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MATH WARS: A RHETORICAL ANALYSIS OF THE TERMS OF DEBATE
By

lrfan Muzaffar

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁlment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Curriculum, Teaching, and Educational Policy

2009

ABSTRACT
MATH WARS: A RHETORICAL ANALYSIS OF'TERMS OF DEBATE
By

lrfan Muzaffar

This study concerns itself with the conﬂict in mathematics education—popularly
known as math wars—in the United States. More speciﬁcally, it investigates the terms
of debate in this conﬂict to develop insights into the varied, and sometimes conﬂicting,
relationships between the perceived nature of mathematics and its pedagogy. It also
extends the use of rhetorical analysis to understand the ways in which the rhetoric of
standards was at odds with the idea of mathematical power for all.

The study centres its analysis on terms—such as mathematical power or
proﬁciency—that became the sites of contestations. It explores the internal logic of
particular clusters of terms, and the relations between them in several signiﬁcant texts
about mathematics and mathematics education. To perform this analysis, I have chosen
the image of drama offered by Kenneth Burke due to its emphasis on conﬂict. Image of
drama is invoked because of its relevance to an understanding of conﬂict. This image
invokes particular scenes, populated with particular agents engaged in purposeful acts
using means available to them As an analytical device to study mathematics education
texts, it implies the recognition that any text having to do with mathematics education
implicitly or explicitly would have some description of background or scene against
which mathematicians, teachers of mathematics, or students [actors, that is] would appear

as engaged in purposive acts of doing, teaching, or learning mathematics.

There are two kinds of texts examined in this dissertation. The ﬁrst kind of texts are
those written long before the current conﬂicts around mathematics education reforms—
i.e. the texts by well-known mathematicians such as Rene Descartes, Bertrand Russell,
and the 19"h century American mathematician Benjamin Pierce. Second kind of texts
examined in this dissertation were generated as part of the mathematics education
reforms, which were constituted in the wake of crisis calls to reform education in the
early to mid 19803. The dramatistic analysis of these texts suggests that terms such as
mathematical power for all and mathematical proﬁciency for all were bound up in two
different conceptions of mathematics and its teaching and learning. While the former
represented a merger of mathematics and pedagogy, the latter emphasized their
separation. The study also discusses the implication of these suggestions for
professional claims of mathematics education.

In the ﬁnal chapter, the study uses dramatistic analysis together with the notion of
American Jeremiad—a reference to perpetual announcements of impending doom and
calls for reform in American history—to develop insights about the relative strength of
reform texts in a scene set up by the Jeremiad of A Nation at Risk. The insights so
developed suggest that mathematics education reforms’ strength as well as their

vulnerability accrued from their reliance on the rhetoric of standards.

Copyright by
lrfan Muzaffar
2009

ACKNOWLEDGEMENTS

I will start by thanking my advisors Helen Featherstone and Lynn Fendler without
whom this dissertation could never be completed. Thank you, Helen and Lynn for being
two amazing mentors. You remained an inexhaustible source of knowledge, critical
comments, ideas, and tons of sympathy through my trials and tribulations as a doctoral
student. The numerous conversations I had with you always helped orient and focus my
intellectual gyroscopes as I groped my way through this dissertation.

My work is also indebted in a great measure to my committee members Suzanne
Wilson and Brian Delany. Suzanne helped me immensely in understanding the
intricacies of mathematics education reforms, always pushing me to think harder and
clearer with her penetrating question. Brian, interested and sympathetic, read the drafts
of my chapters and helped me reﬁne the analysis with his feedback. I also thank Nathalie
Sinclair for being a shaping inﬂuence in the early part of this project as part of my
guidance committee, and also for continuing to provide me with her insights and
comments on my drafts even after leaving the Michigan State for Simon Fraser
University.

I will be remiss if I do not mention Jay F eatherstone who led me to several critical
sources on the history of educational reforms in the United States and responded to my
ideas and drafts. I also want to thank Cleo Cherryholmes for initiating me to curriculum
theory and for remaining supportive of my work even after retirement. There are many
other people to whom I would like to acknowledge my debt. They are Sandra Crespo,
Joyce Grant, Ajay Sharma and Punya Mishra.

I was also immensely fortunate to be part of the Critical Studies Group, an amazing

collection of faculty and graduate students. I would particularly like to thank Don

Moore, Steve Tuckey, Brett Merritt, Kelly Merritt, Jeanne Meier, Ann Lawrence and
Adam Greteman.

I am also indebted to the help provided by the Spencer Foundation Research Training
Grant, which helped me immensely in ﬁnding my way through a maze of theoretical
ideas, many of which are ultimately reﬂected in this work.

My appreciation also goes to my Mom who trusted me and kept me in her thoughts
and prayers, my little brother Kamran who lost his life while I was writing this
dissertation, and other members of my extended clan. And ﬁnally, but in the least, it was
my wife and friend Shamaila, who treated me with indulgence through out this project,
and without whose love, support, patience and care, I could never have completed this

work.

vi

Table of Contents

List of Tables ..................................................................................................................... ix
Chapter I: Introduction ........................................................................................................ 1
The Math Wars at a glance ..................................................................................... 5
Understanding the terms of the debates .................................................................. 9
The Methodological Considerations ..................................................................... 11
Outline of the Dissertation .................................................................................... 17
Assumptions about Relations between Action, Conﬂict, and Drama ................... 20
The Analysis ......................................................................................................... 37
Concluding thoughts: Analyzing the Relations between Terms of the Pentad ..... 40
Chapter III: Mathematics, God, and Pedagogy ................................................................. 43
The Mountain Range: Mathematics as the Scene in Classical Texts .................... 50
The Power to attain Natural Light of Reason: The Agency and the Purpose ....... 68
The Relationship between Mathematics and its Teaching and Learning in
Classical Texts ...................................................................................................... 69
The Drama of Reform Texts ................................................................................. 74
The Reform Texts ................................................................................................. 76
The Scene of Mathematics Education ................................................................... 80
Mathematical Power: Examining Act and Agency in the Reform Texts .............. 95
Purpose: the Mathematically Empowered, Democratic Citizen ......................... 102
Concluding Thoughts .......................................................................................... 105
Chapter V: The Drama of Counter Reforms: Implication for Mathematics Education.. 107
Counter Reforms ................................................................................................. 111
Standards as Pathways and Milestones: The Scene of Mathematical Proﬁciency
............................................................................................................................. 115
Separating Content and Pedagogy: Contesting the Claims of Mathematics
Education ............................................................................................................ 129
Concluding Thoughts .......................................................................................... 132
Chapter VI: The American Jeremiad and Reforms: The Dramatistic Vulnerabilities of
Reform Texts .................................................................................................................. 134
The American Jeremiad ...................................................................................... 136
From Jeremiads to Jeremiads: Transformations from Sacred to Secular .......... 144
Similarities between Theologized and Secularized Jeremiads ........................... 147
American Jeremiad and Mathematics Education Debates .................................. 150
The Scene Described and Acts Called for by A Nation at Risk: The Scene-Act
Ratio for Mathematics Education Reforms ......................................................... 154
Echoes of Jeremiad: The Expressions Scene-Act Ratio in Reforms and Counter-
Reforms ............................................................................................................... 1 59
Reform Texts, Comedy and Tragedy: The Afﬁrmation of scene-act Ratio ....... 171
Concluding Thoughts .......................................................................................... 175

