of chronically
jobless in Detroit increased by 26% even as the city's population
declined by 122 in the same seven-year period (Detroit Free Press,
January 1, 1989). "The city that had once been thought of as being the
land of opportunity for the uneducated and undereducated had suddenly
turned into a land of unemployment, poverty and crime” (Detroit Public
Schools, 1988, p. 3).

The social dislocation of the poor that had begun with urban
renewal and freeway expansions in the 19505 and 605 was exacerbated by
the dramatic increase in the ranks of Detroit's chronically unemployed
in the 19705 and 805. Between 1970 and 1980, the number of persons
living in census tracts with a poverty rate of at least 20 percent

increased dramatically.5 In 1981, one out of three Detroit residents

 

5Wilson (1987) reports that in the nations five largest cities
(New York, Chicago, Los Angeles, Philadelphia, and Detroit) ”population
living in poverty areas grew by 40 percent overall, by 69 percent in

12
was receiving some form of public assistance. Thirty-eight percent
(38.01) of the residents of the city lived below the federal poverty
level in 1986. In 1980, 55.81 of all black female headed families in
Detroit were living in poverty.

There were other indices of the social consequences of joblessness
and poverty. The infant mortality rate in the city was among the
highest in the nation. Young males, lured by the prospect of fast
bucks, were increasingly being recruited as couriers for drug dealers
in the city. In 1987, 336 youth under 17 years of age were shot in
Detroit, 35 fatally (Detroit Public Schools, 1988). The number of
homeless in the city at the time of the study was estimated at over
12,000 with the number of homeless children rapidly growing.6 There
were 10,500 public low-income housing units in 29 developments in the
city. While there was a crying need for additional low-cost housing,
the vacancy rate in these developments was 301! For many families,
even low-income public housing was too expensive. But the high vacancy
rate also reflected the city's inability to adequately maintain the
developments. A considerable number of units, in some cases whole
buildings, were simply uninhapitable.

The impact on the city's schools of economic disinvestment and the

resulting social dislocation cannot be underestimated. It was

 

high-poverty areas (i.e., areas with a poverty rate of at least 30
percent), and by a staggering 161 percent in extreme-poverty areas
(i.e., areas with a poverty rate of at least 40 percent)" (p. 46).

6The Detroit Free Press reported, ”That figure (12,000) reflects
only people who have obtained help from social service agencies.
Groups working with the homeless say the actual number may be as high
as 60,000" (February 24, 1989). Peggy Posa, Director of the Coalition
of Temporary Shelters (COTS) in Detroit, estimated the number of
homeless at 30,000.

l3
estimated that as many as two out of every three children attending
Detroit's public schools came from families receiving public
assistance. The forced moves of families within the city to secure
temporary housing meant that many children might attend several * '
different schools in one year. Students who moved among schools during
the year were often at a disadvantage in their new schools if their
prior learning experiences were a mismatch with those in the new
setting.

Respgnding with new educational ngggrgns. The Detroit economy had
been transformed--from unionized, high-wage goods production to non-
unionized, low-wage service provision. The steady automation of
industrial jobs and the application of new computer technology in
manufacturing meant that more was produced with a smaller work force.
The manufacturing work that remained in the city required skills beyond
those developed by the Detroit schools. These developments had a
profound impact on the employment opportunities for young people in the
city. Black youth who in the past could anticipate taking high-wage
semi-skilled jobs in the plants alongside older family members now
faced a very different future.

The Detroit schools attempted to respond to these developments with
the creation of five vocational technical education centers. Ordered
by the federal court in its 1975 desegregation order, these centers
offered a variety of training programs in health care, protective
services, cosmetology, and food services as well as technical training
for industrial work and the construction trades. But the centers had
failed to meet the expectations of students and the community. All

centers were undersubscribed with enrollments 70-751 of capacity. The

l4

placement rate of vocational center graduates from June, 1983, until
March, 1984, was only 682 and of these, 241 were in "non-related"
areas. A study by New Detroit Inc. (1986) cited major deficiencies in
the vocational programs to develop broader analytical skills as opposed
to specific ”narrow" technical skills. Of concern to employers of
these graduates had been the inability of some to "make good decisions
and judgments, their inability to work without constant supervision,
and their difficulty in communicating effectively with others and
working under pressure" (Monitoring Commission Report, October, 1984;
p. B-54). As Jacobs (1987) pointed out, training at the vocational
centers had focused on giving students short-term low-level job skills
while neglecting the development of long-term basic, analytic and
technical skills that could lead to further training and advancement.

The underutilization of vocational centers by Detroit youth was, in
part, a function of the organization of the programs. Students
enrolled in their neighborhood high school for academic courses. They
were bused to the vocational center for their technical courses and
then bused back to their high school. Transportation from regular high
schools to vocational technical centers was an enduring problem. A
common complaint from teachers and students was that buses were late or
simply did not appear. The U.S. District Court Monitoring Commission
reported that ”Detroit Public Schools buses have recently had a failure
rate exceeding 881 on State Police bus inspections” resulting in
disruption of teaching programs and serious loss of academic time
(Monitoring Commission Report, June, 1984).

Ina grgwing fiscal ggisis. While there was evidence of

questionable management practices, the fact was that the costs of

15
operating the schools continued to rise while the revenues available to
do the job declined. The impact of a growing fiscal crisis was felt
daily in schools and classrooms across the city.

Education in the state of Michigan is financed primarily through
local property taxes. Table 1.1 compares Detroit's effort to finance
its schools to several neighboring communities. The particular
districts were chosen to dramatize the huge disparity in per pupil
expenditures that existed among districts in the Detroit metropolitan
area.

Although Detroit had one of the largest operating millage rates in
the state, it was able to raise only 301 of its revenues through local
taxes. The state, through its school aid formula, provided 622 of the
operating revenue, the federal government the remaining 82. While
Detroit levied six mills more than Birmingham and nearly nine more than
Dearborn, it was able to raise less than one-fourth the per pupil yield
of its two neighbors in local tax revenues. Even with the contribution
of state and federal aid, Detroit had nearly 40% fewer dollars per
pupil than Birmingham and 23% fewer dollars per pupil than Dearborn.
Detroit's shrinking property tax base resulted from economic
disinvestment, industrial relocation, and residential devaluation.

By the 19805, the district was in a near constant state of fiscal
crisis.7 Teachers universally complained about not having sufficient
materials, particularly textbooks and paper. It was common for a

middle school teacher to have one set of textbooks that had to be

 

7In 1975 the district was $73 million in debt. By the end of the
1988-89 school year, the district was $151 million in debt (Detroit
Free Press, June 15, 1989).

16

Table 1.1 SEV Per Pupil, Operating Millage Rates, Local Tax Yield,
State Formula Yield and Operating Expenditures for
Selected Michigan School Districts, 1986-1987

| Local State Operating
SEV Millage Tax Yield Formula Expenditures
District esr_22211 Less esr_nunil nsr_nsnil er u a
Detroit $ 25,375 35.90 $ 911 $2,028 _$3,374
Birminghamb 175,883 29.90 5,453 0 5,453
Dearbornc 154,544 26.98 4,170 0 4,366
Inksterd 20,710 35.65 738 2,182 2,969

Source: Kearney, C. (1987). A primer on Michigan school finance.

8The operating expenditures are the sum of local tax revenue,
state formula aid, state special and categorical aid, and federal
categorical aid.

bVirtually the entire tax base of this nearly all-white, high
income community is residential or commercial.

cDearborn, a city with a long history of public policies designed
to keep out blacks, owes a significant portion of its substantial
tax base to the world headquarters of Ford Motor Company and the
Rouge Steel complex.

dInkster, like Detroit, is a largely black city that has suffered
economic disinvestment and social dislocation.

17

shared by five classes of students. Reductions in the budget for
maintenance resulted in a continual deterioration of the physical plant
of the district, particularly its older structures. Downsizing the
custodial staff meant schools were cleaned less often. Although the
superintendent and the chief financial officer were under increasing
attack for not adequately managing what revenues were available, the
fact remained that the fiscal problems were primarily linked to a
method of school financing in the state that exacerbated the financial
crisis facing districts with shrinking property tax bases.

The growing fiscal crisis was but one matter of extreme urgency.
The other was the crisis in academic performance of children and youth

in the district.

Educational Crises: Academic Achievement and Drop-outs

Student gchievement and testing. Since the mid-19705, the district
had been under continual pressure to improve student achievement on
standardized tests. The test that received most public scrutiny was
the Michigan Educational Assessment Program (MEAP), administered in
early October to all 4th, 7th and 10th grade students in the state's
public schools. MEAP tested sets of objectives in reading and
mathematics. The results were given considerable attention in both the
print and electronic media.

MEAP scores were presented both as the number of objectives
attained and the Category of Achievement. Category 1 indicated mastery
of 0-241 of the objectives, Category 2 mastery of 25-492, Category 3,

50-742 and Category 4, 75-1002. Table 1.1 shows the city-wide

18
achievement results in mathematics for 4th graders for the ten-year

period beginning 1978.

Table 1.2 Percentage of 4th Grade Students City-wide in Each
Achievement Category in Mathematics, MEAP, 1978-1987

 

 

Category 1 Category 2 Category 3 Category 4
1978 10.0 15.8 24.9 49.3
1979 4.7 10.4 21.2 63.7
1980 4.0 13.1 29.2 53.7
1981 3.3 10.1 25.9 60.7
1982 2.2 7.6 24.0 66.1
1983 2.0 8.0 23.1 66.9
1984 1.0 5.2 20.1 73.6
1985 1.0 5.1 19.3 74.5
1986 1.2 6.0 19.9 72.9
1987 0.4 2.5 11.8 83.3

 

Source: Annual Reports prepared by the Department of
Research and Evaluation, Detroit Public Schools

19

The steady improvement in achievement -- lowering the percentage of
students in Categories 1 and 2, increasing the percentage in
Category 4 -- resulted from increasingly focused attention to MEAP
objectives, particularly in the later part of Grade 3 and the first
four weeks of Grade 4. In 1987, for example, 4th grade teachers were
given a set of curricular materials to be used during September to
prepare youngsters for the mathematics portion of MEAP. The packet
included a schedule of instruction for the first four weeks of school,
a pre-test, 16 lessons on place value, addition, subtraction, and
multiplication, a post-test to be administered at the conclusion of the
lessons, one lesson on test-taking skills, and one lesson for
enrichment/review. This set of preparatory lessons may have
contributed to the dramatic increase in student achievement in 1987.

While the city-wide scores showed steady improvement, there was
considerable variation among schools. The disparity in achievement
results made the Board, central and Area administrators, school
principals and teachers vulnerable to the charge that there existed
considerable inequality in education across schools. Many schools in
the poorest neighborhoods of the city had reputations for providing
inferior educational opportunities for the children they served in
comparison to schools located in the city's middle-class neighborhoods.

Table 1.3 gives the 4th grade MEAP achievement scores in
mathematics for two schools in the district. School A consistently
outscored School B in the percentage of children achieving in Category
4 (with the exception of 1987). For the nine-year period 1978-1986,
School A had, on the average, 20.9% more of its students in Category 4

than School B.

20

Table 1.3 Percentage of 4th Grade Students from Two Schools in
Each Achievement Category in Mathematics, MEAP, 1978-1987

 

 

 

Category 1 Category 2 Category 3 Category 4

A B A B A B A B
1978 1.2 15.4 3.6 20.2 10.8 27.9 84.3 36.5
1979 0.8 9.0 5.0 15.7 17.4 14.6 76.9 60.7
1980 1.8 8.0 4.5 10.6 19.6 35.4 74.1 46.0
1981 7.3 2.4 12.2 7.2 19.5 38.4 61.0 52.0
1982 1.4 7.4 9.9 16.8 16.9 40.9 71.8 34.9
1983 0.0 1.0 4.5 10.5 7.5 33.3 88.1 55.2
1984 1.2 A 0.9 11.0 10.1 18.3 27.5 69.5 61.5
1985 2.5 1.0 8.9 11.3 26.6 25.8 62.0 61.9
1986 2.4 5.8 10.7 12.8 19.0 31.4 67.9 50.0
1987 2.2 0.0 4.4 2.4 25.6 7.1 67.8 90.5

 

Source: Annual Reports prepared by the Department of
Research and Evaluation, Detroit Public Schools

21

School A was located in a middle-class neighborhood. The housing
stock, built in the 19205 and 305, was well-maintained, tudor-style
brick, rich in architectural detail inside and out. Neighborhood
residents tended to be two-income families of professionals, managers
and skilled workers, many of whom were employees of the city or the
public schools. Although the neighborhood was racially mixed, most of.
the youngsters who attended the school were black.8 School B was
located in a neighborhood that contrasted sharply with the financial
well-being of the School A community. The signs of poverty were
visible at every turn: boarded up and burned out shells of abandoned
homes; weed-chocked and trash-strewn lots where the city had bulldozed
delapidated and unsafe houses; a commercial strip where rusted security
gates protected empty stores and where black men sat hunched over in
doorways, wearing glazed-over looks of despair and defeat.

Critics of the schools who charged that a quality education was not
available to 511 children found considerable empirical evidence to
support their charge. The demand to improve education for the poor
children who were being underserved was voiced in school board
meetings, in newspaper editorials, and in reports prepared by civic
organizations.

At the secondary level, the district instituted its own minimum
competency testing program in the late 19705. Stopping short of
requiring a minimum competency for high school graduation, the district
created a system of dual diplomas. Those who passed the Proficiency

Exam by the time they were graduating seniors received an ”endorsed”

 

8Exact figures were not available, but a number of residents in
this neighborhood, black and white, chose to send their children to
private and parochial schools.

22
diploma. Those who failed to pass the Proficiency Exam received a
diploma that did not carry the ”endorsed" stamp. The Proficiency Exam
measured basic reading, writing and mathematics skills at the glgngn
de ve .

Teachers at the middle and high schools were supplied with
curriculum materials that had been created specifically around the test
competencies. Teachers were directed to teach to one of the
competencies every week and demonstrate that intention in their weekly
lesson plans. This directive applied to all teachers regardless of the
course level they were teaching. Students took the Proficiency Exam
for the first time in tenth grade. They were permitted to retake any
portion not passed at least once a year until graduation. Although
scores on the Proficiency Exam had improved over the several years of
its administration, nearly one-third of the graduates continually
failed to meet the requirements for an "endorsed" diploma. The fact
that a significant number of students graduated without demonstrating
competency at the eighth grade level was a growing problem for school
administrators and teachers. Increasingly, there were calls to improve
the literacy and numeracy of nll children and youth in the Detroit
schools.

School gggp-gugs. The Detroit schools faced a staggering problem
of youth who were alienated from school, who saw no payoff to remaining
in school or engaging in academic work when they were there. The
district experienced an extraordinary loss of students from the regular
day school program. Less than one-half the students who entered the

ninth grade graduated four years later.

20000

18000

16000

14000

12000

10000

8000

6000

4000

2000

23

20404

13578

0252

7687a
6859b
5019c

 

Grade 9 Grade 10 Grade 11 Grade 12

Source: Profiles of Detroit's High Schools: 1975 toll984. A

Monitoring Commission Report, Detroit School District
United States District Court Monitoring Commission,
October, 1984.

8Students who entered Grade 12.

bGraduating seniors.

cGraduates passing Proficiency Exam.

Figure 1.1 Student Progression Through the Detroit Public High
Schools. Regular Day School Program, 1981-81 9th Grade Class

24

Figure 1.1 shows the progression over succeeding years for the
cohort of students of the 1984 graduating class, beginning at grade 9
in 1980 and continuing through graduation four years later. Over 6800
students did not progress with their classmates from ninth to tenth
grade, a loss of one-third of the cohort. The number of non-
promotions accounted for 13.61 of the cohort, leaving nearly 202 - one
of every five ninth graders - who did not return to the Detroit Public
Schools as tenth graders.

An accurate accounting of dropouts was first undertaken by the
district in 1982. The district's Student Information System Data Base
"tracked” 13,769 students who were new ninth-graders in 1982 to account
for the status of this population four years later.9 Students were
placed into one of three categories: dropouts; high school graduates;
or active--students that remained in high school for more than four
years. A fourth category, involuntary losses, were not included in the
percentages.10 The findings showed that of the 13,769 students who
entered the ninth grade for the first time in 1982, four years later

5,111 (41.31!) had dropped out, 5,959 (48.16%) had graduated, and 1,303

 

9The discrepancy in the size of the cohort "tracked" from 1982
and the size of the ninth grade cohort in Figure 1.1 is explained by
that fact that Figure 1.1 begins with nll students who were enrolled
in the ninth grade in the 1980-81 school year. That is, the
enrollment of 20,404 ninth graders includes students who were
repeating the ninth grade. The 1982 enrollment figure of 13,769 for
1982-83 includes gnly those who were enrolled in the ninth grade for
the first time.

10The district coded the loss of students as follows: dropouts -
non-return5/20 days absent, moved can't locate, suspended/expelled,
employment, left over age, unable to adjust, needed at home, enlisted
in armed services, pregnancy/marriage; involuntary losses - left to
Detroit non-public, Michigan public or non-public, or adult
education, left to other state or country, deceased, special release,
miscellaneous reasons.

25
(10.531) were still attending a Detroit public high school.
Involuntary losses accounted for the remaining 1,396 students. A
similar study of the 14,946 students who were first time ninth graders
in 1983 showed that four years later 5,003 (39.73%) had dropped cut,
5,994 (47.60!) had graduated and 1,596 (12.67%) were still attending a
Detroit high school. Involuntary losses accounted for the remaining
2,353 students (Detroit Public Schools, May 19, 1988). The drop-out

rate had become an educational disgrace and a political problem.

The District's Response to a Growing Crisis

By the mid-19805, the Board and the central administration were
under seige. The evidence was strong that a 'back-to-basics'
educational program based on minimum competencies had not reduced the
alienation and disengagement of kids who were truly at-risk. A
vocational program that emphasized "narrow” technical skills training
was not improving the chances of young adults to find work that was
challenging and rewarding. Every segment of the community was
demanding reform: reduce the number of drop-outs; increase scores in
all schools on the MEAP test; provide the kinds of knowledge, skills
and attitudes that would prepare young people for a changed Detroit and
national economy. The clamor came from the mayor's office, community
and grass-roots organizations, organized labor, business and civic
organizations, and groups specifically organized around improving the
quality of education in the public schools.

In January, 1987, the Board and the superintendent responded with a
curriculum reform initiative. Beginning with the 1987-88 school year,

elementary teachers were instructed to begin a three-year process to

26
implement standardized curricula in seven content area5--language arts,
social studies, mathematics, science, art, music, and health an
physical education. This study investigated the nature of the
mathematics curriculum reform.

The mathematics curriculum was of particular interest. With the
exception of pockets of experimentation with an innovative program in
the early 19805, mathematics education in the Detroit Schools had a
basic skills, minimum competency orientation for more than a decade.
The newly mandated mathematics curriculum was described by those most
intimately involved in its development as "conceptually-based.” It was
organized around ten mathematical "strands” -4 computers and
calculators, estimation and approximation, functions and relations,
geometry, measurement, numeration, operations, patterns, probability
and statistics, and sets and logic -- that were spiraled within and
across the grades. The goals for students as learners of mathematics
were "to think critically, flexibly, cooperatively and
independently...to create and to analyze...to gain understanding and to
develop higher order thinking 5kills...to apply knowledge of
mathematics in solving problems” (Detroit Public Schools, January,
1987).

The reform appeared to be a significant departure from the basic
skills, computational orientation that had previously informed the
district's mathematics program. On the face of it, the reform seemed
consistent with calls from leaders in mathematics education to
reorganize the curriculum around concept development and problem

solving to help students develop their mathematical knowledge and power

27
(An_Aggngn_fn;_An§lnn, 1980; Commission on Standards for School
Mathematics, 1987).

This study inquires about the following: Why did the reform take
the shape that it did? What factors or set of conditions supported the
initiative? What was the potential of the reform to develop the
mathematical knowledge and power of children who were truly
educationally at-risk? What support was given to teachers who were

mandated to implement the new curricula in their classrooms?

STRUCTURE OF THE DISSERTATION

Chapter 2 makes the case for considering an investigation of the
mathematics curriculum, locating the initiative within the context of
the current national agenda to reform the teaching and learning of
mathematics. Three orientations to mathematics education are offered
as a framework for analysis of the Detroit initiative. Those
orientations are teaching for certification of numeracy, teaching for
comprehension of mathematical ideas, and teaching for enabling
mathematical inquiry. I make the argument for providing all
youngsters, and especially those at-risk, with a mathematics education
program in which the goal is to empower students through mathematical
inquiry--where they actively participate in making sense of
mathematical situations; engage in making conjectures, developing
arguments, inventing procedures, building abstractions and
generalizations; and communicate with others about mathematical ideas.
The demands of work and citizenship facing the next generations require

a new definition of what it means to be mathematically literate.

28

Chapter 3 describes the methodology of this fieldwork study. The
chapter describes how immersion in the setting led to a recognition
that the magnitude of the problem and the questions to be asked about
this reform were considerably greater than first formulated. The
chapter further describes the data that were collected and how data
gathered from different sources, at different points in time and in
different contexts were triangulated.

Chapters 4, 5, 6, and 7 are devoted to an analysis of the data to
answer the questions that frame the study. Chapter 4 draws on
historical documents and interviews with key informants to locate the
mathematics reform in a political and organizational context that
shaped its conception and mandates for its implementation. Chapters 5
and 6 examine the formal curriculum documents -- the strands that
organized the curriculum, the objectives that sequenced the content for
instruction -- and the Model Lessons that provided examples of how the
objectives were to be met in the classroom. The analytic framework
developed in Chapter 3 is used to determine the orientation of this
curriculum reform and its potential to address and achieve its
purported goals.

Since teachers are the primary agents of curriculum changes in the
classroom, the way in which they are prepared to implement new
curriculum is fundamental to assessing the potential of a reform to
make a difference in the classroom experiences of children. Chapter 7
examines the inservice opportunities that were provided for teachers in
the district charged with implementing the new mathematics curriculum.

The context and the content of inservice are analyzed for what the

29
inservice provider thought teachers needed to know and how they would
learn what they needed to know.

Chapter 8 returns to the context of the reform. Drawing on the
concept of history as genealogy, I give a genealogical account of the
Detroit reform initiative. I locate the antecendants of the reform in
an instrumental ideology that dominates curricular practice, the
politics of race and school governance in Detroit, thirty-years of
reform agendas in mathematics education, and a recognition of a new
economic order and Detroit's position in it. The genealogy reveals
both consistency and contradiction. An instrumental ideology cannot
inform a curriculum or a practice whose educational goals contravene or
extend beyond the practical and utilitarian.

The question then is raised of what would be required to conceive
and implement a curriculum aimed at empowering students with
mathematical knowledge. I conclude with an attempt to sketch some

answers .

CHAPTER 2

WHY THE CASE OF'MATHEMATICS

INTRODUCTION

Since the late 19505, mathematics education in the United States
has been under continuous scrutiny. Each of the last several decades
has been marked by a new national reform agenda, each with its own
slogan that becomes the shorthand for the substance of the reform. In
the sixties it was ”new math;" in the seventies, "back-to-basics" and
"minimum competency;” in the eighties it is "computer literacy,”
”problem solving," and "conceptual understanding." These reform
initiatives have differed in terms of the students to whom they have
been directed, the mathematics embedded in them and the outcomes they
were intended to achieve. The agendas have reflected a number of
influences: l) changing conceptions of mathematics and the teaching and
learning of mathematics derived from research and deliberation among
mathematicians, mathematics educators, cognitive scientists and
educational psychologists; 2) larger economic and political factors--
challenges to U.S. preeminence in scientific and technological
advances, military superiority, and world markets; and 3) assumptions
about the mathematical learning opportunities that should be made
available to diverse constituencies whether distinguished by "academic
ability,” race, class or gender.

Why is an investigation of the mathematics curriculum reform in

Detroit's elementary schools important? The Detroit initiative occurs
in the context of a current national agenda to improve the teaching and

learning of mathematics for nll youngsters. Despite over two decades

30

31

of attempts to improve the mathematics achievement of students in our
nation's schools, the record shows evidence of only slight improvement
in what students know about and can do in mathematics. Of increasing
concern is the considerable gap in achievement between whites and
minorities. In its report, Evenybody anngg, the National Research
Council (1989) sounds an alarm that ”the widening gap between those who
are mathematically literate and those who are not coincides, to a
frightening degree, with racial and economic categories" (p. 14). What
is becoming increasingly clear is that our urban schools are failing to
adequately educate minorities and the poor to function as citizens and
workers in environments that require increasing quantitative literacy.

The situation is described as a crisis. It might be argued, as
Young and Melnick (1989) do, that schools serving minorities and the
poor have never accepted full responsibility for providing quality
education for these groups of students. The recent ”recognition" of
the problem facing urban schools may result from the increasing numbers
of these students whose educational needs simply can no longer be
ignored.

The current reform agenda is informed by concerns that are
political and social as well as technical and economic in nature.
The growth of technology and its application to virtually every field
of work means that the ability to think mathematically is no longer the
sole province of scientific workers. The loss of manufacturing jobs
that required mid-level skills are being replaced, in part, by jobs
that require not only the ability to compute but the ability to see the
mathematics in problems and to apply mathematical knowledge to their

solution. Responsible citizenship in a democracy requires a citizenry

32
capable of using mathematics to make sense of political and social
issues. Being an informed citizen means being able to understand
positions on issues of public policy that employ mathematical
arguments. Nearly every political issue -- acid rain, abortion, AIDS,
nuclear power, crime, poverty, homelessness -- to some degree is framed
in terms that are numeric. As the National Research Council points
out, ”A public afraid or unable to reason with figures is unable to
discriminate between rational and reckless claims in public policy” (p.
32). Being mathematically illiterate condemns one to second-class
status at work and in the community.

Information about Detroit, its schools, and its children and youth
make real the extent of the crisis facing urban districts. At the same
time, multi-faceted reform efforts in the district stand as testimony
of a history of struggle in the black community to make a difference in
the lives of each new generation. The elementary mathematics
curriculum reform represents a continuation of that struggle. A
question this study attempts to address is ”What is the potential of
this reform to contribute to the empowerment of the next generation of
Detroit youth?"

This chapter provides a framework for examining the nature of this

«.4..-

1atest reform initiative in the Detroit schools. It begins by

\w—a-

exploring the larger context of mathematics reform in which the Detroit

 

project is located. The first section reviews the reforms of the 19605

and 19705 to inquire about intent and outcomes. The second section

o—r—

considers in detail the forces driving the eighties reform rhetoric and

A..—~.W-—«H., '~‘_w> -

how this latest agenda for reform emerged from earlier efforts.

Included here is an extensive discussion of three different

33
orientations to mathematics education -- certification of numeracy,
comprehension of'mathematical ideas, and enabling mathematical
inquiry -- and the mathematical and political interests that each
embodies. The chapter concludes with an argument that, among the three
orientations, only an orientation centered on enabling mathematical
inquiry is likely to provide Detroit young people with the set of
intellectual tools - knowledge, skills, dispositions, critical habits
of mind - to understand, participate in, and shape the world around
them. And while enabling mathematical inquiry may be a necessary
condition to providing Detroit young people with the skills to conceive
the world in which they must survive as problem solvers, it is not
sufficient. There remain more fundamental issues of race, class and
gender that underlie social and economic stratification in American
society and that limit the life chances of minorities, women, the poor

and the working class.

POST-SPUTNIK REFORMS IN MATHEMATICS EDUCATION
I'NewMath"

The reform initiative of the late 19505 and the 19605 was an
undertaking aimed at enriching the content of mathematics for the
“academically talented," those most likely to pursue post-secondary
education in the sciences. The launching of Sputnik, the world's first
unmanned space satellite, by the Soviet Union in 1957 shocked the
scientific and political communities in the United States. The fear of
losing this country's edge in the creation of new technologies and the
implications for world markets and the balance of world power in the

nuclear age led the federal government to embark on a program to reform

34

mathematics and science education. The National Science Foundation and
the National Defense Education Act (passed by Congress in 1958)
provided considerable financial resources for curriculum development
and recruitment of ”talented" youngsters to the study of science,
mathematics and engineering. a

The curriculum materials that were developed emphasized the
internal structure and the unifying concepts of the discipline of
mathematics. They incorporated a broader sample of topics from
mathematics including sets and logic, probability, and the structure of
the real number system. They advocated the use of physical materials
and the discovery and exploration of mathematics by students working
independently on their own. Little attention was paid to the ways in L/'
which teachers taught. Government sponsored teacher institutes
retrained many secondary teachers about "new math” programs. But as
Fey (1978) pointed out, the institute programs concentrated on teaching
teachers more and deeper mathematics while ignoring questions about how
to implement programs in real classrooms. The training available to
elementary teachers charged with implementing "new math" programs was
extremely limited. As Fey commented,

In retrospect it is almost comical that National Science Foundation

teacher education programs were specifically prohibited from

serving elementary teachers. Now nearly every would-be curriculum

reformer acknowledges the crucial role of classroom teachers in

'implementation of new programs (p. 349).

By the beginning of the 19705, there was growing disenchantment
with the "new math" initiative. The biggest concern was the decline in
student achievement in mathematics as measured by national standardized

tests. In addition, questions were being raised about the

appropriateness of the substance of the reform, especially what critics

35
viewed as excesses of abstraction, symbolism and deductive reasoning.
A popular critique was that schools were turning out young people who
knew about union and intersection of sets and binary numbers but who
could not balance a checkbook. One of the most popular books at the
time was Morris Kline's Why Johnny Can'; Adg. Parents complained that
the language and ideas of "sets” and "properties of number systems"
were unfamiliar; their own limitations with the new content made
helping their children difficult at best.

School people, too, became disaffected. Early enthusiasm gave way
to the realities of implemention; many teachers, especially at the
elementary level, did not have sufficient subject matter knowledge to
convey the substance or the spirit of the new curricula. Even
”teacher-proof" curricula conceived by experts to be implemented by
mathematically "unsophisticated” teachers did not insure faithful
presentation of materials as they had been conceived. As Apple (1986)
points out,

...when the material was introduced into many schools, it was not

unusual for the 'new' math and 'new' science to be taught in much

the same manner as the old math and old science. It was altered so
that it fitted into both the existing regularities of the
institution and the prior practices that had proven successful in

teaching (p. 37).

By the mid-1970's, "new math" was being described as a failure at
improving the teaching and learning of mathematics. Fey (1978)
suggested that reformers paid insufficient attention to helping the
public and a broader constituency within the mathematics education
community understand the “fundamental rationale and substance" of the

new programs. But he pointed out another influence, the social context

of the seventies.

36
Furthermore, the criticism coincides with a period of straitened
economic conditions in the schools and for the public at large.
Much of the challenge to make school programs more practical and
accountable for their effectiveness seems to reflect anxiety about
personal economic pressures much more than philosophical
disagreement about educational policy (p. 352).

The call was for a return to the basics.

'Back-to-Basics'

The initiatives of the seventies focused on minimum competencies,
the least that could be expected from youngsters studying mathematics.
Unlike the earlier reform that had been aimed at the ”academically
talented," this campaign was aimed at those presumed to be "less able."
The National Advisory Committee on Mathematical Education (NACOME)
noted the shift:

Mathematics program improvements of the "new math” in the 19605

were primarily motivated and designed to provide high quality

mathematics for college capable students. Today mathematics
curriculum development focuses on issues largely ignored in the
activity of 1955-1970. Attention has now shifted to programs for
less able students, to minimal mathematical competence for
effective citizenship, to mathematical application, and to the
impact of new computing technology on traditional priorities and

methods in mathematics (NACOME, 1975, p. 23).

The "back-to-basics” movement had a profound impact on the
development of curriculum materials. The goals of learning were
expressed in reductive behavioral objectives that defined specific
skill learning with skills narrowly defined. The commercial textbook
market jumped on the bandwagon, producing prepackaged sets of
curricular materials (Apple, 1983b, 1986; Giroux, 1983; Gitlin, 1983).
As Apple noted:

It is nearly impossible now to walk into an American classroom, for

instance, without seeing boxes upon boxes of science, social

studies, mathematics and reading materials ("systems,” as they are

sometimes called) lining the shelves and in use. Here, a school
system purchases a total set of standardized material usually, one

I

37

that includes statements of objectives, all of the curricular
content and material needed, prespecified teacher actions and
appropriate student responses, and diagnostic and achievement tests
coordinated with the system. These tests usually reduce the
curricular knowledge to "appropriate" behaviors and skills (1983b,
p. 149).

The systematic integration of competency-based instruction and
prepackaged curricula was supported by initiatives in nearly 40 states
to mandate some form of state-wide competency testing programs (Apple,
1986). A number of districts instituted minimum competencies for
advancement from one grade to the next (e.g., New York's Promotional
Gates Program; see Labaree, 1984); some required demonstration of
proficiency at minimum skills in reading, writing and computation for
high school graduation. Other districts, like Detroit, fearful of
potential legal challenges to minimum competence as a requirement for
graduation, devised systems of differential diplomas.

These reform efforts did not just ”spring out of nowhere." The
instrumental logic that informed the undertaking was not new. A5
Giroux (1983) has suggested:

It is instructive to remember that the underlying instrumental
logic that infuses educational theory and practice at the present
time is not new. It has simply been recycled and repackaged to
meet the needs of the existing political and economic crisis. For
example, the technological and behaviorist models that have long
exercised a powerful influence on the curriculum field were, in
part, adapted from the scientific management movement of the 19205,
just as the roots of the competency-based education movement were
developed in earlier research work adapted "from the systems
engineering procedures of the defense industry" (Franklin, 1976).
The issue here is that.;.the redefinition of the curriculum in
watered-down pragmatic and instrumental terms cannot be viewed as
problems gglgly due to demographic shifts in the population and
short-term recessional tendencies in the economy. Such a position
not only abstracts the current crisis from its historical and
political roots, it also uses the existing economic crisis to
legitimate conservative modes of pedagogy and to silence potential
critics (p. 44). '

38

The "back-to'basics” movement did not exert a uniform influence
across schools or across districts. Minimum competency programs became
the mainstay in districts with increasing numbers of working-class,
poor and minority youngsters. Educators and employers talked about the
need to provide these youngsters with ”survival skills,” the presumed
knowledge and attitudes needed by the next generation of service and
industrial workers. As Anyon (1981, 1983) found in the fifth grade
classrooms she observed, curriculum and instruction in mathematics (and
other subjects, as well) varied by the social class of the students in
those classrooms.

In the working class schools Anyon studied, children learned basic
facts and skills. They were taught and expected to master steps in
conventional algorithmic procedures. Misapplication of routines by
students was met with teacher comments like "You're confusing
yourselves. You need more practice." Teachers skipped pages in the
text that called for reasoning and inference because they were ”too
hard" for these children. Learning was mechanical, rote and
fragmented. Classroom management was a constant struggle and a lot of
time was spent trying to maintain quiet and order.

In contrast, Anyon found a very different learning environment in
schools that served affluent communities. Instruction was
characterized by individual discovery as well as group work, an
emphasis on creativity and understanding, and the development of
intellectual tools to aid problem-solving and decision-making.

Teachers used questioning techniques that tended to place more
responsibility on students to make sense of problem situations. There

was less reliance on the text for the curriculum.

39

Anyon's analysis suggested that differences in learning
opportunities were, in part, a function of the beliefs held by the
educational community about the assumed abilities and presumed
occupational destinations of children from different class backgrounds.
Teaching in working-class schools denied the capacity for human
creativity. Children learned how to carry out procedures without
explanation. They learned rules "unconnected to thought processes or
decision making of their own” (1981, p. 8). They were being prepared
for the more practical curricula of clerical and vocational training.
On the other hand, children from affluent communities were acquiring
"cultural capital” -- linguistic, analytic and scientific skills, the
ability to analyze, plan and control -- forms of knowledge with an

exchange value in business and the professions.1

Ine fiallune 9f the "bgck-to-basics” movement. The return to basics
had not met the expectations of its advocates for improving
mathematical literacy. Data collected over nearly twenty years showed
the sad results of a focus on minimum competencies. While there was
some evidence that most students were able to compute, the majority did
not understand many basic mathematical concepts and were unable to
apply mathematical knowledge and skills to simple problem solving
situations (Carpenter et al., 1978; Carpenter et al., 1981; Dossey et

al., 1988; Erlwanger, 1973; McKnight et al., 1987; Schoenfeld, 1985).

 

1Others have also noted the relationship between the educational
opportunities available to students from different class backgrounds at
the elementary, secondary and post-secondary levels and the
reproduction of unequal class structures; e.g. Apple, 1979; Bowles &
Gintis, 1976; Karabel, 1972; Karabel & Halsey, 1977; Rosenbaum, 1976;
Young, M.F.D., 1974).

4O

Dossey et a1. (1988) reported on trends in mathematics achievement
as assessed by the National Assessment of Educational Progress (NAEP)
in 1972-73, 1977-78, 1981-82 and 1985-86. Their findings showed that
there had been modest gains in mathematics achievement but the gains
were confined to lower-order skills. Among 9 year-olds, 261 did not
have an understanding of ”rudimentary skills and concepts.” But more
alarming was that less than 25% demonstrated an understanding of the
operations of arithmetic or the ability to apply them to one-step word
problems. They were unable to use charts and graphs to compare
information or to analyze logical relations. One-fourth of the 13
year-olds were unable to demonstrate competency at this level as well.
While the trend had been for Blacks and Hispanics to show improvement,
they continued to achieve considerably below the level of whites.
Furthermore, as the age and level of knowledge and skill increased, so
did the performance gaps among whites and minorities. The Second
International Mathematics Study (SIMS) criticized the repetitive nature
lof mathematics curricula in the United States and the concentration on
computation relative to other important areas of mathematical knowledge

(McKnight et al., 1987).

The Persistence of Traditional Instructional Practices

The reform efforts had little effect on classroom pedagogy. The
profile of a typical elementary mathematics class has remained
virtually untouched by the efforts (NACMOE, 1975; Stodolsky, 1987;
Welch, 1978). The mathematics teacher begins instruction by reading
the answers to yesterday's textbook assignment while children check

their paper (or someone else's to avoid the ”temptation" to "cheat”).

...r..-. -..v

41
If there is a problem that has been particularly troublesome, the
teacher (or one of the ”brighter" students) works the problem at the
board. Then the teacher explains briefly the next piece of material to
be learned, typically what is contained on the next two pages of the
text. The remainder of the time allowed for mathematics is spent
having children work on the next assignment, usually another set of
problems from the text or a workbook. Students work individually at
seatwork while the teacher moves among the desks, answering questions.
As Welch (1978) noted in a National Science Foundation study of
mathematics teaching, "The most noticeable thing about math classes was
the repetition of this routine" (p. 6).