vii

References............................................... ........................................................................ 177

viii

List of Tables

Table 1: Pentad at a Glance—Focus on Classical Texts ................................................... 43

Table 2: Pentad at a Glance—Focus on Reform-based Mathematics Education Texts 73

Table 3: Strands and Unifying Ideas at various levels in the 1992 Framework ............... 79
Table 4: Verbs of Mathematical Power ............................................................................ 98
Table 5: Pentad at a Glance—Focus on Counter-reform Texts ...................................... 107
Table 6: Standard for the Number Sense Strand in grades K-3 ...................................... 120
Table 7: Number of Standards associated with 1999 Framework .................................. 121
Table 8: The Dramatistic Elements in American Jeremiad ............................................ 139
Table 9: Comparison between the Puritan and Secularized Jeremiads ........................... 147
Table 10: Binaries of Math Wars ........................................... . ......................................... 149

Table 11: Comparison of Scene-Act ratio in A Nation at Risk, NC TM Standards, and 1999
Framework ...................................................................................................................... 169

Chapter I: Introduction

If the world around us is at least partially a matter of how it is framed, how the
situation is deﬁned, then the reader, the observer, the audience can best understand

how the procedure operates by getting outside the box of his own logics.

(Gusﬁeld, 1989, p. 7)

Through this dissertation, I study the terms of debate in mathematics education.
Speciﬁcally the study concerns itself with the conﬂict in mathematics education in the
United States in the last two decades or so. In this introductory chapter, I will provide
you with a description of why the terms of debate in mathematics education are worthy of
a dissertation length investigation. This chapter will also describe the research questions
as well as the key methodological decisions taken be me as both a reader of the terms of
the debate and as the writer of this report.

Viewed from outside, especially from a vantage point in a foreign institution, the
United States of the 19903 appear as an exporter of terms such as constructivist teaching
and learning, teaching for understanding, child-centeredness, pedagogical content
knowledge, and professional development schools.l Half understood, but largely
welcomed, these terms are reinterpreted and recontextualized in educational reform
discourses of other cultures without any reference to conditions of their production.

When these words and descriptions produced in one culture traveled to distant cultures—

 

' l have opened this conversation using these terms without highlighting their subtleties and multiple
meanings. For example, while all versions of constructivism see the student as active constructors of
knowledge, there are considerable variations among them (See, for example, Phillips, 1995; Wilson, 2003,
pp. 40-41). My purpose is not to survey these variations in meaning here, but to tell you that to an outsider
these terms come across not as caniers of these subtleties but as recipes for action.

as from the American continent to South Asian Sub-continent where I encountered them
as a mathematics teacher and a teacher educator—they come across as pretty secure
descriptions of teaching and learning generally, and of mathematics teaching and learning
particularly.2

As a physics major in my undergraduate and early graduate work, I was committed to
a view of mathematics—usually attributed to Galileo Galilei (1564-1642)3—as the
language in which the universe reveals its truths. My early encounters with a
constructivist view of learning and teaching did not interfere with this belief about the
nature of mathematics. I encountered terms mentioned above as exclusively about
teaching and learning and not as a set of statements about the nature of mathematics as a
discipline. For example, whereas I understood the term constructivism as a basis for
thinking about how children learn with implications about how they should be taught,
this understanding did not push me to think about what mathematics is or should be.

Not so, after arriving in the United States. Here, the constructivist discourse in the
talk about school mathematics was not just about the teaching and learning of
mathematics. It also impinged profoundly upon beliefs about the nature of mathematics.
In fact, mathematics assumed “many faces” with questions being raised about whether
mathematics educators and mathematicians talked about the same thing when they used

the term mathematics (Sfard, 1998, p. 491). As I was to learn later—and more

 

2 Gita Steiner-Khamsi has used the title “traveling reforms” to talk about the lending and borrowing of
educational discourses across the world (Popkewitz & Steiner-Khamsi, 2004).

3 Galileo thought of universe as “...written in mathematical language, and its characters are triangles,
circles, and other geometrical ﬁgures; without these it is humanly impossible to understand a word of it,
and one wanders around pointlessly in a dark labyrinth”(Galilei & Finocchiaro, 2008, p. 183). This
assumption continues to be widely held by the theoretical physicists (see, for example, Eugene, 1960;
Tegmark, 2007)

thoroughly through the analysis of terms of mathematics education debate in this study—
one could identify various different, but internally coherent,4 texts with each containing
different and conﬂicting descriptions of the nature of mathematics and its relationship
with teaching and learning. The groups adhering to one or the other text about
mathematics and its pedagogy seemed to be hard at work, at times with religious zeal, to
mark themselves off from what they were not (see, Wilson, 2003, pp. 48-49). This
difference appeared as a raging conﬂict in the policy arena.

The conﬂicts were not just in mathematics education. The ﬁgure of war appeared
frequently to refer to conﬂicts in education. Terms such as culture wars, social studies
wars, language wars, and reading wars were commonplace. But, being a mathematics
educator and a believer in the mainstream view of mathematics as certain and a priori, 5 I
found the phrase Math Wars, to echo Alan Schoenfeld, “oxymoronic, a category error”
(Schoenfeld, 2004, p. 253). Yet, here, the proponents and detractors seemed to have
locked horns with each other in this spectacular battle. Stepping into the territory of
mathematics education in the United States was much like becoming part of a territorial
jurisdiction, complete with its citizens, ﬁ’iends, and enemies.

Also, as an outsider, I did not immediately notice that American educational
discourse frequently appeared to be associated with a reform movement. Reforms and
Reformers are a ubiquitous feature of educational discourse in America in a way that they

seldom are outside of the United States. Before I came to the United States, I was not

 

4 My reference to internal coherence merely restates the observations made by other scholars as well that
formal documents have the appearance of a consensus, which masks differences if any between their
authors while appearing to be internally consistent. (See, for example, Wilson, 2003, p. 48.)

5 Hersh (1999) uses the term mainstream to speak of the pervasive view of mathematics as a bastion of
certainty and as consisting of preexisting mathematical forms. I will use this distinction again in Chapter 3.

accustomed to seeing the term reform used for matters considered to be traditionally
under the control of professionals (teachers, teacher educators). As a traveler from a
distant culture it took me some time before I realized that the work I was doing as a
professional mathematics educator in graduate school was also part of a reform
movement. I had done similar work and used the same progressive discourses and
materials, without ever thinking of myself as anything more than a professional
mathematics educator.

Insiders in the ﬁeld of education in the United States may not recognize it due,
perhaps, to their proximity with the term reform, that the commingling of professional
with evangelical as exempliﬁed by the insertion of reforms and reformers within the
discourse of professional ﬁelds such as education is a peculiarly American conjoining of
secular with sacred.6 My reference is not to the form and content of particular reforms.
Rather, I stress the availability of the terms “reforms” and “reformers” as locations to be
inhabited by individuals and groups. In this discourse, the individuals and groups come
across as reformers, anti-reformers, un-reformed, to-beqeformed, or reformed. American
scholarship on educational reforms typically raises questions about factors that promote
or hinder reform, but assumes the ubiquity of reforms without question. To me the idea
of reform is not something familiar that needs to be made strange in order to interrogate
it; it is already strange.

Why have there always been reforms of one kind or the other in the United States?

And why are the reforms and reformers prone to getting into trouble? These questions

 

6 The peculiar mixing of sacred and secular emerges in the unique form of political prose that the Sacvan
Bercovitch terms American Jeremiad (Bercovitch, 1978). The jeremiads produce reforms by directing
public attention toward risk and decline, and calling for reforms to stem decline. 1 will discuss this idea at
greater length in Chapter 6.

 

are not the central questions that guide the inquiry in this project, but they keep
shadowing me throughout this project, pushing me to account for them within the context
of mathematics education debates.

The recent conﬂicts in mathematics education are also battles over reforms. They are
not new disagreements over teaching and learning of mathematics but instead are the
most recent volcanic eruptions from a conﬂict simmering beneath the surface of
American education at least since the middle of the 19th century (Wilson, 2003, p. 5).
Below, I will ﬁrst describe the sequence of events in the math wars brieﬂy. This
description will be followed by my rationale for this study, a discussion of assumptions

that I take to this analysis, and ﬁnally an outline of the dissertation.

The Math Wars at a glance

I do not intend to provide a full description of the recent math wars, but will only
provide a chronology of events and some details that I think are important to develop my
argument. The reader may ﬁnd many important details of math wars omitted in this
drastically reduced summary. For this description and many of my data sources, I have
borrowed heavily from Wilson’s history of the math wars (Wilson, 2003).

“Math Wars” is the media title for the debates around school mathematics that began
in the wake of the calls for reforms made in the report A Nation at Risk (NCEE, 1983). I
will discuss this report and its implications for reforms in more detail in Chapter 6. Here
it suffices to note that this report is widely believe to have spawned the large scale reform
effort to improve student achievement in K-12 mathematics and other school subjects by

developing and implementing national standards—the so called standards-based reforms.

The National Council of Teachers of Mathematics (NCTM) took the lead in
publishing Curriculum and Evaluation Standards (NCTM, 1989) for K-12 mathematics.
Subsequently, the NCTM also published Professional Teaching Standards (N CTM,
1991) and Assessment Standards (NCTM, 1995).7 All of these three volumes were
addressed to teachers, school administrators, teacher educators, policy makers and the
wider public and they responded to the call for change with new ideas about appropriate
goals for school mathematics teaching.

From the perspective of this study, it is important to note that NCTM Standards
assumed that all students could be mathematically empowered. The standards deﬁned
the term mathematical power for all as “an individual's abilities to explore, conjecture,
and reason logically, as well as the ability to use a variety of mathematical methods
effectively to solve non-routine problems.” (N CTM, 1989). The term mathematical
power was repeated and reinforced in the 1992 Mathematics Framework for California
Public Schools8 as well as several other texts produced around the same time (see, for
example, NRC, 1989; NRC, 1993).

Development of mathematical power was seen as associated with an increased
emphasis on learning the meaning of operations, operation sense, mental computation,
estimation and the reasonableness of answers, use of calculators for complex
computation, and thinking strategies for basic facts. At the same time, the standards
deemphasized traditional activities—such as complex paper-and-pencil computations,

traditional algorithms, rote memorization of number facts—as expressions of

 

7 In this dissertation, unless stated otherwise, I will use the shorthand NCTM Standards to refer to
Curriculum and Evaluation Standards (1989).

8 In this dissertation, unless stated otherwise, I will use the shorthand 1992 Framework to refer to
Mathematics Framework for California Public Schools (1992).

mathematical power. Broadly speaking the NCTM standards, attempted to shift
mathematics curriculum and instruction away from ‘drill and kill’ and teaching of ‘paper
and pencil algorithms” towards a vision that placed more emphasis on communication,
problem-solving, and invented instead of already known or standard algorithms for
solving mathematical problems. The standards conceptualized mathematics as a product
of human activity and made the capacities to produce mathematics—the so called
mathematical powers—the objects of pedagogical intervention.

An important feature of the standards and documents associated with them was their
claim to be an expression of professional consensus by the mathematics education
community (Carl & Frye, 1991; Crosswhite, Dossey, & Frye, 1989; Frye, 1990).