By the end of the 19705, there were increasing calls to improve the
quality of mathematics instruction. The argument was that if
mathematics learning was to be improved, mathematics teaching had to
improve. It is in the reform movements of the 19805.that improving the
quality of the mathematics curriculum has been joined by the explicit
recognition of the need to improve the quality of mathematics

instruction.

REFORMING MATHDiATICS EDUCATION IN THE 19808
“Conceptual Understanding“ and 'Prdblem Solving“

The rhetoric of mathematics reform in the 19805 has focused on the
need to develop students' understanding of mathematical ideas and their
problem solving abilities. With its 1980 publication, An_Aggng§_fn;
Agglnn, the National Council of Teachers of Mathematics (NCTM) proposed
a new agenda for mathematics education. NCTM recommended that problem

solving -- problem situations in forms other than traditional textbook

"Mann-9'13. ‘m

42
word problems -- become the focus of school mathematics in the 19805.
In addition, NCTM advocated reducing the emphasis on pencil-and-paper
practice on computational skills and using calculators and computers in
problem solving situations. NCTM was joined by other groups --
National Science Board, Board on Mathematical Sciences, Mathematical
Sciences Education Board, Committee on the Mathematical Sciences in the
Year 2000, American Association for the Advancement of Science -- in
calling for a reshaping of content and focus in the elementary
mathematics curriculum.

A lesson learned from the initiatives of the previous two decades
was that reforming the content of what is to be taught to children was
not sufficient. Improving mathematics education would require
attention to changing traditional instructional practices. However,
the calls for reform early in the decade did not specify how
instruction should be improved. This absence of direction has led to
different interpretations of what instructional practices need to be
changed and how those changes might be achieved.

There has been a tendency to posit a dichotomy between the
traditional approach to teaching mathematics and one focused on
developing understanding of mathematical concepts and processes.

Within a traditional orientation, mathematics education exhibits these
characteristics: the content is static, rule-bounded and linearly-
ordered; students are passive receptacles into which bits and pieces of
mathematical knowledge are poured; teachers are technicians who
dispense knowledge that students must absorb and remember. Teaching
for understanding assumes an opposing set of characteristics:

mathematics is a dynamic body of knowledge connected by a web of

43
relationships; students are active participants in the learning
process; teachers are facilitators of student learning. Unfortunately,
this set of binary distinctions masks what has emerged as two quite
different meanings attached to "teaching for conceptual understanding."
These differences have emerged from diverse theoretical and empirical
studies of teaching and learning and have shaped two distinct
approaches to mathematics education that differ from the traditional
approach. Since this study examines the orientation of the Detroit
curriculum initiative, a discussion of these three orientations is

warranted.

Orientations to Mathematics Education

The framework for analysis used in this study poses three
orientations to the elementary mathematics curriculum. I have labeled
these orientations teaching for certification of numeracy, teaching for
comprehension of mathematical ideas, and teaching for enabling
mathematical inquiry.2 The labels were chosen to represent the
intended outcomes for students who encounter mathematics in these ways.

fifi‘ "‘

These orientations differ in assumptions about how mathematics is

W‘—

represented, what it means to know mathematics and how mathematics is

learned, what should constitute the elementary mathematics curriculum,

 

2Ball (1988) uses the terms ”ordinary mathematics teaching,”
"conceptual mathematics teaching” and ”mathematical pedagogy” to
distinguish three approaches to the teaching of mathematics whose
characteristics are congruent with the categories I have employed.
Madsen-Nason (1988) defines three Levels - l, 2 and 3 - that roughly
correspond to the three orientations. The notion of Levels has also
been used to analyze data from a longitudinal research project with
colleagues Lappan, Lanier and Schram (see Schram & Wilcox, 1988;
Schram et al., 1988, 1989). Our categorization of Levels is also a
map of the orientations employed in this dissertation.

44
the goals for learners of mathematics instruction, and the role of the
teacher and students in the mathematics classroom. They are submitted
as ideal types with the knowledge that any mathematics program might
embody elements of more than one.

anglfilgntion of Numeracy. Teaching for certification of numeracy
corresponds to what is typically called traditional mathematics
teaching. Kliebard (1972) describes this orientation as based on a
metaphor of production, where children are viewed as ”raw material" to
be transformed by ”skilled technicians.” Children are seen as passive
receptacles into which mathematical knowledge is poured and who absorb
what they are told. As Romberg and Carpenter (1986) point out, this
orientation to teaching mathematics assumes that acquiring knowledge
created by others is the end of mathematics instruction. It is an
epistemological orientation where the negngg of knowledge is taken to
Q; knowledge (Dewey, 1916).

Teaching for certification of numeracy has its origins in the
behaviorist learning theories of Thorndike (1922) and Skinner (1968)
and an instructional practice informed by behaviorism. To facilitate
learning, mathematical topics are fragmented into isolated pieces that
are linearly and hierarchically arranged. The curriculum is organized
by scores of discrete behavioral objectives stated a priori, each of
which focuses on a single skill. The implicit assumption is that
learning occurs only through repeated drill and practice and
appropriate reinforcement.

Mathematics is represented as static, as ”discovered” and rule-
bound. The emphasis is on the formal language and symbolic

representation of mathematics. The discipline is portrayed as a system

45
of symbols to be manipulated and a collection of skills, definitions,
rules, procedures and algorithms to be memorized and mastered.
Arithmetic -- addition, subtraction, multiplication and division of
whole numbers, fractions and decimals -- constitutes the bulk of the
topics to be studied in the elementary curriculum. The goal of
instruction is computational speed and accuracy, proficiency with
conventional algorithms and their application to routine word problems.
It is assumed that problem solving requires prior mastery of basic
number facts and algorithmic procedures.

The role of the teacher is to show children how to carry out
procedures and provide ways for them to remember facts and steps in
algorithmic procedures. The teacher acts as a technician who
implements a curriculum conceived by ”experts.” The teacher dispenses
knowledge, the student is expected to absorb and remember it. Learning
mathematics is an individual, solitary effort where children search
their memory for accumulated facts and algorithmic solutions. Teacher
and text act as the source of epistemological authority--teacher as the
judge, textbook as the standard for judgement (Lampert, in press, a).

Gangrenension of mathematical ideas. Teaching for comprehension of
mathematical ideas aims at developing understanding for mathematical
concepts and processes. It has much in common with certification of
numeracy. The content continues to be organized around scores of
behavioral objectives stated a priori, each of which focuses on a
single skill or procedure. The curriculum divides mathematics into
”topics, each topic into studies, each study into lessons, and each
lesson into specific facts" (Romberg, 1983, pp. 133-134). The

curriculum may represent mathematics as a body of related ideas, but it

46
does so within a linear, rule-bounded, hierarchical arrangement.
Arithmetic continues to dominate but may be supplemented by topics from
geometry, statistics and probability, and number and set theory.

Teaching for certification of numeracy and comprehension of
mathematical ideas differ in the goals for instruction and the
assumptions about how students learn and what instructional practices
enhance learning. Teaching for comprehension of mathematical ideas
grows out of an interest in "improving traditional mathematics
instruction by making it more efficient and effective" (Romberg &
Carpenter, 1986, p. 860). It has been shaped by two research agendas:
l) to demonstrate the efficacy of beginning mathematics instruction at
the concrete level, and 2) to identify the teacher behaviors associated
with effective teaching.

The goal of instruction is to develop among students not only
proficiency with mathematical procedures but also an understanding of
why they work. It is believed the instructional route to realizing
this goal requires a particular sequence of development. The
assumption is that learning occurs by first manipulating concrete
materials that model a procedure, then linking concrete models to
pictorial representations, and, finally, using pictorial
representations as the bridge to abstract or symbolic representations.
This developmental sequence derives from several theories of learning,
in particular, Piagetian stages of development and individual
differences in learning styles. A considerable body of research has
been conducted to demonstrate the power of beginning instruction at the
concrete level, particularly in developing understanding of algorithmic

procedures (see, for example, Dienes, 1970; Fennema, 1972a, 1972b;

47
Hynes, 1979; Merseth, 1978; Payne & Rathmell, 1975; Prigge, 1978;
Stevenson, 1975; Suydam & Higgins, 1977; Wheatley, 1978).

This orientation has also been shaped by more than two decades of
"process-product” research that has sought to identify specific teacher
behaviors that correlate with student achievement (see Brophy & Good,
1986, for a review of this research). Most of this research has
focused on generic teacher instructional behaviors. Examples of
teacher behaviors studied include teacher praise, criticism and
reinforcement (Flanders, 1970); the pattern of teacher questioning,
student response, followed by teacher feedback (Stallings & Kaskowitz,
1974); quantity of instruction-~time spent in whole-group, direct
instruction (Stallings et al., 1978); clarity of teacher presentations
(Rosenshine, 1968; Hiller et al., 1969); "withitness"--techniques for
monitoring the class, minimizing time spent on transitions and student
conduct (Kounin, 1970; Brophy & Evertson, 1976; Anderson et a1, 1979);
pace of instruction and time-on-task (McDonald, 1976); expectations of
students communicated by the teacher (Berliner & Tikunoff, 1976); the
use of teacher questioning (Soar & Soar, 1979). In most of these
research efforts, the behaviors of learners and the specific nature of
the content being taught have been ignored or considered outside the
scope of inquiry. In virtually all of them, the dependent variable has
been student achievement scores on standardized tests.

Good and Grouws' work (1975, 1977) -- an elaboration on process-
product inquiry -- investigated effective instructional practices of
mathematics teachers. Their initial studies in mathematics classrooms
found that the quality of the developmental portion of a mathematics

lesson correlated with student achievement scores on standardized

48

tests. Using this finding, they developed and tested a teaching model
for whole-class instruction. The model prescribes a sequence of
instruction that includes daily review, development of the topic of the
lesson for about 20 minutes, seatwork, and homework (Good, Grouws, &
Ebmeier, 1983). The development portion of the lesson is key. The

' teacher does not just tell students what they are to learn but

f incorporates lively explanations and demonstrations. During this

‘portion of the lesson, the teacher assesses student understanding with
”product" and ”process” questions and controlled practice, providing
immediate feedback. Seatwork is assigned where students are held
accountable for completing a set of exercises.

Combining results from these areas of research, a picture of
"effective instruction" in the mathematics classroom emerges. The role
of the teacher is to begin instruction with a review or mental
arithmetic activity, tell students the objective for the day's
lesson -- what they are expected to learn -- and introduce the lesson
with an activity that draws on previous experience or prior knowledge.
The teacher develops the topic for the day through direct instruction,
showing students what they are to learn by demonstrating to the entire
class with the appropriate concrete materials, pictures or symbols.

The teacher poses ”product" questions (i.e., ”What answer did you
get?') and ”process" questions (i.e., ”How did you get that answer?"),
evaluating whether student responses are correct. Students are
assigned exercises for controlled or guided practice, where the teacher
is available to provide immediate feedback and repeat or elaborate on

instruction as necessary. Each lesson ends with independent

49
"successful” practice, a homework assignment, and closure--where
students tell what they learned about the day's topic.

A distinguishing feature of this orientation to the teaching and
learning of mathematics is the assumption that student achievement is
directly related to the instructional technique and technical
competence of the teacher. An effective teacher maintains a brisk pace
of instruction, keeps students on task and holds them accountable to
complete assigned work. Effective mathematics teaching is a procedural
and managerial matter. The teacher plans for instruction by selecting
suitable materials to model a mathematical concept or procedure and
developing questions to assess student understanding. Within the
structure imposed by the teacher, students pursue teacher defined
explorations with concrete, pictorial and symbolic representations,
answer teacher questions and complete practice exercises. Authority
for knowing rests with the teacher who decides if student answers are
correct and explanations sufficient.

Enabling mathematical inguigy. An orientation centered on enabling
mathematical inquiry derives from recent research and deliberations in
mathematics, mathematics education, cognitive science and educational
psychology (see Romberg & Carpenter, 1986, for a review of this
research). There are several basic assumptions that underlie this
orientation and that distinguish it from the others. One of the
assumptions is that children come to school with a rich informal
knowledge of mathematics and demonstrate a natural capacity for and
interest in understanding mathematical ideas (Ginsberg, 1977).

Further, children are not passive learners, but actively construct,

50
interpret and put structure on new mathematical learning (Carpenter et
al., 1982; Resnick & Ford, 1981; Steffe et al., 1983; Wittrock, 1974).

This orientation to mathematics education has an explicit goal to
empower students. This requires an environment where students become
active participants in the construction of mathematical knowledge,
making sense of mathematical situations and communicating with others
about mathematical ideas. It means to engage in the 'nxssgiss of
mathematics, learning what it means to do mathematics“ (Ball, 1988, p.
16). A learning community is created in which students learn to make
conjectures, develop arguments to validate assertions, invent
procedures, build abstractions and generalizations, and apply
quantitative and spatial reasoning to real-world situations (Romberg,
1983).

Mathematics is seen as dynamic--the creation of human activity,
derived from abstractions of empirical observations in the real world,
bounded by culture and history, changing over time. Mathematics is
seen as a body of knowledge -- skills, concepts, propositions --
connected by a rich web of relationships.

Teaching and learning take on characteristics quite different from
the other two orientations. The traditional emphasis on arithmetic
shifts to include a broad range of content. Instruction is ”chunked"
around key mathematical ideas that are explored over time to develop an
understanding of concepts and procedures and the relationships among
them. The intention is to develop a set of intellectual tools--
mathematical thinking, higher-order reasoning skills, problem solving
abilities, and natural and symbolic language to communicate

mathematical ideas to others.

51

The teacher plays a critical role as a facilitator of student
learning. The teacher affords opportunities for students to engage
collectively and cooperatively in mathematical inquiry, providing
problem situations in real-world contexts that exemplify mathematical
ideas so that students can experience mathematics as a language and a
tool for understanding the world around them. The teacher provides a
variety of representations - physical, pictorial, numeric, geometric,
graphical, algebraic - and helps students to see the power of each
representational form to enrich the understanding of the mathematics
embedded in a situation. And the teacher acts as a guide, listening to
students, posing questions for further reflection, responding to
questions by deciding which ones to pursue and which to put aside for
the moment. The locus of epistemological authority shifts from the
teacher and text toward the community of ”teacher and students as
inquirers who have the power to use mathematical tools to decide
whether an answer or a procedure is reasonable” (Lampert, in press,
a).3

The politisal interest embedded in each orientation. Each of these
orientations embodies a political interest as well as a mathematical
interest although the former is not often made explicit. A political
interest is found in the different pgwgr relations that exist in the
classroom between teacher and students in relation to the locus of
epistemological authority. A political interest inheres also in a
moral and ethical sense in the intended outcomes for students--

certification, comprehension, or enablement. Gordon (1983) labels

 

3The recent NCTM publication Curriculum snd Evaluation Sgangszds

is; Ssnosl Maghematics provides a vision of teaching for
mathematical inquiry.

52
these interests teaching for control, teaching for understanding, and
teaching for liberation.4

Certification of numeracy embodies an interest in control.
Learners are dependent on the teacher to create meaning. They lack
comprehension of fundamental ideas. Mathematics is offered as
”objective, lucid, and unequivocal" (Gordon, p. 361). Students have no
sense of connectedness, among mathematical ideas or their application
to life outside classrooms. Students do not question; they come to
believe that much of mathematics -- the "clear demonstrations, logical
relationships, and solutions that are errorless, if not elegant” -- is
beyond comprehension (Gordon, p. 361). What students learn is that
”mastery of truths has to do with getting the appropriate beliefs"
(Scheffler, 1976, p. 205).

Comprehension of mathematical ideas embodies an interest in
understanding. ”The emphasis is on relational knowledge in the sense
that the student comes to see why 'things are the way they are'"
(Gordon, p. 374). Students no longer just fill in blanks; they solve
problems. But the problems have been posed by others; other's
formulations are taken as given. The emphasis is on "adjustment” so
that one can ”function objectively within the paradigm of the status
quo" (Gordon, p. 374).

The third orientation, enabling mathematical inquiry, embodies an
interest in liberation. Students become problem posers. They commit

themselves to inquiry and in creating mathematical knowledge, come to

 

aEnglish (1983) describes three values around which curriculum
decisions are made that parallel Gordon's categories. English terms
the positions ”control,” ” consensus,“ and "emancipation.”

53

know more about themselves and others. The language of mathematical

inquiry itself can be liberating.

Consider that all mathematics begin with expressions such as ”what
if?,' ”suppose," '1et“--suggestive of the way in which mathematics
provides for its own liberation, creation and aesthetic
excitement...we could replace these expressions as follows: “what
if” with "what if I," ”let" with ”if I let” or ”if we let” (Gordon,
p. 378).

Power relations shift as authority for knowing moves from teacher and

text to the community of teacher and learners. Being a part of a

collectivity that has the power and the intellectual tools to create

knowledge is emancipatory. It is an act which Gordon describes as
paramount in providing personal, adequate evidence for knowing, not
only evidence for substantiation but, more importantly, evidence
that clarifies personal (as versus a weak or surface) understanding
and tests the taken-for-granted reality (p. 362).

In arguing for mathematics as liberation, Gordon quotes Paulo Freire.
Any situation in which some men prevent others from engaging in the
process of inquiry is one of violence. The means used are not
important; to alienate men from their own decision-making is to
change them into objects (p. 361).5

It is an orientation that embodies the principles of equality and

justice.

THE IMPERATIVE FOR ENABLING MATHEMATICAL INQUIRY
The Commission on Standards for School Mathematics (1989) argues
for the necessity of reforming the teaching and learning of mathematics
that is consistent with the aim of enabling mathematical inquiry. They
assert that the demands of work in the new economy require a degree and
kind of mathematical literacy that go well beyond certification of

numeracy and comprehension of mathematical ideas. The Commission

 

5While I find this quote powerful, Freire is guilty of
committing another kind of violence - making women invisible.

54
summarizes the views of Henry Pollak, a noted industrial mathematician,
about the needs of the next generations of workers.

* The ability to set up problems with the appropriate operations.

* Knowledge of a variety of techniques to approach and work on
problems.

* Understanding of the underlying mathematical features of a
problem.

* The ability to work with others on problems.

* The ability to see the applicability of mathematical ideas to
common and complex problems.

* Preparation for open problem situations, since most of real
problems are not well formulated.

* Belief in the utility and value of mathematics (p. 4).

But there are other important reasons to aim for enabling beyond
this key practical and utilitarian one, reasons of a social and
political nature. Again, the Commission speaks to the issue.

In a democratic country in which political and social decisions

involve increasingly complex technical issues, an educated,

informed electorate is critical. Current issues--such as
environmental protection, nuclear energy, defense spending, space
exploration, and taxation--involve many interrelated questions.

Their thoughtful resolution requires technological knowledge and

understanding. In particular, citizens must be able to read and

interpret complex, and sometimes conflicting, information (pp. 4-

5).

For too long, mathematics has played the role of gatekeeper. Those
who have managed to penetrate the ”finely polished, impersonal
mathematics curriculum" (Gordon, p. 362) have secured a key to
opportunity. Most who have acquired the key have had to suspend sense-
making in favor of mastery of mathematical knowledge as presented. A
few have persisted in getting behind the facade. Their efforts have

revealed the jumps of intuition, the stumblings and false starts, the

“terror and triumph" of constructing mathematical knowledge. Those who

 

6A colleague has quoted Pollak as admonishing the mathematics
education community with the following: ”When will teachers
understand that in the real world you don't know which page of the
text the problem is on!”

55
have given up trying to make sense, who have concluded that they are
incapable of mastering mathematics as given, ”try to escape the
situation, the feelings, and the knowledge that one does not know"
(Gordon, p. 372). Gordon continues,
These beliefs lead to fear--the fear is not of mathematics qua
mathematics, as I understand it; rather, this self-destructive
condition derives from the fear of not knowing, believing that one
should know, and from the fear of being discovered, of not being
able to defend oneself (pp. 373-373).
Escaping the situation diminishes one's options for further study,
future employment and full participation in society as an enlightened
and mathematically literate citizen.
Mathematics education must have as a fundamental goal the
empowerment of all learners to create knowledge and meaning.
Orientations to the teaching and learning of mathematics that turn

children into anonymous learners and teachers into mere technicians are

not liberatory--they enslave. Mathematics education must assume an
*' r“ 4.-.. w , .

‘u-W‘" "

orientation that promotes mathematical inquiry. As the Commission on
Standards for School Mathematics has argued, teachers and
administrators can no longer tolerate practices that distort what it
means to know mathematics and that delimit the full participation of
far too many students in our classes.

Teaching for enabling mathematical inquiry holds the potential for
what Aronowitz and Giroux (1985) call ”creative and reflective
discourse and action.”

[It] takes the issues of community and liberation seriously, and in

doing 50, gives new meaning to the pedagogical and political

necessity of creating the conditions for emancipatory forms of self

and social empowerment among both educators and students (p. 43).

Teaching for enabling mathematical inquiry is a radical agenda,

mathematically and politically. It is a struggle worthy of engagement.

CHAPTER 3

METHODS OF DATA COLLECTION AND ANALYSIS

INTRODUCTION

As is often the case with field-based studies that grow out of an
ethnographic paradigm, the nature of the questions that initially
inform a research study may change with immersion in a setting. The
collection and analysis of data proceed jointly throughout the study.
As new materials are collected and analyzed, new questions may emerge,
conceptual frameworks may prove to be inadequate and, in some cases, a
study may be radically redirected (Bogdan & Biklen, 1982; Eisenhart,
1988; Erickson, 1986; Hammersley & Atkinson, 1983). Initial questions
may fade in significance, to be replaced by new questions that, given
circumstances not anticipated, take on greater importance and even
urgency. Such was the case with this study. The study reported here
and the questions it considers represent a considerable departure from
its initial conceptualization.

This chapter describes the evolution of the study - the emergence
of new questions, the recognition of a much larger problem, the need
for different conceptual and analytic frameworks. In addition, the
methods of data collection and analysis used to carry out this study
are described. The intent of the chapter is to report on the

methodology and to tell the intellectual autobiography.

56

57
EVOLUTION OF THE STUDY
The Beginning: Acting to Influence Teacher Knowledge and Beliefs About
Conceptually-Oriented Mathematics through Inservice

The project was initially conceived to be an observational case
study of Marilyn Miller,1 an elementary mathematics supervisor in the
Detroit schools, as she planned and conducted inservice opportunities
for teachers, as she worked with teachers in their classrooms, and as
she developed and disseminated curriculum and resource materials. The
purpose was to investigate the nature of inservice and instructional
support she provided for teachers charged with implementing the newly-
mandated, ”conceptually-based," elementary mathematics curriculum. The
main question with which I began the study was, "In what ways and to
what extent does the supervisor act to influence teacher knowledge and
beliefs about mathematics, the elementary mathematics curriculum,
mathematics teaching and mathematics learning?"

Ine pzoblem to be studied. Leaders in mathematics education had
been calling for a reorganization of the mathematics curriculum around
concept development and problem solving since the beginning of the
19805. At the same time, there was the recognition that implementation
of a conceptually-based, problem solving approach to mathematics
instruction posed several substantial problems. Chief among them was
the limitation of teachers' knowledge about mathematics and the
teaching and learning of mathematics. There was a widely held belief
among reformers that teaching conceptually-oriented content required

teachers to have a conceptual understanding of mathematics, to know why

 

1Marilyn Miller is a pseudonym. In Chapter 7 I provide
background information on Marilyn's history as a mathematics
teacher, consultant, and supervisor.

58
understanding concepts is important, and to know how to help students
gain that understanding (Devaney, 1983; Shulman, 1986; Lampert, 1986b;
Resnick, 1983). If teachers are to overcome these limitations, it was
reasoned, they must have opportunities for high-quality professional
development programs.

A number of recommendations were made. School districts should
provide inservice on the mathematical content and pedagogy that develop
conceptual understanding and problem solving ability. Districts should
provide incentives for teachers to invest in inservice. School systems
should employ well-qualified mathematics specialists to coordinate
inservice efforts. Inservice should provide opportunities to integrate
theory and research with applications and practice in classrooms (An
Agenda for Action, 1980; NCTM, 1985). These guidelines were useful in
planning inservice but they seemed to underestimate or fail to
acknowledge the complexity of changing teacher knowledge and beliefs.

The literature on inservice education overwhelmingly pointed to the
ineffectiveness of most efforts to facilitate and sustain change in
teacher beliefs, attitudes and classroom practices (Wood & Thompson,
1980; Guskey, 1986). In an attempt to map the terrain leading to
teacher change, wide-ranging research and development agendas had been
mounted. A sampling suggests the diversity: defining models for more
effective staff development (Gall & Renchler, 1985; Wood & Thompson,
1980; Rauh, 1978; Jacullo-Noto, 1981); examining the institutional
context in which inservice occurs (McLaughlin & Marsh, 1978; Little,
1984; Barth, 1985); attending to teacher concerns (Lieberman, 1985;
Swanson-Owens, 1985; Hall & Loucks, 1978); investigating the role of

the teacher as learner, advisor and translator of research (Chism,

59
1985; Greene, 1978; Clark & Lampert, 1985; Little, 1985; Doyle &
Ponder, 1977; Day, 1985).

The missing link in this research was the inservice provider and
her/his role in the implementation process. Even when developers of
distinctive mathematics curriculum materials acknowledged that a
skilled and influential local coordinator was essential for training,
implementation and management, their research on the effectiveness of
innovative programs paid little attention to the specific ways in which
the district specialist influenced the course of the implementation
efforts.2

This study, as originally conceived, aimed to provide insights and
knowledge about planning and conducting inservice for teachers charged
with implementing an innovative mathematics curriculum. In particular,
how did one of Detroit's elementary mathematics supervisors, Marilyn
Miller, act to influence teacher knowledge and beliefs about
mathematics, teaching mathematics and learning mathematics? What
teacher knowledge bases -- subject matter knowledge, pedagogical
content knowledge, strategic knowledge -- did Marilyn act to influence
and in what ways? In what ways did she attempt to influence teacher
beliefs about curriculum content and methods of instruction and what
sources of authority - research findings, the "wisdom of practice”
(Shulman, 1987), district mandates, equity - did she invoke?

There was also a concern about the ways in which contextual factors
might affect implementation efforts. The structure of existing

curricular materials (particularly textbooks), the pressure to increase

 

2See, for example, the evaluation of the Comprehensive School
Mathematics Program, CEMREL, Inc., conducted by the CSMP Review Panel
for submission to the Joint Dissemination Review Panel, 1982.

60

test scores, the burdens of size, numbers and space, the movement of
children in and out of classrooms (or from school to school), may
frustrate a teacher's efforts at implementation. Most of these factors
are beyond the control of classroom teachers and staff developers. In
fact, some scholars have argued that changes in teacher knowledge,
skills and beliefs will not suffice to overcome the environmental
constraints (Cuban, 1984; Sarason, 1971; Jackson, 1978). Of concern in
this case was the extent to which Marilyn attended to contextual
constraints: the degree to which they were acknowledged; the degree to
which she was sympathetic to the demands that constitute the worklife
of teachers; the examples she provided of instructional practices and
curriculum in use under conditions of contextual constraints.

Collecting ghs data. The first phase of data collection involved
interviewing the elementary mathematics supervisor and attending a
series of inservice workshops she conducted for teachers in the
district. Over a period of four months, I observed nearly 50 hours of
inservice, recording the events with fieldnotes and audio-tapes. I
accumulated materials that were part of the inservice curriculum--
research articles, instructional activities focused on subject matter
and cooperative learning, model lessons, a quiz on effective
mathematics teaching practices. I also collected samples of teachers'
work on various activities. I secured copies of teachers' evaluations
of inservice - a written "Workshop Feedback" distributed by Marilyn and
completed by participants at the end of the three-day series.

I interviewed nearly a score of classroom teachers representing a
broad range of tenure with the district, from newly hired to 30-year

veterans. My interest in talking to teachers was three-fold. The

61
first was to inquire about how they were making sense of this new
”conceptually-based” mathematics curriculum and the district-wide
mandate to implement. Second was to learn what influenced their
decisions on whether to attend inservice. The third was to find out
from those who had attended inservice their assessment of the sessions.
Some of the interviews were tape-recorded; all were recorded with
fieldnotes. Some of the teachers with whom I spoke had taken advantage
of the inservice opportunities provided by the supervisor, others had
not. The interviews were partly structured. There was a set of
questions asked of all teachers and an additional set asked of those
who attended inservice. At the same time, the interviews were
conducted so as to allow for probes to pursue further issues raised by
an individual teacher. The Appendix contains the standard interview
schedule that formed the core of teacher interviews.

I also collected curriculum materials that had been developed by
the district and distributed to teachers and Area mathematics
specialists. These materials included: a document describing the
district's philosophy regarding mathematics education and the
mathematical strands that organized the K-5 curriculum; the sets of
objectives that constituted the intended curriculum and sequenced
instruction at each grade level; the sets of model lessons that
provided examples of how the curriculum was to be enacted in the
classroom as teachers taught to the instructional objectives.

5 w ve a s 5' Emer i eme . Erickson describes
the analytic process in ethnography as "repeated trials at
understanding recurrent events" (1986, pp. 143-44). My focus in the

first "trial" was on what Marilyn thought teachers needed to know and

62
be able to do to implement the new curriculum and how she thought they
would acquire the necessary knowledge and skills. At this point in the
study I had conducted two formal interviews with Marilyn, both prior to
any of the inservice offerings. I had observed two different 9-hour
workshop series for K-l teachers and collected artifacts from those
workshops. In addition, I had accumulated most of the formal
curriculum documents that had been distributed to Area mathematics
specialists and classroom teachers.

I transcribed each of the interviews with Marilyn. In analyzing
these interviews, I looked for the following: 1) the goals Marilyn had
for teachers as she planned inservice; and 2) what she intended to
focus on -- teachers' subject matter knowledge, classroom activities,
the strands and objectives, instructional strategies, instructional
materials, classroom management, research, testing -- and why. I did
not do a full transcription of the inservice workshops. Rather, I
listened to the audio recordings and elaborated on fieldnotes that I
had taken during inservice. I listened to one of the recordings a
second time, making a list of activities/events and the number of
minutes spent on each one.3 For each event/activity I noted the
following: 1) where the event was centered-~Marilyn ”lecturing" or
giving a demonstration where teachers listened and observed; Marilyn
and teachers acting together as one large group; teachers working
individually or in small groups on an activity; 2) what the topic of

the event was--curriculum strands, research, using manipulatives,

 

3One of these workshops was conducted on December 7, 8 and 10,
the other December 14, 15 and 17, 1987. Both were for teachers in
grades K-1. There was only slight variation in the two series and
that was caused by extremely bad weather on DecemberlS that resulted
in a late beginning and an early dismissal.

63
grouping for instruction, teaching to an objective, understanding
concepts, problem solving; 3) how the event/activity was carried out--
lecture, demonstration, reading, 'jigsawing," using manipulatives,
taking a test, making lists.

I compared what Marilyn had talked about as goals for the inservice
with the activities she conducted in the workshops. I wrote an
analytic memo (Hammersley & Atkinson, 1983, pp. 164-167), noting
several emerging themes: 1) there seemed to be a heavy emphasis on a
selected, fairly narrow body of research findings on teacher
effectiveness and effective mathematics teaching as a rationale for
instructional strategies; 2) there seemed to be no getting away from
the emphasis on testing in the district, either in practice or in
teachers' minds; 3) contextual factors seemed to be a significant
element in the inservice and implementation effort; 4) there seemed to
be a strong technical, prescriptive orientation to the inservice
design; 5) there seemed to be a contradiction between a rhetoric of
”teaching for conceptual understanding” that Marilyn used and the
choices she made for the curriculum of inservice.

These last two points became increasingly significant as I tried to
make sense of what I was hearing and what I was seeing. At the time, I
was assuming the dichotomous traditional/conceptual orientations to
mathematics and mathematics teaching and learning that I described in
Chapter 2. It appeared to me that a substantial part of inservice was
not aimed at any substantive change in teaching practice. Despite the
rhetoric of ”teaching for conceptual understanding," inservice seemed
to be more closely aligned to a traditional approach but aimed at

making it more effective and efficient. I puzzled over this apparent

64
contradiction, wrote an analytic memo which I asked several colleagues
to comment on, and read (and re-read) some theoretical and empirical
works suggested by them (Ball, 1988; Good, Grouws & Ebmeier, 1983;
Hiebert & Lefevre, 1986; Lampert, 1986a, 1986b, in press a, in press b;
Madsen-Nason, 1988). What I began to see was that the rhetorical goal
of developing conceptual understanding of mathematical ideas had become
commonplace in the discourse of the mathematics education community.
But the meanings contained in this goal and the means to attainment of
it were matters of disagreement and even debate. What was becoming
apparent was that the framework I had brought to the study in terms of
orientations to mathematics education, traditional vs. conceptual, was
inadequate. It simply did not reflect two very different meanings
attached to "teaching for conceptual understanding." The meaning I
attached to ”teaching for conceptual understanding" reflected an
orientation aligned with enabling mathematical inquiry. There had been
an assumption, on my part at least, that informant and researcher
shared a common meaning for a common language. That assumption was
being called into question. Another possible interpretation for the
"apparent" contradiction between Marilyn's rhetoric and her practice
was that for her, teaching for conceptual understanding was more
aligned with an orientation of teaching for comprehension of

mathematical ideas.

What Does 'Teaching For Conceptual Understanding“ Mean in This Setting?

The ssssnd wavs sf snalysis; New sategsgiss. The second wave of

analysis began with a new framework for thinking about orientations to

the teaching and learning of mathematics. The new framework posed

65
three orientations as ideal types: 1) certification of numeracy; 2)
comprehension of mathematical ideas; and 3) enabling mathematical
inquiry (see Chapter 2).“ I returned to the data for a second ”trial,”
shifting the focus to the question, "What does teaching for conceptual
understanding mean in this setting?”

I distinguished the three orientations to mathematics education
along the following categories: 1) the goals for students as learners
of mathematics; 2) the representation of mathematics; 3) the selection
of and emphasis on topics to be studied; 4) the verbs that describe
student actions; 5) the role of the teacher in instruction; 6) the role
of the student in instruction; 7) how learning is portrayed; and 8) the
source of authority for knowing.

These categories provided a useful framework for data-source
triangulation-~the interrogation and comparison of data gathered at
different points in time, in different contexts and from different
sources (Hammersley & Atkinson, 1983, pp. 198-200). The categories are
presented below with brief descriptions for each orientation to

illustrate the differences among them.

Goals or Students as arne o themat
Certification computational speed and accuracy; proficiency with
of Numeracy conventional algorithms and their application to

routine word problems.

Comprehension proficiency with and understanding of mathematical
of Mathematical procedures - knowing how to use them and why they
ideas work; understanding basic concepts such as place

value and operations.

 

4These orientations have been described in detail in Chapter 2.
I am particularly indebted to Deborah Ball for helping me to think
less simplistically about this matter.

Enabling
Mathematical
Inquiry

Certification
of Numeracy

Comprehension
of Mathematical
Ideas

Enabling
Mathematical
Inquiry

Certification
of Numeracy

Comprehension
of Mathematical
Ideas

Enabling
Mathematical
Inquiry

66

empowering students - to learn £222; mathematics
and to learn to ss mathematics.

WW2:

a system of abstract symbols to be manipulated and
a collection of skills, definitions, rules and
procedures to be memorized and mastered; static,
“discovered," and rule-bounded.

a body of related ideas within a linear, rule-
bounded hierarchical arrangement; an abstract
system of concepts and procedures for which there
are appropriate concrete models and pictorial
representations.

dynamic, the creation of human activity; a body of
skills, concepts and propositions connected by a
rich web of relationships.

Topics 10 fig Sgngisg

arithmetic - addition, subtraction, multiplication
and division of whole numbers, fractions and
decimals - constitutes the bulk of the topics to be
studied; emphasis is on formal language and symbolic
representation.

arithmetic dominates but is supplemented by topics
from geometry, statistics and probability, and
number and set theory.

shift from arithmetic to a broad range of content
and processes--numeration and number sense,
computation and estimation, concepts of whole number
operations, measurement, geometry and spatial sense,
statistics and probability, fractions and decimals,
patterns and relationships; mathematics as problem
solving, communication, reasoning; mathematical
connections.

 

‘ 5See Cnggisnlnn ans Evslustisn Sgsndszgs i0; Sshsol usgnenatiss,

1989, from which these were drawn.

67

Verbs That Descgibe Sgngsn; As§12n§6
Certification read, identify, name, recall, recognize, add,
of Numeracy subtract, multiply, divide, state the rule,

demonstrate accuracy.

Comprehension relate, extend, demonstrate, complete, estimate, of
Mathematical measure, compare, apply strategies, determine

Ideas reasonableness of answers.

Enabling investigate, explore, predict, invent, generalize.
Mathematical

Inquiry

The Role Of The Teaches In lnsgnnsgign

Certification shows students how to carry out procedures and
of Numeracy provides ways for them to remember facts and steps
in algorithmic procedures.

Comprehension shows students what they are to learn by
of Mathematical demonstrating with appropriate concrete materials
Ideas and letting students practice what they have been

shown; monitors student learning during instruction
with "product" and "process” questions; evaluates
student responses and provides immediate feedback;
maintains a brisk pace of instruction, keeps
students on task and holds them accountable to
complete assigned work.

Enabling creates a total environment where teacher and
Mathematical students engage with one another in inquiry; poses
Inquiry interesting, challenging and non-routine problem

situations that lead to learner inventions; provides
a variety of representations; acts as a guide,
posing questions for further reflection, responding

 

6The schema used here represents an alteration of categories of
student actions proposed by Avital and Shettleworth (1968) who
adapted Bloom's Taxonomy (1965) to describe levels of mathematical
performance more precisely. Avital and Shettleworth provide three
levels: (1) recall-recognition (knowledge in Bloom); (2) algorithmic
thinking (comprehension and application); and (3) open search
(analysis/synthesis). The first category calls for recall of facts,
definitions and procedures in exactly the way they were presented.
The second category requires transfer of knowledge. Most common
would be application of algorithms to solve routine story problems.
The third category calls for using knowledge in nonroutine
situations to discover relationships and make generalizations.

Certification
of Numeracy

Comprehension
of Mathematical
Ideas

Enabling
Mathematical

Inquiry

Certification
of Numeracy

Comprehension
of Mathematical
Ideas

Enabling
Mathematical
Inquiry

Certification
Numeracy

Comprehension
Mathematical
Ideas

68

to questions by deciding which to pursue and which
to table.

e Ro e O The 8 at n as t o
absorbs and remembers the knowledge that a teacher
dispenses.

pursues teacher defined explorations, answers
teacher questions and completes practice exercises
within structure imposed by the teacher.

becomes a mathematical risk-taker—-makes guesses and
pursues hunches, offers hypotheses, accepts
challenges to hypotheses and marshalls arguments in
support of them, yielding to the force of an
argument that is mathematically more reasonable.

How Learning Is Portrgyeg

an individual, solitary effort where students
search their memory for accumulated facts and
algorithmic solutions; students are passive
receptacles into which mathematical knowledge is
poured.

requires a particular sequence--concrete to
pictorial to abstract/symbolic; is related to the
instructional technique and technical competence of
the teacher.

children are not passive learners; students
actively construct, interpret and put structure on
new mathematical learning; requires a learning
community that supports individual and collective
efforts to make conjectures, develop arguments,
invent procedures, build abstractions and
generalizations, apply quantitative and spatial
reasoning.

Source Of u ow

teacher and text - teacher as judge, text as the of
standard for judgment.

teacher - decides if student answers are correct of
and explanations sufficient.

69

Enabling classroom community of students and teacher who

Mathematical have power to use mathematical tools and standards

Inquiry of the mathematical community to decide about the
reasonableness of processes and the results of
investigations.

There were several “settings" to which this framework was applied.
The first that I interrogated was the set of formal curriculum
documents. These included: a booklet Qgg121§_£nblig_§gh921§_212;

ad' :a_ e - _ad- ,_ _ r. u u' t as-‘ a o, ec v‘ - 1a s‘ua
Egngggign that contained the district's philosophy regarding
mathematics education and defined the ten mathematical strands and
learning objectives for each strand at each grade level; the
Instructional Sequence that specified the behavioral objectives for
each grade and the sequence in which teachers were to teach to those
objectives; and Model Lessons to Eromote Thinking, a set of materials
that included an elaboration of each of the ten curriculum strands
followed by a set of model lessons for specific objectives at each
grade level. The purpose in examining these materials was to determine
what warranted describing the curriculum as ”conceptually-based" and
whether it aimed for comprehension of mathematical ideas or enabling
mathematical inquiry. Chapters 5 and 6 are devoted to the analysis of
the formal curriculum documents.

The second setting I examined was the set of interviews I had
conducted with Marilyn. The third was the setting of inservice
workshops. Chapter 7 scrutinizes these settings.

e amew k 0 3 es t e tat th fo ' ew
gnggtigng. The schema just described provided a useful framework to

analyze the nature of this elementary mathematics curriculum reform.

70
The analysis of the formal curriculum documents revealed a tension
between an orientation toward teaching for certification of numeracy
and an orientation toward teaching for comprehension of mathematical
ideas. The inservice workshops seemed to be oriented toward teaching
for comprehension of'mathematical ideas. As I began to get a sense of
the orientation of this initiative, another set of questions emerged.

How would implementation of this mathematics curriculum make a real
difference in the lives of the substantial number of at-risk children
in the Detroit schools? The District's publicly expressed goals for "
students as learners of mathematics were to help them ”think
critically, flexibly, copperatively and independently,” to develop
”higher-order thinking skills of analysis, synthesis and evaluation,”
”to apply their knowledge of mathematics in solving problems," and ”to
develop, to create, and to analyze.” These seemed like goals that
required an environment, a curriculum and modes of teaching aimed at
enabling mathematical inquiry. There appeared to be a new
contradiction, a contradiction between these intended goals and the
nature of the reform itself and how teachers were being educated for
its implementation.

I began to sense that the magnitude of the problem embedded in this
study was far greater than simply how the supervisor was using
inservice to influence teacher knowledge and beliefs toward
implementation of this reform. Why was the reform conceived as it was?
What factors shaped the design of the strands and objectives, the
instructional sequence, and the set of Model Lessons? What set of

conditions supported mandates for its implementation? With these

71
questions, I returned to data already collected, shifting the focus of

features to attend to.

THE OOHTEXTS OF EDUCATIONAL.POLICYMAKINC

An interrogation of the data with these questions as the lens
produced new insights. An important element in shaping this reform was
a district-created test called the Assessment of Basic Curriculum
Skills (ABCS). I set out to learn more about its development and its
influence on this reform. A second insight was the frequency of
references to ”increasing administrative authority." I wanted to learn
how this was manifested and to what extent it was a factor in this
reform initiative. Put broadly, I wondered about the contexts in
which this reform was conceived. The search led me back nearly two
decades to events that were to shape this initiative. The inquiry
brought into clear focus the historical, sociopolitical, economic and

organizational situatedness of this reform effort.

Historical, Political and Organizational Contexts

A number of sources provided data relative to the contexts that
influenced the course of this reform. I interviewed several key
figures with a long history of involvement in Detroit school politics
and policymaking. One such individual was a veteran, high-ranking and
highly-respected central administrator in the district. He provided an
account of factors that led to the decision to adopt the city-wide
curricula, going back to the decentralization of the district in the
early 19708. He made available his extensive files on the U.S.

District Court desegregation orders, including in-house memoranda as

72
well as formal documents prepared for and submitted to the Court and
the Monitoring Commission. He provided his perspective on the tendency
in the district to centralize authority over curriculum.

A person with a unique vantage point from inside as well as outside
the schools was a member of the U.S. District Court Monitoring
Commission. The Monitoring Commission was established as an arm of the
Court to oversee the federal desegregation order. It was to function
as a citizen's committee, auditing the District's efforts and assisting
the Court with reports and recommendations regarding implementation of
the components of the desegregation plan. He provided a perspective on
the central administration's response to the court order to include
educational components to wipe out the effects of past discriminatory
practices in the district. He also linked the desegregation efforts
with changes in the administrative structure of the district and the
attempt of the central administration to wrestle control over
instructional matters from the decentralized regional boards and
superintendents. He provided documents prepared by the Monitoring
Commission including press releases, status reports and lengthier
Profiles submitted to the District Court, N.A.A.C.P., the Detroit
Schools, the State of Michigan, the Detroit Federation of Teachers and
made available to the public.

A third key individual whom I interviewed was the education
director of a civic leadership organization representing a coalition of
corporate, business, labor and community organizations. The
organization was conceived following the 1967 rebellion to combat the
urban social problems that had contributed to the uprising. Its

education director at the time of this study was a man with a long and

73
distinguished history of educational activism at a number of levels --
from teaching in Mississippi Freedom schools to grassroots community
efforts in Detroit to the Michigan Department of Education. He, too,
provided another perspective on decentralization, the struggle of the
black community for control over its schools, and the recent trend to
recentralize increasingly more authority over curriculum and
instruction at the central administration level.

An interview with an administrator in Research and Evaluation
provided a history of the development of ABCS. Interviews with
additional personnel in the district -- Area supervisors, district
curriculum and instructional specialists, -- added considerably to the
unfolding story of the conceptualization of this reform.

This rich set of data was subjected to data reduction out of which
emerged a number of categories: school governance; racial politics;
equity; testing; curriculum decision-making; goals for learning. I
looked for instances of these categories in the various interviews I
had conducted and documents I had collected. In some cases these
categories intersected. For example, understanding school governance
involved understanding the influence of the politics of race in
Detroit. Curriculum decision-making intersected with testing and goals
for learning. In addition, sub-categories emerged that were important
to consider. School governance subsumed several themes: administrative
authority over the District--decentralization, then recentralization;
administrative authority over curriculum; administrative authority over
testing; city-wide mandates. The interrogation of the data helped to
reveal in what ways and to what extent these contexts were significant

in explaining the nature of this reform. Context also became a lens

74
for further reflection on teacher inservice. A reexamination of
inservice data revealed how context -- the structure of the inservice
sessions and the settings of participating teachers' practice --

influenced the content of inservice.

Economic Contexts

This reform was conceived in a district faced with an incredible
set of problems, many of which grew out of conditions in the larger
community over which schools had no control. I collected data to
assess the impact of economic and social dislocation on the district's
ability to provide a quality educational program for its youngsters.
Nearly every person in the schools with whom I spoke made some
reference to the impact of the increasing poverty and social
disorganization on the system and in individual classrooms.

I interviewed experts on the changing structure of the Detroit
economy. They provided important demographic and economic data for the
city that graphically portrayed the extent of social disorganization
that resulted from two decades of economic disinvestment and industrial
dislocation. They portrayed a grim future of further impoverishment
and reduced life chances for the next generation of its citizens. They
also voiced a sense of urgency about the need for the Detroit schools
to prepare young people with the knowledge, skills and dispositions to
find and hold steady, decent paying jobs in a changed Detroit economy.
Additional data were gathered on school financing, student achievement,
school drop-out rates, and student mobility within the district.

Coincidentally, The Detroit Free Ezgss, in 1988, dedicated the "op

ed” page once a week to its investigations of the state of education in

75
the city's schools. The weekly published reports served as another
data collection point, providing valuable information about
organizational politics in the schools and the impact on schools of

economic and social disorganization in the city.

.A.PLAN FOR PRESENTING THE FINDINGS

In the chapters that follow I describe the results of the analysis.
Chapter 4 describes the influence of sociopolitical and organizational
contexts that ultimately led to and shaped this curriculum reform
initiative. Chapters 5 and 6 examine the curriculum reform through its
formal documents to determine the orientation of the reform and to
inquire about its potential to empower Detroits' youngsters with
mathematical knowledge and power. Chapter 7 examines the inservice
provided for teachers charged with implementing this reform. The
inquiry focuses on what the inservice provider thought teachers needed
to know to implement the reform and how she thought they would learn
what they needed to know. As in the previous two chapters, the
ultimate concern is the extent to which inservice provided teachers
with the knowledge, skills and dispositions to develop mathematical
literacy among the next generation of youth, particularly minority
youth, to participate fully as citizens and workers in a democratic
society.

In Chapter 8, I return to the idea of context. Using Foucault's
notion of history as genealogy, I provide a ”genealogical analysis”
(see Noujain, 1987) of the Detroit elementary mathematics curriculum
reform. It is a narration of the set of ideas, events, practices and

people that combined to form the initiative. The basic antecedants

76
were an instrumental ideology and a technical rationality that dominate
American education and the curriculum field; the politics of race and
school governance in Detroit; the recognition of a new economic order
and Detroit's position in it.

I conclude by arguing for a further transformation of mathematics
education in Detroit toward enabling mathematical inquiry. To do
anything less means that mathematics will continue to be a gatekeeper,
denying the possibility of the good life to far too many of our

citizens, particularly those of color.

CHAPTER 4

THE INFDUENCE OF SOCIOPOLITICAL.AND ORGANIZATIONAL
CONTEXTS IN SHAPING THE REFORHL

INTRODUCTION

This chapter examines the contextual factors that had a direct
influence on the nature of the Detroit mathematics curriculum reform
initiative. This reform was shaped and supported, in part, by factors
located within the school context. The Federal Court desegregation
order of 1975 included a set of educational components that would
ultimately lead to this curriculum reform effort. In addition, two
decades of changes in organizational and authority structures and
recent efforts to recentralize administrative control over curriculum
would influence the nature of the mandate for its implementation. This
chapter addresses the first question of this study: In what ways and to
what extent had political and organizational factors shaped the design

of this curriculum and mandates for its implementation?

THE TESTING COMPONENT OF THE DESEGREGATION ORDER

The Assessment of Basic Curriculum Skills

The development of 5505. The testing component of the
desegregation order has particular relevance to this study of the
elementary mathematics curriculum reform. The testing component was
established by orders of the Court on May 11 and June 1, 1976. The
orders directed the Detroit Board of Education and the State Board of
Education to jointly plan a comprehensive testing program ”consistent
with the goals of a desegregated system.” It specified four elements:
1) review all tests and testing procedures for racial, ethnic and

77

78
cultural bias; 2) provide inservice training to all staff on non-
discriminatory administration of tests and uses of test results; 3)
develop an objective-referenced city-wide testing program in reading,
writing and mathematics at all levels; 4) establish an evaluation
program to assess the effectiveness of instructional programs and use
the evaluations for curriculum development and planning and policy
determinations. The district's accomplishments in developing a city-
wide objective-referenced testing program and using the test results to
evaluate the effectiveness of instructional programs ultimately led to
the curriculum reform that is the object of this study.

At the time of the Court order, the city-wide testing program at
the elementary grades used the Stanford Achievement Test, the Iowa
Tests of Basic Skills, and the Cognitive Abilities Test, all norm-
referenced standardized tests. In July, 1977, the Detroit Testing Task
Force appointed a Test Review Team to conduct an examination of all
tests used in the city-wide testing program. At the conclusion of
their review, the Team made the following recommendations: "1) the
California Achievement Tests (CAT, 1977 edition) be adopted for all
pupils in benchmark grades (to be determined by the Central Board of
Education); 2) objective-referenced tests (ORT's) be given city-wide in
grades 1 through 12 as they are developed; 3) until ORT instruments
were ready, the CAT be used to fill any voids in the new city-wide
testing program; and 4) no norm-referenced group aptitude tests be
administered" (Detroit Public Schools, June, 1983, p. 10). By 1980,
CAT had replaced all previously used standardized tests of student

achievement in grades 1-11.

79

The district launched it's objective-referenced test development
program in 1977. The first test developed was the Ninth Grade
Communication Skills Test, administered as a pre-test and post-test to
assess reading, writing, and study skills. The second test was the
High School Proficiency Program Examination (see Chapter 1). Both
tests were developed in collaboration with a private test development
company. The development of a criterion-referenced program for the
elementary and middle grades was an in-house effort. From 1977 until
1982, the Research and Evaluation Department, Test Development Team,
and Curriculum Development and Services staff worked to develop a
program to measure student achievement in reading, writing and
mathematics for grades l-8. The program was called Assessment of Basic
Curriculum Skills (ABCS). Teams of curriculum specialists, mathematics
supervisors, mathematics instructional specialists, testing specialists
and classroom teachers worked on the development of the mathematics
portion of ABCS for grades 1-8. They drafted sets of learning
objectives, specified mathematical competencies, developed scope and
sequence charts of objectives, wrote test specifications and test items
for each competency and field tested some of the math items (Detroit
Public Schools, June, 1982; Detroit Public Schools, April, 1984).

The mathematics portion of ABCS represented a compromise of
contending views about what it means to know mathematics and how that
knowledge should be assessed. Mathematics specialists argued that the
test should assess student understanding of mathematical concepts and
their application in problem solving and that each competency be
introduced in a problem solving mode. Teachers, on the other hand,

wanted conventional standardized tests emphasizing computational

80
proficiency with some word problems. To reach a compromise, a group of
mathematics supervisors, teachers and testing specialists were brought
together and charged with designing a test to strike a balance between
the two positions. An administrator in Research and Evaluation
provided the following account of the conflict surrounding the effort
to create the mathematics portion of ABCS.

The math portion of ABCS was the most difficult. We worked closely

with the elementary mathematics supervisor who had been very active

in the NCTM movement. She wanted more than just to have students
compute. We hired math teachers to develop test items. When the
math supervisor and curriculum people saw the first test, they
rejected it. It looked too much like a standardized test. Problem
solving was not real problem solving, just computational problems
using words. 80 curriculum told the math people to go off and
develop their own. So they did. When we saw their test, 22 said,

'No way!’ It was just too difficult - real hard. So curriculum

hired a third group of teachers and got the math supervisors to

work with us and to supervise the teachers. But there was still
daily conflict between the two groups trying to strike a balance.

So what you see in ABCS is a balance between traditional math tests

and the far extreme problem solving. This is in the middle. When

I get comments about how difficult it is I think, 'You should have

seen what we had before.’

The test development team identified nine "competency strands" that
provided the organizing framework for test objectives and test items--
operations, numeration, estimation, patterns, geometry, measurement,
sets and logic, functions and relations, and statistics and
probability. Drawing on skills selected from textbooks adopted by the
district and skills endorsed by the NCTM (An Agenda for Aggign, 1980),
the test-makers created two sub-tests, one assessing computational
proficiency, the other conceptual understanding and application. In
its final report to the Court on implementation of the testing
components, the Board noted that ”several revisions of the competencies

were considered before agreement was reached...Members of the

mathematics department strongly urged that each of the mathematics

81

competencies be introduced as word problems or in problem solving modes
and in line with the goals of the National Council of Teachers of
Mathematics“ (Detroit Public Schools, June, 1983, p. 25). The final
form of the Assessment of Basic Curriculum Skills was administered for
the first time in spring, 1983.

the est 0 d ve the cu . In the first
administration of ABCS, the results on the mathematics portion were
dismal. Table 3.1 shows the percentage of students at each grade level
who achieved 60! or above on each competency.

Of the 45 cells in the table, 18 show percentages of 501 or less.
These cells indicate that at a particular grade for a specific
competency, fewer than half the children were able to answer at least
601 of the items correctly. First and second graders did relatively
well. At least half the children in Grades 1 and 2 scored at or above
601 on all nine competencies. But beyond second grade, the level of
achievement tended to drop. The lack of achievment was acute at grades
4 and 5. At the fourth and fifth grade, the majority of children did
well on only two comnetencies--operations and measurement at Grade 4,
numeration and sets and logic at Grade 5. The poor performance of
children on the mathematics portion of ABCS raised a question about the
degree to which the test items were correlated with what children were

being taught.

82

Table 3.1 Percent of students at Each Grade Achieving At or Above
601 on Each Mathematics Competency, ABCS, May 1983

 

COMPETENCY GRADE

1 2 3 4 5
Operations , 61 84 61 57 48
Numeration 73 62 71 34 66
Estimation 55 56 38 41 50
Patterns 64 50 58 28 39
Geometry 69 83 60 44 24
Measurement 81 88 64 64 47
Sets and Logic 92 78 77 35 58
Functions and Relations 53 75 47 37 30

Statistics and Probability 80 91 39 44 38

Source: Report from 1 to Area Superintendents,
November 14, 1983.

 

The Board had informed the Court in its final report on the
testing component -- which was submitted before results of the first
administration of the ABCS test were available -- that the mathematics
portion of ABCS correlated ”highly" with the "new curricular emphasis"
on problem solving. It is unclear what was used for the correlation
study since the district did not have a mathematics curriculum per se.
Apparently the scope and sequence of objectives that had been developed
for ABCS was distributed to Area offices and may have made its way to

some teachers' hands, but it had not been defined as the explicit

 

1The name of the person who released the memo is not included
for confidential reasons.

83
mathematics curriculum. A later study, conducted in 1985 by
mathematics specialists from one of the district Area offices, assessed
the degree of correlation between the scope and sequence of ABCS
objectives and three of the four textbook series adopted by the Board
for use in the district. That assessment revealed a significant number
of objectives at each grade level that were absent from or inadequately
attended to in one or more of the texts.

One element of the testing component stipulated that test results
be used to improve the effectiveness of educational programs, inform
policy and planning decisions and assist curriculum development. Given
the considerable support of mathematics supervisors and testing,
evaluation and curriculum personnel for ABCS, the decision was made to
reexamine the K-5 mathematics curriculum. A writing team was charged
with developing an instructional sequence and a guide that reflected
the ”new and changing needs with emphasis on higher level thinking
skills and future employment" (Detroit Public Schools, 1984).

The administrator in Research and Evaluation confirmed that key
personnel involved in the test development effort viewed ABCS as a
vehicle to promote changes in the mathematics curriculum.

There is no question that people were thinking about ways the test

could lead to an improvement in the curriculum. In fact,

(a highly respected veteran administrator in
the district who was a moving force in the test development effort)
talked about a test driven curriculum. We all recognized the power
of tests. If you look at ABCS, the writing program in the city has
changed drastically because of testing. The High School

Proficiency Exam with its paragraph writing - that made us put it
in ABCS.

 

In January, 1987, the Superintendent and the Board approved a K-S
mathematics curriculum derived from the scope and sequence of

objectives that had been developed for the competency strands of ABCS.

84
The curriculum consisted of a set of one-hundred or more objectives for
each grade level. The objectives were sequenced by month and teachers
were told to teach the objectives in the order in which they were
listed. The Board mandated city-wide implementation of the curriculum

over a three year period beginning with the 1987-88 school year.

THE MANDATE FOR.CITY4WIDE IMPLEMENTATION
The decision to adopt a standardized mathematics curriculum
district-wide reflected two administrative concerns: 1) improving
educational equity for all children; and 2) increasing central

administrative control over curriculum matters.

Improving Educational Equity

The adoption of a city-wide mathematics curriculum was an attempt
to equalize learning opportunities across schools, regardless of their
size or location within the city. The Detroit School Board noted in a
September, 1975, response to the Court's stipulation that the testing
program would provide information on the comparability of student
achievement across schools in the district. The comparison of MEAP
scores showed considerable variation across schools (see Chapter 1).
Children attending schools in some of the poorest neighborhoods of the
city were scoring substantially below children who attended schools in
the city's middle-class neighborhoods. The district was under
considerable pressure to improve the quality of mathematics instruction
for n11 children in the district. Adoption of the standardized
curriculum announced to the community -- teachers, administrators,

students, parents -- that every youngster in the city was entitled to

85
quality instruction in mathematics. As a high level central
administrator in the district commented:

The city-wide curriculum says there are common goals for 511

students. Teachers can still tend to individual needs and teachers

can still be creative...We are going along with a national movement
to define a core curriculum for gxggxgng.

The mandate to implement the curriculum city-wide was also a
reponse to the problem of multiple transfers within the district. The
dramatic increase in the ranks of the city's chronically unemployed and
the forced moves of families within the city to secure temporary
housing meant that many children might attend several different schools
in one year. For many teachers this meant that a significant subset of
their students changed over the course of the year. Some teachers
reported that as many as half the students in their class might change
in one school year. A third grade teacher in a combined
elementary/junior high school told me of visiting a sixth grade room in
her school that fall.

I walked into her room and I looked around expecting to see some

familiar faces. But there was not a single youngsge: in that

classroom that I had had in my class just three years before. You

see, nearly all our children are bussed in and many come from a

low-income housing project. These poor kids move around from

school to school.
The educational problems associated with a mobile student population
was mentioned by every person at every level of the school organization
that I interviewed. The hope was that standardization of curricula
across the district would reduce the educational disruption for
children who moved among schools during the year. In this respect, the
district initiative was an attempt to deal with a set of educational

problems rooted in social and economic inequalities within the larger

community.

86
An egaliatarian impulse was not the only force driving the
standardization of curricula. The move to mandate district-wide
implementation of curricula in seven content areas reflected a
continuing attempt by the central administration to exert increasing

control over curriculum and instruction.

Increasing Central Administrative Control Over Curriculum

Over a period of nearly two decades, the district had undergone
significant changes in organizational and authority structures. The
decentralization of the district in 1970 had shifted considerable
administrative authority to regional boards and the superintendents

they appointed. Under the decentralized arrangement, regional

administrators assumed control over curriculum and staffing decisions

in their schools. The Court overseeing the federal desegregation order

found the decentralized structure problematic. The member of the

Monitoring Commission provided the following account.

The Court and the Monitoring Commission saw decentralization as an
obstacle. The Court was determined to deal with all the effects of

segregation including educational programs which were different

among Regions. For example, the reading programs were different in
the Regions. With bussing imminent, the Court and the Commission

felt the district needed to standardize the reading program.

The educational components stipulated in the 1975 federal

desegregation order, to which the central board was obliged to respond,

restored some administrative authority over educational programs to the

central administrative level. Again, the Monitoring Commission member:

The central administration at the time of the desegregation order
didn't have much power. The Court order gave them the opportunity

to consolidate power. The stipulation to include educational
components gave the central administration the opening to take

recommendations from earlier task force reports and propose them to

the Court. They said, ”We want these programs,” and the Judge

looked for a rationale to support them that was consistent with the

87

intent of the order. Generally they wrote proposals that would get

funding from the Feds. Reading, inservice and testing were going

to get ESEA funding...Actually what they came in with was not a

reading program. It was a reading mgnhggngn; program, a testing

program. The move to ORT (objective-referenced testing) was
supported by the state and they got sufficient funding from ESEA
and the state...Those programs that were most successful was where
cooperation with Region administrators was solicited.

The desegregation order began the transfer of power over
instructional programs from the regional to the central level. The
recentralization of the district in 1981 further altered the balance of
authority over educational programs but the transfer of power from
Areas to central administration was not complete or automatic. There
remained considerable ambiguity about what schools should be teaching
and who should make those decisions. According to key personnel who
led the way in the reform initiative, the decision in 1987 to adopt a
standardized curriculum and mandate its implementation city-wide
reflected an attempt to address two important issues; 1) providing
educational leadership at the administrative level, and 2) defining the
essential learnings in mathematics for grades K-5.

Egoviging educational leadership. One concern of central
administrators was the lack of educational leadership exhibited by Area
superintendents and school principals. There was the belief that these
administrators lacked knowledge about curriculum matters. As a central
administrator commented:

Unfortunately, many or our (Area) superintendents and principals

are not very knowledgeable about curriculum. They have seen their

responsibility as administration and supervision, not instruction
or curriculum. Principals never were expected to be great
instructional leaders. We haven't articulated this expectation of
principals.

With respect to mathematics, he offered:

There has been a lot of resistance on the part of principals. They
think math is computation. There is a lot of resistance to

88

estimation, geometry and problem solving. They want to keep it

simple--train' em. They would rather have teachers teach the black

box. We have been getting a lot of flack on ABCS.
A district elementary mathematics supervisor put it this way:

We had the ABCS objectives on paper but when we would go from one

Area to another, we saw different things. One Area mandated this,

another mandated that. Some Area superintendents had a reputation

for insisting that their teachers teach to the tests. They put a

heavy emphasis on improving MEAP test scores. Even some of the

Area curriculum administrators had their math specialists preparing

materials just aimed at the tests. And mostly that was at a

computation 1evel...Some teachers say to me, "I have been told to

teach the test. You tell me I'm supposed to follow the
instructional sequence. I'm told I have to spend one week on this
skill, one week on that skill. Practice, practice, practice.” Or,

”My principal doesn't like the noise that's coming out of my room,”

when that is indeed good noise. That's a very legitimate

complaint.

Key administrators and mathematics supervisors at the central level
felt the educational leadership provided by some Area administrators
and building principals had been inappropriate or inadequate. In the
case of mathematics, they felt the dominant interest in improving test
scores resulted in an inordinate emphasis on developing low-level
computational skills while ignoring other important mathematical
understandings. It was not the case that those promoting the new
mathematics reform were not concerned about test scores. Indeed they
were. However, their belief was that the new objectives, if followed,
would not only better prepare children for the tests but would enhance
their understanding of a broader range of mathematical ideas. By
mandating implementation of a uniform curriculum city-wide, central
administration was stepping in to take control of educational matters
where it felt local leadership had failed.

e th u enta e in a . In the process

of developing the ABCS test, those working on the mathematics portion

identified nine "mathematical strands" in which all students should be

89
competent. With this set of mathematical topics as an organizing
framework, the team defined specific learning objectives for each
strand. These objectives were made available to teachers several years
before the curriculum initiative of 1987. Some teachers said they
looked at the objectives occasionally as they prepared their children
for ABCS but what they taught day-to-day was pretty much defined by
what they found in their textbooks.

Textbook adoptions were centrally approved but selections for
classroom use resided at the Area and individual school level. Given
the considerable differences that might exist across textbook series
for a particular subject matter, the city-wide adoption of texts did
not insure a common orientation to the teaching of a content area.

In the case of mathematics, the district had approved four textbook
series for elementary classrooms--Harper & Row, 1982 edition, Holt,
1981 edition, Open Court, 1981 edition, and CEMREL. Both Harper & Row
and Holt represented a fairly traditional approach to mathematics
instruction. Open Court and CEMREL, however, were rather innovative
programs that promoted a more conceptual orientation and emphasized
problem solving and higher-order thinking. While the proponents of the
curriculum initiative would have preferred to see Open Court or CEMREL
used in classrooms, it was estimated that as many as 9 out of 10
teachers used Harper & Row or Holt. The mandate to implement a
standardized curricula defined by ”mathematical strands” and sets of
behavioral objectives was an effort to specify what should be the
outcomes of instruction in all schools, regardless of what text an

individual Area or school chose to use.

90
The set of common objectives for all classrooms was also intended
to define for teachers the fundamental learnings in mathematics. The
high-ranking central administrator commented on teachers' subject
matter knowledge:

A lot of people don't understand the fundamental concepts of the
subject matter they are teaching. The strands and objectives tell
teachers what are the fundamental ideas that are the basis of the
elementary mathematics curriculum. We have identified for them the
fundamental learnings for K-S. Our teachers are good at teaching
low level skills. But when you look at the failure of our kids on
test scores at grades 4 and 5, the instruction may not be good
enough. They (teachers) may not understand what they should be
teaching. We have tried to help teachers here. Not only have we
specified the objectives at each grade level, we have developed
some lessons.

The elementary mathematics supervisor spoke of teachers' subject matter
knowledge and their assumptions about what is appropriate for children
to learn.

Many teachers admit they don't have the knowledge. At one

inservice, one teacher got really angry. She thinks estimation is

too hard for first graders. She didn't understand what we were
requesting. She didn't understand at what level we wanted to have
kids doing estimation. The same goes for functions and relations
and statistics and probability. Many teachers see those words and

think children can't do that. But really, they have been doing a

lot of this. One teacher was surprised to discover that the

graphing she had her children doing was statistics. Calculators is
another. We even get complaints from parents that their kids won't
be able to add, subtract, multiply and divide.

Organizing the mathematics curriculum around "mathematical
strands," defining the fundamental learnings for each strand with
scores of specific objectives, and mandating the implementation of the
instructional sequence in every classroom in the district was perceived
as essential to improve the quality of mathematics instruction. The
strands refocused the curriculum around a broader set of mathematical

ideas. The curriculum objectives defined the outcomes of instruction.

The instructional sequence provided the order in which the objectives

91
were to be covered in the classroom. Considering the beliefs of
administrators and supervisors about teachers' limited knowledge and
understanding of mathematics, the decision to specify a uniform
curriculum for all classrooms was understandable. But those who were

prime movers in the initiative also expressed some reservations.

The Dilemmas in Increasing Centralization and Standardization

The centralization of decision-making about curricular matters
created some dilemmas. The high-ranking central administrator worried
about how teachers would implement the objectives in the classroom. He
acknowledged the potential "risks” inherent in this curriculum reform

agenda.

There is the risk that implementation will just be mechanical and
that we are taking away decisions of teachers.

The elementary mathematics supervisor worried about this as well.

I worry that we are taking away too much of the creative juices of
teachers. But I see this curriculum is intended as a support. For
teachers who can do differently and are capable and can show me a
plan and rationale, fine.

The central administrator expressed a desire for an approach that
would provide for greater teacher decision making.

What I would like to see is a limited number of objectives for each
strand. These would not be the whole curriculum but the
fundamental curriculum. I would like to see us not prescribe the
learning activities. That would be the teacher's choice. They
could draw on mastery learning, Madeline Hunter, Socratic
questioning, whatever. I would like to see us write some
objectives in a more constructivist manner. I would like to keep
our function in the district at the level of general objectives -
essential learnings - and let them (teachers) figure out how to do
that. I would like to see us link this with a program of staff
development. We really don't have any staff development agenda in
the district.

At the same time he talked about efforts currently under way at the

92
central administrative level to create whole sets of materials for
instruction and evaluation for each objective.
We are working on some other ways to help teachers who don't have
the skills or the strategies or don't use them the way they ought
to be taught. We are developing some lesson plans, resource
materials and test items that are correlated among the curriculum
objectives. If we are successful, teachers will be able to pull up
on the screen (computer monitor) a specific objective, and with it
a set of suggested resources, a lesson plan developed by teachers,
and some test items. We know this can become very restrictive and
controlling or it can be freeing. It depends on the use that is
made of it. But I guess everything worth doing presents a dilemma.
Despite their concerns and perhaps misgivings about the extent to
which the reform denied teachers the opportunity to make instructional
decisions based on the real needs of their kids, the plan for the next
three years was to work toward total implementation in every classroom.
As the central administrator summed it up:
The main thing that led to the decision to adopt a city-wide
curriculum was gisargay. There were so many different things going
on that it was hard to help the schools. Maybe it's a function of
size, maybe specialization. But others have had to step in and

assume instructional leadership. When things aren't working, when
the district is under pressure, there is the tendency to tighten

up.

SUMMARY

The initiative to reform the elementary mathematics curriculum in
the Detroit Public Schools was situated in and grew out of a broader
context of sociopolitical and organizational factors. Over a period of
two decades, structural changes in the city's economy, intervention of
the federal court in the schools through its desegregation order,
decentralization, and then recentralization, of the administrative
organization of the district had interacted to shape this curriculum

reform agenda.

93

The stipulation of the court to include a testing component
provided the occasion for key personnel at the central administrative
level to influence the mathematics testing program to reflect the
recommendations issued by the National Council of Teachers of
Mathematics--expanding basic skills beyond computational proficiency
and incorporating problem solving. The test then became the means to
instigate curriculum change. Under different circumstances, the poor
showing of students on the ABCS test might have resulted in abandonment
of the test. However, the substantial support it enjoyed at the
highest levels of the central administration produced another outcome.
An instructional sequence was defined for each grade level to reflect
the mathematical objectives tested by ABCS and was mandated for
district-wide implementation as the K-5 mathematics curriculum.

The NAACP had filed suit in federal court on April 6, 1971,
charging the Detroit schools with intentional and official racial
segregation. Sixteen years later, as a direct outcome of the orders
issued by the court in hradley v, Millihgn, the Detroit schools
initiated a city-wide curriculum reform agenda.

The development and implementation of a standardized district-wide
curriculum required a shift in the locus of power over curricular
matters away from the decentralized Regions that had been created by
state legislative action in 1969. The desegregation order facilitated
a partial shift since it was the Central Board and administration that
were required to respond to the court's orders creating the educational
components. The decision by city voters in 1981 to recentralize the
district provided the opportunity for further consolidation of power at

the central administration level. With the power over curricular

94
decision-making for the district in its hands, central administration
used its authority to respond to a number of educational problems that
resided within the system itself.

First, there was the belief that Area administrators and building
principals lacked sufficient knowledge about curricular matters to
provide proper educational leadership. Central administrators from a
variety of departments stepped in and assumed leadership over what they
perceived to be "disarray” and "chaos” in educational programs.

Second, it was widely assumed that many elementary teachers did not
understand the central ideas of the subjects they were teaching,
especially mathematics. Mathematics instruction had emphasized
computational proficiency and basic skills to improve test scores to
the exclusion of teaching a broader range of content. Teachers tended
to teach what came next in the text. The city-wide curriculum
represented an attempt to make explicit to Area and building staff the
fundamental concepts and the essential learnings at each grade level.
The instructional sequence was intended to discourage teachers from
using the traditional texts that dominated classrooms as the sole
determinant of what they would teach and what children would have an
opportunity to learn.

The mandate to implement a uniform curriculum in all classrooms in
the city was also an attempt to deal with a set of educational problems
rooted in social and economic inequalities within the larger community.
The social dislocation that resulted from structural changes in the
Detroit economy was felt by the schools. It was hoped that the
standardization of curricula would equalize learning opportunities

across schools, regardless of their location within the city, and would

95
assure that students who transferred during the year could do so with
little disruption in learning.

Several administrators expressed some reservations about the effect
a reform of this nature would have on teacher creativity and
professionalism. But at the same time, the dominant concern seemed to
be standardization. And that concern went beyond defining curricular
objectives. Teachers had been given an instructional management system
in reading. They were required to keep a profile sheet for each
student on which they were to record each reading competency mastered.
The district was deeply invested in an instructional and classroom
management model that specified a set of generic teacher behaviors
that, it was claimed, would lead to effective and efficient instruction
in any classroom. The district had committed to put every staff member
through 30 hours of training on those behaviors and how they should be
implemented. The mathematics supervisor promoted a model of ”effective
mathematics instruction" that incorporated the generic teacher
behaviors with a model for "active mathematics teaching."

The explanations given by key personnel for mandating a city-wide
elementary mathematics curriculum were certainly reasonable. But there
was another possible explanation that merits consideration. Given the
extent of efforts to standardize educational practice through mandated
curricula, an instructional model, testing programs, and management
systems, I would posit that these initiatives were also designed to
serve administrative convenience and accountability. For example, the
set of teaching behaviors associated with the instructional model in
which staff were trained was used regularly by school administrative

and supervisory personnel when they observed teachers in their

96
classrooms. The behaviors had become a criteria by which teacher
effectiveness was assessed. These criteria had the potential to become
powerful regulators of teacher and student behavior during instruction.
The instructional sequence that defined the mathematics curriculum held
the same potential to regulate teacher behavior and student learning.
It appeared that decisions about curriculum and instruction were
increasingly made at higher levels in the system hierarchy with little
input from the classroom teachers charged with implemention.

The education director of the civic leadership organization, a
proponent of investing greater authority for decision-making at the
local school level, was highly critical of what he called the
administration's ”lock-step" approach to education.

The district has a reponsibility to define goals and objectives for

all its children and young people. But these behavioral objectives

have nothing to do with the real needs of kids. It has to do with
the need to quantify, the public relations need to show numbers
like improved test scores. This district has got to trust the
profession to make appropriate decisions. Right now, the way
things are handed down to teachers, there is no sense of ownership
and therefore no sense of responsibility. The administration says,

"We know what's in the best interest of our children so we will

make the curriculum city-wide." Their effort to put into lock step

a uniform curriculum and an instructional practice denies teachers

at the local school level the right to decide how they can get to
where they need to go.

In the chapters that follow, the inquiry will examine in detail the
content of the mathematics curriculum reform and the nature of

inservice provided for teachers charged with its implementation.

CHAPTER 5

THE FORMAL CURRICULUM DOCUMENTS

INTRODUCTION

The city-wide elementary mathematics curriculum that had been
adopted by the Detroit Public Schools was described by those most
intimately involved in its development as ”conceptually-based.” This
chapter examines how teaching for conceptual understanding was
represented in the formal curriculum documents that had been adopted by
the district and distributed to its teachers. The formal curriculum
documents were of particular interest. First they represented a choice
of knowledge selected from a much larger universe of knowledge (Apple,
1979). Curriculum materials described what it was that children should
learn in the study of mathematics. They also represented the
discipline itself through the selection of content and the organization
of that content. That is, curriculum documents represented the
substance and the nature of the discipline to those who used them. The
inquiry undertaken here examines the explicit and implicit goals that
were the foundation of the formal curriculum documents, the
mathematical strands that broadly defined the curriculum content, and
the behavioral objectives that organized and sequenced that content for
instruction.

This chapter is divided into three sections. The first section
describes the district's goals for mathematics education and the ten

mathematical strands that organized the K-5 curriculum as elaborated in

the document, peggoit Enhlig fighggls Egg-kingezggrten-Qrgdg 12

97

98
Qngriculnm; figrands Q Objectives - Maghemhgics Edughgign (Detroit
Public Schools, January, 1987).

The second section examines in detail the objectives in the Grade 3
curriculum. This grade level was chosen for several reasons. In Grade
3 children typically study the four basic operations on whole numbers
and the use of conventional algorithms to perform computations. Often
the content is a recycling of processes introduced in earlier grades
but applied to larger numbers. The goals of instruction are usually
weighted toward children becoming proficient in symbol manipulation.
The introduction of new ideas appears late in the school year (if at
all). An examination of Detroit's Grade 3 curriculum reveals the
extent to which this district-wide curriculum reform aimed to develop
conceptual understanding of whole number operations. Further, a close
examination will show the extent to which the curriculum represented a
reshaping of content over what had been typical fare. This section is
divided into three parts. One interrogation examines a subset of the
Grade 3 objectives to explore the extent to which they were linked to
enhance conceptual understanding of numbers, numeration and operations
with numbers. The second interrogation examines the entire set of
objectives, attending to the verbs that described student actions to
determine the degree to which skill proficiency, interpretation and
application of information, or mathematical inquiry were emphasized.