The NCTM Standards were received warmly by the advocates and criticized
scathingly by the critics. Through its publication of 1992 Framework, California took the
lead in aligning its K-12 mathematics education policy with NCTM Standards. Fierce
debates followed the efforts to align the practices of teaching and learning in school
districts with the precepts of the NCTM Standards and 1992 Framework. The opponents
emphasized what was deemphasized by the reform texts; the traditional ways of teaching
and learning mathematics such as drill, paper and pencil computation, learning standard
algorithms, and automatic recall of basic mathematical facts.

These debates, which spanned most of the 19905, turned into a bitter controversy that
by the middle of 19903 had embroiled parents, math educators, mathematicians,
education reformers, local school boards, federal and state policymakers, and political
pundits in a struggle over the content and teaching of mathematics. The ﬁgure of war

began to dominate these debates as they turned more and more acrimonious. As Suzanne

Wilson puts it: “. . .the language of war crept into many debates about mathematics
education” (Wilson, 2003, p. 2). A scene dominated by the ﬁgure of war tends to divide
humans into warriors, peace makers, and spectators. The language in which the debates
were projected by the ‘warriors,’ especially those resisting the reforms, sometimes
mirrored the language of civil war. This language of war permeated up and down and
across the system, as evidenced from the calls by the US. Secretary of Education Riley
for cessation of hostilities and a return to “civil and constructive discourse” in his address
to the Mathematical Association of America in 1998.

Meanwhile, in California, the math wars led to a revision of 1992 Framework and
publication of a new set of standards to replace the NCTM standards. In 1999 the revised
Mathematics Framework for California Publics Schools together with revised Content
Standards (CDE, 1999) were endorsed by Califomia’s legislature.

Important from the standpoint of my project is the shift in vocabularies between the
1992 and 1999 Framework. The 1999 Framework replaced the language of
mathematical power for all with that of mathematical proficiency for all. It deﬁned
mathematical proficiency as a measurable construct whose presence/absence could be
indicated by the students’ scores on standardized tests. The new standards were an effort
at deﬁning in unambiguous terms what the students needed to know and be able to do in
order to be declared proﬁcient. They came across as precise statements of several
desirable strands for school mathematics and of precise benchmarks on each one of those
strands. Becoming mathematically proficient, according to the 1999 Framework,

required all students to cover all the benchmarks in all the strands at each grade level.

Thos shift and its implications for mathematics education are discussed more fully in
Chapter 5.

What needs to be mentioned in this introduction is that after the shift in vocabulary of
mathematics education in California, a different kind of consensus emerged and found its
way into the major NCTM and other publications. For instance, in the year 2000,
NCTM published the Principles and Standards of School Mathematics (PSSM) (N CTM,
2000). PSSM dropped the term mathematical power from its repertoire of terms in its
description of the principles and standards of school mathematics. Likewise, the most
recent NCTM publication titled Focal Points for Curriculum (N CTM, 2006) also makes
no reference to the term mathematical power. Meanwhile, several documents have
emerged at the national level reﬁning the concept of mathematical proﬁciency as the new

consensus (Ball, 2003; Kilpatrick, Swafford, & F indell, 2001; USDOE, 2008).

Understanding the terms of the debates

I began this chapter with an account of my encounter with a blizzard of terms—such
as constructivist teaching and learning, teaching for understanding, child-centeredness,
pedagogical content knowledge—in different contexts. The point of this brief narrative
of my encounter with similar terms in different contexts was to highlight that the meaning
of these terms—some of which, such as mathematics, we take for granted—are not stable
and same across different texts. I have come to believe that terms do not exist in a
necessary and inextricable relationship with particular worldviews. When the terms that I
have talked about, left their discursive places of origin and traveled to the distant lands
where I encountered them, they did not necessarily carry with them the trace of their

origin. In their new contexts, ideas such as child-centeredness or pedagogical

constructivism were not part of a progressive worldview as they are taken to be in the
United States. They remained the same, yet were different. In general, we may assume
that when terms travel from one discursive community to the other, they are
recontextualized in their new discursive habitats.

I believe it is also important to recognize that we use terms that are perpetually
constituted by the conﬂicts. When I began my work as a mathematics educator, terms
such as constructivism, teaching for understanding, mathematical empowerment, worked
as guides for our practice. But, as I have observed in the previous section, the discourse
deﬁned by these terms was challenged. For the workers in the ﬁeld of mathematics
education it is important to recognize that terms which shape their work are not always
expressions of scientiﬁc progress, but residues of rhetorical conﬂicts in education. This
project—much like cartography—is about understanding conﬂict by drawing maps of
speciﬁc relations between the terms the terms used in mathematics education in the
conﬂicting discourses.

The understanding of terms in conﬂicting discourses does not assume a necessary
relationship between the use of particular terms and the worldviews of the users as a
strategy to avoid what Wilson refers to as “facile simpliﬁcations.” (Wilson, 2003, p.xii).
In her documentation of the math wars in California and, more generally, in the United
States, Wilson suggests the need to avoid facile simpliﬁcations in thinking about reforms
and counter reforms. About the math wars, Wilson concludes that:

This wasn’t a debate between the Republican-Conservative-Traditionalist-
Positivist-Math-as-Skills-Direct-Instruction-Social—Eﬂiciency Camp and the

Democratic-Progressive-Constructivist-Interpretivist-Math-as-Conceptual-

10

Understanding—Child-Centered-Instruction-Democratic-Equality Camp (Wilson,
2003,p.l65)

I interpret Wilson as suggesting that it is not easy, not even desirable, to draw the
lines in ways that declare conservatives as against democratic equality or progressives as
essentially against social mobility. The facile simpliﬁcations tend to conﬂate political
and pedagogical orientations in misleading ways. An example from a Californian
mathematician documented in the Notices of the Mathematical Association of America
(MAA) is in order here. A mathematics professor who was concerned about some
mathematical aspects of the reform texts was reported as having been stunned when
invited to speak at a local Republican convention. “Mathematicians tend to jump into
such issues with both feet,” she says, “and then they ﬁnd themselves labeled as right
wing conservatives. And it’s pretty hilarious. I don’t know any mathematicians who are
right-wing conservatives” (Abigail Thompson, quoted in Jackson, 1997, p. 820). Yet,
when we direct our attention to competing goals of education and on conﬂict in terms of
worldviews signiﬁed by such terms as progressives and conservatives, 9 we foreclose

other ways of thinking about debates in mathematics education.