In the third instance, the examination focuses on the relative emphasis
placed on each strand at Grade 3 and how the sequence of objectives

portrayed the development of new ideas over time.

99
The third section in this chapter examines the Grade 3 textbook
most commonly used in the district to assess the extent to which the

text was aligned with and supported the Instructional Sequence.

The Analytic Framework

In an earlier chapter I proposed a framework for analyzing the
orientation of this curriculum reform. Within an orientation of
certification of numeracy, arithmetic constitutes the bulk of topics
studied. The goal of instruction is computational speed and accuracy
and proficiency with conventional algorithms. Mathematics is portrayed
as a system of abstract symbols to be manipulated and a collection of
skills, definitions, rules and procedures to be memorized and mastered.

An orientation that aims for comprehension of mathematical ideas
emphasizes understanding procedures -- knowing how to use them and why
they work. Mathematics is represented as an abstract system of
concepts and procedures for which there are appropriate concrete models
and pictorial representations. Arithmetic continues to dominate the
curriculum but is supplemented by topics from geometry, statistics and
probability, and number and set theory.

A mathematics curriculum centered on enabling mathematical inquiry
has an explicit goal of empowering students -- to become active
participants in the construction of mathematical knowledge, to make
sense of mathematical situations, to engage in doing mathematics, to
communicate with others about mathematical ideas, and to use the power
of mathematics to understand the world around them. The content is
”chunked” around key mathematical ideas and problem situations provide

the context in which students engage in inquiry.

100
These orientations are ideal types and any mathematics program is
likely to contain elements of more than one. The intent in this
chapter is to determine the orientation of this curriculum reform

effort as it was represented in the formal curriculum documents.

THE R25 MATHEMATICS CURRICULUM
Goals for Students as Learners of Mathematics
In January, 1987, the Office of Curriculum Development and Services
presented to the Superintendent and School Board a document that

defined the strands and objectives in mathematics education for all

Detroit Public Schools. The booklet, Dgh191;_£nh11§_§gh921§_£;g;
de a ten-Grade l u icu um' St ds b c ves - m c
Egnggglnn, contained a statement of philosophy regarding mathematics

education. The statement is presented here in its entirety to
demonstrate the ways in which the district viewed mathematics education
and the goals for students as learners of mathematics.

Never before has a generation of Detroit students had a greater
need for enjoyable, understandable, and relevant experiences in
mathematics. The primary focus of mathematics instruction in
Detroit Public Schools is to improve the ability of students to
think systematically and apply their knowledge of mathematics in
solving problems. The advent of calculators, computers, and other
high technological advances is making it necessary for students to
develop a facility in mathematics far superior to that possessed by
previous generations. Space exploration, international competition
and fiscal economics have created new and eliminated old jobs and
are challenging the youth of tomorrow to think critically,
flexibly, cooperatively, and independently. The quest for new
medical breakthroughs complemented with the wonders of
microelectronics challenges our youngsters to develop, to create,
and to analyze. The rapidly changing routine procedures for
everyday experience calls for effective problem solving to lead a
successful life.

A mathematics program built on success looks to the future and
provides background experiences necessary to prepare children for
varied career choices. It prepares students to function
effectively, intellectually and independently in all areas of life.

101

It provides a multitude of experiences to foster a positive

attitude towards mathematics and provides exposure and enrichment

in geometry, measurement, estimation and other basic skill areas in
mathematics.

The program's strengths lie in providing concrete aids and/or

models for young people to gain understanding and to release

student potentials by developing higher order thinking skills of
analysis, synthesis, and evaluation. The mathematics program
spirals concepts and promotes pupil proficiency in all basic skill
areas from pre-kindergarten through grade twelve.

This statement described a set of goals for students as learners of
mathematics that seemed to be aimed at helping them develop their
mathematical knowledge and power. This was evidenced in the following
passages: ”to think systematically and apply their knowledge of
mathematics in solving problems...to think critically, flexibly,
cooperatively and independently...to develop, to create, and to
analyze...to foster a postive attitude towards mathematics...to gain
understanding...by developing higher order thinking skills of analysis,
synthesis, and evaluation."

It is of note that these expanded goals for a reformed mathematics
curriculum were tied almost exclusively to technological revolution,
competition in the international marketplace and structural changes in
the U.S. economy. For example, "The advent of calculators, computers
and other high technological advances is making it necessary for
students to develop a facility in mathematics far superior to that
possessed by previous generations...Space exploration, international
competition and fiscal economics have created new and eliminated old
jobs...The quest for new medical breakthroughs complemented with the
wonders of microelectronics challenges our youngsters.” The context

for reforming mathematics was technological advance and the increased

use of quantitative methods in business, industry, and social,

102
biological and physical sciences. This is not to suggest that our
information-age, advanced manufacturing economy should not have an
influence on the shape of mathematics education in the schools. Indeed
it should. Mathematical literacy is becoming essential for a growing
number of occupations within the new economy, particularly in the
widening sector of knowledge workers. Mathematics is practical and
utilitarian to the extent that it trains a workforce. But there are
other reasons beyond this key one for transforming the elementary
mathematics curriculum and mathematics instruction. The goals were not
linked to the explosion of mathematical knowledge that is altering
conceptions of what fundamental knowledge should be selected and
emphasized for all learners.1 The goals of engaging in mathematical
inquiry, of understanding the historical and cultural evolution of
mathematics, of learning to reason and communicate mathematically, of
becoming confident in one's ability to g9 mathematics (see, for
example, Commission on Standards for School Mathematics, 1989) were

absent from the District's philosophy.

The Curriculum Strands and Objectives
The Detroit Public Schools mathematics curriculum was organized
around ten mathematical strands that were spiralled within and across
the grades. The strands were described in DetgoL; Public SchggTs Eze-
de e -G ur lum' Strands Ob t v s - a m
Egnghgign (Detroit Public Schools, January, 1987) as ”major concept

themes whose objectives increase in scope and complexity" from Pre-

 

1 Davis and Hersh (1981) claim that over half of all mathematics
has been invented in the last half-century.

103
kindergarten to Grade 5. The curriculum strands, as defined in this
document, are presented below.

CALCULATORS and COMPUTERS enables students to use modern
technologies to solve problems at a level of difficulty beyond
paper and pencil skills.

ESTIMATION and APPROXIMATION helps students to make intelligent
judgmental decisions when given situations where exact answers are
not necessary. This includes rounding, checking for reasonableness
and using mental arithmetic strategies to formulate estimates.
FUNCTIONS and RELATIONS empowers students to form comparisons of
set elements or numerical quantities that have defined
relationships.

GEOMETRY permits students to organize, construct, and apply
abstract concepts to physical models in the real world.

MEASUREMENT equips students to relate numerical quantities to
physical models.

NUMERATION allows students to form understandings of the structures
of number systems.

OPERATIONS provides students with facility in computational skills.

PATTERNS simplifies problems enabling students to see relationships
between visual and abstract information.

PROBABILITY and STATISTICS helps students read, interpret, make
simple tables, charts, graphs and to use mathematics to determine
the likelihood of future events.

SETS and LOGIC assists students to organize and interpret

information, to use sequential thinking, and to draw conclusions in
solving problems that arise in an ever changing society.

The organization of the curriculum around central mathematical
themes as well as the inclusion of some of these strands represented a
clear departure from the traditional emphases of the elementary
mathematics curriculum. The range of content that comprised the newly
adopted curriculum -- estimation, operations, measurement, geometry,
statistics and probability, patterns, relations and functions, and sets

and logic -- was consistent with recommendations from the National

104
Council of Teachers of Mathematics to reshape the content and refocus
the intent of the elementary mathematics program (Commission on
Standards for School Mathematics, 1987). However, the language used to
describe the strands in this document seemed more symbolic than
mathematical. Each strand was elaborated with a statement of a student
outcome. In some cases, the outcome was quite narrow. For example,
the operations strand was defined as 'provid(ing) students with
facility in computational skills." Yet, an understanding of operations
involves considerably more--knowing properties of and relationships
among operations, having a sense of the effect of operating on numbers,
and recognizing real-world situations that exemplify an operation. The
measurement strand "equips students to relate numerical quantities to
physical models.” But the study of measurement should also help
students to understand attributes that one measures and the process of
measuring. The description of each strand did not capture the range of
mathematical ideas -- concepts, skills, procedures -- to be explored,
the ideas to be emphasized, and the multiple ways in which students
might encounter these ideas.

In addition to representing the ten ”central themes” rather
narrowly, the elaboration of each strand did not suggest the
interrelatedness of the themes. For example, estimation is essential
to children's understanding of measurement but the statements for these
two strands did not indicate this linkage. Geometric ideas contribute
to the development of number and measurement concepts but these
connections were not made explicit. Patterns -- the regularities that

are the essence of mathematics -- are central to the ideas of number,

105
geometry and measurement, but as in the other cases, linkages among
these strands were, at best, ambiguous.

Written texts serve varying purposes. These strand statements were
both political and symbolic in that they defined for the school
community a new framework around which the elementary mathematics
curriclum was to be organized. However, they served a limited
mathematical purpose. They defined a set of general student outcomes
but they provided little sense of the range of mathematical ideas
embedded in each strand, how those ideas might be connected, and why
those linkages might be important. One place to examine the
mathematical content of these curriculum strands is the set of grade
level objectives. The next section provides an examination of the

City-Wide Instructional Sequence for Grade 3.

GRADE 3 INSTRUCTIONAL SEQUENCE

Objectives for student learning about numbers, numeration and
operations with whole numbers

Traditionally arithmetic has constituted the bulk of the elementary
mathematics curriculum. The central aim of instruction has been
proficiency - accuracy and speed - with computational algorithms and
their application to routine word problems. Recent calls to reform
mathematics education have advocated a revision in teaching the concept
of whole number operations in the elementary curriculum. The National
Council of Teachers of Mathematics (NCTM) has recommended that
considerably less time be spent on drill and practice on paper-and-
pencil computations. Instead there should be an expanded emphasis on

developing the concepts of addition, subtraction, multiplication and

106
division--the properties and relationships of each, the relationships
among them, and problem situations in real-world contexts that
exemplify the operations. In addition, considerable time should be
spent developing children's informal experiences with the operations,
providing opportunities to use concrete materials and invent techniques
to solve problem situations before introducing symbolic work. Clearly
computation should continue to have a key place in the elementary
mathematics curriculum, but instruction in computation should emphasize
a variety of algorithms, mental arithmetic strategies, estimation and
the use of calculators. The move should be toward conceptual
understanding of whole number operations and away from memorizing rules
for carrying out the conventional algorithmic procedures (Commission on
Standards for School Mathematics, 1987, 1988, 1989). This section
examines a subset of the Grade 3 objectives to determine the extent to
which the organization and presentation of these objectives was
consistent with the NCTM recommendations for reduced attention to
paper-and-pencil computations and increased attention to developing
children's number and operation sense.

The Grade 3 instructional sequence was composed of a series of
behavioral objectives organized in sets under one of the ten strands.2
Strands were revisited several times over the course of the school year
and occasionally a particular objective was revisited. A total of 116
objectives constituted the Grade 3 curriculum. Of these, 55 (472) were
devoted to developing ideas about number, numeration and operations

with whole numbers. Figure 5.1 shows the strands devoted to these

 

2The City-Wide Instructional Sequence of Grade 3 appears in the
Appendix.

107
concepts, an example of an objective from each strand, and the number
of objectives in that strand that addressed these ideas.

The objectives in these strands fell into four categories: (1)
reading, writing and computing with whole numbers; (2) demonstrating
the meaning of operations and the effects of operating on whole
numbers; (3) understanding relationships among numbers; and (4)
applying operations to solve word problems. The first category,
reading, writing and computing with whole numbers, included writing
numbers in standard notation, demonstrating place value, and recalling
facts and using the conventional algorithms with increasing speed and
accuracy. Nearly half the objectives related to developing number and
operation ideas (25 of the 55) were concerned with increasing
computational proficiency.

The second category, demonstrating the meaning of operations and
the effects of operating on numbers, included using properties in
computation,3 relating addition and subtraction as inverses, relating
multiplication to repeated addition and division to repeated
subtraction, and applying a given rule to a set of numbers to determine
the missing input or output. Less than one-third of the
objectives devoted to whole numbers (16 of 55) were aimed at developing
conceptual understanding of whole number operations. There was,

however, some ambiguity about several of these objectives. For

 

3This included using the property of zero in addition and
subtraction and zero and one in multiplication, and applying the
commutative and associative laws to addition and multiplication.

NUMERATION

FUNCTIONS AND
RELATIONS

OPERATIONS

PATTERNS

ESTIMATION

CALCULATORS
AND COMPUTERS

108

W
QBJECTTVE

Demonstrate accuracy in
reading and writing whole
numerals through 999.

Identify numbers as multiples
of 2, 3, 5 or 10.

Interpret and complete
addition, subtraction or
multiplication number
sentences using one of these
symbols, <, >, -, 4.

Recall the addition and
subtraction facts through
sums of 18 with or without
the use of aids (pictures,
number lines).

Apply addition, subtraction,
multiplication or division
strategies to solve money or
other word problems.

Interpret and complete a
number pattern (using numbers
0-99; starting at any number)
in ascending or descending
order counting by ones, twos,
fives or tens.

Round any two-digit number
to the nearest multiple of
ten .

Round the individual problem
numbers to the nearest multiple
of ten or hundred and estimate
the sum or difference.

Check computation using a
calculator.

Predict number to be displayed
and verify its results.

17

17

Figure 5.1 The Strands and Objectives Devoted to Number,
Numeration and Operations with Whole Numbers

109
example, the objective ”apply a given rule to a set of whole numbers
and determine the missing whole number output" could be met in a way
that provided an opportunity for students to explore the effects of
operating on a set of numbers and the patterns and relationships that
emerge. On the other hand, this objective could simply provide another
occasion for computational drill and practice. How this objective was
treated in instruction was, in part, a function of its presentation in
textbooks, a matter that will be examined in a later section of this
chapter.

The third category, understanding relationships among whole
numbers, included identifying rules when given input and output
numbers, identifying numbers as odd or even and as multiples, ordering
and classifying numbers, completing number patterns, and using mental
arithmetic strategies. Ten of the 55 objectives fell in this category.
It is of note here that the objective "apply mental arithmetic
strategies to solve addition, subtraction, multiplication or division
problems” was met only once and that was in late May. One might have
expected a consideration of mental arithmetic strategies earlier in the
school year to help children develop more efficient ways to recall
basic facts and answer fact problems, particularly given the emphasis
at this grade on computational proficiency.“

The fourth category was composed of one objective that was visited
four times during the year: "apply addition, subtraction or

multiplication strategies to solve money, time or other word problems

 

l'During inservice conducted by a district mathematics supervisor,
teachers were urged to devote the first five minutes of each class to
mental arithmetic activities that might or might not be connected to
the topic of the day's lesson.

110
(one or two steps).' Work on word problems was not integrated
throughout the curriculum. Instead, word problems tended to come after ’
a series of objectives devoted to computational proficiency. The
hierarchical arrangement of the objectives suggested that work with
problem solving was appropriate only after children had mastered basic
facts.

These objectives reflected a tension between mastering basic facts
to enhance computational proficiency and developing conceptual
understanding of place value and whole number operations. Two
objectives called for relating "the inverse of addition and
subtraction” and "the inverse of operations.” Yet children were
required to learn nearly two-hundred subtraction and division facts by
Grade 5. At third grade, children were expected to ”state the
subtraction facts through sums of 18.” At fourth grade, they were
expected to ”state the 90 basic division facts.” By Grade 5 children
were expected to recall these facts ”with greater accuracy and speed."
It appeared that memorizing hundreds of facts for addition,
subtraction, multiplication and division took precedence over using the
concept of inverse to derive subtraction facts from addition facts and
division facts from multiplication facts. Learning conventional
algorithmic procedures to enhance computational proficiency seemed to
be an end in itself. No objective suggested that students might
explore or invent alternative algorithms. Long multiplication --
”three digit times a two digit” -- and long division -- '4 digit
dividends by 2 digit divisors" -- were practiced even though the
curriculum stressed the importance of the calculator in the elementary

mathematics classroom.

111

The tension was also reflected in the lack of integration of
calculator objectives throughout the sequence. Calculators and
computers was one of the ten mathematical strands of the curriculum.
This suggested that calculators would be used throughout the study of
mathematics. There were four objectives devoted to the use of
calculators in the Grade 3 sequence. These objectives appeared once,
consecutively at the end of January. Three of the objectives had
children use the calculator to solve problems, check the accuracy of
predicted answers, determine the reasonableness of displayed answers
and check paper-and-pencil computation. With the fourth, children
used the calculator to predict a number to be displayed. The single
appearance of these four objectives was curious, leaving one to wonder
about the extent to which calculators were integrated in the
curriculum. Only one of these objectives exploited the power of the
calculator as an investigative tool to explore number concepts. It
might be argued here that what is not explicitly stated is implied.
The matter of whether and to what extent calculators were integrated
into the curriculum will be revisited in the next chapter when another
source of evidence, the Model Lessons, is examined.

Few of the objectives devoted to numbers, numeration and operations
with whole numbers were concerned with decision-making. Where
decision-making was involved, the process was mechanistic. For
example, the exclusive focus of the estimation objectives was having
students apply a rule to round numbers to the nearest ten or hundred,

use the estimated numbers to calculate sums and differences, and then

112

use that answer as a check of the reasonableness of the "real” answer.5
Making judgments about whether a situation required an exact answer or
if an approximation would be sufficient was not considered. Deciding
whether and under what conditions calculations could be performed more
efficiently mentally, with paper-and-pencil, with calculators or
computers was not addressed in any objective.

The objectives were only statements. They did not assure or even

predict the behavior of teachers, the performance of students, or the

 

5The potential difficulties embedded in this mechanistic, rule-
bounded approach were evidenced in a demonstration lesson conducted by
a district mathematics specialist in a third grade classroom. The
lesson involved rounding individual problem numbers to the nearest
multiple of ten and using estimation to check the reasonableness of an
answer. The first example the specialist demonstrated at the.board was
42 + 59. Children estimated 42 as close to 40 and 59 as close to 60
and said the sum should be close to 100. They computed the sum of 42
and 59 and got 101. The specialist asked, ”Do you think 101 is a good
answer?” The children answered in unison, "Yes." She asked one
youngster to explain. "Because 101 is close to 100,” was the child's
response. After several more examples, the specialist assigned
problems from the textbook with the directions to ”find the estimate
before you do your problem." Almost immediately, a youngster found a
problem that ”didn't work" and asked for help. The specialist put the
problem, 26 + 27, at the board. Children estimated 26 as close to 30
and 27 as close to 30 and the sum as close to 60. Then they added 26
and 27 getting 53. "Is 53 a good answer?" the specialist asked. The
youngsters replied loudly in unison, "NO.” Seeming surprised, the
specialist said, ”Sometimes estimates are not real close. This happens
when these numbers (circles 6 and 7) are close to five. Estimation
isn't a sure fire thing. I would say, 'This isn't close, maybe I
should go back and check.'” She then turned to the classroom teacher
and said, "I try to stay away from ones like this but they keep
creeping in."

The children had been taught a rule and a procedure and were
applying it as they had been shown. But sometimes it ”didn't work”
like it was supposed to. The specialist's response to the children was
simply to say that it wasn't a "sure fire” rule. After the class, I
asked her about a process that did not always ”work." Her concern was
not about how to help children understand the limitations of the
process in deciding if answers are ”good.” Instead, she said to me,
"I should have looked more carefully at the problems in their book
before I gave them the practice. I wouldn't have given them the ones
that didn't work." Her concern was to do a better job of planning so
that children would not encounter situations that defied the rule.

113
actual life of the classroom. The claim here is not that these
objectives -- the limited mathematical goal of each one individually,
the manner in which they were sequenced, or the absence of objectives
that would contribute to further understanding of numbers, numeration
and operations with whole numbers -- precluded teachers from pursuing
an expanded, linked set of objectives and goals. Rather, the argument
is that these objectives represented a view of mathematics and an
approach to mathematics teaching that atomized content, isolated
computational proficiency from problem solving contexts, and ignored
decision-making and judgment.

These objectives are but one source of evidence to bring to the
question of the orientation of this curriculum reform initiative and
the degree to which it might foster conceptual understanding and higher
order thinking in learners of mathematics. I turn next to the entire
set of Grade 3 objectives to examine the verbs that described student

actions.

verbs That Described Student Actions

The second interrogation of the Grade 3 curriculum involves an
examination of the verbs used in the objectives to describe student
actions. A mathematics curriculum that aims to develop higher order
reasoning and conceptual understanding provides opportunities for
students to explore problem situations, make conjectures, validate
assertions, and communicate the results to others. Problem situations
provide the context in which students build abstractions and
generalizations about mathematical properties and relationships. To

determine the extent to which the Grade 3 instructional sequence

114
promoted conceptual understanding and higher order reasoning, the
entire set of objectives was analyzed.

The schema used here represents an alteration of the categories
proposed by Avital and Shettleworth (1968) as described in Chapter 3.
Three categories or levels of actions are described. These levels are
distinguished by the degree to which skill proficiency, interpretation
and application of information, or mathematical investigation and
inquiry are emphasized. Low-level objectives aim at skill proficiency
and emphasize recall of basic number facts, speed and accuracy in
conventional algorithmic procedures, and a reliance on memory to attain
the objectives. Examples of verbs that describe student actions at
this level would include read, identify, name, recall, recognize, add,
subtract, multiply, divide, state the rule, and demonstrate accuracy.
Mid-level objectives require the application of mathematical
information. Examples of verbs that describe student actions at this
level include relate, extend, demonstrate, complete, estimate, measure,
compare, apply strategies and determine reasonableness. High-level
objectives emphasize mathematical investigations and open searches. At
this level student actions would include investigate, explore, predict,
invent, and generalize. Figure 5.2 indicates the number of Grade 3

objectives that fell into each category.6

 

6There is some degree of interpretation that has gone into
placing these objectives within categories. A single verb was not
always the determining factor. For example, ”identify a circle graph"
and "identify and state the rule which indicates the relationship
between whole numbers when given input or output" seemed to be
objectives aimed at different levels of thinking, the former a matter
of recognition, the latter a matter of application. And as noted
earlier, some objectives could be met as simply occasions for
additional drill-and-practice of computational algorithms rather than
opportunities for application to nonroutine situations. Where there
was some ambiguity, I placed the objective in the higher category.

115

Wm WM
Low level - Skill proficiency 58 (50.01)
Mid level - Application 52 (44.81)
High level - Open searches 6 ( 5.21)

and investigation

Figure 5.2 Number of Objectives by Category
of Level of Actions

Half the Grade 3 objectives were devoted to memorization and recall
of basic number facts, computational proficiency with conventional
algorithms and manipulation of symbols. Most of the remaining
objectives aimed at application. Here students constructed graphs,
classified numbers and objects by their attributes, completed patterns
of numbers and objects, estimated measures of length and weight,
related and compared units of measure, investigated ways to use a hand-
held calculator, applied addition, subtraction, multiplication and
division strategies to word problems, demonstrated the relationship
between operations and compared fractional parts of numbers and
objects. Only six objectives approached fitting the category of
investigation. In these instances, students were expected to interpret
graphs and interpret and create patterns of numbers and objects.

None of the objectives captured the spirit of open-ended inquiry
where students make conjectures, validate assertions, abstract
generalizations or invent procedures. Content was fragmented. What
was stressed was the individual objective that was to be accomplished
in a day, or at most two. Even where several objectives appeared in

sequence under one strand, they did not support open searches. For

116
example, the first four objectives in the month of September were
related to probability and statistics. They required students to
'1) read and interpret a pictograph using ratios of 1:1 and 1:2 using a
key; 2) read and interpret a bar graph using calibrations of units of
l, 2, and 5, using a key; 3) construct a bar and pictograph using data
with ratios: 1:1 and 1:2; 4) identify a circle graph." None of these
objectives involved collecting and organizing data or making decisions
about appropriate displays for different types of data. There was no
objective devoted to formulating questions and solving problems that
would involve collecting and analyzing data. Action verbs such as
"investigate, explore, predict, make decisions and discuss findings”
were absent from this set of objectives. It was a significant omission
because the emphasis on reading and interpreting graphs ignored the
importance of using statistical ideas as tools to solve problems and to
describe and interpret the world around children.

The absence of objectives that are explicit about opportunities for
open-searches does not preclude such explorations in classrooms. How
these objectives are met in the classroom is a function, in part, of
how these ideas are treated in textbooks and how teachers interpret the
range of concepts, skills and procedures embedded in each objective.

In a subsequent section I will examine the most commonly used textbook
to determine if activities were included that asked students to
generate questions about which they might collect data, collect data,
make decisions about appropriate displays, and use data to make

decisions or predictions.

117
The Relative Emphasis Placed On Each Strand And The Development Of New
Ideas

A second way to interrogate the entire set of Grade 3 objectives to
determine the orientation of this curriculum is to examine two related
notions: 1) the relative emphasis placed on each major theme as
represented by the distribution of objectives among the ten strands and
2) the extent to which the curriculum included new content and how
these new ideas were developed over time.

The giggribhgion of objectives among she ghgangs. The strands
included content that traditionally had not been emphasized in the
elementary school mathematics curriculum. The ideas of functions and
relations had certainly been embedded in elementary mathematics, but
making these concepts one of the central topics of instruction was a
departure from the traditional curriculum. Patterns, probability and
statistics, and estimation and approximation had been given the status
of central themes of the curriculum. Figure 5.3 describes the number .
of objectives in the entire Grade 3 sequence devoted to each curriculum
strand. The interest here is to determine the extent to which this
curriculum represented a reshaping of content from the traditional
emphasis on arithmetic.

Two strands, numeration and operations, were devoted to developing
an understanding of place value and the meaning of operations, to
reading, writing and ordering numbers symbolically and to developing

computational proficiency. The following objectives were

92rri2212a_§£rang

CALCULATORS AND COMPUTERS

ESTIMATION

FUNCTIONS AND RELATIONS

GEOMETRY

MEASUREMENT

NUMERATION

OPERATIONS

PATTERNS

PROBABILITY AND STATISTICS

SETS AND LOGIC

118

be

11

5

22

21

24

6

8

( 4.
( 5.
( 9.
( 4.
(18.
(18.
(20.
( 6.

( 5.

32)
22)
52)
32)
92)
12)
72)
92)

22)

( 6.92)7

Figure 5.3 The Distribution of Grade 3 Objective

Among the Curriculum Strands

 

7The objectives at the other grade levels showed a somewhat

similar distribution.

Strand

Calc. & Comp.
Est. & Approx.
Func. & Rel.
Geom.

Meas.

Numer.

Oper.

Patt.

Prob. 6 Stat.
Sets & Log.

Grade 1
( 2.12)
( 2.12)
( 8.32)
( 5.22)

15 (15.62)

34 (35.42)

15 (15.62)
8 ( 8.32)
4 ( 4.22)
3 ( 3.12)

U'IQNN

Numbe; gt theggivgg
Grade 2 Grade 4
5 ( 5.62) 6 ( 4.52)
5 ( 5.62) 5 ( 3.82)
10 (11.22) 12 ( 9.12)
7 ( 7.92) 6 ( 4.52)
10 (11.22) 18 (13.62)
21 (23.62) 26 (19.72)
12 (13.52) 38 (28.82)
6 ( 6.72) 7 ( 5.32)
5 ( 5.62) 6 ( 4.52)
8 ( 9.02) 9 ( 6.82)

8
6
l4
7
21
20
29
6
8
11

Grade 5

( 6.
( 4.
(10.
( 5
(16.
(15
(22.
( 4.
( 6.
( 8.

22)
62)
82)

.42)

22)

.42)

32)
62)
22)
52)

The large number of objectives for operations at Grade 4 reflected a
total focus during September to prepare students for the state-wide

MEAP test administered in October.

119
typical: "demonstrate accuracy in reading and writing whole numerals
through 999;" ”interpret and/or illustrate the meaning of whole numbers
through 999;” ”subtract whole numbers through 3 digits (with and
without regrouping);' ”demonstrate the meaning of multiplication as
repeated addition;' ”recall the multiplication facts through 9 times
9;" "identify a number that is 10 or 100 more or less than a given
number;' ”sequence up to 4 counting numbers, least to greatest or
greatest to least;' “apply the property of zero to addition and
subtraction.” Several of the objectives in these strands called for
the use of physical aids, concrete models and pictures as ways to
represent the mathematical ideas in the lesson: ”determine and/or
demonstrate place value through 999 using aids;” "recall the addition
and subtraction facts through sums of 18 with or without the use of
aids (pictures, number line);" "identify positive and negative integers
on a number line;" "determine one-half of an even whole number through
18; even multiples of 10 and 100, with or without aids."

In addition, several objectives from other strands seemed to be
aimed at computational proficiency. One objective in the measurement
strand called upon students to ”identify, compare, write and compute
money values through $9.99.” An objective of the calculator and
computer strand called upon students to "check computation using a
calculator." Under functions and relations, students were to ”apply a
given rule to a set of whole numbers and determine the missing whole
number output” and "identify the relationship of a pictured sum of
money to a given value in a problem involving money through $9.99.”
Assessing the reasonableness of results of computation was relegated to

the mechanical process of rounding off addends and finding their sums:

120

”round the individual problem numbers to the nearest multiple of ten or
hundred and estimate the sum." Two objectives from the patterns strand
represented a form of skip-counting: "interpret and complete a number
pattern (using numbers 0-999; starting at any number) in ascending or
descending order counting by ones through sixes, tens and hundreds.”
In all, 59 of the 116 objectives (50.82) were devoted to place value
understanding, symbol manipulation and computational proficiency with
whole numbers, fractions and decimals.8

Geometry was given scant attention in the Grade 3 sequence. The
five objectives in this strand focused narrowly on naming, identifying
and drawing shapes, knowing their characteristics, and memorizing terms
and symbols: ”identify geometric figures, symbols and their written
words: square corner, point, ray, line segment, line and angle;”
”identify and recognize cube, cone, cylinder and sphere by word names
and models in the everyday environment;' ”draw points, line segments,
rays, angles and lines using appropriate tools;' ”identify and
reproduce lines of symmetry.” None of the objectives was designed to
foster development of students' spatial sense - how shapes are related,

the effects of changes made on shapes as they are rotated and flipped,

or as shapes are combined.9 There were no objectives to help students

 

8Analysis of the objectives for Grades 4 and 5 revealed a similar
concentration: 66 of the 132 Grade 4 objectives (50.02) and 66 of the
130 Grade 5 objectives (50.82) were devoted to symbol manipulation and
computational proficiency with whole numbers, fractions and decimals.

9At Grade 4, 6 of the 132 objectives were devoted to the geometry
strand, at Grade 5, 7 of 130 objectives. At these grades, there were
a few objectives devoted to developing spatial sense. Grade 4:
”complete a drawing applying the concept of symmetry.” Grade 5: "use
geometric tools to construct circles, angles of a given measure or
other known geometric figures;' "identify and relate two dimensional
shapes to three dimensional figures;“ “construct models of solids;” and
"identify two dimensional pictures of three dimensional shapes.” No

121
see how geometrical ideas can contribute to an understanding of
measurement and number concepts.

Some objectives in the measurement strand seemed to be aimed at
developing an understanding of the attributes of length and weight.
Linear measure was introduced by using non-standard units and then
progressing to standard units. One objective called for using non-
standard units to measure the weight of objects. Several objectives
appeared to be aimed at deciding on appropriate units of measure
depending on the size of the object to be measured: ”relate and compare
metric units of measure, centimeter and meter;” ”identify the
reasonable customary or metric unit to measure the length of an object,
height or distance;” "identify the reasonable customary or metric unit
to measure weight (mass).” Estimation of quantities was included in
several of the measurement objectives: ”estimate and determine the
length of an object or distance using non-standard units;” ”estimate
and determine the length of an object using whole metric units.”
Likewise, measurement ideas were incorporated in some objectives in the
estimation strand: "identify and estimate length when given a non-
standard unit.” This was the only instance of an explicit link between
strands.10

Of the six objectives in the probability and statistics strand,
only one aimed to develop probabilistic ideas: ”demonstrate the
probability of a simple event with six or fewer outcomes, using aids."

The sets and logic strand lacked coherence as evidenced by a jumble of

 

objective explored the changes in shapes as they are rotated or combined.

10At Grade 4, the link between estimation and measurement
included estimating the perimeter of a polygon and at Grade 5,
estimating the area of a polygon on a grid.

122
randomly placed, discrete objectives: ”identify the position of objects
according to left/right, top/bottom, over/under, above/below;"
"identify the correct word statement, picture, place, thing or number
from 2 or more statements;” ”read, interpret and supply elements in a
matrix;” “read and interpret Venn diagrams;" ”decide the truth or
falsity of a number of statements involving the terms all, every, each,
exactly, and no."11
e eve e t ew d . Of the 116 objectives in the Grade 3

sequence, 30 (25.92) were review from prior grades, 24 (20.72)
represented extensions of ideas first introduced in an earlier grade,12
and 62 (53.42) were presented for the first time. More than half the
objectives represented the introduction of new mathematical content.
Simple calculation indicated that this schedule of instruction allowed
at most two days to consider an objective that introduced a
mathematical idea for the first time. To explore the implications of
this schedule, consider two measurement ideas that were introduced for
the first time at Grade 3.

In January, three objectives related to measurement were introduced
for the first time: "determine and record the area of a fighzg on a
grid in square units;" "use a nghgig or customary £2121 to measure the
giggg of a pglyggn or pictured object;” ”determine and record the

perimeter of a pglyggn or ggig following the lines on the grid”

 

11The single Grade 4 probability objective read ”determine the
probability of an event and express as a ratio (spinner, die, etc.)."
The two probability objectives at Grade 5 were "analyze the probable
possibilities of a coin, die, deck of cards and various kinds of
spinners" and "determine the probability of a simple event and express
it as a ratio (fraction)."

12Most of these objectives extended the numeration strand and
addition and subtraction to three digit numbers.

123
(emphasis in curriculum document). Measuring objects was not new. An
objective in PK-K called upon children to determine the length of a
given object using nonstandard units. At Grade 1 students determined
the length or height of an object using a standard unit. However, this
was the first appearance of an objective that specifically called for
using metric or customary rulers. In addition, this was the first time
the concepts perimeter and area were included.

Perimeter and area are key measurement concepts that require
considerable exploration with a variety of activities to build student
understanding (Commission on Standards for School Mathematics, 1989;
Driscoll, 1981; Nelson & Reys, 1976). The two objectives devoted to
these concepts employed only one way - figures on grid paper - for
students to investigate these powerful ideas. Students, prior to these
objectives, had considerable experience taking linear measurements of
objects. Extending the idea of measuring length to finding perimeter
was a reasonable objective. What was interesting was that the
objective did not link this measurement concept to determining the
perimeter of common objects. Instead, the medium of instruction was
the grid. The notion of perimeter was removed from the real world and
abstracted to a piece of paper. The objective for determining and
recording area was also problematic. The objective did not speak to
the notion of area as covering but as simply counting squares on a
grid. As with perimeter, the concept of area was separated from the
world of real objects and abstracted to drawings on a piece of paper.

Key mathematical ideas were not "chunked" to explore relationships
and make connections. For example, in Grade 2, objectives called for

estimating and measuring the lengths of objects with non-standard and

124
standard units. At Grade 3, these objectives were revisited again,
twice in November and twice in December. However, the work on
perimeter and area to be carried out in January was not conntected to
these four objectives on linear measure. The objectives on linear
measure, perimeter and area were isolated rather than being a part of a
larger unit on measurement where prior learning could be extended by
applying length to the concepts of perimeter and area. The
organization and placement of objectives related to length, perimeter
and area portrayed these key mathematical ideas as distinct from each

other rather than part of a rich network of connected ideas.

SUMMARY

The Grade 3 instructional sequence embodied a tension between
certification of numeracy and comprehension of mathematical ideas. The
tendency toward the certification of numeracy was evidenced in a number
of ways. The topics to be studied were organized around scores of
behavioral objectives, each of which focused on a single skill or
procedure. Computational proficiency with whole numbers, fractions and
decimals dominated the set of objectives. Although the organization of
objectives around ten central themes suggested that mathematics is a
body of related ideas, the sometimes disparate objectives that were
grouped together within a strand failed to capture the essence of the
connections. The hierarchical arrangement of objectives continued to
perpetuate the notion that mastery of basic facts and development of
skills and procedures must precede work with word problems. And the
sheer number of over one-hundred objectives precluded in-depth study

over time of fundamental, yet complex, mathematical concepts.

125

At the same time, there was some evidence of comprehension of
mathematical ideas. The concept of place value was emphasized,
contributing to an understanding of number and the base ten system as
well as to computational speed and accuracy with conventional
algorithms. Estimation strategies were taught to enhance mental
computation and to provide a technique to assess the reasonableness of
answers, although there were limitations in the specific strategy of
rounding off. The objectives suggested that mathematics is a system of
concepts and processes for which there are appropriate concrete models.
Over a dozen objectives stated explicitly that physical aids or models
were to be used to recall number facts, carry out measurements, compare
fractions, identify geometric figures, and interpret and extend
patterns. Several objectives were explicit about using pictorial
representations.

These formal curriculum documents did not embody attributes of
enabling mathematical inquiry. Key mathematical ideas were presented
as isolated skills and processes to be covered in a day or two rather
than topics that demand exploration over time to develop an
understanding of concepts and procedures and the relationships among
them. The actions that would signify an orientation toward
mathematical inquiry -- investigate, explore, predict, formulate
problems, make conjectures, develop arguments to validate assertions,
invent procedures, build generalizations, discuss findings -- were
totally absent from the formulation of these objectives.

In reference to the orientations to teaching mathematics, these
objectives moved between certification of numeracy and comprehension of

mathematical ideas. Taken separately, some objectives seemed to fall

126
in the first category while others seemed more appropriately located
within the second. Taken together, they seemed to portray an attempt
to shift teaching and learning from an exclusive focus on the
acquisition of facts and rules and the manipulation of symbols toward
understanding key ideas about number, numeration, operations and
measurement. While arithmetic tended to dominate the objectives, there
was the inclusion of objectives representing a broader range of
mathematical topics. Despite the broad goals presented in the
district's statement of philosophy to develop higher order reasoning
skills and problem solving abilities, these curriculum objectives
seemed aimed at a lesser goal--to understand mathematical processes and

their application to routine word problems.