The Methodological Considerations

When exploring a particular cluster of terms, I do not work like a social scientist.
That is to say, 1 am not using a particular conceptual repertoire and applying it to conﬂict

in mathematics education. Nor am I developing a grounded theory of particular conﬂicts

 

9 For an explanation of conﬂicts in terms of worldviews, see, Apple, 1982, 1986, 1993; Shor, 1986

11

in the tradition of anthropologists. Rather, I read the texts that I chose to study in a
manner similar to literary analysis.

The argument of this dissertation is based on exploring the internal logic of particular
clusters of terms, and the relations between them. I expect my readers will apply a
similar mode of reading to follow my argument. To perform this analysis, I have chosen
the image of drama offered by Kenneth Burke (Burke, 1945) due to its emphasis on
conﬂict. Kenneth Burke’s dramatism—whose theoretical underpinnings I develop in
more detail in Chapter 2——-works for me as it takes ambiguity in the meaning of terms
describing human action as central to the conﬂict. As Burke puts it: “Since no two things
or acts or situations are exactly alike, you cannot apply the same term to both of them
without thereby introducing a certain margin of ambiguity, an ambiguity as great as the
difference between the two subjects that are given the identical title” (Burke, 1945, p.
xix). This dramatistic ambiguity is echoed in Sfard’s question about whether
mathematicians and mathematics educator mean the same thing when they use the term
mathematics (Sfard, 1998).

As someone who has acquired an awareness of the shifting meanings of the terms
through actually going back and forth between different linguistic communities, I saw
terms—such as the ones designating the subject matter of mathematics and its
relationship with teaching and learning mathematics-"within particular texts as the very
sites of conﬂict in the mathematics education debates. For example—as I mentioned in
my brief history of the math wars—the term mathematical power for all was a site of
conﬂict inasmuch as the opponents of the reform texts containing it sought to substitute it

with mathematical proficiency. An exploration of internal logic of the discourses reveals,

12

as I will show in more detail subsequently, that these terms are bound up with very
different notions of what constituted mathematics and mathematical ways of knowing.
Thus, considering the terms of mathematics education debates as Sites of the conﬂict, I
am interested in ways in which these terms function to provide us with a structure of
understanding the subject matter of mathematics and of motivation for doing, teaching,
and learning mathematics.

There are two kinds of texts examined in this dissertation. First kind consists of texts
that came into being long before the present conﬂict between reformers and counter
reformers, texts created by well-known mathematicians within a broad time frame
ranging from the 17th to late 19th century. These texts give us a chance to see some
continuities as well as discontinuities of perspective between the discourse associated
with some iconic ﬁgures in the history of mathematics and the current debates in
mathematics education.

In choosing these texts, I follow the distinctions between mainstream and humanist
mathematical texts offered by Reuben Hersh (1999) in his book What is Mathematics,
Really? Hersh’s distinctions work for me because they are not set in ﬁxed time periods.
What he calls Mainstream can be seen as distributed from ancient Pythagoreans to
modern day Platonist mathematicians (Davis & Hersh, 1981). As he puts it:

For the Mainstream, mathematics is superhuman—abstract, ideal, infallible, eternal.

So many great names: Pythagoras, Plato, Descartes, Spinoza, Leibniz, Kant, F rege,

Russell, Camap. . .Humanists see mathematics as a human activity, a human creation.

Aristotle was a humanist in that sense, as were Locke, Hume, and Mill. Modern

philosophers outside the Russell tradition—mavericks—include Peirce, Dewey, Roy

13

Sellars, Wittgenstein, Popper, Lakatos, Wang, Tymoczko, and Kitcher (Hersh, 1999,

p. 92).

Like Hersh, I do not assume progress from mainstream to humanist notions of
mathematics. I imagine these distinctions as existing in texts across times and spaces.
From a rhetorical standpoint, there seems to be no difference in the motivation of a
Pythagorean or a modern day mathematicians who believes—like Pythagoras—that
mathematical objects are not constructed by humans but exist independently of them.

Speciﬁcally, I explore the descriptions of the nature of mathematics in René
Descartes’ (1596-1650) Meditations, in archived letters of Bertrand Russell (1809-1880),
and the writing of Benjamin Peirce (1809-1880). I then make use of these descriptions to
think about the possibilities of acts of teaching and learning mathematics that become
thinkable in relation to those conceptions of mathematics.

The second kind of texts examined in this study are those that directly relate to the
conﬂict in mathematics education in the last two decades. As described earlier in this
chapter, I call these the reform and counter reform texts in this dissertation. The reform
texts examined in this dissertation are the NCTM Standards and the 1992 Framework.
The NCTM Standards and 1992 Framework may be imagined as a single textual complex
which was ultimately replaced by the California Mathematics Standards and 1999
Framework (CDE, 1999), which I refer to as the counter reform texts.

On what grounds can we compare the texts so disparate and distant in nature and
what insights can we gain from such comparison between the terms of debate in
mathematics education? To constitute the grounds of comparison between the texts I use

the image of drama as employed by rhetorician Kenneth Burke (Burke, 1945). Why

14

drama? Drama suggests organization of texts that deal with human actions and their
motivation in terms of descriptions of actors—in my case reformers, mathematicians,
counter reformers, and learners—as engaged in purposive acts in speciﬁc scenes using
agencies [means]. Using the ﬁgure of drama as an analytical device implies the
recognition that any text—whether Pythogorean, Cartesian, or Reform texts of the last
two decades—would have some description of background or scene against which
mathematicians, teachers of mathematics, or students [actors, that is] would appear as
engaged in purposive acts of doing, teaching, or learning mathematics.

When I am occupied with ﬁnding out what school mathematics might mean in
particular texts, I do not see—following Wilson’s phrase that I cited above and repeat
here—it as deﬁned primarily in terms of a conﬂict between Republican-Conservative—
Traditionalist-Positivist-Math-as-Skills-Direct-Instruction-Social-Eﬂiciency Camp and
the Democratic-Progressive-Constructivist—Interpretivist-Math-as-Conceptual-
Understanding—Child-Centered-Instruction-Democratic-Equality Camp. My reference
point becomes the drama rather than the particular worldview or a theoretical notion that
may be seen to be at the heart of it. By using drama as a reference point, I hope to Show
that the texts about mathematics attributed to Descartes, Peirce and Russell may be
centuries apart but may still be dramatistically similar inasmuch as they contain similar
descriptions of nature of mathematics and actions of mathematicians. Similarly,
mathematics education counter reform texts may bear no direct relation to Cartesian
texts, and yet be seen as enacting a similar drama. As far as the drama is concerned, what
matters is not the traditional label of the text but the ways it describes the context, the

agents, their purposive acts, and the means with which they act.

15

A rhetorical analysis of the texts that I describe above is offered in response to the
following question:

What can we learn about the terms of mathematics education debates of the 19905
by looking closely at some of the terms that ﬁgure prominently in key
mathematics texts, mathematics education reform texts, and mathematics
education counter-reforrn texts?

To respond to this I take the following set of questions to each set of texts”):

1. Scene: How is the context of mathematical activity described in classical
mathematical texts and in key reform and counter reform texts??

2. Act: How are mathematical acts described in these different texts?

3. Agent: Who is described as entitled to act, that is, to do mathematics in these
different texts?

4. Agency: What means are available to act, that is, to do mathematics in these
different texts?

5. Purpose: What is described as the purpose of mathematical activity in these
different texts?

The ﬁve categories mentioned before the statement of question are the elements that
organize descriptions in dramatism. Dramatism, as I have mentioned earlier, borrows the
metaphor of drama to organize descriptions of actors—in my case mathematicians,
reformers, counter reformers, learners, and so on—as engaged in purposive acts in

speciﬁc scenes using agencies [means] made available to them.

 

'0 The questions provided here are adapted from ﬁve questions that Kenneth Burke takes to descriptions of
human actions and their motivations. As Burke put it, “...any complete statement about motives will offer
some kind of answers to these ﬁve questions: what was done (act), when or where it was done (scene), who
did it (agent), how he did it (agency), and why (purpose)” (Burke, 1945, p. xv).

l6

Outline of the Dissertation

Chapter 2 provides a primer to Burke’s dramatism. My readers should know at the
outset that Burke uses terms of dramatism in ways that may sound unfamiliar. Chapter 2,
therefore, is my attempt to clarify the terms that will organize the analysis throughout this
dissertation.

In Chapter 3, I take the research questions mentioned above to selected classical texts
on mathematics. In particular, I reach for descriptions of mathematics and mathematical
activity in Rene Descartes’ meditations, in the writing of the 19th century American
mathematician Benjamin Peirce, and in excerpts from the letters of logician Bertrand
Russell. By examining the texts that are temporally distant from the reform and counter
reform texts, I want to make a rhetorical point about the vast differences in meaning that
inhabit the same words. Terms such as mathematics, mathematical powers, and
mathematics teaching and learning carry quite different meanings and implications when
articulated in recent reform documents and in these classical mathematical texts.

The drama of the classical mathematical texts unveiled in this chapter will become a
dramatistic reference point for comparison with the more contemporary discourses.

In Chapter 4 and 5, I will examine the reform (Chapter 4) and counter-reform
(Chapter 5) texts using the same set of questions. In all of these texts, I observe the ways
in which the texts constrain the meaning of mathematics and of what it means to learn
and teach mathematics.

Chapter 6 shifts the scene and views of both reforms and counter reforms as
competing acts on a stage prepared by the calls for reforms. The argument in this chapter

highlights the ways in which the language of standards embedded in the calls for reform,

17

and, by corollary, in the responses to such calls, works to undermine the progressive

reforms.

18

Chapter II: Understanding Dramatism

If man-as-symbol-user, then action; if action, then conﬂict, if conﬂict, then drama. And if drama,
then you must ﬁnd a critical language that deals with drama.
(Booth, 1974)

This chapter is designed to introduce the reader to key ideas of Dramatism (Burke,
1945), which is a technique of analysis of language and thought as basically modes of
action rather than as means of conveying information. Burke’s oeuvre is huge. In this
chapter, I will only be concerned with the ideas that I believe my readers must know
before they proceed to read this argument. Speciﬁcally, the following points will form
the body of this chapter:

1. I will describe assumptions about connection between language, conﬂict and
drama. Next, I will describe the ﬁve elements of what has been called dramatistic
Pentad. These are stated succinctly by Booth in the passage with which I begin
this chapter, and I will repeat and expand them in the ﬁrst part of the chapter.

2. I will describe the elements of Dramatism, which are mapped onto the set of
questions that I take to each text in Chapters 3, 4, 5, and 6 and the relations
between them.

In what follows, I will ﬁrst discuss the assumptions that undergird Dramatism and
then I will explain the technique of analysis Dramatism provides and the key terms that

will help in understanding the subsequent analysis.

19

Assumptions about Relations between Action, Conflict, and

Drama

Understanding the idea of action and how Burke distinguishes it from motion is
central to understanding Dramatism. Burke makes a distinction between Action and
Motion. For him the terms explaining motion describe unambiguously stated concepts,
such as, for example, those dealt with in physics. However, human action is different
from physical [according to Burke, even biological or behavioral] motion, and involves
ambiguities. For example, descriptions of human comedies, tragedies, and conﬂicts defy
unambiguous conceptualizations in the manner of science. The broad question to which
the language of Dramatism responds is posed by him in these terms: “What is involved,
when we say what people are doing and why they are doing it”(Burke, 1945, p. xv). The
answer to this question is not a theory of motives, but a rhetoric of forms. Five forms of
thought, he argues, are necessarily present in any description of action: “...any complete
statement about motives will offer some kind of answers to these ﬁve questions: what was
done (act), when or where it was done (scene), who did it (agent), how he did it (agency),
and why (purpose)” (Burke, 1945, p. xv).

Wayne Booth explains the need for Burke’s dramatistic method succinctly: “if man-
as-symbol-user, then action; if action, then conﬂict, if conﬂict, then drama. And if
drama, then you must ﬁnd a critical language that deals with drama” (Booth, 1974, p. 8).
This phrase contains the assumptions about language as a mode of action and the conﬂict
and drama that becomes visible when we assume language as such. Dramatism, as will
become clearer in the following discussion, cannot work without these assumptions. That

is to say, if we do not assume the language to be symbolic action, we lose sight of the

20

ambiguity and the possibility of conﬂict that characterizes drama. Below, I take each part
of Wayne Booth’s assertion and discuss the assumed relations between the action,

conﬂict, and drama suggested by it.