THE TEXTBOOK AND THE INSTRUCTIONAL SEQUENCE

The Board of Education had approved four textbook series for use in
elementary mathematics classrooms in the district. The two series most
commonly used were Holt Maghematigs (1981 edition) and flazper Q ng
hathematlgs (1982 edition). One or the other of these series was used
in an estimated 902 or more of the classrooms in the district.13 This
section examines the extent to which the most commonly used Grade 3
textbooks provided an instructional program aligned with the district's
Instructional Sequence.

Both textbook series represented a traditional orientation to

mathematics education. Computational proficiency with conventional

 

13Also approved for use in the district were Real flggh (Open
Court, 1981) and QQmQIehgnslve School hahhematlcs Pgogghn (CEMREL,

1979), two textbook series that were significant departures in both
content and orientation from traditional textbook series.

127
algorithms was overwhemingly emphasized. Twelve of the 16 chapters in
the Harper & Row Grade 3 text were devoted to operations on whole
numbers. Word problems were found at the end of chapters with an
occasional set of two or three word problems following a page of
computational exercises. The hierarchical arrangement suggested that
problem solving was an appropriate activity only after children had
mastered basic facts and operational algorithms. Virtually all the
word problems were of a routine nature. The method of solution was
obvious from the statement of the problem or its placement in the text.
Few of the problems involved open searches or provided for multiple
answers.

The pages included here from the Harper & Row text were typical.
Lesson 3 from Chapter 6 on subtraction with regrouping provided
children with examples, a set of computational exercises, and a set of
word problems that could be solved by simply following the example.
Lesson 9, the last regular lesson in the chapter did ask children to
decide if a problem required addition of subtraction to solve. Most
chapters in this text had a "challenge” at the end. Some were logic
activities, some provided atypical environments in which to work on
basic facts, like "magic squares." Several ”challenges” incorporated
geometric representations of numbers such as ”Building Numbers--
Rectangular Arrays." One-third of the “challenges" involved problems

with money, as in I'You be the Clerk!” These problems allowed for

LESSON

128

5 More Renaming

   
 

334 students go to the Emerson School. 3 3 4
193 are boys. How many are girls? _. 1 g 3
@8882
Step 1
3 3 4 Can you subtract ones? Yes.
—193 4—3=1
—_—1_ Write 1 in the ones place.
2 13 Step 2
3 3 4 Can you subtract tens? No.
— 1 9 3 Rename a hundred.
1
213 Step 3
33 4 Subtract tens.
—193 13tens—9tens=4tens
"'_—_ Write 4 in the tens place.
4 1
2 13 Step 4
3 3 4 Can you subtract hundreds? Yes.
— 1 9 3 2 hundreds - 1 hundred
1 41 = 1 hundred

Write 1 in the hundreds place.

i/ Check. 193 + 141 - 334
141 girls go to the Emerson School.

 

TRMI' Capy and subtract. Check.

314

A. 418
-263

 

116

518
a. 680 c. 275 D. 367
-190 —185 —176

 

 

 

Harper & wa Mathematics, 1982.

129

Exercises
Copy and subtract.
Check the first row by adding.

 

515 317
1. 669 2. 470 a. 867 4. 740 5. 345

—185 -290 —387 —390 —174

 

6. 568 7. 458 a. 650 9. 507 10. 806
—197 —288 ~390 —325 —374

 

 

 

11. 702 12. 608 1a. 415 14. 518 15. 614
- 281 — 216 — 283 — 392 — 281

 

16. 719 17. 632 1a. 957 19. 607 20. 818

 

 

—275 —191 —394 —135 -136
Solve.
21. 864 seats in the 22. 500 people at the game.
baseball stand. 260 for the home team.
192 are empty. tor the visitors.

people at the game.

23. 338 seats in the gym. 24. 450 seats in the gym.
184 people in seats. 290 seats are empty.
seats are empty. seats are not empty.
25. 627 students at the 26. 953 tickets for the
Green School. football game.
132 are new this year. 162 are left

students are not new _ have been sold.
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Subtraction/Problem Solving

Harper 6 Row Mathematics, 1982

117

Exercises
Write add or subtract. Then solve the problem.

1.

900 people are at the
beach.

360 came in cars.

23 did not come in cars.

520 rafts are for rent.
120 are rented.
'2 are still left.

544 people came to
the beach on Sunday.
325 came on Monday.
:5 came both days.

Mark has $8.25.
A raft costs $9.15.
He has .3 too little.

Tom has. $7.25.
He spends $3.00
on ride tickets.
He has .2; left.

247 people bought
ice cream.

139 people bought
hot dogs.

3 people bought
something.

    
   
  
    

. Ts"
, l:
l

\r
*3

 
  
   
    
  

 

130

2. Pam found 362 shells.
Ann found 197 shells.
They found ‘_1 in all.

4. 475 came to the beach.
296 went on rides.
I did not go on rides.

6. Jane has $5.25.
She needs $8.19
for a beach ball.
She needs 3 more.

a. A hat costs $6.02.
A beach coat is $9.00.
Together they cost '.'.‘.

Jane had $7.25.
She spent $2.25
on a sand pail.

She has "i left.

Gene brought 8 dimes.
Sid brought 10 nickels.

    
    
     
  

  

._ HI'K, How much more money
/ 5‘9“ did Gene bring than Sid?
. g'

 

5

 

    

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131

CHALLENGE

Building Numbers—
Rectangular Arrays

Use 12 squares to build these arrays.

 

2 rows

 

 

 

 

 

 

 

 

 

6 columns
4 rows 6 rows

 

   

3 columns
3 rows 2 columns

 

 

 

 

 

 

 

 

4 columns

Use squares to build as many different arrays
as you can for each number:

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.
Each array must have at least 2 rows and 2 columns.

Keep a record like this:

 

Number 4 6

 

 

 

 

 

 

 

Arrays

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rows 2 2 3

 

Columns 2 3 2

 

 

 

 

 

harper & Row Mathematics, 1982

 

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132
multiple answers. However, problems of this sort were not incorporated
into regular lessons but were suggested as optional activities for more
able students.

Each series had a chapter devoted to geometry located near the end
of the text. The focus of these chapters was on naming, identifying
and drawing shapes, memorizing terms, adding numbers marked on sides of
figures to determine perimeter, and counting squares inside figures to
determine area. Harper 5 Row had one lesson on symmetry. Probability
was treated formally in only one of the texts and there it was listed
as an optional topic.

The use of calculators in the classroom was deemphasized.
Occasional "calculator activities" were included at the end of
computational exercises but often were unconnected to the topic at
hand. The calculator activity shown here followed the chapter on
measurement in Harper & Row. It was one of four calculator activities
in the Grade 3 text.

In many cases, neither text provided material responsive to a
particular objective in the Instructional sequence. The Harper & Row
text provided one lesson on bar graphs but none on pictographs or
circle graphs. In that single lesson students were expected to read
and interpret a bar graph but not to gather data or construct a bar
graph. The Holt text did suggest kinds of data that students might
themselves collect and display. But even here, the activity was text
or teacher directed. It was not suggested that students might generate
questions about which they would collect data or make decisions about

how to display that data.

133

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UES

4O 50

1. Use the word MATHEMATICS. See how many

smaller words you can make from those letters.
Find the values of the smaller words.

Samples:

T 40
H 20
E + 100

160

 

. A puzzle:

Enter a number in
your calculator.

I17
{310
[33
BS
.14
E119
E

The display shows
the number you
first entered.

40
100

T+4O

 

180

3. Another puzzle:

Enter a number in
your calculator.

100
El 50

25
Number in display
‘3 50

E] The number you
entered first.

E] The number you
entered first again.

The display shows
200.

Harper 8 Row Mathematics, 1982

g-

:w‘ 3’

." ..“ir

. -'
... ~- 0" u - ... _
'/ O o' 'l ..
..o.‘ .7 sL"M 3“

 

 

134
In July, 1985, the Area D office published finrighrng_rhg_m§rh

gurriculum; A textbook correlation to rhe marhematics instructional
sequence. Their analysis found that of the 116 Grade 3 objectives, the

Harper & Row student text did not include material perinent to 51 of
the objectives, Holt did not include material relevant to 36 of the
objectives.14 Neither text provided materials to meet the following
objectives from the functions and relations strand: "apply a given rule
to a set of whole numbers and determine the missing whole number output
and input;" "identify and state the rule which indicates the
relationship between whole numbers when given input or output;" and
”interpret and complete addition, subtraction or multiplication number
sentences using -, -, <, >." These are key objectives in developing an
understanding of relationships among whole numbers. Of the eight
objectives from the sets and logic strand, only one was covered in
either text.

The two most commonly used textbooks in the district supported
those objectives in the Grade 3 Instructional Sequence that were aimed
at computational speed and accuracy, proficiency with conventional
algorithms and their application to routine story problems. They were
far less effective in meeting many of the objectives related to
understanding the meaning of operations, the effects of operating on
whole numbers, the relationships among whole numbers, and the

application of whole number operations to problem situations. In some

 

14Although a math instructional specialist from another Area
indicated some disagreement with the textbook correlation, a number of
teachers with whom I spoke complained that they could not find material
in their textbooks to tach to many of the objectives.

135
cases, they completely failed to provide material pertinent to
objectives in the Instructional Sequence.15

The lack of alignment between the content of the most commonly used
textbooks and the objectives of the instructional sequence raises some
serious questions. The objectives represented the intended city—wide
curriculum. But what of the enacted curriculum? Was it driven by the
set of instructional objectives, the content in the textbook, teacher
decisions or some combination? How were objectives to be met if
appropriate material was not contained in the approved textbooks?
Which of the three -- textbook, instructional sequence, or teacher
decision -§ determined the order in which mathematical ideas were
studied? How could teachers and students be held accountable for
teaching and learning mathematical topics that were inadequately
treated in or absent from teacher and student texts?

Given the discrepancy between some of the objectives in the
instructional sequence and the available material in the commonly used
textbooks, supplemental materials had been provided to teachers. These
materials included ”Tips for Using the Instructional Sequence,” a
further elaboration of the range of ideas within each of the ten
strands, and a set of model lessons, many of which focused on
objectives for which the treatment in the commonly used textbooks was

lacking or was inadequate. The next chapter examines the set of model

 

15The textbook adoptions were made in 1982 at the same time that
the objectives for the ABCS test were being developed. I was unable to
determine the extent to which these objectives were used as criteria
for the selection process. The district had a six-year cycle for
textbook adoption. The mathematics texts were eligible for review and
new adoptions during the 87-88 school year. I was told by the
mathematics supervisor that a decision was made to delay new
adoptions, reflecting dissatisfaction with any texts they had reviewed
to adequately meet the district's objectives.

136
lessons that accompanied the instructional sequence for Grade 3. Such
an investigation will provide a sense of how the curriculum objectives
were to be enacted in the classroom as teachers taught to these

objectives.

CHAPTER 6

THE MODEL LESSONS

INTRODUCTION

In 1986, the elementary mathematics supervisors in the district
brought together a team of math instructional specialists, classroom
teachers, a building principal and a city-wide program coordinator for
a research and writing project. The team was charged with the task of
creating sets of model lessons for each grade level, drawing on two key
areas of research. The first domain concerned findings from research
in mathematics education, cognitive science and educational psychology
about how children learn mathematics. The particular finding that
influenced the team's work was using concrete materials to introduce
mathematical ideas, linking concrete models to pictorial
representations, and then using pictorial representations as the bridge
from the concrete to the abstract/symbolic. The second domain
concerned findings from effective teaching research-- those teacher
behaviors that seem to be effective in increasing student achievement
on standardized tests and that result in more efficient teaching. The
lessons were intended to develop the critical thinking skills and
problem solving abilities of youngsters. The lessons were also
intended to address the mismatch between objectives in the
Instructional Sequence and available material in the commonly used
textbooks in the District.

Committees of five to seven teachers at each grade level piloted
the lessons during the 1986-87 school year. In September, 1987, the

Mathematics and Science Department of the District published uodel

137

138
Lessons to Promote Ihinking. This set of materials included an
elaboration of each of the ten curriculum strands followed by a set of
model lessons for specific objectives at each grade level. The number
of model lessons ranged from 22 for Grade P1 to 40 for Grade 4. At
each grade, some of the model lessons focused on the use of calculators
and were designated ”hand-held calculator lessons.” The model lessons
provided teachers with examples of mathematical ideas that fit within
the strands as well as particular instances to meet specific
objectives.

In Chapter 4 the argument was made that the goal of developing
students' critical thinking and higher order reasoning skills and
problem solving abilities by reorganizing the curriculum around ten
mathematical strands was in tension with the objectives that defined
content and sequenced instruction. The instructional objectives
coupled with the commonly used textbooks tended to perpetuate a
traditional view of mathematics curriculum and instruction that
atomized content, isolated computational proficiency from problem
solving contexts, and ignored inquiry, decision-making and judgment. At
the same time, there was evidence of an interest in and attempt to
develop an understanding of numeration and whole number operations.
Place value was emphasized, concrete models to develop number facts
were incorporated, relationships between operations were suggested, and
estimation strategies to enhance mental computation and assess the
reasonableness of answers were included. Yet, while the reorganization
around mathematical strands suggested a reshaping of content from the

traditional emphasis on arithmetic, less than a quarter of the

139
objectives were devoted to geometry, probability and statistics,
patterns, and sets and logic.

But the objectives alone, as simply statements of expected student
outcomes, provide scant information about how these objectives were to
be met in the classroom as teachers taught to the objectives. This
chapter examines the set of model lessons for Grade 3. The model
lessons afford the opportunity to investigate how teaching to specific
objectives was portrayed to teachers. The model lessons provide
another source of evidence about the potential of this curriculum
reform to develop youngsters' critical thinking and problem solving
abilities.

The chapter is divided into three sections. The first section
examines the introduction to uodel Lessons to Eromoge Thinking. The
inquiry concerns the goals for student learning as expressed in this
document and the explication of the range of ideas encompassed in each
of the curriculum strands. The second section examines the content of
the Grade 3 model lessons -- the distribution of the lessons among the
strands, an example of a model lesson that embodied elements of
developing understanding, and an example of a model lesson that did
not. The third section examines the form of the model lessons to
determine the extent to which curricular form enhanced or constrained

teaching for understanding.

The Analytic Framework
The framework for analysis of the model lessons continues with the
three orientations developed in Chapter 2. The analysis inquires about

the role of the teacher in instruction, the role of the student in

140
learning mathematics, assumptions -- explicit or implicit -- about how
math is learned, what teachers take as proxies for learning, and where
authority for knowing rests.

Within certification of numeracy, the role of the teacher is to
show students how to carry out procedures. The teacher gives
directions, works examples and assigns seatwork for students to
practice a skill or procedure. The role of the student is to absorb
and remember the knowledge that a teacher dispenses. Learning
mathematics is an individual, solitary effort where children search
their memory for accumulated facts and algorithmic solutions. Task
completion, right answers and periodic paper-and-pencil tests for
recording a mark serve as proxies for student learning. Teacher and
text stand as the source of epistemological authority.

Within comprehension of mathematical ideas, learning is portrayed
as requiring a particular sequence of development moving from concrete
to pictorial to abstract. The explicit/implicit mode of instruction
draws on effective teaching research. The role of the teacher is to
show students what they are to learn by demonstrating to the entire
class with the appropriate materials and then letting students
practice. The teacher monitors student learning during instruction by
posing "product" and ”process” questions, evaluating student responses
and giving immediate feedback. Opportunities are provided for students
to work in groups on ”fun” -- though generally routine -- activities.
Within the structure imposed by the teacher, students pursue teacher
defined explorations with concrete materials, answer teacher questions
and complete assignments on practice exercises. More than task

completion and right answers are sought. Students are expected to

141
provide explanations for procedures they have used. However, authority
for knowing continues to rest with the teacher who decides whether
student answers are correct and explanations sufficient.

An orientation aimed at enabling,mathematical inquiry makes several
important assumptions about learners: they actively construct
mathematical knowledge; their prior knowledge and current conceptions
affect how they make sense of new knowledge; they are able to construct
arguments about why mathematical ideas are true and are able to figure
out if strategies and solutions are reasonable. The role of the
teacher is to create a total environment where teacher and students can
actively engage with one another in inquiry. That requires posing
interesting, challenging and non-routine problem situations that are
rich in mathematics and that lead to learner inventions. Assessing
student understanding is a continual, on-going process in instruction,
seeking evidence by listening to students and asking questions to
determine what they are nnning to know. Students learn to use multiple
representations - concrete, numeric, graphic, algebraic, spatial - to
describe a mathematical situation. In a supportive environment, they
become mathematical risk-takers--making guesses and pursuing hunches,
offering hypotheses, accepting challenges to hypotheses and marshalling
arguments in support of them, and yielding to the force of a better
argument. The locus of epistemological authority shifts from the
teacher and text to the classroom community of students and teacher.
Together they have the power to use mathematical tools and standards of
the mathematical community to decide about the reasonableness of

processes and the results of investigations.

142
This chapter examines the set of model lessons for Grade 3 to

determine their orientation.

MODEL LESSONS TO PROMOTE THINKING
The Introduction to the Model Lessons
The introduction to the set of model lessons placed emphasis on
developing students' ability to reason.

Skill development of the fourth (r)easoning has been designated
by the general superintendent for the students in the Detroit
Public School System. The mathematics core curriculum
emphasizes the teaching of thinking skills. To help students
attain reasoning (thinking) skills, it is of utmost importance
that students understand the concepts being taught...Thinking
skills have been interwoven into each lesson to develop/improve
the problem solving abilities of students...Increasing the
emphasis on critical thinking reduces the traditional
difficulties students have had with organizing, presenting and
interpreting data, estimating results and calculating mentally
(Detroit Public Schools, September, 1987; p.ii).

In keeping with the District's philosophy regarding mathematics
education, the introduction described developing critical thinking and
reasoning skills and problem solving abilities as goals that informed
the lesson writing effort. Three pedagogical tools were specified as
key to helping students understand mathematical concepts--questioning,
using concrete materials, and using hand-held calculators.

Teaching by means of posing questions is to be valued over
expository instruction...Above all, many concrete and/or semi-
concrete models have been used to help the learners gain
understanding before symbols/abstraction. The most recent research
results regarding how a child learns has been incorporated into
these lessons...Hand-held calculator lessons using the problem
solving approach are included for certain strand objectives. A
summary of numerous studies indicates that achievement scores of
student (sic) using calculators in the classroom are as high or
higher than those of students not using calculators in their
instruction (Detroit Public Schools, September, 1987; p. ii).

The introduction also provided an explication of each of the

curriculum strands beyond the brief description in the Strands and

143
aneeeigee and, in some cases, a rationale for including the strand in
the curriculum. For example, the value of estimation in the curriculum
was linked to the fact that ”Most people use estimation every day, more
often than they use pencil and paper computation. Therefore, it is
imperative that estimation be given much emphasis...'(p. ix). The
range of activities suggested for this strand included rounding, front
end estimation, averaging, and compatible numbers as well as estimating
with measurement. The development of linear measurement ideas was
portrayed as requiring a particular sequence.

In order to help students develop the concept of a unit, each

lesson has as its primary objective the perception of length and

then its measure using 0 - andar un ts e ore standard units are
introduced...The experience with a non-standard unit help (sic) to
develop ideas common to all measurement situations... the process
by which a child learns to measure is sequential in development
from perception, use of non-standard units, comparison and finally
to the use of a standard unit of measure. Children should have
continuous experiences with the hands-on activity of measurement

throughout the grades (p. xiii).

Numeration was described as ”basic to mathematical proficiency" and
computational proficiency dependent on a "thorough knowledge of place
value” (p. xv). Several concrete materials -- beans and beansticks,
Dienes blocks and money -- were suggested as models for place value.

It is noteworthy that while each of these materials embodies a
different model as a representation of place value, this was not
acknowledged nor were teachers given a sense of the mathematical
situation for which a particular model might be appropriate.1 A
particular developmental sequence for place value was defined.
All ideas children internalize fully must be met first in the world

of three-dimensional things - gne manipulative wezlg. Later, the
same ideas can be encouraged and understood in sketches, pictures,

 

1For a discussion on the “thoughtful use of manipulatives," see
Schram, Feiman-Nemser and Ball, 1989.

144

and diagrams - ehe nedel er renneeentetienel werlg. Eventually,

children work with these same ideas expressed in symbols — the

W (p. XV).2

The operation strand emphasized ”facility with the basic facts of
arithmetic” and “thinking mathematically.” More than drill and
practice was demanded; posing questions "where the use of memory is
just the beginning” was suggested.

Are there 6 different ways of finding parts or addends that equal 3

in the form ___ + ___ - ___? Are there more? Can you be sure that

you have found them all (p. xxvii).

A direct link between operations and another strand, patterns, was made
explicit. It was suggested that to answer questions like those posed
above, students should make a chart and obserVe the patterns that
emerge.

Sets and logic encompassed organizing and interpreting data, using
”sequential steps to analyze and conceptualize problems," and drawing
conclusions. The purpose of including probability in the curriculum
was "to familiarize students with intuitive ideas and to enrich the
mathematics program, as well as provide practice work in whole numbers,
graphing, percents and decimals...make predictions and...discover new
information" (p. xxxiii). Statistics developed skills in "counting,
measuring, recording, ordering, classifying, comparing, displaying and

interpreting information" and reinforced ”basic arithmetic skills in an

interesting setting” (p. xxxiv).3

 

2In a subsequent paragraph, the "model or representational world"
was referred to as the pictorial level and that ”picturing numeration”
was a ”prerequisite to long term memory and success at the abstract
level" (p. xvii).

3There was no elaboration on the geometry strand in this document
— presumably an oversight in putting the materials together.

145
Unlike the brief definition of the curriculum strands in Stgends

d 0 e v (described in Chapter 5), the elaboration of each strand
in flegel_Leeeene_;e_£19ne§e_1hinking suggested a range of mathematical
ideas -- concepts, skills, procedures -- to be studied, ideas to be
emphasized, and various ways in which students might encounter them.
The elaborations also provided instances in which ideas from one strand
would be met in another. Estimation was essential to developing
computational proficiency and measurement concepts. Understanding the
four arithmetic operations contributed to developing the ability to
solve simple equations and find their inverses, ideas from the
functions and relations strand. Developing an understanding of place
value and operations with whole numbers was linked to discovering
patterns and finding relationships.

While the strand definitions in Strands end Objectivee appeared to
serve a political and symbolic purpose, their elaboration in Lessens to
Ezomote Ininking served a mathematical purpose. The explications
provided a sense of a range of mathematical ideas embedded in each
strand, how some of those ideas were connected, and why those linkages
were important. The elaborations also served an instructional purpose.
Teachers were instructed to follow a specific developmental sequence
for some ideas. They were alerted to special problems children had in
developing some concepts. The importance of ”asking questions to
promote student understanding and check comprehension of concepts” was
emphasized. It was recommended that some lessons be taught and then
revisited throughout the school year, ”with added insight." Teachers
were told that while it was important for youngsters to master basic

facts for instant recall, recall alone was insufficient.

146

...it is important to realize that students need more than just

basic knowledge. They must begin to internalize the relationships

that exist...to develop higher order thinking skills. These higher

levels include: analysis, synthesis and evaluation (pp. xxviii-

xxix).

It appeared that developing higher order thinking skills and problem
solving abilities was as important a goal as developing computational
speed and accuracy and proficiency with conventional algorithms. The
document seemed to be oriented toward comprehension of mathematical
ideas.

The next section inquires about the extent to which the model
lessons themselves represented a shift from the traditional emphasis on
arithmetic by examining the distribution of model lessons among the
curriculum strands and special features of lessons that could
contribute to developing conceptual understanding. The investigation
then moves to a consideration of several specific lessons as examples
of the unevenness in which the broad goals of developing understanding,

higher order thinking skills and problem solving abilities were

embedded in instructional activities.

The Content of the Grade 3 Model Lessons
istr but on f mode esso o a ds. Table 6.1 shows
the distribution of model lessons among the curriculum strands and
notes particular features of the lessons.
Seventeen of the lessons focused on objectives for which the

treatment in one or both of the commonly used textbooks was lacking or

was inadequate, as assessed in Engiching ghe nath curriculnn; A

W
Estimation and

Approximation

Functions and
Relations

Measurement

Numeration

Operations

Patterns

Probabiity and
Statistics

Sets and Logic

”(DO-00‘”

147

E

fififiifi fifififi fifififi 5E5

:13
t-‘r‘

E5

ML

ML
ML
ML

uses concrete materials.

provides extensions.

incorporates games.

BC

BC
BC

BC
BC

HO
BC

BC

8

3
f

\Juxh:
c‘a‘a‘
m

mu

teacher demonstration only; no student activities.

series of lessons covering 3-6 days.

objective is absent from or inadequately treated in commonly
used texts.

Figure 6.1 Distribution of Model Lessons Among Curriculum
Strands and Features of the Lessons

148
textboo cor elatio to the athemat nst ctiona se uen (see
Chapter 5). These objectives included

* students will be able to construct, read and interpret a bar
graph/pictograph (ML 1).

* the student will be able to l) interpret and/or extend a pattern
of shapes, pictures or objects using a combination of any three
of the following attributes: color, shape, size, thickness,
position, texture; 2) state the rule used in forming a given
pattern and extend a pattern; 3) create and state the rule of a
pictorial pattern (ML 6).

* students will be able to demonstrate their perception of their
length/height (ML 17).

* the student will be able to read, interpret, and supply elements
in a matrix (ML 20).

* students will demonstrate the probability of simple events with
two outcomes (ML 22).

Fifteen of the 31 lessons required the use of concrete materials,
supporting a view that learning must begin with concrete experiences.
The six model lessons from the numeration strand used concrete,
pictorial and symbolic representations. Beansticks, hundreds squares
and Dienes blocks were suggested as possible models to demonstrate
place value. Pictorial representations included drawing bundles of
sticks and coloring centimeter grid paper to illustrate numbers. The
symbolic representations included place value charts, expanded
notations (e.g., 300 + 20 + l and 3 hundreds 2 tens and 1 one) and
standard notation (e.g., 321). Other lessons employed counters, dice,
attribute blocks, and non-standard measuring devices.

Suggestions for extending an objective were included in 10 lessons.
The nature of these extensions included expanding work to larger
numbers, broadening the set of operations to be performed, and
increasing the complexity of the activities. The use of games as a
context to practice computation and regrouping were incorporated into

three lessons. The games in Model Lesson 11 required players to be

able to represent whole numbers concretely, pictorially and

149
symbolically and tell from each representation the relative sizes of
whole numbers.

In several instances a set of consecutive lessons was devoted to
the same objective, showing how a topic might be pursued over several
days of instruction. Model Lessons 2 and 3 were devoted to having
students ”determine and/or demonstrate place value through 999 using
aids and identify a number 100 more or less than a given number.”
Together, these two lessons were intended to be covered over a period
of at least four days. Model Lessons 7, 8 and 9 had students ”identify
the missing relationship or operational symbol in an incomplete number
sentence.” Model Lesson 13 from the patterns strand incorporated a
series of lessons to be carried out over a six day period. Hundreds
charts were used to discover number patterns. This work was expanded
at the abstract level to patterns that arise when counting by hundreds.
Patterns were used to help reinforce basic number facts and find sums
and differences of two digit numbers.

Interestingly, no model lessons had been created for the geometry
strand, a noteworthy absence considering the limited treatment of this
topic in the commonly used textbooks. However, two lessons in other
strands employed attribute blocks in classification activities and
geometric pattern explorations.“

The analysis here has taken a broad sweep of the entire set of
Grade 3 model lessons, attending to their distribution among the
strands and certain features of the lessons. The analysis suggests

that the model lessons seemed oriented toward comprehension of

 

1‘There were no model lessons for the geometry strand at any of
the other grade levels. One model lesson from the sets and logic
strand involved classification activities with geometric shapes.

150

mathematical ideas. Developing an understanding of number, numeration
and whole number operations was given particular attention.
Mathematics was represented as an abstract system of concepts and
processes for which there are appropriate concrete models and pictorial
representations. Learning was portrayed as requiring a particular
sequence of development moving from concrete to pictorial to abstract.
At the same time, numeration and operations dominated the model
lessons. Nineteen of the 31 model lessons (61!) were devoted to
objectives related to place value, computation and operations.
Probability and statistics and sets and logic received scant attention.
The geometry strand was ignored.5 The dominance of lessons devoted to
topics on numeration and operations tended to perpetuate a view of
elementary mathematics as primarily arithmetic.

This broad sweep has given a sense of the orientation of the model
lessons as a whole. There appeared to be a tension between a
traditional emphasis on topics of arithmetic and an attempt to develop

understanding of mathematical ideas about numeration and operations.

 

5The chart shows the distribution of model lessons among the
curriculum strands for the other grades.

Senand humbe; of hodel Leesone
Grade Pl Grade P2 Grade 4 Grade 5
Est. & Approx. 1 3 3 3
Func. & Rel. 4 5 6 8
Geom. O O O O
Meas. 4 4 8 3
Numer. S 4 5 4
Oper. 3 4 7 8
Patt. 2 3 2 2
Prob. & Stat. 1 l 5 6
Sets & Log. 2 l 3 l

The increase in lessons in the probability and statistics strand
represented more attention to displaying and interpreting data. Only
one lesson was devoted to probabilistic ideas. Roughly one-third of
the Grade 4 and 5 lessons had been introduced at an earlier grade.

151
But the investigation has examined elements of lessons taken out of
context; the investigation has been limited and partial. Further
probing at the level of a specific lesson in its entirety is warranted.

The model lesson is another piece of evidence about the nature of
this reform. It exemplifies the way in which mathematics and
mathematics teaching and learning were represented to both teachers and
students. And it provides the occasion for further inquiry about what
this reform would help students to learn and know. The inquiry
continues with an analysis of several of the Grade 3 model lessons.

The first model lesson was chosen because it embodied a number of
elements that could contribute to developing understanding of
mathematical ideas. It stood out from all the other lessons as an
exemplar in connecting mathematics to the real-world of children, using
interesting problem situations and multiple representations, and
extending the world of mathematics beyond numbers and computation. The
other lessons were selected because they typified the unevenness that
existed among the majority of the lessons in terms of
conceptualization, mathematical substance, and student activities.

A model lesson to develop understanding. Model Lesson 6 was
created for an objective of the patterns strand. The lesson begins
with a statement of the objectives.

OBJECTIVE: The student will be able to: l) interpret and/or extend

a pattern of shapes, pictures or objects using a combination of any

three of the following attributes - color, shape, size, thickness,

position, texture; 2) state the rule used in forming a given

pattern and extend the pattern; 3) create and state the rule of a

pictorial pattern.

The lesson describes materials the teacher would need to carry out

instruction and the vocabulary to be developed.

152

MATERIALS: Attribute blocks (or railroad board or construction
paper) for teacher and students; teacher made pattern cards.

VOCABULARY: attribute blocks, attributes.
Instruction begins with a motivation activity. In all instructional
phases, teachers are provided with examples of what to say during
instruction, what questions to ask, and responses to accept.

MOTIVATION: "Let's imagine that you were invited to your friend's
birthday party last week. One of the games the children played was
”Find a Pattern.” The rule was to look around your friend's home
and see how many patterns you could find and then to write down the
pattern and the rule used to make the pattern. Let's imagine that
you won the game by finding the greatest number of patterns. We're
going to play that game in class today to see how many patterns we
can find."

The next component of the lesson is called instructional input, the
direct instruction phase of the lesson.

INSTRUCTIONAL INPUT:
1. Location of patterns in the environment

Discuss patterns in the environment and the rule used to form
them: placement of windows, cupboards, doors and hardware; flag
stripes (alternating red and white stripes); clothing designs;
fencing (chain link); black keys on a piano (2 space 3);
designs on towels; patterns in furniture; wallpaper designs;
brick walls; floor tile; pairs of shoes, mittens.

Compile a list of patterns found and add to it ghrenghen; the
school year.

II. Pattern formation and extension

Pass out attribute blocks to students and discuss them by
asking:

1. "Describe this shape.” (note color, shape, size)

2. “What is the same about these two shapes?” (Pick out common
attributes; use the term attribute)

3. ”What is different about these two shapes?"

4. "How many ways are these two shapes the same?”

5. "How many ways are these two shape different?”

6. ”Find another piece that has the same common attribute.
What is it?“

Hold up one attribute block and say:

1. "Hold up a piece that is different from this in two ways?"
2. "How is your piece different, Johnny?”

3. "Let's make a pattern of pieces that are different from each
other in two ways."

153

4. Form the pattern using student input. Discuss how the
pieces differ.

AA AA

State the rule used. (change in color and size)

Form a pattern of pieces that differ by three attributes: e.g.,

(915: B

”How do these pieces differ?" (Color, shape, size)

 

 

 

 

The next phase of the lesson, guided practice, provides an
opportunity for children to practice what the teacher has taught.

GUIDED PRACTICE:

Play "Guess My Rule” with attribute blocks.

A. "I am going to put some attribute blocks on the chalkborad.
Each time I put one up, I must follow a rule. See if you can
guess what my rule is.”

B. Display a pattern. Say: ”What piece do you think I could put
up next? Why?” Answer by saying: "That piece does follow the
rule I am thinking of" or ”That piece does not follow my rule."

C. Continue asking students to select blocks following the
pattern. When you think most students know the rule, ask
someone to verbalize it.

Debugging an incorrect pattern.
A. Provide students with patterns, some of which are correct and
some with one incorect piece.

8. Ask questions such as: Is this pattern correct? How do you

know? How are these pieces different? What rule was used to
form the pattern?

Challenge
Ask: ”Can we make a pattern of pieces that differ in four
ways?” How?” (Change of position)

-..R A ® W

The lesson ends with closure.

 

 

 

 

154

CLOSURE:
1. Provide the students with an opportunity to develop their own
patterns.

2. Other students should guess the rule used to make up the
pattern. They may also extend the pattern.

This lesson was one of three in the set that incorporated
suggestions for ways in which a teacher might extend the ideas.
INSTRUCTIONAL.INPUT:
111. Logic extension of patterning:
A. Review the meaning of attributes. Select two blocks and
discuss how they are the same and how they differ from one

another.

8. Make "one difference” trains. Each piece must differ from the
one next to it by l attribute: e.g.,

A .am...’

Explain the differences as the pieces are selected.

C. Provide the children with a grid (matrix) such as the
following.

The pieces must differ from each other vertically and horizontally
by 1 difference: e.g.,

®® [—9116
Aél [33:19_

Other answers are possible.

«___-3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EXTENSION - INDEPENDENT PRACTICE:
Tonight for homework play ”Find a Pattern" at home. Write down
the patterns you find and state the rule used to make the pattern.
You may also find patterns somewhere around your school.

The analysis of this lesson focuses on the mathematics children

have an opportunity to learn and the intellectual abilities being

155
developed. The lesson begins in the concrete world of children.
Finding patterns in their environment could help children see how
mathematical ideas are a part of the world in which they live. More
than recognition is expected. Children are to describe the "rule" used
to form the patterns. In their own language, children could begin to
describe the regularity and repetition they find in elements of their
surroundings.

The lesson continues at the concrete level with attribute blocks.
Teacher questions direct children to attend to the attributes of each
piece: color -- red, blue or green; size -- big or little; and shape --
square, circle or triangle. Teacher questions then ask children to
compare a block with pieces in their set. Responding correctly to
teacher questions requires knowing the ways in which attribute pieces
are alike and ways in which they differ. The activity allows for more
than one correct response. For example, if the teacher displays a
small red triangle, children could select several pieces that differ by
one or two attributes. This series of activities helps children
develop ways to classify the blocks in their set and see relationships
among the various pieces.

The several versions of "Guess My Rule” provide a problem solving
context in which children reason about what features seem to be
important as a pattern is being created, when a set of blocks appears
to repeat, and what the nature of the pattern is. The context also
provides an opportunity for patterns within patterns (although this is
not mentioned at this point in the scripted lesson). For example,

children might see several different patterns in the following

 

156
arrangement depending on the attribute around which they think the

pattern has been created.

R G 6)

One pattern is red, green, red, green. A second pattern is little,

 

 

 

 

 

 

 

 

big, little, big. A third pattern is square, square, circle, circle.
Adding a small red triangle as the next piece in the sequence would
invite children to see if their initial idea needs to be revised. The
activites provide an interesting context in which creative thinking and
logical reasoning are encouraged.

This initial set of activities focuses on the three attributes ~-
size, shape and color -- that distinguish each block from every other.
The ”challenge” later in the lesson provides a problem situation in
which children consider if it is possible to create a pattern where the
pieces differ in four ways. This could lead children to consider the
position or orientation in which the pieces are placed on the table--as
is suggested in the lesson.

Following the teacher led activities where children discover
patterns and the ”rules" by which they are created, youngsters are
given the opportunity to create their own patterns. As one child
creates a pattern, another guesses the "rule" and then extends the
pattern. Such an activity could provide the occasion for some very
creative pattern-making while helping children develop their ability to
classify and to reason mathematically. It provides a context in which

children could talk about mathematical ideas. The ”rule” by which one

child creates a particular pattern might not be unique (see the example

157
above). In seeing another "rule" and justifying it by extending the
pattern, youngsters are actively engaged in constructing meaning in a
situation where more than one ”right" solution is possible.

The extensions -- making ”one difference” trains and completing a
two-by-two matrix -- provide further problem situations in which to
explore relationships. Introducing the matrix affords youngsters
another form in which to organize and present information. Whereas
previous activities employ a linear organization of the attribute
pieces, the matrix suggests a different representation in which
relationships can be explored vertically, horizontally and along the
diagonals.

This lesson embodies elements of teaching for enabling mathematical
inquiry. The conceptualization of the lesson, its connection to the
real-world of children, the use of interesting problem situations and
the use of different forms in which to classify, organize and present
information attempts to develop a range of intellectual tools to
enhance children's mathematical abilities. The lesson draws on
youngster's abilities to reason, to communicate about mathematical
ideas, to invent, generalize and create. The lesson extends the
child's conception of the world of mathematics beyond numbers and
computation to include the patterns and regularities that are part of
the naturally geometric world they encounter, both in and out of the
classroom.

Unfortunately, few of the model lessons approached this one in
terms of its potential to develop children's abilities to reason
mathematically. More often than not, the model lessons seemed more

focused on instructional technique and "fun” activities than the

158
substance of the mathematics. Model Lessons 8 and 4 and 12 exemplify
those model lessons that seemed unlikely to develop a set of
intellectual tools that could help youngsters become self-confident

problem solvers.

od e so 1 ke to devel et nte e tua o
pleblem eeiving. Model Lesson 8 was created for an objective in the

functions and relations strand. The lesson begins with the statement
of the objective.