If Man-as-symbol-user Then Action

I will ﬁrst discuss what is entailed in regarding language as a mode of action. The
ﬁrst point in that direction is to recognize the ubiquity of language. As Burke puts it:
...can we bring ourselves to realize. . . just how overwhelmingly much of what we
mean by “reality” has been built up for us through nothing but our symbol
systems? Take away our books, and what little do we know about history,
biography, even something so “down to earth” as the relative position of seas and
continents? What is our “reality” for today (beyond the paper-thin line of our own
particular lives) but all this clutter of symbols about the past combined with
whatever things we know mainly through maps, magazines, newspapers, and the
like about the present? In school, as they go from class to class, students turn from
one idiom to another. The various courses in the curriculum are in effect but so
many different terminologies. And however important to us is the tiny sliver of
reality each of us has experienced ﬁrsthand, the whole overall “picture” is but a
construct of our symbol systems. To meditate on this fact until one sees its full
implications is much like peering over the edge of things into an ultimate abyss
(Burke, 1966, p. 5).
An example from sociologist Joseph Gusﬁeld will make Burke’s point clearer. While
conceptualizing his research on auto deaths and alcoholism, Gusﬁeld points toward two

distinct terminologies, the drinking-driver and drinking-driving. The ﬁrst, as he puts it,

21

“directs attention to the agent (driver) as the source of the act, the second frames the
experience as an event...The ﬁrst, drinking-driver, is a call to transform the motorist. The
second, drinking-driving, directs attention to the auto, the road, the event” (Gusﬁeld,
1989, p. 15). Extending this point to mathematics education debates, I take terms—
which will be discussed more fully in the next three chapters—such as mathematical
power and mathematical proficiency as associated with particular descriptions of learning
and teaching of mathematics.

Burke tells us that symbols, or terms as I call them, are all we have available to
apprehend things and motives. The identities we assume, the roles we accept, the reality
of the stars we gaze at, all reach us as words about reality. The support for this claim is
not restricted to Kenneth Burke. For example, in the 19505, linguists Edward Sapir and
Benjamin Whorf put forward their eponymous hypothesis that “language shapes
thought.” Also, the philosopher of language, John Searle agrees that language is
constitutive of thought:

In order that something can be money, property, marriage, or government, people
have to have appropriate thoughts about it. But in order that they have these
appropriate thoughts, they have to have the devices for thinking those thoughts,
and those are essentially symbolic or linguistic devices (Searle, 2006, p. 95).

Similarly, Ian Hacking claims that dividing people into categories in order to count
them (as in the case of censuses) created human kinds. He writes about the emergence of
ways of counting people, as constitutive of human types:

Even the decennial censuses in the different states amazingly Show that the

categories into which people fall change every ten years. This is partly because

22

social change generates new categories of people, but I think the countings were
not mere reportings. They were part of an elaborate, well-meaning, indeed
innocent creating of new kinds of ways for people to be, and people innocently
“chose” to fall into these new categories (Hacking, 2002, p. 49) .

Philosopher of linguistics, J .L. Austin (1962) made a similar distinction between
performative utterances and statements. Austin said, “the issuing of a [performative]
utterance is the performing of an action—it is not normally thought of as just saying
something” (Austin, 1962, pp. 6-7).

Hacking, Searle, Austin, and Burke all make the same point about language as
providing individuals with ways of acting—seeing, saying, doing, and thinking etc.
Finally, I exemplify the inﬂuence of terms that we use by applying this idea to my self-
perception as initially simply a mathematics educator, and eventually a reformer
mathematics educator.

The idea of language as doing something is also exempliﬁed beautifully in Garry
Wills’ rhetorical analysis of the Lincoln’s famous Gettysburg Address. Speaking about
the meaning of the Civil War, Wills writes, “The Civil War is, to most Americans, what
Lincoln wanted it to mean” (Wills, 1992, p. 35). And Wills attributes Lincoln’s success in
deﬁning the meaning of this conﬂict for future generations to the Gettysburg Address.
Lincoln’s speech, then, is action, or a performative utterance inasmuch as it alters [or
constructs] the reality of civil war for most Americans.

Let me elaborate this further by giving two examples from the mathematics education
debates in the United States. First, consider the ways in which language of counter-

reform texts worked to undermine the term “progressive” or reform by associating it with

23

disparaging terms such as faddist, dogmatic, unscientiﬁc, touchy feely, un-rigorous, fuzzy,
and permissive. At the same time, the term “traditional” in those texts was loaded with
an attitude of flexibility, balance, rigor, and discipline. While there is no necessary
relationship between any label and the terms used to deﬁne an attitude toward it, such
relationships are strategically constructed in the public discourse to achieve an effect,

and, therefore, ought to be examined as modes of action.

If Action then Conﬂict

The assumption about language as a mode of action can be extended to an
understanding of conﬂict in general, and conﬂicts in education particularly. For example,
consider two statements about mathematics and their signiﬁcance as modes of action.
When we want teachers to act in certain ways in their math classrooms, we say,
‘mathematics is reasoning.’ This statement, together with the rhetoric explaining and
supporting it, is a way of acting on teachers, schools, and curriculum. Conversely, when
we push our audience to accept mathematics as consisting of apriori objects then we push
them to act in a way that makes them appear to be in search of those apriori objects.
Conﬂict may be thought of as an engagement between these two performative utterances,
to use J.L. Austin’s phrase, to deﬁne the meaning of mathematics.

In these terms, Chapters 3, 4, and 5 in this dissertation delineate three different modes
of action and spell out the ways in which they attempt to deﬁne the meaning of
mathematics and its teaching and learning.