OBJECTIVE: The student will be able to identify the missing
relationship or operational symbol in an incomplete number
sentence.

The vocabulary for the lesson includes the following terms; open, true,
number sentence, relative, table, greater than, less than. The lesson

begins.

MOTIVATION: Ask the students to tell you what the word ”relative"
means. Have them give you examples. (mother, aunt, uncle, etc.)
Say: ”Today we are going to make a table that is 'related' to an
open number sentence. (Note: You may have to review the words:
table, true and open number sentences from the previous lessons.)

INSTRUCTIONAL INPUT:
Write on the board: 04- 3 -A

Ask: ”What do the Band Arepresent?" (numbers)
Say: ”Let's put numbers in the and to change our open sentence

to a true sentence. Let's write our true sentences and then make a
table to represent our sentences."

 

 

 

 

TABLE

D+3- DHA
m+3- 1 4
m+3- 0 3
%+3- 7 W
+3- 4 9
[3:]+3-& _3 6

 

 

159

Do several other sentences with the students, using addition and
subtraction.

In the guided practice, inequalities are introduced.

GUIDED PRACTICE:
Say: ”Let's look at some open sentences that contain the greater
than or less than symbol."

Write on the board: [3+ 3 >A

.Ask: "What are some true sentences?" (Examples are given.)

Say: ”Let's make a table related to our sentences." (See below)
_1 3
3 I 3

Divide the class into pairs and ask them to write true number
sentences using less than or greater than. They are to write a
related table for each as demonstrated.

Select several students to write the true sentences on the board
and the related table - sharing various strategies and answers.

Independent practice calls for the following.

The

INDEPENDENT PRACTICE:

Have the students make up three (3) open sentences, giving at least
five (5) true sentences and constructing a "related” table. Or use
these samples.

+D-7
etc-5
A'U'3
6-D-A

lesson ends with closure.

CIDSURE:

Ask: ”How do these tables differ from a function table?” (The
output number is unique in a function table but not in a 'related'
table.) ”How are they alike?” (input, output, rule and
relationship)

Model Lesson 8 is poorly conceived. As in most lessons, the single

attempt to link the objective with the world of children is the

motivational phase of the lesson. In this part of the lesson, ”related

160
to” is linked to youngsters' understanding of the word "relative.” The
use of the terms ”relative" and "related to” in the context of this
particular lesson is rather peculiar. It is not clear how calling on
children's prior knowledge about family relationships is likely to
facilitate their understanding of mathematical relations.6
The activities in the lesson do not seem to follow from the stated

objective. The objective is to ”identify the missing reieeienehin_e;

ene;e§iene1_eynhel in an incomplete number sentence.” But problems for

 

6The motivational part of the lessons was curious. In most cases
it tended to be a teacher statement telling students what they would be
doing during the lesson. In some instances it was a story or problem.
For example, Model lesson 2, from the numeration strand begins with the
following motivation.

Teacher presents the following problem: " (name a
student in your class), wanted to buy a Big Mac. She/he had 75
cents. The Big Mac cost 85 cents. Could she/he buy the Big Mac?
(wait for yes or no responses) If she/he had 95 cents, could
she/he buy it? Today we are going to explore an easy way of
solving problems like these.

Model lesson 10 from the numeration strand focuses on writing three
digit numbers in expanded notation. For motivation, the teacher is
instructed to say,

Today we are going to expand numbers like a rubber band.

Model lesson 18 on estimating by rounding numbers to the nearest
multiple of ten, begins with the following:

Review by counting by tens. Say: ”There was a little town
call(sic) Ten-town. No other numbers but tens lived there, or so
they thought. One day 38 came to Ten-town. All the tens told
him/her to leave because he wasn't a ten. Thirty-eight saw the
King of Ten-town and ran over to him. The King whispered
something in his ear. Thirty-eight belongs to Ten-town. What was
the King's rule? Today we are going to find out what the King's
rule was.”

It is not clear how these motivational statements might inspire
children to engage in the upcoming lesson. Nor is it clear how they
provided ”hooks" into prior knowledge or experiences. In some cases,
there seemed to be no logical link between the motivation and the
central idea of the lesson, Model lesson 8 being a case in point.

161
teacher demonstration and student practice involve supplying missing
nnnhele in an open sentence. The lesson is marked by inconsistency in
the use of terms ”relation,” ”related to,” "related table," and
“function table.“ In the introduction to the functions and relations
strand, teachers were advised that the terms functions and relations

not be defined "MW”

Function: Special or typical work or purpose or thing or person -

to do its workzact, e.g., a table can function as a desk or the

function of the brakes is to stop the car or...+5 means to add five

to a number you started with...Relations are simply 'connections.’

Example: Brother is related to sister (p. xi).
In this lesson, the distiction made between a function and a relation
is that a function yields a unique output for a given input whereas a
relation may yield more than one output. For example,
x + 5 - y is a function because a given value for x gives a unique
value for y. The open sentence x + 5 > y is a relation since for a
given value of x there are many values for y that make the statement
true. The term ”related to" is applied only to those tables
constructed from inequalities. These tables are distinguished from
”function tables," constructed from equalities. The directions for
independent practice call for constructing a ”related table” for a set
of open sentences. et ve 0 en t e et e ua o
amass!

The notion of beginning with the concrete and using pictorial
representations as a bridge from the concrete to the abstract seemed to
be treated rather simplistically in this and other model lessons.

While this lesson represented a mathematical idea solely at the

abstract/symbolic level, it followed a lesson that employed concrete

materials (counters) and pictorial representations. In Model Lesson 7,

 

162
devoted to the same objective as Lesson 8, the instructional input
phase begins with a story.

We had five puppies at home. Two ran away. How many puppies are
still at home?

The lesson moves from illustrating the story with counters to drawing
five dots and crossing out two of them to then writing the problem
symbolically, 5 - 2 - 3. The pace with which the lesson moves from one
representation to the next suggests that making connections among
multiple representations comes easily and naturally for children.
Creating appropriate pictures for stories -- or stories for
pictures -— and then translating those forms into the symbolic language
of mathematics is a complex set of ideas for children (see Hiebert,
1980). The confusion that youngsters might be expected to encounter,
especially in moving from the pictorial to the symbolic, is exacerbated
in Model Lesson 7 by the examples themselves and what constitutes
"right” answers. The two drawings below are given for teachers to
demonstrate. The shapes are to be ”replaced by unique numbers" to make
each open sentence a true sentence that reflects the mathematical idea

in the drawing.

.4 D‘A'L—J 6 D'A‘CZr

163

As in most model lessons, "correct” answers are provided for
teachers. The "correct” true sentence for drawing A is '5 + 3 - 8.”
The ”correct" true sentence for drawing B is “4 - l - 3.“ It is not
clear why those and not others are the "right" answers. For example, 5
+ 8 - l3 and 5 - l - 4 are interpretations that children might
reasonably and logically make given the pictorial representations of
the situations. In fact, it is probably likely that children would
have considerable difficulty making sense of the lesson's ”right
answers."

Model Lesson 8 typified most of the other model lessons in that it
treated a mathematical idea outside a real-world context. No
situations were provided from the real-world of children which would
give rise to the mathematical concepts of the lesson. In contrast to
Model Lesson 6 where a broad range of intellectual tools were being
developed to enhance children's mathematical reasoning abilities, Model
Lessons 7 and 8 pursued a more limited set of goals. Getting to the
symbolic representation and becoming proficient in symbol manipulation

seemed to be the goals that drove the lessons.

Model Lessons 4 and 12 are devoted to recall of addition and
multiplication facts. They are included here as examples of missed
opportunities to engage in further inquiry about the mathematical ideas
embedded in the context. Both lessons use cross number problems as the
environment in which children practice recalling addition and
multiplication facts. A cross number problem is one in which sums (or
products) can be obtained using three different sets of addends (or

factors). For example, the following cross number problem in Model

164
Lesson 4 "works” because adding across gives 4 and 6 whose sum is 10,
adding down gives 5 and 5 whose sum is 10, and adding diagonally gives

7 and 3 whose sum is 10.

 

3 I
024

+

 

 

 

 

 

 

 

 

In some problems, the sum or partial sums are given and children fill

in the missing pieces.

 

 

 

 

 

 

8 + IO

 

 

 

 

A similar approach is used in Model Lesson 12 with multiplication.

72 5
l 5
Y ___J.

Because the objective of the lessons is simply to recall facts,

 

 

 

 

 

 

 

 

 

 

interesting investigations are not pursued. For example, can a cross
number problem with some elements supplied be completed in more than
one way? What is the nature of numbers that ”work” for cross addition
or cross multiplication? How many "clues” are required to complete a
cross number problem? Can those clues be provided in any of the eleven

places? Is there a systematic way in which to solve these problems?

165

Cross number problems could have been used to extend children's
understanding of addends and factors but the opportunity is missed.
Rather than using this as an occasion to extend students' learning
about numbers and operations with numbers, becoming more proficient in
the recall of basic facts is an end in itself.

Model Lesson 12 incorporated an ”challenge" activity for “children
proficient in division.”

1. "As a challenge, find out if this one works." (no)

‘5' l

M 6
LI a?

 

 

 

 

 

 

 

 

 

 

 

2. ”Can you make it work?" (It will work if you plan ahead or if '
you multiply the diagonals to avoid ”ear trouble."

24 6 - 4, 4 2 - 2 [horizontal]

24 4 - 6, 6 2 - 3 [vertical]

24 x 2 - 48, 6 x 4 - 24 [diagonal]

0.. 0'.
H. on

3. ”Let's prove our results."

4 § 2 - 2 [vertical]

6 § 3 - 2 [horizontal]

48 § 24 - 2 [diagonal]

This optional activity underscores the extent to which manipulating
symbols took precedence over "sense-making" in some of the lessons.
There are no questions posed to inquire about what ”plan ahead” might
mean or how one would go about it so cross division problems would
"work.” Nor are there questions about why multiplying along the

diagonal might make sense or if that would always "work." Recall of

basic facts is removed from the typical format of rows of problems for

drill-and-practice. But the narrow focus of the objective results in a

 

166
missed Opportunity to explore the richness of the mathematics embedded
in the situation.

The majority of model lessons were limited in terms of the
conception of mathematical content and the goals for student learning.
While a few were exemplary in their attempt to enhance children's
abilities to reason mathematically and to make connections between the
world of mathematics and the real-world of children, many were mediocre
attempts to wrap computational proficiency in new packaging. My
discussion of the limitations of these lessons as manifested in their
text does not imply that teachers would not be able to do something
other than what the lesson, as text, does. As I stated in Chapter 5,
curriculum texts dod not assure or even predict the behavior of
teachers or students in the classroom. What I am arguing is that
nearly all these model lessons represent a view of mathematics and an
approach to mathematics teaching oriented toward certification of
numeracy and comprehension of mathematical ideas.

The potential in these lessons to promote higher-order thinking and
problem solving abilities was limited and was, in part, a function of
the way in which mathematical topics were conceived and objectives and
activities selected. But the limitation of these lessons was also
embedded in the instructional model that informed the design of the
lessons. The next section examines the instructional model to
determine the extent to which curricular form enhanced or constrained

teaching for understanding.

167
The Form of the Model Lessons

Each model lesson was constructed using the same format; grade,
strand, objective, materials, vocabulary, motivation, instructional
input, guided practice, independent practice and closure. The
curricular form was influenced by research on teaching, in particular,
the work of Hunter (1976, 1984) and Good, Grouws and Ebmeier (1983).
Hunter's work has aimed at categorizing the acts of teaching and
learning -- performing task analyses, sequencing instruction,
developing a ”professional vocabulary" to describe it all -- and
identifying the "scientifically-based” cause and effect relationships
that exist between teaching and learning. The aim is to increase the
speed and efficiency of teaching and learning (Costa, 1984). To that
end, Hunter has turned a select set of findings from effective teaching
research into a recipe for "effective lesson design." Hunter's
elements of an effectively designed lesson include anticipatory set,
objective and purpose, input, modeling, checking for understanding,
guided practice and independent practice. Hunter has been the
inspiration for a number of educational entrepreneurs who market staff
development workshops modeled on her program for lesson planning,
instruction and classroom management.

The Detroit schools had invested heavily in a version called
Essential Elements of Effective Instruction (EEEI), known locally as
”triple E I." In 1983, the district committed to provide 30 hours of
inservice for all its administrative, supervisory and instructional
personnel. The program was to be conducted over a period of five
years. Initially, the inservice providers were staff development

personnel from the intermediate school district. Over the years, in-

168
house personnel were trained as trainers and they replaced the original
inservice providers. EEEI had become a powerful regulator of teacher
behavior. Not only were elements incorporated in these Model lessons,
they were also used by supervisory personnel as criteria for assessing
teacher competence.7

The second influence on the form of these model lessons was the
work of Good, Grouws and Ebmeier referred to in Chapter 2. Using
results from process/product and experimental studies, these
researchers defined a teaching model for mathematics. The elements of
an effective lesson from this perspective include beginning a lesson
with a mental arithmetic activity, devoting at least 20 minutes to the
development portion of the lesson and increasing student participation
through guided practice before assigning independent practice. The
model calls for active teaching where the teacher gives clear
explanations, monitors student understanding with product and process
questions, provides immediate and corrective feedback, and allows for
distributed and successful practice.

There is a clear compatability between the models of teaching
derived from Hunter and Good et al. The first model, grounded in
research on teaching not directly related to mathematics, prescribes a
set of generic teacher behaviors that are assumed to be applicable to
any content area. The second model aims to "improve traditional
mathematics teaching by making it more efficient or effective” (Romberg

and Carpenter, 1986). Both offer a ”technology” of instruction. Both

 

7Teachers and administrators used the term "triple E I” without
explanation-~it was a common schoolhouse term in the district.
Nearly every teacher with whom I talked could recall without
difficulty the teacher behaviors associated with EEEI.

169
are direct instruction models that aim for efficiency in teaching and
learning. Neither model is definitive about the specific content of a
lesson. The emphasis is not on what should be taught - content is
ignored or is taken for granted. The emphasis is on controlling the
pace of instruction, demonstrating what students need to learn,
attending to student understanding through questioning, and holding
learners accountable for work.

Every model lesson bore the stamp of the merging of elements from
these singular approaches to instructional design. Interviews with
mathematics specialists in the district who had worked on the model
lessons acknowledged the influence of ”EEEI' and ”Good and Grouws.”
They felt that Good and Grouws had not given much direction about the
development portion of the lesson and therefore the lesson writing team
had tried to be specific about what teachers should do during the
instructional input phase of the lesson. The instructional input of
each lesson instructed teachers how to conduct the lesson:
demonstrations and concrete materials to model an idea; problems to use
for demonstration purposes; charts and diagrams to write at the board
or overhead to facilitate instruction; questions to ask students (along
with the correct student response); activities to model and then have
students practice. The role of the teacher was to show students what
they needed to know by demonstrating to the whole class and then
providing exercises for guided and independent practice.

Teachers were to monitor student learning by posing questions
during the instructional input, guided practice and closure phases of
the lesson. Teachers were told to ask product and process questions.

Product questions required a simple number or word response (e.g., What

170
are the sums when we add across? What did we do to 73 to get 83?)
Process questions, which tended to be asked at the end of work on a set
of problems, required more. Some required the interpretation of data
(e.g., How many more/less birthdays are in May than June?). Some
allowed for a variety of responses (e.g., How are these numbers alike?
How are they different?) A few were intended to elicit a discussion
about solutions and strategies (e.g., Where would you start to solve
this problem? What did you do first? Why? Did you do it a different
way? Tell us about it?) Despite the variety in form, process
questions served mostly as another check of correctness.

Lessons ended with closure. In some lessons, teachers were
instructed to ask students to tell what they learned from the day's
lesson, occasionally sharing their learning with a neighbor. In
others, teachers were instructed to discuss solutions and strategies
with children. In a few, teachers asked students to compare the topic
of the day's lesson with a topic covered earlier. And sometimes the
teacher was instructed to review or summarize for the children.

The instructional model assumed that learners absorb what has been
taught. The emphasis was on managing instructional delivery --
maintaining a brisk pace of direct instruction, structuring information
and presenting it to students, monitoring their performance and
providing corrective feedback, keeping them on task and holding them
accountable to produce completed exercises. The model supported a view
of mathematics as ”well-organized and analyzed” knowledge and sets of
skills that ”can be presented (explained, modeled) systematically and
then practiced or applied during activities that call for student

performance that can be evaluated” (Brophy and Good, 1986; p. 130). It

171
supported a view of learning mathematical concepts and processes by
attending to teacher demonstrations and practicing what has been shown.
Questions occasionally asked children to be reflective about the
process they had used. But questions were never posed that invited
children to make conjectures, pursue further hunches or make convincing
arguments.

The Introduction to the Model Lessons described developing critical
thinking and reasoning skills and problem solving abilities as goals
that informed the lesson writing effort. The foregoing analysis
suggests there were several constraints toward achieving those goals.
One set of constraints was mathematical: objectives that focused on
narrow bits and pieces of mathematics; the superficial treatment of
topics rather than extended explorations to capitalize on the richness
of the mathematics embedded in a lesson; inconsistencies in content
among objective, motivation, instructional input and practice
activities within a lesson; a naive faith that the sequence concrete to
pictorial to abstract would automatically lead to student understanding
of mathematical ideas.

The second set of constraints resided in the management approach to
instruction embedded in the instructional model. There is a growing
body of research on mathematics teaching and learning that is calling
into question the efficacy of the direct instruction model to promote
higher-order reasoning and critical thinking (Doyle, 1983; Peterson,
1979; Peterson and Fennema, 1985). Peterson (1988) suggests that

Higher-order thinking may require a less direct instructional

approach that transfers some of the burden for teaching and

learning from the teacher to the student and promotes greater
student autonomy and independence in the teaching-learning process

(pp-S-G).

172
The model lessons and the instructional delivery system did not
facilitate a shift away from the teacher as the sole source of
epistemoligical authority. The teacher continued to be the authority
for knowing, deciding if student answers were correct and explanations

sufficient.

SUMMARY

The model lessons appeared to grow out of an orientation that moved
between certification of numeracy and comprehension of mathematical
ideas. Learning was portrayed as requiring a particular sequence of
development moving from concrete to pictorial to symbolic. Concrete
and pictorial representations were used with some consistency to model
situations and operations. In several lessons where work was primarily
at the symbolic level, i.e., writing number sentences, constructing
input/output tables, writing numbers in expanded notation, teachers
were reminded to return to the use of concrete materials if children
were having difficulty.

It was clear from the design of the lessons and the directions for
”instructional input” that the role of the teacher was to show students
what they were to learn by demonstrating to the whole class and then
providing exercises for ”guided” practice. Within a structure imposed
by the model lessons, students were expected to pursue activities,
answer teacher questions and complete assignments for independent
practice. In many instances not only were they required to provide
correct answers but also explanations for procedures used. A few
lessons incorporated games as a fun activity but these were of a fairly

routine nature. They did not require logical reasoning or mathematical

173
knowledge to develop game winning strategies. Rather, they were games
of chance adapted to include practice in basic skills.

A few lessons attempted to enhance children's abilities to reason
mathematically and to see connections between the world of mathematics
and the real-world around them. But the majority made scant effort to
develop children's higher-order thinking and problem solving abilities.
This was a function, in part, of the way lessons were conceived in
terms of the mathematical content, the learning objectives, and the
instructional activities. But it was also a function of a model for
teaching that emphasized management of instruction, that assumed
learning required breaking subject matter into discrete pieces for
students to master, and that divorced mathematics education from

inquiry.

CHAPTER 7

INSERVICE FOR.TEACHERS CHARGED WITH IMPLEMENTATION

INTRODUCTION

The previous two chapters focused on the formal curriculum
documents-~the mathematical strands that organized the curriculum, the
objectives that sequenced the content for instruction, and the model
lessons that provided examples of how objectives were to be taught in
the classroom. The analysis suggested that the reform embodied a
tension between teaching for certification of numeracy and teaching for
comprehension of mathematical ideas. The argument was made that the
potential to develop children's critical thinking and higher-order
reasoning and problem solving abilities was constrained by several
factors: scores of objectives that atomized content and focused on
narrow bits and pieces of mathematics; a pace of instruction demanded
by the need to cover over one-hundred behavioral objectives that
limited the time available to explore new and complex ideas; a direct
instruction model that embodied a management approach to instruction.

However, curriculum reform remains political and symbolic until
consideration is given to what is required for its implementation. The
reform represents a commitment to the community to improve the
education of its children. If implementation is compromised, the
reform exists only as political rhetoric and a curriculum on paper.
Since teachers are the primary agents of curriculum change in the
classroom, the ways in which they are prepared to implement new
curricula are fundamental to assessing the potential of a reform to
make a difference in the learning opportunities for children. This
chapter examines the inservice opportunities that were provided for

174

175
elementary teachers in Detroit charged with implementing the newly-
mandated mathematics curriculum.
The two elementary mathematics supervisors in the Mathematics and
Science Department planned and conducted a series of workshops to help
teachers implement the mathematics curriculum. This chapter examines

1 one of the supervisors.

the inservice conducted by Marilyn Miller,
The investigation takes at look at both the context and content of
inservice. The first section introduces Marilyn Miller by briefly
describing some of her history as a mathematics educator, how she
thought about her role as supervisor, and what ideas were central to
how she thought about inservice. The second section examines the
context of inservice in terms of the structure of the inservice
sessions and the settings of participating teachers' practice. The
third section examines the content of inservice to determine how
"teaching mathematics for conceptual understanding” was made manifest
in the enacted curriculum of inservice. Finally, the inservice
programs are analyzed as opportunities for teacher learning. Using a
framework developed by Ball and Wilcox (1989), the discussion inquires
about (1) assumptions about what teachers needed to know to implement
the curriculum; (2) assumptions about how teachers would learn what

they needed to know; and 3) the model of changing teachers' practices

that underpinned inservice.

THE PROVIDER OF INSERVICE
Marilyn Miller had worked in the Detroit schools since the mid-

1970s as a teacher, a teacher consultant and most recently as an

 

1Marilyn Miller is a pseudonym.

176
elementary mathematics supervisor. In the early 19808, she served as a
consultant to elementary teachers in several schools who were
implementing an innovative instructional program from CEMREL called the
Comprehensive School Mathematics Program (CSMP). CSMP had a number of
distinctive features: 1) it included content from probability, set
theory, relations and functions; 2) problem solving provided a context
for generating topics and developing computation skills; 3) the
curriculum was spiraled-~a topic was treated briefly for one lesson and
then returned to several days later; 4) the program provided teachers
with a set of highly detailed, scripted lessons that specified the
sequence of tasks and the questioning techniques to carry out whole
group instruction. Marilyn trained teachers using CSMP in both content
and pedagogy. Her staff development work with CSMP ended when the
district lost the resources that had funded the experimental program.

With funding for CSMP teacher consultants no longer available,
Marilyn returned to the classroom where she herself used CSMP materials
in instruction. In the mid-803, she took a leave and entered graduate
school to pursue doctoral studies in mathematics education and
curriculum. She returned to the district in January, 1987, as an
elementary mathematics supervisor. Part of her responsibilty was to
oversee mathematics instruction in over 89 elementany eehoele in the
district!

During her doctoral studies, Marilyn encountered research on
effective teaching and effective mathematics instruction. This was to
have an important influence on how she thought about her role as a
supervisor.

I did a lot of reading during the Ph.D on effective instruction.
What I don't like about Madeline Hunter is that she really doesn't

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talk much about higher-level thinking and actually have people talk
about how they are thinking. In this system you are not authentic
unless you go through the (EEEI) training so I went through the
training. It was exactly what I had read. And the criticisms I
had still stuck. Her model doesn't help you where you really need
it. It doesn't tell you how to teach the lesson, how to develop
the idea of the lesson. Neither can Good and Grouws. This is
where the supervisor's job comes in - inservice, working with
teachers, showing them the steps in how to develop a lesson.

Sometimes I feel like I'm a salesperson. I'm selling mathematics,
selling the proper teaching of mathematics. That's why I want to
do inservice. The other supervisor is having Area instructional
specialists provide inservice. I think my doing this shows more of
a bottom up rather than top down. They (teachers) need to feel
some kind of stake in change...l'm new so I need to establish my
credibility. So I try to deal with the affective as well as the
cognitive domain. I want them to know me as a person, as someone
who has this information and is willing to share.

Marilyn was deeply committed to changing the typical practice of
teachers in mathematics classrooms. She described Detroit as "ahead of
a lot of other systems." She felt the organization of the elementary
curriculum around the mathematical strands was innovative. But she
viewed the objectives and the instructional sequence as transitional
until there were textbooks that fit a more conceptual approach to
mathematics teaching and learning.

The objectives and the instructional sequence don't promote
conceptual understanding. Teachers can read these and they might
not be able to pick up the concepts. Hopefully the model lessons
provide a model in each strand. But you have to spend time on a
concept. Like in subtraction, maybe two weeks worth of lessons.
That's one of the problems with our instructional sequence. That's
why I'm not a real stickler. My vision does not fit with our
instructional sequence at all. It is a band-aid approach until we
get a new book. Then we can revisit the sequence in light of the
new text...I'm trying to model in inservice that lesson, what good
mathematics teaching is, from top to bottom. I'm trying to model a
caring kind of feeling which I think teachers should have to kids
so that I realize there are negative forces that exist, but I try
to maintain a positive attitude toward them and the kids.

Marilyn considered inservice her strength and a key element in
reforming the teaching and learning of mathematics in the district. In

the short time she had been a supervisor, she had established a

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reputation among top administrators as a dynamic, intelligent
mathematics educator. One senior administrator said he would pay

careful attention to anything she had to say.

THE CONTEXT OF INSERVICE

The Structure of the Inservice Sessions

Marilyn and her colleague planned a nine-hour workshop series to be
conducted during this first year of the implementation effort.
Inservice was organized by Area and by grade level with the following
pairings: K-l; 2-3; and 4-5. Each supervisor planned a total of nine
workshops among the three Areas for which she was responsible. As the
junior supervisor, Marilyn acquiesed to her colleague's suggestion that
the nine hours be scheduled in three sessions in a single week-oas
opposed to being offered over a more extended period of time. Although
Marilyn thought such an arrangement would not provide an optimum
environment, she had been in her position less than a year and was
reluctant to press the matter with a senior member of the department.
Each series of workshops was held after school on Monday, Tuesday and
Thursday from 4:00 - 7:00 p.m. Inservice for four of the Areas was
conducted at an educational center at a local private college.
Attendance for many teachers in these Areas required a considerable
drive from their schools. Workshops for the remaining two Areas were
held in a centrally located middle school.

Teachers self-selected to participate and were paid the contractual
stipend of $12.64 per hour. Attendance at each workshop series was
limited to roughly 45 participants. Every staff member's request to

participate was honored. While the total number of participants pre-

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registered for Marilyn's nine workshops was near 400, the number who
attended was less than 300.2 Several principals and assistant
principals also took advantage of these inservice opportunities. One
assistant principal explained her participation.

I come to these workshops to become better acquainted with the

curriculum strands and objectives. I think as an administrator I

need to be knowledgeable about the new curriculum. By being here I

show support for teachers in my building who are trying to

implement the instructional sequence. And it's a message to my
people that I think this deserves the best efforts of all of us.

At the conclusion of the first series for K-l teachers, Marilyn
expressed extreme dissatisfaction with the schedule to which she had
reluctantly agreed.

This scheduling is bad. These teachers are already tired when they

come after a full day at school. And this way doesn't give them

time to think about what we have done or try some things and then
come back and talk about it. You need more time to pound in this
conceptual development. I'm nee going to agree to this next time.

I'm tired too by the third day. She (her colleague) organized this

but she doesn't give the inservice. She gets three specialists to

do it. Well, I don't want to do that.

Marilyn was not alone in her assessment of the constraints imposed
by the schedule. At each series of workshops, attendance fell off on
the third day. Some teachers found it difficult to muster the energy
to engage thoughtfully in some of the activities. One instance called
for teachers to read two short articles on research in mathematics and
then tell a partner what they had learned. At one workshop, a teacher

rolled her eyes, stifled a yawn, and commented to a colleague at her

table,

 

2This discrepancy between pre-registration and attendance was
partly explained by the fact that two of the workshop series were held
in December in the two weeks prior to the holiday break. Teachers who
did attend reported that this was an especially "hectic" time in their
buildings given the additional activities for observing and
celebrating the holidays.

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These articles are just too heavy for this time of day. I don't
want to do this. I'm just going through the motions.

The others at her table agreed.

Some teachers who chose not to attend the workshops said they made
their decision based on when the inservice was offered. At one school
where I met informally at lunch with a group of teachers, not a single
faculty member planned to take advantage of the inservice program. As
one teacher put it,

Why are teachers expected to attend these meetings on their own

time? If this is so important why don't they provide for this

during the school day? They send teachers to EEEI training and
give them five full days of released time. This just says to me
that they don't think this is all that important.
There was an additional complication for those in this school who might
otherwise have been interested in the inservice. The school was on a
late schedule to accommodate student bussing. As a consequence,
teachers were not released from their buildings until after 3:45,
making attendance at a 4:00 workshop held elsewhere difficult, at best.

The structural features of the city-wide workshops were a
constraint on Marilyn's ability to provide a set of experiences that
engaged and challenged teachers to explore new ideas, reflect on
current practice, try new activities and arrangements in their
classrooms, and talk with others about attempts at these new efforts.
Teachers came to these workshops after a full day's work--as did

Marilyn herself. Teacher fatigue was evident in stifled yawns,

drooping eyelids and nodding heads, and, in some cases, minimum

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engagement in some of the workshop activities, especially those that

required more intellectual investment.3

The Settings of Participating Teachers' Practice

This inservice effort must be understood within a broader set of
contextual factors that had an impact on the worklife of teachers in
the district. Nearly two decades of industrial dislocation and
economic disinvestment from the city had significantly reduced the
district's ability to provide a quality education program. A universal
complaint from teachers was the lack of materials in their classrooms,
especially textbooks and paper. The new curriculum embraced the use of
calculators in the mathematics classroom, yet few teachers who attended
the workshop series had a set of calculators for their students. The
mathematics supervisor promoted the use of concrete materials to
introduce mathematical ideas, in particular, attribute blocks, base 10
materials, and geoboards. But the district did not allocate funds to

schools or individual teachers to purchase these materials. The

 

3This was in sharp contrast to the level of energy and
engagement of the math lab teachers from one Area at their quarterly
meetings. These meetings were held in the Area office and were
scheduled during the school day. Math lab teachers were given
released time from their schools to attend. The Area curriculum
director described this group of teachers as often feeling like "the
stepchildren in the area," unappreciated for their special efforts
with children, not included in professional development activities.
He had invited Marilyn to three of their meetings to demonstrate the
use of manipulatives. She had chosen attribute blocks, base 10
materials, and geoboards as the concrete materials she thought would
be most useful. During the morning sessions, teachers were immersed
in explorations with the materials, working with colleagues, asking
questions about the multiple uses of each set of materials. At the
end of the first workshop, one participant said to me, "Isn't she
(Marilyn) wonderful! She makes you feel like you can conquer the
world.” And with that she rushed down to the materials center and
made her own set of base ten materials to take back to her lab.

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conditions of teachers' work was further affected by the social
dislocation that accompanied economic disinvestment and increased
impoverishment of families. Many teachers found themselves trying to
teach a classroom of youngsters whose population was transient. Some
teachers reported that they had experienced years where only half the
class membership remained stable.

Structural changes in the economy were coupled with significant
changes in the district's organizational structure. The recent history
of decentralization, then recentralization, had left considerable
confusion about who was in charge and to whose directive teachers were
bound. While the central administration was telling teachers to follow
the sequence of instructional objectives that accompanied the new
curriculum ”in the order listed," some Area and building administrators
were ordering teachers to concentrate on a narrow set of computational
skills assessed by the various tests administered city-wide.

Teachers had been mandated to implement city-wide curricula in
eegen content areas--language arts, social studies, mathematics,
science, music, art, and health/physical education. This initiative
represented not only an attempt to equalize learning opportunities
across the district but was a response to considerable pressure to
increase student scores on state-administered tests of educational
achievement. The mandate also reflected a continuing effort by central
administrators to further reduce the authority of Area superintendents,
exert increasing control and centralization over curriculum, and
fashion curriculum in a way that supported instruction to improve test

scores .

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The response to the mandate among teachers with whom I spoke was
mixed. Some welcomed the reshaping of mathematics content around
strands as evidence that the district was moving away from a “back-to-
basics, minimum competency, teach-to-the-test' orientation. Others
felt the content had not really changed, that the strands were “just
fancy names for things we have always taught.“ Others felt some of the
new objectives were 'too sophisticated for the kinds of kids we get.“
Regardless of their opinion about the appropriateness of the new
mathematics curriculum, the majority who attended inservice felt
overwhelmed by the demand to simultaneously implement newly mandated
curricula in ell content areas. And some displayed skepticism about
the district's long-term commitment to the new curriculum. At one
inservice, a teacher questioned Marilyn:

How long are we going to do this (use the strands and objectives)?

Are we going to be doing something else in three years? I'm trying

to learn this but if I do are you just going to snatch this away in

a couple of years and then we'll have to learn something new?
Many teachers wanted assurances that their efforts to become
comfortable with the new curriculum strands and competent in following

the instructional sequence would not be met by a new and different

initiative a few years hence.

THE CONTENT OF INSERVICE
The city-wide elementary mathematics curriculum was described by
Marilyn as "conceptually-based." The meaning attached to ”teaching for
conceptual understanding” was evidenced by the ways in which Marilyn
talked about mathematical concepts and concept development, what she
believed teachers needed to know and do to teach for understanding, and

how she modeled teaching for conceptual understanding in workshops.

184
What Did Teaching for Conceptual Understanding Mean?

Marilyn, to use her own words, was “very model oriented." At every
point where she talked about ”teaching a concept,” the notion of
modeling was key.

Teaching for conceptual understanding is the most important

process. It is not unique to math but it is the groundstone of

1earning...I am very model oriented - this is the model I want to
see. The second day of the workshop we talk about concept
understanding and I model it.

Model here had three meanings. First, Marilyn believed that
teachers should introduce a mathematical idea with a manipulative that
is a concrete representation - a model - of the idea. As she put it,

Everything (in math) is a concept and you need that before other

processes. Like you need to understand numeration before addition

and before the algorithm. For every topic there is a model to

develop that concept. Take place value and regrouping. Base ten
materials are better than bundles of sticks because you can't take
one from the ten strip...Sometimes it's hard to find the model for

a particular concept but I know there is one. You need to find the

right manipulative, the right material to model a concept. Then

you need to go to the pictorial so that when kids are led to the
symbolic they have a picture in their memory bank.

Marilyn incorporated a number of concrete materials. With lower
elementary teachers, she used plastic objects for classification
activities and pattern explorations. She used ”squared materials" (a
base 10 model) to develop an understanding of addition with and without
regrouping. With upper elementary teachers she used attribute blocks
to develop an understanding of the critical attributes of geometric
shapes and as another environment for pattern explorations. She also
used base 10 materials to develop the conventional long multiplication
and division algorithms. At every inservice session, teachers used
some kind of concrete materials as a mathematical model.

A second meaning Marilyn attached to the term model referred to the

process by which she believed students learn concepts. On the second

185
day of inservice, Marilyn introduced teachers to the 'Rathmell Model
for Mathematics Concept Development.” The model specified five steps
to concept development requiring multiple opportunities for children to
l) generate the idea and give multiple examples; 2) recognize instances
of the idea -- what it is, what it is not; 3) represent the idea in
three forms -- a concrete model, oral language and written symbols; 4)
represent the idea from one form to another; and 5) learn the
properties of the specific concepts.

The third meaning of the term model refered to lesson design and a
mode of teaching the lesson. Marilyn described planning a lesson in
the following way.

The key to developing conceptual understanding is lesson

development. I want to have teachers appreciate the need for more

lesson development when they are planning to teach a concept. Look
at the concept. Know what they are getting across. What is the

model and what modeling needs to be done and what are the steps I

need to take to model a concept. What are the questions I should

ask to get at understanding. What materials do they need to
practice with...Teachers should have a single objective in mind.

It should be clear and stated to the learner in a way so they know

what they are expected to learn. The focus of the lesson should be

on that one specific piece of information.

Once a lesson had been planned, Marilyn believed there was a
particular sequence that instruction should follow. She called it ”the
model for effective math instruction.” The instructional components
were 1) mental arithmetic -- five minutes of mental computation at the
beginning of class; 2) motivation -- getting students into a
"mathematical mode," recalling previous learning and relating it to the
day's lesson; 3) lesson development/instructional input -- a minimum of
20-25 minutes where the "teasher_i§_teashins”. demonstrating with

manipulatives that model the concept or process, asking product and

process questions to check for student understanding; 4) guided

186
practice; 5) independent practice; and 7) closure -- students say what
they have learned from the day's lesson. Marilyn explained to me how
she came to this model of instruction.

Parts of it came through my reading in research on math education.

It comes a lot from those models already existing like Good and

Grouws and Madeline Hunter which I know Detroit teachers are being

inserviced in, so you have to tap into something that is already

being pushed by the system.“ Those kinds of models which research
says makes effective instruction and my own experience with kids
regarding manipulatives and the kinds of questions I know get
responses from kids.

Marilyn told workshop participants,

This is a model of instruction based on research. Research has

identified what effective teachers do that lead to increased

student achievement on standardized tests.

The ability to ”teach for conceptual understanding" did not seem to
require a sense of what makes a mathematical idea a concept. Nor did
it require distinguishing between mathematical concepts, skills and
processes. Everything tended to be a concept. Conceptual
understanding was not linked to a notion of seeing connections among
"chunks” or ”nodes” of mathematical ideas. Rather, developing
conceptual understanding was a technical and procedural matter: finding
a concrete material to model the process; motivating students to engage
in the lesson; demonstrating how to use the materials; asking two kinds
of questions -- product and process -- to check for understanding;

providing guided and independent practice; closing each lesson by

having students tell what they did.

 

“Marilyn did not mean this as a point of criticism of the
District's investment in EEEI, its version of Madeline Hunter. In
fact, she found most of the techniques of the training compatible with
her favored model from Good and Grouws. Her criticism of both was that
neither model gave teachers enough direction on exactly how to carry
out the direct instruction phase of the lesson.

187
The Curriculum of Teaching for Conceptual Understanding

Marilyn had three main objectives for the city-wide inservice.

First, I want to develop their understanding of what concept

understanding is - the use of manipulatives and hands-on activities

with lessons. I want them to become familiar with the research on

effective math instruction. And I want to introduce them to

cooperative learning, give them an opportunity to work in groups to

see how that arrangement contributes to concept understanding.
While there were variations in teacher activities among the inservices
depending on the grade level, there was a standard format that Marilyn
used to organize each day's workshop.

Dex one. The first day of the three-day inservice was devoted to
two main topics; (1) defining the mathematical strands that organized
the newly mandated curriculum and (2) introducing a specific model of
cooperative learning. Teachers created their own definitions of each
strand -- calculators and computers, estimation and approximation,
functions and relations, geometry, measurement, numeration, operations,
patterns, probability and statistics, and sets and logic --and then
compared with Marilyn's version. A subsequent activity employed a Venn
diagram. Teachers were given a handout that identified the mathematics
strands that were tested by ABCS, MEAP and CAT along with the
mathematical strands of the Detroit curriculum (DPS). Teachers were to

locate the strands for each test and the DPS curriculum within the

appropriate numbered section of the intersecting circles.

A
AVA

DPS 7

 

 

 

 

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For example, measurement was a strand in ABCS, MEAP, CAT and DPS so it
belonged in the section numbered 8. Algebra was a strand only in MEAP
so it was correctly placed in section 2. Marilyn's purpose in
selecting this activity was for teachers to realize that the new
curriculum encompassed all topics assessed on the three major tests
administered in the district. The message she wanted to convey to
teachers and administrators was that if teachers taught to the
objectives of the new curriculum, they would simultaneously and
automatically be preparing youngsters for the various tests.

Marilyn produced a handout on different goal structures --
cooperative, competitive and individual -- and then provided a specific
model of cooperative learning and the four characteristics of this
model. Marilyn told teachers,

Over 700 research studies support this structure for learning...

Cooperative learning leads to higher achievement, more intrinsic

motivation, better relationships, better attitudes toward teachers

and the school, higher self-esteem, peer group support, more on-
task behavior and better collaborative skills.
Interestingly, less than half-an-hour the first day was given to
teacher cooperative group activities.

Day Eve. The content of the second day was a mixture of various
activities. Marilyn told me before inservice the focus would be on
”How shall we learn mathematics and problem solving?” She began with a
review of the curriculum strands and the three learning goal
structures. Then she handed out a "cooperative logic activity." Each
group of four teachers was given an envelope with a bar graph and four
clue cards, one for each group member. The clue cards read, 1) there

are a total of 15 pets; 2) 6 pets are dogs; 3) there are 4 less gerbils

than dogs; and 4) there are twice the number of cats than gerbils. To

189
solve the puzzle, the group had to make a bar graph to show the number
of dogs, cats, fish and gerbils. Members were not allowed to show
their cards to each other. Solving the problem led to a discussion
about cooperative learning and problem solving strategies.

The next activity was 'jigsawing.” Marilyn provided copies of six
articles from Beeeereh_fliehin_geeeh. The articles focused on
estimation and mental arithmetic, problem solving, and manipulatives.
Each teacher read two articles and then ”taught” others at the table
what she/he learned from the research. Then Marilyn produced a
transparency that identified "five learning processes: technological
applications, algorithmic procedures, mental strategies and estimation,
problem solving and concept development." Workshop participants were
asked to make connections between these learning processes and what
they had just read in the articles on research.

As the final activity, teachers were introduced to the Rathmell
Model of Concept Development and Marilyn elaborated the meaning of each
step by demonstrating how a teacher could use the model to help a
youngster understand the meaning of the number five.

Day ghzee. The final day of inservice was devoted almost
exclusively to ”effective mathematics instruction." Marilyn spent
about 30 minutes telling teachers about effective schools and effective
teaching research. The researchers whose work she referenced were
Edmonds, Hunter, and Good and Grouws. She indicated that these
researchers had conducted their studies by observing teachers in
classrooms and recording teacher behaviors. As a result, common
teacher behaviors that correlated with increased student scores on

standardized tests had been identified. Marilyn did indicate to

190
teachers that she had a problem with using standardized tests as the
sole measure of teaching effectiveness.

I have a problem with scores on standardized tests. I'm not sure
standardized tests are the only or the best tool.

While she questioned the measure of teaching effectiveness in the
research to which she referred, she embraced the set of "effective"
teacher behaviors that derived from it.

After introducing the subject of this body of research, she gave
teachers a "Quiz on Effective Math Teaching” (see Appendix). Marilyn
gave the correct answers and entertained discussion about each of the
23 items. Then she made a list of the common teacher behaviors
associated with effective mathematics instruction.

- assess prerequisite skills

- state the objective

- warm-up - motivation - anticipatory set

- develop concept/idea - discussion, questions

- time-on-task

- systematic review and maintenance

- high teacher expectations

- comprehension/understanding

- practice - multiple ways besides paper—and-pencil

- informative feedback

- independent practice

This listing led to a discussion of what teachers needed to
consider when planning a lesson. Teachers offered the following:
objective; materials; procedure; time-line; evaluation; why are you
teaching something; practice; and review. Marilyn accepted these as
important considerations. Then she identified the components of an
effective mathematics lesson and demonstrated by teaching a model
lesson.

A considerable portion of inservice was spent acquainting teachers

with research. Marilyn had found research to be a powerful influence

on her own thinking about teaching and learning. She believed the new

191
curriculum was based on a solid foundation of research and she wanted
to communicate that to teachers. In one of our many conversations, she
told me,
Some teachers tell me they don't like the curriculum. Well, they
don't know the research. Other teachers are going back and saying
this is really worthwhile. We're not just making these decisions
out of the clear blue because we want to mandate. There is
something based on solid ground as to why these decisions are being
made - based on research, experience and a true understanding of
how children learn. Educators are more involved in collecting that
research and it is becoming more believable and being used more in
schools.

This matter of the presentation of research findings will be taken up

again later in the chapter.

What Did Teaching For Conceptual Understanding Look Like?

Marilyn considered it essential that she model effective
instruction by demonstrating with a model lesson. She had been
critical of the shortcomings of the sets of model lessons distributed
with the strands and objectives. She felt the lessons did not give
adequate attention to the development/instructional input phase of a
lesson or the kinds of process questions that would "get kids to
think.” She also was interested in demonstrating how a teacher should
begin at the concrete level, procede to the pictorial representation,
and then move to the symbolic level. This section describes three
model lessons that Marilyn taught. These lessons will be examined for
their potential to help teachers think about how to teach for
understanding and their potential to develop children's higher-order
thinking and problem solving abilities.

A ieseon on numezetien. The objective of the model lesson Marilyn

demonstrated for K-l teachers was, "Given a place value model (base 10

192

materials), the learner will group ten ones as one ten to rename a
number."5 She began with a mental arithmetic activity called "You are
my ten friend.” Marilyn stated a number less than ten and then called
upon a teacher to give the number that together with hers would make
ten. She told teachers they should do this with children until each
one had a chance to respond.

She then introduced the lesson with the motivation.

Suppose you wanted to make a phone call. You reached in your
pocket and pulled out thirteen pennies. What would you do?

The teacher responses varied. Marilyn pressed until one teacher gave
her the answer she was looking for: ”I'd see if I could trade ten
pennies for a dime." Marilyn continued;

Today we are going to use the same idea and practice trading ten
ones for one ten.

Each pair of teachers was given a zip-lock bag with a set of
plastic materials; one large orange square, 16 orange strips and 20 or
more small white squares.6 Marilyn held up each shape and identified
it.

We are going to call this (a small white square) a unit. This (the
orange strip) we will call a long. This one (the large orange
square) is going to be called a flat. This is a universal model
and we use this language because you may want to use these
materials later with decimals.

 

5This model lesson appeared in a packet of 20 model lessons
entitled ”Success Strategies for Mathematics.” These lessons were
designed to be taught to fourth graders during the month of September
in preparation for the state-wide MEAP test given in October. The
introduction to the set of lessons stated ”In developing the lessons,
attention was given to research on how students learn and understand
mathematical concepts and algorithms. A variety of methods, questions
and concrete materials are needed to provide the learner with a rich
conceptual background, as well as, provide for the various learning
styles of the students.”

6An orange strip was equivalent to ten small white squares. The
large orange square was equivalent to 10 orange strips.

193
She demonstrated the use of product questions to get at student
understanding.
Show me with your materials how many units it takes to make a long.
Show on your fingers how many units it takes to make a long. Show
me with your materials how many longs it takes to make a flat. Now
I want you to figure out how many units it takes to make a flat.
When teachers were ready to answer the last question she posed, she
added:
I want you not only to tell me how many units it takes to make a
flat but how you figured it out. This is a process question,
asking kids how they got an answer, not just what their answer is.
The lesson continued with Marilyn demonstrating the Bankers Game.
As she spun a spinner, she set out the number of units corresponding to
the number pointed to on the spinner. She told teachers the rule of
the game: whenever a player has ten units, those units must be traded
for a long. She continued the play, first spinning a 6, then a 5.
With the second spin she asked:
How many units do we have now? Do we have enough for a trade. How
many units do we need to trade? How many units are left? How many
longs do we have?
For the guided practice portion of the lesson, teachers duplicated her
moves at their places, raising their hands to indicate when a trade was
necessary. For independent practice, teachers played the Bankers Game
(teachers used a die rather than a spinner) until one player had five
longs.

After all groups had completed the game, Marilyn said:

Every lesson needs to end with closure. You need to ask children
to tell you in their own words what they did. Tell me what we did.

Teachers answered almost in unison, ”We traded ten units for one long."
The entire lesson was conducted with concrete materials only. As

Marilyn told the participants:

194

We did this whole lesson without writing a single abstract symbol.

This kind of work is necessary for the addition algorithm. You can

also see that there is no reason to teach regrouping separate from

non-regrouping.

A_1eeeen_en_enere§iene. For upper elementary teachers, Marilyn
worked with base ten materials in a somewhat different context. Rather
than teach a model lesson devoted to a single objective, she
demonstrated how these materials could be used to model and teach the
four conventional algorithms of arithmetic. As she manipulated the
materials for addition and subtraction, making trades, she wrote the

problem symbolically saying,

Writing this with numbers is simply making a record of what we have
done with the concrete materials. This is what the algorithm
means. You're showing them what this process means. Many times
kids don't know what this means. They've heard the rules. Now
they know why the regrouping actually took place... Another thing
you could do is make a record with pictures. A large square could
represent a hundred, a bar could represent a ten, and a small
square could represent a one. So if they don't have the concrete
materials, they can draw representations of the problem and show
the trading.

Marilyn deliberately manipulated the arrangement of the base ten
materials to exactly replicate the procedural steps in long
multiplication and long division. To model "seventeen times fourteen,"
teachers were instructed to start with ”ten times ten" (which they
represented with a hundreds square) and then ”build up and out,"
creating the array shown below. Then they were instructed to draw a

horizontal line above the hundred square.

 

 

 

 

195
As Marilyn wrote a record of what they had done, she told them,
Above the line we have seven rows with 14 in each row. We have
seven tens and we have seven rows of four. We can trade twenty of
the ones for two tens (she drew circles grouping two sets of ten
dots) and have eight ones left. So seven fours is twenty eight, we
trade for two tens, write the eight and carry the two which is
combined with the seven other tens giving us nine tens. Below the
line we have ten rows with fourteen in each row, ten times fourteen
is one-hundred forty.

This is what she wrote at the overhead.

14
98

;a_0
238

Teachers seemed very confused about how to arrange the concrete
materials, uncertain which number represented the number of rows and
which told the number of elements in each row. Noticing their
confusion but aware that little workshop time remained, she said:

You're going to have to practice this before going to kids. Be

sure and plan the enae; problems you are going to use with them.

You can see I have my cards. I always know exactly what problems

I'm going to do whether it's with teachers or kids.

And one teacher responded loudly, "I guess so!!!"

These two lessons exemplified a key element in Marilyn's approach
to teaching for understanding-~introducing new ideas at the concrete
level. Beginning with the concrete meant manipulating physical
representations of a mathematical idea or procedure. Marilyn was
deeply concerned about developing youngsters' understandings of place

value, numeration and operations. She emphasized providing a physical

196
model and manipulating it so there was a clear map from the model to
the paper-and-pencil algorithm.

Although teachers, as well as Marilyn, lapsed into using the
language of ones and tens rather than units and longs in the first
lesson, there was no discussion about how or when to introduce this
language with young children. Nor was there discussion about how to
connect the concrete representation embodied by the base ten materials
with the symbolic representation of numerals. Marilyn expressed
concern on several occasions that early elementary teachers were in a
rush to get to the symbolic. She wanted to show them that they could
develop children's understanding of place value and addition without
writing a single number. Teachers physically made trades of ones for
tens and tens for ones. They appeared to enjoy using the base 10
materials and talked among themselves about how they could use these
with young children. They seemed to be persuaded of the effectiveness
of using concrete materials to develop competency with addition
problems that involved regrouping.

The teachers who participated in the second lesson also enjoyed
using the materials to model addition and subtraction with regrouping.
They were shown how to move from manipulating physical materials to
making a picture of them. And then they were shown how to ”make a
written record” of what they had done with the physical model. But
there was considerable confusion and skepticism about the use of the
materials for long multiplication.

This set of lessons raises a question about the choice of
mathematical goals embedded in lessons. Numeration is a fundamental

mathematical concept and developing an understanding of place value is

197
a significant goal of instruction. Similarly, the four operations are
fundamental processes and developing an understanding of and
proficiency with them is a significant goal. In addition, developing
an understanding of operating on large numbers and using physical
representations to generalize about computational processes is a worthy
goal of instruction. But is mastery of the conventional long
multiplication algorithm a significant goal?

The second lesson as conducted was not likely to develop higher-
order reasoning and problem solving abilities, although such potential
existed. For example, there were many ways in which teachers could
have shown 14 x 17 with the concrete materials. Various arrangements
could have led to different ways to "record the answer” and perhaps
alternative algorithms. The objective was not "opening the way to
invention"7 of algorithms or abstractions. Marilyn demonstrated how to
manipulate base 10 materials to exactly genlieeee the paper-and-pencil
steps in the long multiplication algorithm. The procedural steps of
the conventional algorithm were the ends of instruction and the
physical materials were manipulated to that end.

A lesson from measnrement. This lesson was taught to upper
elementary teachers. The topic was rectangles and areas of rectangles.
Marilyn began with a mental arithmetic activity on estimation where
teachers judged the reasonableness of answers to a set of computational
exercises. Then came the lesson objective.

My objective, and again I would tell this to kids, the purpose of
this lesson is to be able to identify a rectangle as shapes with

 

7This is a phrase Glenda Lappan used in a conversation about the
importance of symbolic development and our shared belief that getting
to the ”abstraction" is necessary for completeness. But that
requires that the abstraction itself be significant.

198

four square corners and four straight sides and find many
rectangles with the same area.

The lesson took the following course. As the motivation for the
lesson, Marilyn showed the teachers how to fold a sheet of paper to
make a square corner and then they used it to find examples of square
corners around the room. Then she presented at the overhead the

figures drawn below and asked which were rectangles and why.

[EL—’— 0

 

 

 

 

 

 

 

 

 

 

 

 

 

When she pointed to D she realized she had forgotten the additional
condition that a rectangle is a closed figure and added that condition.
Next she asked teachers to tell which ones had at least two square
corners. When there was uncertainty, she had them check it out with
their square corners. Then she proceded to the area of rectangles.

She distributed a sheet of centimeter grid paper with one square
darkened in. She explained that this square was one centimeter on a
side and ”so we say that this square has an area of one square
centimeter." She told the teachers to draw on their grid paper a
rectangle with an area of 12 square centimeters and said,

That means it is going to be twelve of these squares. After you

have drawn that, I want you to draw as many rectangles as you can
that have an area of 12 square centimeters.

199
One teacher asked, ”Do we count it laying down and standing up as the
same thing?" Marilyn responded with:

Good question, Cynthia. Sure, for right now it is the same thing.
After teachers had a chance to make all the possible rectangles they
could, Marilyn indicated that with students she would have them come to
the overhead and draw their various rectangles. She then proceeded to
draw rectangles that teachers suggested: 1 x 12, 2 x 6, and 3 x 4. One
teacher suggested 24 x 1/2 which Marilyn accepted and said:

That's very good. And some kids might come up with that. That is
very good thinking."

After the first three rectangles had been drawn, Marilyn asked the

teachers:

Does anybody know a multiplication sentence that we can give to
each rectangle?

One teacher responded with "An array, one times twelve." Marilyn

repeated the teacher.

That's right. One times twelve. Now I would have taught
understanding of multiplication already using that so they would
have had this knowledge to be able to say that this is one times
twelve. And this is two times six (pointing to the 2 by 6
rectangle and writing ”2 x 6” at the overhead). And this is three
times four (pointing to the 3 by 4 rectangle and writing '3 x 4”).

Next Marilyn handed out another sheet of grid paper on which was
drawn a 1 x 12 rectangle and a 12 x l rectangle, each with the lower
left corner at (0,0) and said

You will see that two of those rectangles have already been
graphed. We're going to graph the rest of them and we're going to
see something happen. I want you to graph the other ones that we
saw, and what they would be if you turned them, and you're always
going to start with the left-hand corner of your shape at zero.
Then put a dot on the upper right-hand corner of each rectangle.

Marilyn drew the various rectangles at the overhead, including the 1/2

by 24, as teachers worked at their tables.

200

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T

 

Figure 7.1 Rectangles With Area 12

 

 

 

201

Okay, now we're going to draw, very carefully we're going to
connect those points. What happens? (See Figure 7.1).

Several teacher answered, "We got a curve."

And Marilyn replied,
Okay, we made a curve. And I'll tell you that any rectangle that
has an area of 12 square centimeters, its right hand corner will be
on that curve that we just drew. And that is where I would end the
lesson for the kids.

But then she proceeded,
Okay, so does anybody know what this curve is called. (Short
pause, no teacher offers a guess). This is called a hyperbolic
curve. But, when you know about the area. This is a graph of this
function (writes xy) length times width, x times y. Whatever that
would be would give me a graph of that. So when you graph x times
y you're going to get this kind of curve. Now we're not talking
about this with kids, but we're giving kids experience starting to
see some of the patterns so that when they get to that higher level
mathematics in high school they say, ”I remember that. We drew a
picture of that.” And there's a lot of calculus when you start
talking about the area of the curve and all kinds of other stuff
built into this kind of an idea. So, I'd like you to tell your
neighbor what a rectangle is.

In unison the teachers told each other that a rectangle is a closed

shape with four square corners and four straight sides. At this point

the model lesson was concluded.

This lesson was considerably different from a traditional textbook
approach to area of a rectangle where youngsters are given the formula,
A - l x w, and a set of problems for calculation. In fact, the formula
was never introduced. Instead, teachers were asked to draw as many
rectanges as possible with an area of 12 squares on a piece of
centimeter grid paper. Part of Marilyn's intention was to have
teachers see that area involves covering a surface and counting the
number of units, in this case centimeter squares, that cover it. But

she went beyond a simple draw and count exercise. She had teachers

202
explore all possible rectangles with a given area. This opened the
activity to consideration of rectangles whose sides were not whole
numbers. She connected the calculation of the number of squares
covering a rectangle with prior learning--the array representation of
multiplication. And she used a graphical representation to show all
the rectangles with an area of 12 square centimeters. The lesson
demonstrated an approach to teaching youngsters about rectangles and
area of rectangles that was a considerable improvement over typical
“plug in the formula, grind out the answer" approaches. At the same
time, the lesson could be described as a missed opportunity.

This lesson was rich in mathematical ideas that were not explored.
Time was a factor. The lesson was taught at the end of the last day of
inservice and time was running short. Teachers were tired after a long
week and so was Marilyn. The context of inservice -- the limited
amount of time, offered after a full day's work, nine hours compressed
into three after school sessions in one week -- was a constraint on
opportunities to pursue a topic over a more extended period of time.

At the same time, the missed opportunity reflected a more
fundamental issue -- defining objectives for instruction and goals for
student learning. The assumption underpinning lesson development, as
evidenced in the objectives for the Model Lessons and those that
Marilyn taught at inservice, was that a lesson should focus on, to use
Marilyn's words, ”one specific piece of information.” This orientation
to defining instructional objectives seemed unlikely to create in
teachers a disposition to look for connections among mathematical ideas

or help children to see those connections.

203

Consider some of the linkages that could have been made to other
important ideas by posing additional questions for investigation. ”How
can we be sure we have found all the possible rectangles with a given
area if the sides are whole numbers?“ Such a question would invite
learners to consider ways to systematically account for all
possibilities, an important problem solving strategy. It could also
provide another context in which to think about factors of whole
numbers. ”What happens to the perimeter of these rectangles as we
change the lengths of the sides?" 'Is it possible to find a rectangle
that has the smallest perimeter?" ”Is it possible to find a rectangle
that has the largest perimeter?" These questions would invite learners
to look for patterns, to make some guesses and pursue some hunches.
Learners could explore the idea of limits at an intuitive level. A
complimentary problem could be posed to investigate how area changes
when perimeter is held constant.

The ideas in this lesson could have been explored by locating them
in a problem situation connected to the real-world of youngsters. Ball
(1988) describes a problem she has used in her third grade classroom to
explore a range of ideas about area and perimeter.

Suppose you had 64 meters of fence with which you were going to

build a pen for your large dog, Bozo. What are some different pens

you can make if you use all the fence? Which is the pen with the
nee; play space for Bozo? Which pen allows him the leeee play

space (p. 4)?

Problems of this kind suggest to children that there is a connection
between school mathematics and their world outside the classroom.

It was not at all clear if or how‘teachers made sense of the curve

they constructed by connecting the upper right corners of the

rectangles. Marilyn did not ask them to make conjectures about what

204

they thought the curve represented. Instead she told them what it
represented. It was not clear how Marilyn's brief ”lecture" about
hyperbolic curves and their algebraic representation contributed to
teacher understanding. Up until teachers were told to connect the
corners of their rectangles, there was spirited conversation at the
tables. But when Marilyn mentioned "a hyperbolic curve,” "function”
and ”calculus,” teachers looked at each other, some shook their heads,
and there was not another question asked or comment made.

But the objective had been accomplished - teachers were able to

tell what a rectangle is and find several with the same area.

INSERVICE AS TEACHER EDUCATION
This section examines the inservice programs as opportunities for
teacher learning. The analysis uses three frames to examine inservice
as teacher education: 1) assumptions about what teachers needed to know
to implement the curriculum; 2) assumptions about how teachers would
learn what they needed to know; and 3) the underlying model of changing

teachers' practices.

Assumptions About What Teachers Needed To Know

Marilyn believed that teachers lacked essential research knowledge
about effective mathematics teaching and the pedagogical skills derived
from that research. Her goal was two-fold: 1) to equip teachers with
that knowledge by introducing them to a body of research on "effective
mathematics teaching," and 2) to demonstrate how to ”teach for

understanding” by teaching a model lesson. For Marilyn, the essence of

205
conceptual understanding was captured in the model of ”effective
mathematics instruction."

Learning was portrayed as requiring a specific sequence of
development; first manipulating concrete models, linking concrete
models to pictorial representations, and then using pictorial
representations as the bridge to the abstract. The use of base 10
materials suggested that correctly arranging the materials in the
specified configuration, moving to drawings with squares, lines and
dots, and then ”making a record of what had been done” automatically
brought conceptual understanding to procedural routines. However,
there is growing research evidence that making meaningful connections
among these representations is not automatic. At the concrete level,
the ability to manipulate materials may lie not in children's
conceptual understanding but in the structure of the materials
themselves and the ease with which they can be manipulated to get right
answers. Moreover, even when the physical materials are manipulated to
exactly map the steps of paper-and-pencil algorithms, linking
conceptual knowledge with procedures has not been consistently
effective (see, for example, Carpenter, 1986; Driscoll, 1981; Resnick,
1982). As with the sets of model lessons that had been given teachers,
there was an assumption that if teachers began instruction with
concrete materials then student understanding would be automatic. But
as Hiebert (1980) has pointed out,

...if concrete materials are going to be useful, frequent, explicit

links must be made between the physical and symbolic

representation...lt is not just the use of concrete materials that
improves mathematical understanding, but rather the explicit

construction of links between understood actions on the objects and
the related symbol procedures (p. 509).

206
Virtually untouched in the inservice was teacher subject matter
knowledge. Although Marilyn acknowledged that many teachers, by their
own admission, were uncomfortable with the content of some of the
strands, attending to this teacher concern was not an objective.
Instructional techniques and technical competence were at the heart of

inservice.

Assumptions About How Teachers Would Learn.What They Needed To Know

Marilyn used the program's content as pedagogy. She modeled in
inservice what she expected teachers to do in their classrooms: tell
students what they needed to know, demonstrate with concrete materials,
give opportunities to practice what they had been shown, ask product
and process questions to assess understanding, and have students tell
what they learned. At the conclusion of the series of workshops,
Marilyn wondered about how successful she had been.

I'm not sure I help them make the connection between the

objective, the concept in the objective and the manipulative to

model it. The way you write your objective you would try to
represent that concept. I'm not sure I help them make that
connection. They may walk away with a manipulative but maybe not
an understanding of the concept that it models.

Having teachers teach each other was another pedagogical tool.
Marilyn provided six articles on research in mathematics education and
had triads of teachers at a table read two of the articles and then
teach the other groups. In several of our conversations Marilyn spoke
of the importance of having small group interaction, where learners
(whether teachers or students) could "bounce ideas off one another."

Marilyn believed that change took place over time. But the limited

resources, financial and human, available to her precluded an extended

inservice program. The organization of three after-school sessions in

207
a single week produced increasing levels of disengagement among many
participants by the last session. The time for beginning a workshop
overlapped dismissal time at several schools thereby limiting
attendance for those teachers who might otherwise have been interested.
The distance teachers had to drive between individual schools and the
inservice site reduced potential participation. In this inservice, the
context itself was a considerable constraint on teachers' participation
and opportunities for teacher learning.
Marilyn told teachers she would be happy to respond to individual
requests for classroom visits, school-site workshops, and resource
materials. Inservice provided the techniques. What teachers needed
was time in their classrooms to practice delivering instruction in the

mode prescribed.

The Underlying Model of Changing Teachers' Practice

The inservice embodied a technical rationality. Teachers were
presented with "systematic knowledge,” -- specialized, scientific and
standardized -- that ”ignore(d) complexity, uncertainty, instability,
uniqueness and value-conflicts” (Schon, 1983; p. 39). The
instructional model, the research out of which it grew, and the
practices it prescribed were presented as objective facts rather than
as something to be questioned, analyzed or negotiated. Teachers were
not told that there is considerable disagreement about the suitability
of the direct instruction model to teach for conceptual understanding
and higher-order thinking. They were not told that realizing the
potential of manipulatives to increase student learning has been shown

to be exceedingly complex and far from automatic. What was portrayed

208
was a consensus model of research. The dichotomy between the expert --
the researcher and the supervisor -- and the practitioner -- the
classroom teacher -- was maintained.
In three days of inservice, teachers were taught about the

10 strands of the K-5 mathematics curriculum.

goal structures for learning in the classroom.
elements of cooperative learning.

rules for ”groups of four” in cooperative learning.
outcomes of cooperative learning.

mathematical learning processes.

steps to concept understanding.

characteristics of a problem situation.

steps in problem solving.

l6 problem solving strategies.

11 behaviors that characterize effective teachers.

11 components to planning and delivering an effective
mathematics lesson.

*fifl-fl-fl-fi’bfl-fifl-fl-X-
bbmu‘owaw

Teachers were "inserviced" - a term that Marilyn herself used to
describe her efforts - on a model for grouping children for
instruction, a model specifying a sequence of steps to develop an
understanding of mathematical concepts, a model for problem solving, a
model for planning and teaching an effective lesson. The emphasis was
on ”methodological refinement," giving teachers models and sets of
instructional procedures to effectively and efficiently implement a
newly mandated, pre-determined curriculum. Aronowitz and Giroux have
called this management pedagogy, "forms of pedagogy that routinize and
standardize classroom instruction," and that reduce teaching to "lock-
step" behaviors (1985, p. 28).

Underlying the orientation of this inservice program was what
Zeichner has called "a view of teaching as an 'applied science' and a
view of the teacher as primarily an 'executor' of the laws and
principals of effective teaching" (1983, p. 4). The topics that

constituted the curriculum of inservice suggested that it is possible

209

and desirable to quantify effective teaching practice. As Marilyn

said, ”I'm selling the ngene; teaching of mathematics."8

SUMMARY
Inservice focused on providing teachers with a set of "research-
based” instructional strategies and techniques. As Ball & Wilcox
(1989) have argued, there is a logic in selecting a specific focus for
inservice.

Choosing a focus for inservice makes sense; it suggests a strategy
of engaging teachers in changing aspects of their practice by
targeting a single aspect or ingredient of their teaching. A
broader sweep, especially under conditions of limited time, seems
likely to do little more than ripple gently the waters of practice.
Yet choosing a singular focus has its drawbacks, for the
interactions among the threads of teaching are not well-understood.
Focusing on techniques of teaching without, for instance, engaging
teachers in considering their assumptions about learning, may prove
a futile intervention (p. 34).

Take the use of manipulatives as an example. When teachers encounter
this as a research-based pedagogical strategy, but do not reexamine

their view that operating at the symbolic level is "not only preferable

 

8Teachers who attended the workshops completed a ”Workshop
Feedback” at the end of the session. I collected data from 109 of
these forms. Teachers indicated they had a better understanding of the
mathematics strands and how they were related as a result of the
workshop. The components of the workshop they mentioned as liking best
were 1) the use of manipulatives -- 351 of the teachers mentioned this
specifically, 2) cooperative learning/small group work, and 3) sharing
thoughts, ideas with other teachers. Teacher suggestions included
having more materials available -- more manipulatives, model lessons
for specific objectives on basic facts and regrouping, lessons relating
to the tests, more "make-and-take." They also commented on the desire
for more inservice on instructional strategies such as cooperative
learning and motivating activities. And they voiced their concern
about time - the limited time available for inservice and the
inconvenient time at which the workshops were scheduled. There was
overwhelming praise for Marilyn. These comments were typical: ”She
makes me feel motivated to change how I teach;” "The instructor knew
her material and was enthusiastic'" "...the professional treatment of
the presenter, the equalitarian mode.”

210
but also a sign of understanding” (Schram et al., 1989), their use of
manipulatives in instruction may be superficial. If teachers are
encouraged to use manipulatives without attention to what are
appropriate models and what mathematics are embedded in the physical
material, the use of concrete materials may contribute to
misunderstanding and misconceptions. Without explicitly considering
the connection between mathematical ideas and appropriate concrete
representations, manipulatives have the potential to be merely ”fun
things to do” to break the routine/monotony in the mathematics
classroom. Or they are made available as "crutches," to be abandoned
in favor of symbols as soon as possible.

The choice of focus for this inservice made sense in light of the
reform "writ small.” As I have argued in earlier chapters, the reform
appeared to grow out of an orientation aimed at teaching for
comprehension of mathematical ideas. The curriculum for students and
the curriculum of inservice emphasized management of instruction,
assumed learning required breaking knowledge -- mathematical and
instructional -- into discrete pieces to be mastered, and divorced
mathematics education from inquiry.

But when considering the reform ”writ large," the inservice did not
make sense. Simply improving teachers' instructional technique will
not provide the knowledge -- about mathematics or mathematics teaching
and learning -- that is required if Detroit children are to have
opportunities to develop higher-order reasoning, critical thinking and
problem solving abilities. These youngsters need environments in which
they can engage in mathematical inquiry. And creating such

environments is not just a matter of perfecting an algorithmic approach

211
to mathematics teaching. Teachers need a deeper knowledge of
mathematical ideas beyond arithmetic. They need opportunities
themselves to experience I'doing" mathematics, to see mathematics as the
creation of human activity. Without a richer understanding of subject
matter, their efforts to guide student inquiry -- listening to
students, posing questions for further reflection, responding to
questions by deciding which to pursue and which to defer to a later
time -- will be meager. Like the model lessons that accompanied the
formal curriculum documents, inservice provided minimal help to
teachers about how to develop children's higher-order thinking and
problem solving abilities. Effective mathematics teaching and learning
was tied to technical competence and efficient management of
instruction.

The matter of the constraints imposed by the context of inservice
was serious and contributed to the limitations of these workshops as
opportunities for teacher education. At the same time, the technical
orientation that dominated the inservice leaves me to wonder how and to
what extent inservice might have been different had contextual
constraints not been so dominant. In the final chapter, I return to a
further look at the context in which this curriculum reform was

embedded.

CHAPTER 8

THE GENEALOGY OP‘THE CURRICUUUM REFORM

INTRODUCTION

This final chapter brings me full circle, back to the context in
which this curriculum reform was conceived. In the preceeding three
chapters, I have argued that despite a rhetoric of conceptual
understanding, developing higher-order analytic and critical thinking
skills, and applying knowledge to problem solving, this curriculum
reform was likely to fall far short of providing young students in
Detroit with the mathematical tools they will need as citizens in the
21st century.

One interpretation that could be made is that school people were
saying one thing but doing another. The problem with this
interpretation is that it is too simplistic. Further, it casts doubt
on the intentions of reformers to meet the real needs of Detroit
children. To draw this conclusion would be both a mistake and a
disservice to the dedicated professionals in this study who have
committed many years to improving Detroit's schools. The limitations
of this reform do not, I think, represent personal shortcomings or
failures. Instead, I will argue that the conception of this reform
grew out of the efforts of well-intentioned people who were doing the
best they could given a particular context, the dominance of a

particular ideology and a particular moment in history.

212

213
A.GENEALOGICAL.ACCOUNT
Introduction

Noujain (1987), drawing on the historical and philosophical works
of Michel Foucault, argues that "history possesses a genealogical
character” (p. 158). The past is a series of intersecting,
discontinuous elements that are linked like a chain. The elements may
be any identifiable entity--events, ideas, practices, people. In
undertaking a genealogical analysis of a particular object, the
historian looks for the elements that preceeded the object and out of
which it emerged. Noujain stresses that the relationship between
elements is not a causal relation but is a relation of succession.
However, he goes on to say that once having established the succession,
the historian may proceed to investigate causal relations, but then the
historian turns into a "social scientist of sorts" (p. 160).

In this final chapter, I assume the role of historian and critical
social scientist. I provide a genealogical account of the Detroit
mathematics curriculum reform to show how it can be understood»when
placed in a broader historical context.

Providing a genealogical account of the Detroit curriculum reform
has required searching for the antecedents of the reform, those
components of events, ideas, institutions, practices, and persons that
intersected and combined to form it. Figure 8.1 gives the schematized
genealogy. Earlier chapters have been devoted to some of these
elements in detail, in particular the politics of race and school
governance in Detroit, the reform agendas for mathematics education,
and the recognition of a new economic order and Detroit's position in

it. What has received less explicit attention is the instrumental

214

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215
ideology that has dominated American education and the field of
curriculum.

I begin this chapter with a discussion of the domination of an
instrumental ideology on the field of curriculum development and its
extension to other aspects of educational life, for teachers and
students. From that, I show the influence of instrumentalism on the
Detroit reform initiative. I then revisit the other antecedents of the
reform to show the linkages among events, ideas, practices and people
that shaped this initiative.

I conclude this study with a look at a new challenge to the
dominance of instrumentalism in mathematics education and I suggest
changes that would need to take place for Detroit to transform its

curriculum toward teaching for enabling mathematical inquiry.

Instrumental Ideology

o inati n enta eo . Giroux,
in his essay "Literacy, ideology and the politics of schooling,"
identifies a number of assumptions that define an instrumental
ideology.

The major premises of instrumental ideology are drawn from the
logic and method in inquiry associated with the natural sciences,
especially the principles of prediction, efficiency, and technical
control....Central to the logic of instrumental ideology and its
view of theory is the notion that all social relations shall be
subject to quantification....Knowledge in this view is seen as
objective, outside of the existence of the knower...neatly severed
from the world of values....Since knowledge in this perspective is
valued for its utility and practical application, there is little
room for questions concerning the ethical nature and consequences
of the use of knowledge (1983, pp. 209-210).

Giroux goes on to argue that an instrumental ideology has dominated

American education in general and the curriculum field in particular.

216

Instrumental ideology, in both historical and contemporary terms,
has had a powerful and pervasive influence on American education.
Curriculum theory and practice as well as more specific pedagogies
of literacy have been largely structured by the values and
assumptions inherent in instrumental ideology....The strength and
pervasiveness of the logic of instrumental ideology in the
curriculum field can be seen historically in the early work of
educators like Franklin Bobbitt (1918), W.W. Charters (1923), D.S.
Sneeden (1921), and others....That this view is deeply ingrained in
the ideology of schooling can be seen in the powerful support given
by educators to competency-based systems of instruction,
behaviorist models of pedagogy, and the various versions of systems
theory approaches to curriculum theory and policy development (p.
211).

The birth of the field of curriculum is generally associated with
the publication of Bobbitt's Ihe_Qn;1ienlnn in 1918. It was initially
and powerfully dominated by administrative imperatives to organize and
manage time and activities according to sound business principles.
Almost from the beginning, the curriculum field was influenced by the
principles of scientific management promoted by Frederick Winslow
Taylor. The basics of scientific management -- efficiency, control,
prediction and technical expertise -- were appropriated by those
responsible for curricular matters. Kliebard (1975) quotes Elwood
Cubberley, an early advocate of the adoption of the tenets of
scientific management to school management and curriculum development.

Every manufacturing establishment that turns out a standard product
or a series of products of any kind maintains a force of efficiency
experts to study methods of procedure and to measure and test the
output of its works. Such men ultimately bring the manufacturing
establishment large returns by introducing improvements in process
and procedures and in training the workmen to produce larger and
better output. Our schools are, in a sense, factories in which the
raw products (children) are to be shaped and fashioned into
products to meet the various demands of life. The specifications
for manufacturing come from the demands of twentieth-century
civilization, and it is the business of the school to build its
pupils according to the specifications laid down. This demands
good tools, specialized machinery, continous measurement of
production to see if it works according to specifications, the
elimination of waste in manufacture, and a large variety in the
output (p. 32).

217

The ideology and practices of scientific management were highly
compatible with behaviorist psychology. The efficient management of
curriculum was bolstered by defining the goals of learning with
behavioral objectives and observable and measureable competencies.
Curriculum development was a practical matter, a systematic attempt at
guiding learning in schools. Tyler's (1949) four questions were the
quintessence of the dominant tradition.

(I) What educational purposes should the school seek to attain?

(2) How can learning experiences be selected that are likely to
be useful in attaining these objectives?

(3) How can learning experiences be organized for effective
instruction?

(4) How can the effectiveness of learning experiences be
evaluated?

The traditionalists,1 who have dominated the field, have been overly
concerned with practical considerations and classroom applications at
the expense of curriculum theory and research. Service to
practitioners, guided by principles, has been more important in
directing curriculum development and implementation than theory. The
Tyler legacy lives on in later curriculum texts by Taba (1962), Saylor
and Alexander (1966), Popham (1969), Tanner and Tanner (1980), Zais
(1976) and Glatthorn (1984).

During the curriculum reform movements of the 1960s, the
atheoretical, unscientific traditional approaches to curriculum
development were challenged by a new breed of curriculum worker,
conceptual-empricists. Heavily influenced by mainstream social science
and behavioral science, they sought more valid principles for

curriculum decisions based on theoretical development and research.

 

1The terms traditionalists, conceptual-empiricists and
reconceptualists are used by Pinar (1978) and Giroux et al., (1981).

. w.¢-—-—-

218
But as Pinar (1978) and Giroux et al. (1981) have argued, ultimately
their differences with traditionalists were less a matter of kind than
degree. Their efforts resulted in research-based prescriptions for
ways to structure the intended and enacted curricula. The classroom
became the site to categorize and investigate problems using input-
output schemes, and devise models for improved practice. The efforts
resulted in such innovations as mastery teaching (Block, 1980) and
information-processing models of teaching (Gagne, 1976). Short (1986),
in a brief historical look at curriculum design, acknowledged that the
field was dominated by conceptions of a "measured" curriculum, where
the key feature is "the reduction of their intended outcomes to
prespecified elements, which, when 'taught' and 'learned,’ can be
measured, and on which a definitive report of 'results' can be made
public" (p. 6). The instrumental ideology had been given a legitimacy
derived from quantitative research modeled on the natural sciences.

A consequence of a technical and instrumental rationality2 in
education is that curriculum decisions have been removed from the
classroom teachers who implement them. The effect has been a
deskilling of the nations' teaching force. There has been a separation
of the conception of curriculum from its execution. Teachers often are
asked to do little more than carry out plans, goals and activities
conceived by someone else. At the same time, teachers have been
reskilled to be more efficient managers of classroom instruction and

record-keeping (Apple, 1983b; Gitlin, 1983). The standardization of

 

21 use the term rationality as Giroux has defined it; "a specific
set of assumptions and social practices that mediate how an individual
or group relates to the wider society. Underlying any one mode of
rationality is a set of interests that defines and qualifies how one
reflects on the world” (1981, p. 8).

\—
"m.

3

219
curricula, the focus on minimum competencies and scores of behavioral
objectives, the integration of testing and teaching to the test makes
possible more efficient planning, in the district and in the classroom.
The attraction of integrated systems to efficiency-minded
administrators, especially in inner-city schools, is captured in a
statement by some Chicago school administrators.

Providing materials that were centrally developed and successfully
field tested would: 1) reduce greatly the time needed to prepare
and organize materials; 2) require little inservice time; 3) be
economical for schools in Chicago and elsewhere to implement; 4)
standardize the definition, sequencing, and quality of instruction
necessary for mastery of each objective; 5) reduce greatly the time
needed for developing lesson plans; and 6) be easy for substitutes
to use (quoted in Aronowitz & Giroux, 1985, p. 29).

But the consequences for teachers and students should not to be
applauded. Professional decision-making on matters of curriculum and
instruction is removed from the hands of professional teachers. As
Apple (1983a) has argued,

When individuals cease to plan and control their own work, the
skills essential to these tasks atrophy and are forgotten. Skills
that teachers have built up over decades of hard work -- setting
curricular goals, establishing content, designing lessons and
instructional strategies, individualizing instruction from an
intimate knowledge of each student's desires and needs, and so

on -- are lost. In the process, the very things that make teaching
a professional activity -- the control of one's expertise and

time -- are also dissipated. There is no better formula for
alienation and burnout than the loss of control of the job (p.
323).

At the same time that teachers are being deskilled, children are being
turned into anonymous learners. Their behavior is preselected before
instruction is carried out or learning activities engaged in. That

behavior becomes that basis on which effectiveness of teaching and

 

3M. Katims & B.F. Jones, "Chicago Mastery Learning Reading:
Mastery Learning Instruction and Assessment in Inner City Schools.”
Paper presented at the Annual Meeting of the International Reading
Association, New Orleans, 1981, p. 7.

220
learning is judged (Apple, 1979). The logic of a technical rationality
removes teachers from participating in a critical way in curriculum
planning and curriculum evaluation. Teacher competence gets defined as
refinement of technical skills. Students are not encouraged to
generate their own meanings or to participate in evaluating their own
classroom experiences. Student competence gets defined as mastery of
discrete objectives. The principles of order, control, and certainty
preclude other approaches. Teaching for certification of numeracy and
teaching for comprehension of mathematical ideas emerge from an
instrumental ideology.

Cha e e o n n t umenta d . A challenge to the
dominance of an instrumental ideology in the curriculum field has come
5 from the reconceptualists. Theorists such as James Macdonald, Dwayne
f Huebner, Herbert Kliebard, Maxine Greene, William Pinar, Michael Apple,
. Madeleine Grumet, Henry Giroux, and Jean Anyon have played a major role
in reconceptualizing the issues, interests and modes of inquiry that
inform curriculum theory and practice. The singular theme that has
united their various interests is opposition to the instrumental and
technical rationality that dominates traditional curriculum theory and
practice. Their work has been aimed at offsetting the apolitical,
ahistorical and technical orientations that dominate the field.

Drawing on the intellectual traditions of existentialism,
phenomenology, psychoanalysis, neo-Marxism and critical theory, they
have made the human subject the primary focus of a theory and practice
aimed at emancipation from exploitation and oppression. Instead of
asking how knowledge can be organized so that teachers can more

efficiently deliver it and students more effectively master it, they

221

raise a different set of questions about knowledge. Whose knowledge is
it? Whose meanings are attached to it? Who has access to what kind
of knowledge? How do forms of evaluation legitimize certain ways of
knowing? Questions of this nature expose the ideological underpinnings
of curriculum design, implementation and evaluation and reveal the
complex relationship between schools and the larger social order.

Despite a growing body of theoretical and empirical literature that
has emerged from the work of the reconceptualists, they have not posed
a significant threat to the dominance of an instrumental rationality.
The effects of their work have yet to be felt in any substantial way in
the classrooms of America's schools. As one of their main proponents
has pointed out, in the history of the curriculum field from its roots
in scientific management to its present domination by systems
management schemes, there is an ideological commitment to
instrumentalism that seems particularly resistant to challenge, even
when it is contradictory to broader goals of learning (Giroux, 1981).
One reason for this may be that the work of reconceptualists has tended
to focus on theory development and research but not the practical
implications of this study for classroom teachers. The critical work
of reconceptualists has been to uncover the structures and practices of
schooling that deny justice and equality to all learners. They have
not devoted their efforts, by and large, to providing a vision of a new

school environment except at a fairly abstract level.

The Inheritance Of An Instrumental Ideology In The Detroit Reform
The Detroit reform contained unmistakable elements of an

instrumental ideology. The most obvious manifestation was the set of

222
one-hundred plus behavioral objectives at each grade level that defined
what children should learn. The noble goals for Detroit children, as
stated in the District's philosophy, were reduced to forms of overt
behavior that could be observed, quantified and evaluated. An
investigation of the verbs that described student actions showed the
dominance of actions that could easily be measured--read, identify,

name, recall, state the rule, measure, apply strategies. Verbs that

would describe student actions more difficult to quantify
investigate, explore, invent, predict, generalize -- were absent from
the formulation of learning objectives.

An instrumental ideology informed the sets of model lessons. The
form of the lessons and the prescribed teacher and student actions
embodied a technical and management orientation. They specified what a
teacher had to know, say and do. They specified the materials to be
used during instruction. They even specified correct student
responses. With these lessons in hand, a teacher's limited knowledge
of mathematics would not stand in the way of efficient and effective
instruction. I

The process of deeision-making that shaped the reform was
centralized among curriculum, testing and mathematics specialists. The
teacher representation on committees did not alter the fact that
experts conceived the reform. The majority of teachers in this
district were asked to do little more than ”to execute someone else's
goals and plans and to carry out someone else's suggested activities”
(Apple, 1983a, p. 323).

The technical rationality embedded in an instrumental ideology was

evident in the inservice opportunities. Refining teachers' technical

W- '1.—

223
competence in instruction was at the heart of inservice. Teachers were
provided with research-based techniques for planning and carrying out
instruction and for evaluating student understanding. This was true
not only of the inservice conducted specifically around this reform but
also the much larger effort to train all district personnel in EEEI.
Inservice in this district aimed not to improve teachers' subject
matter knowledge or to improve their decision making. Research was not
offered as something to be questioned or held up against one's own
experience. It was not used to build the reflective capacity of
teachers. Research was presented as having revealed the cause-effect
relationships that could be turned into skills for teachers to use in
improving their practice. Research was being ”translated" into
practice.

It is almost certain that an instrumental ideology shaped the
professional education of the agents of this reform -- those who
created ABCS, conceptualized the curriculum, created Model Lessons,
provided inservice for teachers charged with its implementation. I am
not claiming that they were unwitting servants to the domination of an
instrumental ideology, although perhaps they were. It seems to me it
is a matter of not questioning the taken-for-granted, of not being
reflective about what interests guided their own behavior. An
instrumental ideology saturates consciousness and shapes daily
behavior. It reifies hierarchical forms of decision-making. It aims
to find the/best means to deliver knowledge, not to inquire why this
knowledge and not others. The curriculum planners, the test makers,
the inservice providers were acting reasonably given the dominant

ideology. They believed they were on "the cutting edge” of mathematics

224
education reform. What they did not seem to be asking themselves was
how and to what extent this initiative would make a real difference in

developing children's mathematical literacy.

The Inheritance of the Politics of Race and school Governance in the
Detroit Reform

There are structural and institutional determinants in school
systems that sustain and uphold efficiency-oriented management
techniques. But at times there are forces that pull in an opposite
direction. In Detroit, the demand for community control by the parents
of black children led to the decentralization of adminstrative
authority over the schools. Curriculum decisions were decentralized to
the local Region. Region Boards were more visible and more likely to
consider demands from organized groups of parents, students and
teachers in the community.“ But the politics of race and the politics
of school governance were soon in tension. Desegregation and
decentralization produced administrative conflicts.

The Judge overseeing the desegregation order was not satisfied that
simply ordering the reassignment of students and teachers to achieve
racial integration would end the effects of past discriminatory
practices. To that end, he ordered the implementation of eight

educational components. The stipulation of educational components to

 

“Although the evidence is anecdotal, the early 1970s appear to
have been a time of increased studies in Black history and literature
in the Detroit schools. In an earlier study of one of the Detroit high
schools, I found that a number of such courses had been taught,
including Swahili. The English Department also published a Black
Literary Journal of student writing. A veteran teacher in the school
told of going with a group of parents and students to the Region Office
to petition for these courses and finding a sympathetic Board.

225
be administered by the Central Board and its administrative staff
produced an uneasy mix of Region and central administrative authority.
As the Court provided the opportunity for greater centralization of
authority and management over educational programs in the District,
there was increasing confusion about who had ultimate authority.

At the same time, the city was experiencing a significant downturn
in the economy and the schools felt the fiscal effects. In addition,
Detroit had become a majority black city. The confluence of tighter
economic times, administrative disorder in the schools, and a black
majority in the city led to calls to recentralize the schools. The
concerns of community control had diminished, to be replaced by demands
for efficiency and accountability.

Recentralization did not completely resolve the struggle for
control over school administration. But the corner had been turned
toward restoring centralized control over the District. City-wide
testing had been mandated by the Court. Recentralization provided the
opportunity to use the test to develop curriculum, to standardize it
for easier management and order its city-wide implementation for easier

accountability.

The Inheritance of Reform Agendas for Mathematics EducationaAnd
Changing Economic Realities

The testing component of the desegregation order specified that the
District undertake a review of its city-wide testing program and
develop an objective-referenced testing program in reading, writing and
mathematics. The first efforts at test development began in 1977. At

the time, the slogan of the national agenda for mathematics education

226
was ”back-to-basics.” It was not surprising that one of the first
tests developed was the High School Proficiency Exam. The test,
created to determine if a graduate would receive an “endorsed” diploma,
measured basic reading, writing and mathematics competencies at the
eighth grade level.

By the time work began on ABCS (the test from which the new
curriculum was derived), the reform agenda for mathematics had changed.
The National Council of Teachers of Mathematics was calling for a
redefinition of "basics.” Computational proficiency was deemed no
longer sufficient. Problem solving must become the focus of school
mathematics and calculators and computers should be available to all
children. The recommendations for reforming mathematics education were
a recognition that technology was changing patterns of production.
Basic skills would no longer provide the knowledge workers would need
in an increasingly complex, technological economy.

The mathematics portion of ABCS represented a compromise between
those who held to testing computational proficiency and those who
advocated assessing student understanding of ideas and using word
problems as the context for computation. Once the test had been
developed, the next step was to determine the degree to which the
curriculum - in this case, textbooks - were aligned with the test. The
curriculum reform that emerged was an effort to assure that what
teachers taught and what children learned correlated with the tests.

The desegregation order, a city-wide testing program, a national
agenda to reform the mathematics curriculum in recognition of a new

economic order had come together to influence the Detroit initiative.

227
Summary

This reform emerged from an ideology, a set of agents and
practices, time and space focused events, and events more global and
long term. An instrumental ideology was the antecedant for the
systematic integration and alignment of testing, curriculum planning,
instructional materials development, and inservice opportunities. The
politics of race and school governance, and the interaction of school
governance with management concerns embedded in instrumentalism were
antecedants of city-wide mandates for testing and curriculum
implementation. The recognition of a new economic order and the
mathematical knowledge required for participation in it influenced
national reforms for mathematics education that in turn influenced the
Detroit initiative.

This reform was situated in and created out of a set of social,
political, economic, organizational and educational contexts. By
seeing it in relation to these contexts, the reform becomes
understandable. It becomes understandable how well-intentioned
professional educators who wanted to make a real difference in the
lives of Detroit children conceived a reform of this nature. There is
a consistency of practice that emerges from the analysis. But there is
also a fundamental contradiction. An instrumental ideology cannot
inform a curriculum or a practice that empowers. To paraphrase Giroux
(1983) -- who was speaking about the literacy of writing -- an
instrumental approach to mathematical literacy does not treat

[mathematics as "both the medium and product of one's experience in the
1l‘world." Mathematics is stripped of its ”normative and critical

dimensions" and is reduced to the mastery of skills and the

228
understanding of concepts and processes (1983, p. 213). Likewise, an
instrumental approach to improving teaching practice reduces the
knowledge teachers need to utilitarian and practical forms.

The common curriculum Detroit children would encounter as it was
organized cannot develop the abilities they will need to succeed when
they leave school. Citizens and workers must be able to work with
others, defining problems in the real-world that are messy and not
well-formulated, seeing the mathematics in them, and bringing their
"math sense" to solve problems. It is difficult to see how the Detroit
curriculum reform, with its emphasis on covering hundreds of discrete
behavioral objectives, could create this kind of learner.

An instrumental ideology is dominant in American education, but it
is never total or exclusive. In this reform we have seen struggles
; over the control of school administration, compromises over the content
S of ABCS and a shift away from an exclusive focus on minimum
computational competencies, an exemplary Model Lesson where children
would have the opportunity to investigate a mathematical idea and begin
to see the power of multiple representations. The District had
approved two distinctive textbook series for use in mathematics
classrooms. These texts embodied elements of teaching for
A comprehension of mathematical ideas and teaching for enabling
E mathematical inquiry. Although they were used in few classrooms, their
availability suggested an interest in moving beyond certification and
comprehension as goals for student learning.

Curriculum is not static. In the past three decades we have seen
changes in national agendas for reforming mathematics education. The

19803 have been marked by calls to improve the content, curriculum and

229
instruction for all learners. The recommendations in the early part of
this decade called for more problem solving and the use of calculators.
But as this decade draws to a close, a very different campaign is being
waged. There has emerged a new challenge to the domination of an

instrumental rationality.

A.NEW CHALLENGE TO THE DOMINANCE OF INSTRUMENTALISM

An.Agenda For Enabling,Mathematical Inquiry

The 1989 publication of the National Council of Teachers of
Mathematics, Cugrieulum and Evalnation Standezds in; Seheel
Mathematiee, represents a challenge to the instrumentalism that has
dominated mathematics education. The document represents three years
of development in which draft versions have been presented to the
members of NCTM, classroom teachers across the country, state policy
makers, textbook publishers, supervisors of curriculum, supervisors of
mathematics instruction, and mathematics educators in teacher
preparation programs. The presentations were intended to promote a
vision of mathematics education aimed at enabling children and youth
with the power of mathematical knowledge, to judge the degree to which
these constituencies supported such a vision, and to raise questions
about what would be required to implement the vision. The document is
in some ways unique. It is not simply a statement of broad goals for
reforming mathematics education. Each standard provides an elaboration
of mathematical content and mathematical instruction. Each standard
emphasizes knowing and doing mathematics. Each standard describes
content that is appropriate for ell learners. And each standard is

accompanied with some activities as examples of what knowing and doing

230
mathematics can look like in the classroom. To provide an example of
the vision, portions of the first standard, mathematics as problem

solving, for grades K-4 are presented.

In grades K-4, the study of mathematics should emphasize problem
solving so that students can--

* use problem-solving approaches to investigate and understand
mathematical content;

* formulate problems from everyday and mathematical situations;

* develop and apply strategies to solve a wide variety of problems;

* verify and interpret results with respect to the original
problem;

* acquire confidence in using mathematics meaningfully.

Problem solving is not a distinct topic but a process that sould
permeate the entire program and provide the context in which
concepts and skills can be learned (p. 23).

Each standard provides examples of activities appropriate to the
content of the standard and the vision. These involve activities that
children work on individually, in small groups and as a whole class.

Project problems, which often require several days of class time,
provide an opportunity for children to become immersed in problem-
solving activity. Some situations allow children to be
particularly creative in their formulation of problems. Here is
one such situation:

The class is given the opportunity to plan and participate in an
all-school ”Estimation Day.” The children, in pairs or threes, are
to design estimation activities to be completed by children in
other classes. Each group will supply all the necessary materials
and monitor the activities. The activities might include guessing
children's heights, the number of candies in a jar, the lengths of
various pieces of string, the weight of a bag of potatoes, the
length of the room, the number of times they can write their names
in a minute, or the length of time required for an ice cube to melt

(p. 25).

The vision portrayed in the goals for learning and the types of
learning activities stand in stark contrast to what is typically found
in traditional textbooks and what existed in the formal curriculum

documents of the Detroit reform. It is a vision toward teaching for

231
enabling mathematical inquiry. The question is, what would be required

to create this vision in classrooms in the Detroit Public Schools?

What is Required to Create This Vision in Classrooms?

My effort here is to sketch out some of the key issues that are are
likely to be raised concerning this new reform agenda.

Ieeehe;_kneglegge. The dominance of a typical routine in
mathematics classrooms results from teachers' own experiences, as
learners of mathematics, as students in elementary teacher preparation
programs, and as practicing classroom teachers. As students in
elementary and secondary mathematics classrooms, the "apprenticeship of
observation” has provided teachers with a view of how mathematics is
taught. Preservice programs typically require a methods course in
teaching elementary mathematics that is likely to focus on improving
technical competence, to make the typical routine more efficient and
effective. Their experiences as students, in elementary, secondary and
university mathematics classes, have taught them what it means to know
mathematics -- being able to recall appropriate rules and algorithms to
apply to problems. As practicing teachers, the curricular materials
they may be mandated to use and the instructional practices that are
endorsed contribute to the continuation of business as usual.

This is a problem that must be addressed at both the preservice and
inservice level. If teachers are to incorporate materials and
practices aimed at enabling mathematical inquiry they must have
opportunities to deepen their knowledge about mathematics and what it
means to know and do mathematics. They must have opportunities to

deepen their knowledge of how children learn mathematics. And they

232
must have opportunities to consider what kinds of instructional
practices promote mathematical inquiry.

This requires districts to take a hard look at what constitutes
inservice in their systems. The continuing professional development of
their staffs should become a priority. The one-shot workshop, whether
offered after-school by in-house people or at all day sessions provided
by nationally recognized, charismatic educational entrepreneurs selling
their wares, is notoriously ineffective in changing what goes on in
classrooms. The practice needs to be reconsidered. Where offerings
are long-term, as Detroit's EEEI training, districts need to look
carefully at what these programs are intended to do. Refining teacher
techniques at delivering direct instruction will not develop ways of
thinking, acting and questioning that promote mathematical inquiry in
classrooms. Just as students need knowledge beyond that which is
technical and practical, so do professional teachers.

legging. Districts, especially those in urban settings with large
minority and poor populations, have been under intense pressure to
improve student scores on achievement tests. It is not likely that
will subside. But tests need to change because a curriculum aimed at
enabling mathematical inquiry is very different from most curricula and
the tests with which they are integrated. Using standardized test
scores, whether norm- or criterion-referenced, as the sole indicator of
the effectiveness of instructional programs may simplify reporting but
it has limited value in telling us what children are coming to know.

The NCTM Seengelge suggest that increased attention needs to be
paid to assessing not only what students know but how they think about

mathematics, multiple methods to assess student understanding, and
-..v/ r

233

multiple schemes for scoring and recording what they know. They also
argue that the purposes of testing need to be reexamined and broadened.
Giving periodic paper-and-pencil tests for the purposes of recording a
mark and then averaging a set of marks to give a course grade represent
the narrowest view of assessment. Giving multiple choice tests but not
fasking why students answer questions incorrectly ignores the importance
'of meaning that students attach to situations and how it influences the
ways in which they think about and solve problems.

Rethinking testing and evaluation is no small matter. Having a
single score to report to the community can make for good public
relations. It fits well with administrative concerns about
accountability and standardization.5 The public has a right to be
informed about the progess of the students in its schools and the
effectiveness of instructional programs. But educational professionals
need to help the public realize how little they learn by simply asking,
”How much have our scores gone up?”

School edministratien end ozganizatien. Superintendents, district
administrators and building principals are often trained in
organizational theory and systems management where they learn elaborate
language systems, management controls and accountability systems. The
ideas embedded in this instrumental and technical approach to school
administration led to further centralization of decision making and

standardization of instructional programs. If mathematics education is

 

5Administration of ABCS was discontinued in the Detroit schools
after 1987 because students continued to do poorly and because the test
results were not reported as a single score. Area superintendents
found the public relations value of the test to be negative. They
applied sufficient pressure on central administrators that the decision
was finally made to abandon the testing program that had been the basis
on which the curriculum was conceived.

234
to be transformed, fundamental changes need to occur in the ways
schools are organized and administered.

Teachers need to be freed from lock-step curricula and pedagogies
that reduce the work of teaching to managing instruction. If we want
to develop a set of intellectual tools in our students, then we need to
redefine the work of teachers as intellectual work, not technical work.
This means teachers must have a role in shaping the purposes and
conditions of mathematics education in their schools and in the
district.

Teachers also need to be continual learners; schools should become
sites of thecontinuing professional development of their staffs. That
requires rethinking how schools are organized, what time is available
for the planning and reflection that intellectual work demands, the
norms and structures that shape relations among the staff.

The tendency toward increasing centralization of control is
: contradictory to the diffusion of decision making that the new vision
of mathematics education suggests. The recent interest in site-based
management schemes may be the opening to restoring decisions about the
specifics of educational programs to local schools.

The velue efi mathemaeiee. If teaching mathematics for enabling
mathematical inquiry is to become a reality in elementary classrooms,
we need to confront the extent to which mathematics is seen as having
only utilitarian value. In Exer1b291_fleun§s. the National Research
Council calls mathematics ”the invisible culture of our age” because it
influences in multiple but often hidden ways the practical, civic,
professional, leisure and cultural layers of our lives. Mathematics

education has a responsibility to reveal the layers and to do it in

235
interesting ways so that the familiar lament, "Why do 1 have to know
£h1§?" no longer rings in classrooms.

' I argued earlier that mathematical literacy needed to take on new
meaning, not just in a limited economic and technical sense, but also
in a political and social sense. Classrooms must become places where
intellectual inquiry is promoted, where children working together
engage in purposeful activity, where they learn to make reasoned
judgments about their own thinking and that of others. This does not
deny the practical and utilitarian value in training a workforce.
Rather it recognizes that the mathematical needs of workers in a new
economy require precisely these skills. But there are reasons beyond
this key one. A new kind of quantitative literacy is required for
people to engage thoughtfully and responsibily as citizens in a
democratic society. And this requires not only changing the content of
mathematics instruction but providing environments where students
working together experience the ways in which collective efforts at

problem solving enhance understanding.

A Necessary but Not Sufficient Condition for Liberation

This reform initiative was a response, in part, to problems that
were rooted in the larger context of social, political and economic
factors. Economic disinvestment in the city of Detroit has meant that
black youth face a very different future from earlier generations who
were able to find high-wage, semi-skilled work in the plants. Some of
these manufacturing jobs lost to the city have been replaced by

manufacturing and service work that requires increasing levels of

236
technical expertise and quantitative literacy. But many of the lost
jobs have been replaced by low-skill, low-wage work in the service
sector. The problems that confront Detroit youth and young adults are
not just school-based. Consequently, the solutions are not to be found
solely in educational reform. Proposing technical solutions to
problems that are, in part, politically and economically rooted ignores
the relationship between schools and society. There is a much larger
agenda of reform.

Transforming mathematics education in the Detroit schools will not
change the structure of the Detroit economy. Transformation of the
mathematics curriculum toward enabling mathematical inquiry cannot by
itself overcome the forces of racism, sexism and class bias that
structure inequality in this society. But mathematics education can no
longer play a gatekeeping role, limiting the opportunities of many
people to function in a world that is demanding increasing mathematical
literacy of its citizens. Students who have the opportunity to
experience mathematics education in an environment that promotes
inquiry, where others' formulations are not taken as given, where the
subject matter itself becomes the vehicle to investigate the structured
inequality in society, will have developed a set of intellectual tools
to participate more critically and fully as workers and citizens.

The transformation of mathematics education toward enabling
mathematical inquiry is bold and will be expensive and disruptive.

This may prove to be a sizeable hurdle for urban districts like Detroit
that are already overburdened by contextual factors over which they
have no control. But we have come to recognize that our society is

becoming increasingly multi-racial, multi-cultural and multi-lingual.

237
Urban districts that serve increasing numbers of students of color must
implement policies and programs that do not leave their students
marginalized in society, as workers or as citizens. It is required in

the new economic order and it is appropriate to the good society.

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APPENDIX A

Teacher Interview Stbedule

1. In what ways is the DPS curriculum -- strands and objectives --
different from the mathematics curriculum you previously taught?

2. Are there some mathematical topics you will be teaching for the
first time?

3. What would help you to implement the strands and objectives?
4. What materials do you have that you find helpful?

5. What influenced your decision about whether or not to attend
inservice?

6. (For those who attended inservice) What did you hope to learn
from inservice? How has inservice influenced the way you think about
the mathematics curriculum and ways of teaching mathematics?

7. What support, in addition to inservice workshops, has been made
available to you?

8. What factors enhance/constrain your efforts to implement the
curriculum strands and objectives?

9. What goals do you have for your students as learners of
mathematics?

10. Do you think the goals of the district are the same as your?
11. Using manipulatives has received considerable emphasis in
inservice. What purpose do you think manipulatives serve? Do you
use manipulatives? When, how often, to teach what? With what

results?

12. What do you think about when you hear the term "conceptual
understanding?”

250

APPENDIX B

CITYeHIDE INSTRUCTIONAL.SEQUENCE - GRADE 3

Septemhep PROBABILITY AND STATISTICS
Read and interpret a pictograph using ratios of 1:1 and

1:2, using a key.

Read and interpret a bar graph using calibrations of
units of 1, 2 and 5, using a key.

Construct a bar and pictograph using data with ratios of
1:1 and 1:2.

Identify a circle graph.

SETS AND LOGIC
Identify the position of objects according to left/right,
top/bottom, over/under, above/below, on/off, first/last,
first through tenth, before/between/after,
inside/outside, on and both.

MEASUREMENT
Identify the value of dozen(s).

Identify and record time by the hour, half-hour and
quarter hour (non-digital).

Name the days of the week and the months of the year in
order.

CALCULATORS AND COMPUTERS
Boot and interact with appropriate computer assisted
learning software (if available).

NUMERATION
Interpret and/or illustrate the meaning of whole numbers

through 999.

Demonstrate accuracy in reading and writing whole
numerals through 999.

Determine and/or demonstrate place value through 999
using aids.

FUNCTIONS AND RELATIONS
Interpret and complete addition, subtraction or
multiplication number sentences using one of these

symbols, <, >, - or +.

OPERATIONS
Recall the addition and subtraction facts through sums of

18 with or without the use of aids (pictures, number
line).

251

Octobep

252

FUNCTIONS AND RELATIONS
Relate the inverse of addition and subtraction.

PATTERNS
Interpret and/or extend a pattern of shapes, pictures or
objects using a combination of any three of the following
attributes: color; shape; size; thickness; position;
texture.

OPERATIONS
Add whole numbers through two digits with and without
regrouping.

Demonstrate the meaning of multiplication as repeated
addition within reasonable limits, and vice versa.

FUNCTIONS AND REIATIONS
Apply a given rule to a set of whole numbers and
determine the missing whole number output.

Apply a given rule to a set of whole numbers to determine
the missing input numbers using addition, subtraction or
multiplication.

GEOMETRY
Demonstrate ways to partition a circle, square, rectangle
or equilateral triangle as appropriate into a given
number of congruent parts (2, 3, 4, 6, 8, or 16).

FUNCTIONS AND RELATIONS
Identify and state the rule which indicates the
relationship between whole numbers when given input or
output.

OPERATIONS
Add and subtract whole numbers through two digits.

NUMERATION
Recognize and write number words through 999.

Identify the value of a given digit up through a 3 digit
number.

Identify a number that is 10 or 100 more or less than a
given number.

Write numbers in standard notation given in words (orally
or written) or expanded notation and vice versa.

Write money amounts using standard money notation, given
dollars, dimes or pennies or given the money notation
select the monetary amount.

November

253

OPERATIONS
Apply addition, subtraction strategies to solve money or
other word problems.

MEASUREMENT
Identify, compare, write and compute money values through
$9.99.

OPERATIONS
Addition and subtraction of money numbers.

Recall the multiplication facts where both factors are 0-
5.

NUMERATION
Identify numbers as :odd or even; two times as much;
twice as much; half of a given set through 18.

Use the property of zero and one in multiplication.

SETS AND LOGIC
Identify the correct word statement, picture, place,
thing or number from 2 or more statements.

NUMERATION
Use the commutative law in addition to solve equations.

Use the associative law for addition to solve equations.

OPERATIONS
Apply addition, subtraction or multiplication strategies
to solve money, time or other word problems (one or two
steps).

Add and subtract whole numbers through three digits.

NUMERATION
Using place value analyze two numbers through 999 using
<, >, or -.

PATTERNS
Interpret and complete a number pattern (using numbers 0-
99; starting at any number) in ascending or descending
order counting by ones, twos, fives or tens.

MEASUREMENT
Read and interpret thermometers Celsius and Fahrenheit of
various calibrations.

OPERATIONS
Multiply whole numbers through 3 digit times 1 digit (no
regrouping).

Decembe;

Januapy

254

MEASUREMENT
Estimate and determine the length of an object or
distance using non-standard units.

Estimate and determine the length of an object using
whole metric units.

Relate and compare metric units of measure (cm and m).

GEOMETRY
Identify geometric figures, symbols and their written
words: square corner; point; ray; line segment; line;
angle.

Identify and recognize cube, cone, cylinder and sphere by
word names and models in the everyday environment.

Draw points. line segments, rays, angles and lines using
appropriate tools.

Identify and reproduce lines of symmetry.

MEASUREMENT
Compare and identify the lengths of objects.

Draw line segments for given whole unit lengths.

PATTERNS
Interpret and complete a number pattern using numbers 0-
999 (starting at any number) in ascending or descending
order counting by ones through sixes, tens and hundreds.
Include multiplication patterns.

ESTIMATION
Identify and estimate length when given a non-standard
unit.

OPERATIONS
Subtract whole numbers through three digits (with and
without regrouping).

NUMERATION
Identify numbers as multiples of 2, 3, 5 or 10.

ESTIMATION
Round any two-digit number to the nearest multiple of
ten .

Round any three-digit number to the nearest multiple of
hundred.

Round the individual problem numbers to the nearest
multiple of ten or hundred and estimate the sum or
difference: incomplete number sentence with results of

Eebruapy

255

less than a thousand; money problems using pennies,
nickels, dimes, dollars or ten dollars; word problems.

Use estimation to check the reasonableness of answers.

MEASUREMENT
Determine and record the area of a figure on a grid in
square units.

Identify the reasonable customary or metric unit to
measure the length of an object, height or distance.

Use a metric or customary ruler to measure the sides of a
polygon or pictured object.

Determine and record the perimeter of a polygon or grid
following the lines on the grid.

PATTERNS
State the rule used in forming a given pattern and extend
the pattern.

Continue a pattern (objects or pictures) or equivalent
ratios by adding to the series.

CALCULATORS AND COMPUTERS
Investigate ways to utilize the hand-held calculator to
solve problems: basic facts; two-step problems; check on
accuracy of predicted answers; two and three digit
computation.

Determine reasonableness of displayed answers.
Check computation using a calculator.

Predict number to be displayed and verify its results.

NUMERATION
Classify a number by its attributes.

Sequence up to 4 counting numbers, least to greatest or
greatest to least.

FUNCTIONS AND RELATIONS
Determine the appropriate operation needed to solve
addition, subtraction or multiplication word problems.

SETS AND LOGIC
Read, interpret and supply elements in a matrix
(vertical/horizontal format).

March

256

PATTERNS
Interpret and complete a number pattern where two
function rules repeat to form a pattern.

Create and state the rule of a pictorial and/or number
pattern using whole numbers.

NUMERATION
Identify positive and negative integers on a number line.

OPERATIONS
Apply addition, subtraction, multiplication or division
strategies to solve money or other word problems (one or
two steps).

Recall the multiplication facts through 9 times 9.

Multiply whole numbers through 3 digit times 1 digit
(with or without regrouping).

ESTIMATION
Round the individual problem number to the nearest
multiple of ten or hundred and estimate the sum,
difference or product: incomplete number sentence with
results of less than a thousand; money problems using
pennies, nickels, dimes, dollars or ten dollars; word
problems.

OPERATIONS

Recall the addition and subtraction facts through sums of
18.

FUNCTIONS AND RELATIONS
Identify the relationship of a pictured sum of money to a
given value in a problem involving money through $9.99.

PROBABILITY AND STATISTICS
Demonstrate the probability of a simple event with six or
fewer outcomes, using aids.

SETS AND LOGIC
Read and interpret Venn diagrams, with 2 or 3 sets.

Construct Venn diagrams.

Use synonyms for comparative statements, i.e., ”not
large” means ”small."

Use negations to describe new sets.

Decide the truth or falsity of a number of statements
involving the terms: all;, every; each; exactly; no.

April

I?

257

MEASUREMENT
Estimate and determine the length of an object using
whole numbers of metric and customary units.

Determine and record the weight of objects using
customary and metric units.

Identify the reasonable customary or metric unit to
measure weight (mass).

Compare and identify the weight (mass) of objects.

NUMERATION

Identify positive and negative integers on the number
line.

PROBABILITY AND STATISTICS

Read a pictograph using data of ratios of l, 2, 5 and
10:1.

NUMERATION
Apply the commutative and associative laws to addition
and multiplication (single digit).

OPERATIONS
Demonstrate the meaning of division as repeated
subtraction.

Divide whole numbers through 2 digit dividends by 1 digit
divisors (limit to division facts).

FUNCTIONS AND RELATIONS
Manipulate models to order fractional parts.

OPERATIONS
Demonstrate ways to partition a circle or square into 2,
4, 8 or 16 congruent parts.

Compare halves through sixteenths of various physical
wholes.

Identify models of halves through fourths, sixths,
eights, twelfths, and sixteenths.

Compare fraction "families” using models (halves,
fourths, eighths, sixteenths; thirds, sixths and
twelfths).

Add and subtract like fractions with aids.
PATTERNS

Interpret, complete and extend a pattern when time, money
or temperature is used to develop the pattern.

258

FUNCTIONS AND RELATIONS
Order a set of 3 to 5 two and/or three digit numbers.

Order two or three sets of objects or pictures: more to
fewer; most to fewest, fewest to most; ascending or
descending order.

OPERATIONS
Apply mental arithmetic strategies to solve addition,
subtraction, multiplication or division problems.

NUMERATION
Apply the property of zero to addition and subtraction.

FUNCTIONS AND RELATIONS
Relate the operation of addition to multiplication.

June MEASUREMENT

Investigate the volume of a figure in cubic units using
aids.

 

Investigate and compare the capacity of various
containers.

NUMERATION

Read and write Roman numerals using the symbols 1, V and
X.

MEASUREMENT

Measure and describe the weight of objects using non-
standard units.

Relate and compare customary units of measure (in., ft.,
yd., mi.).

OPERATIONS

Determine one-half of an even whole number through 18;

even multiples of 10 and even multiples of 100, with or
without aids.

APPH'DIX C

Dr. Douglas Grouws
Univ. of Missouri

QUIZ ON EFFECTIVE MATH TEACHING

There is data available to support the statement...

T

F

l.

10.

ll.

l2.

l3.

14.

Teachers make a difference in the amount of learning that
takes place in classrooms.

Some teachers are more effective than others.
Some math teachers are more effective than others.
Effective and less effective teachers can be identified.

Effective and less effective mathematics teachers have been
identified and studied.

A teacher's impact on pupil learning is relative stable
from year to year.

Characteristics of teachers such as number of math courses,
number of years of teaching experience, and so on, are
highly correlated with how much pupils learn in math
classrooms.

Teacher behaviors (i.e., what a teacher does in the
classroom) are highly correlated to how much pupils learn
in math classrooms.

Highly effective math teachers outperform their
counterparts by getting greater gains from the high ability
students in their classes.

Time variables correlate highly with effective teaching.

Periodic observation of teachers increases student learning
in those classrooms.

List three teaching behaviors associated with effective
math teaching.

 

 

 

Similar teaching behaviors are associated with effective
math teachers and effective English teachers.

Math teachers can be trained to teach differently.

259

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

260

Teachers that teach according to prescribed conditions for
effectiveness perform little better than their counterparts
who teach in their regular ways.

When classroom observers rank math teachers according to
their effectiveness their rankings are very similar to the
rankings one would get from examining standardized
achievement test scores.

Instructional pace (i.e., rate of movement through the
curriculum) is positively correlated to student
achievement.

Spending more math time on developing ideas than on
practice is associated with better student math scores.

Frequency of teacher lecturing or explaining to students is
positively related to students' perceived usefulness,
enjoyment, and importance of mathematics.

The average amount of time teachers spend on development in
mathematics classrooms is in the 15-301 range.

Instruction during mathematical problem solving units is
very much like regular instruction.

High effective mathematics teachers tend to individualize
instruction with regard to pace and content than do less
effective teachers.

There is a best way of teaching mathematics.

A good reference on this topic is the book, Selecped leeues
in nephematice Edneetien edited by Mary Lindquist.