Two aspects of language as a mode of action contribute to conﬂict. The ﬁrst is the
need to come up with deﬁnitions to work with. For example, as I will consider in more

detail in the subsequent chapters, mathematics is conceptualized and deﬁned. Particular

24

conceptions of mathematics are then associated with descriptions of mathematical
practices or its teaching and learning. That is to say, a performative utterance or a mode
of action involves constructing deﬁnitions or, working with existing ones. Second,
deﬁnitions are inhabited with ambiguity. To understand this, consider the example of the
term standards in education. As I will discuss in detail in Chapter 6, what counted as the
meaning of term standards varied depending on its location in various competing
discourses. Different actors deﬁned standards based on the concepts available to, and
valued by, them. Below, I will discuss the conceptual connection between deﬁnitions,
ambiguity, and conﬂict, which becomes available when we think of language as action.
Definitions and Antagonisms: Deﬁning a term marks off its discursive boundaries.
Setting the boundaries implies an inside and an outside: certain meanings and
implications are included in the term — they are declared to be part of its meaning — while
others are excluded. Exclusions may be thought of as a prerequisite for conﬂict. It
follows that conﬂicts accompany the process of setting up clear deﬁnitions. Burke—and
almost three centuries before him, Spinozal l—acknowledged this “inevitable paradox” in

setting up deﬁnitions. As Burke puts it:

To tell what a thing is, you place it in terms of something else. This idea of
locating, or placing, is implicit in our very word for deﬁnition itself:
to define, or determine a thing, is to mark its boundaries, hence to use terms that

possess, implicitly at least, contextual reference (Burke, 1945, p. 24).

 

H This notion of any deﬁnition of a thing as making sense only in terms of what it is not, i.e. in terms of its
context in which it appears to be rooted is associated with Spinoza’s notion of “determination is
negation”(For a discussion on this, see, Burke, 1945; Morgan, 2002, p. 892).

25

Setting up deﬁnitions, standpoints, and perspectives is, as the above quote suggests,
also an act of marking oﬂ the boundaries, thus creating an inside and outside. This can set
up the inside and the outside as each others’ antagonist. The point I am making is that
deﬁnitions, standpoints, perspectives etc. become possible through deﬁning an outside.
As Burke puts it, “To tell what a thing is, you place it in terms of something else. This
idea of locating, or placing, is implicit in our very word for deﬁnition itself: to deﬁne, or
determine a thing, is to mark its boundaries, hence to use terms that possess, implicitly at
least, contextual reference. We here take the pun seriously because we believe it to reveal
an inevitable paradox of deﬁnition...” (Burke, 1945, p. 24).

Given above is indeed a very geometric description of deﬁnition. That is to say, just
as in coordinate geometry speciﬁc coordinates in space are deﬁned in relation to a frame
of reference which positions them in relation to a ﬁxed set of coordinates. We also place
the coordinates of language as action in relation the coordinates of language as not

action. Burke describes, and also complicates this relation as follows:

Contextual deﬁnition might also be called "positional," or "geometric," or
"deﬁnition by location. " The embarrassments are often revealed with particular
clarity when a thinker has moved to a high level of generalization, as when
motivational matters are discussed in terms of "heredity and environment," or
"man and nature," or "mind and matter," or "mechanism and teleology," where
each of the paired terms is the other's "context" in the universe of discourse

(Burke, 1945, p. 26),

Take for example the ﬁrst pair, “heredity and environment”. Burke seems to be

saying that these binaries reveal embarrassments, because of what I interpret to be,

26

mutual exclusions. That is, when one generalizes human behavior in terms of heredity,
excluding environment, then one forgets that environment is needed for the deﬁnition of
heredity. That is to say, it is only through excluding the environment [Read NOT
heredity] that heredity becomes possible. Hence, in a certain important sense, heredity is
because of what it is not. Likewise, the progressive/conservative binary may also be
treated similarly. Progressive deﬁnes or positions itself in relation to conservative, and
would lose its identity if there was no such reference point. That is to say, progressive is

deﬁned partly in terms of what it is not.

To wrap up this discussion on the effects of deﬁnition on the possibility of conflicts:
Deﬁning something is construed in this discussion as an action that works to set the
boundaries of a concept. Setting up boundaries, Burke lets us see, inevitably involves
exclusions and inclusions. The exclusions haunt the deﬁnition from the outside. It is
interesting that any deﬁnition—including deﬁnitions of identities such as progressives or
conservatives—becomes possible only by marking off, and thus deﬁning an outside. But
their condition of possibility also works to undermine them. The conﬂict is only to be
expected from what is excluded in the process of formulating a deﬁnition or an identity.
For example, while it is impossible to even think about progressive without marking it
off from conservative, the former is also threatened by the latter and vice-versa.

Since they are necessarily based on selections, deﬁnitions both reﬂect and deﬂect aspects
of what they attempt to deﬁne. Burke refers to these clusters of terms as terministic
screens. “Terrninistic screen” is Burke’s term for a sort of grid of intelligibility,
consisting of a cluster of terms, through which we make sense of the world. When it is

deﬁnite, it would, Burke argues, select some aspects of the reality it is describing and

27

deﬂect others. That is to say, Burke is denying the possibility of totalizing theoretical
constructs about human motives. In this dissertation, the descriptions of relations
between mathematics and its teaching and learning in mutually agonistic discourses are
treated as terministic screens.
Men seek for vocabularies that will be faithful reﬂections of reality. To this end,
they must develop vocabularies that are selections of reality. And any selection of
reality must, in certain circumstances, function as a deﬂection of reality. Insofar
as the vocabulary meets the needs of reﬂection, we can say that it has the
necessary scope. In its selectivity, it is a reduction. Its scope and reduction
become a deﬂection when the given terminology, or calculus, is not suited to the
subject matter which it is designed to calculate (Burke, 1945, p. 59).
Thus, conﬂict appears to follow from earnest efforts to deﬁne, reﬁne, and perfect a
discourse, which ends up in exclusions and inclusions. As Burke puts it, “we are rotten
with perfection” (Burke quoted in, Booth, 1974, p. 12). I interpret “rottenness” here to be
indicating the ultimate paradox of positioning oneself rigidly and attempting to perfect
that position. It reminds me of the plot of the movie Quills (2000). The plot [or scene to
use the language of Dramatism] contains the famous French character Marquis de Sade
held as an inmate in a French insane asylum in the 17908 and his reformer, a priest. The
scene unfolds as a long engagement between the reformer and the to-be-reformed. But
the perfect act of reform is symbolized by elimination of Marquis de Sade from the scene,
signifying the elimination of all the evil he stood for. However, ironically, as the priest

eliminates de Sade, he himself assumes the character of de Sade. In the Burkean sense,

28

the priest “rots with perfection,” through a paradoxical operation involving the
elimination of what he loathed.

As mentioned above, the Chapter 3, 4, and describe three different modes of action.
By the same taken, they are also descriptions of three different ways of deﬁning
mathematics and its teaching and learning. The argument in these chapters will
emphasize these texts as presenting terministic screens about the reality of mathematics
and its teaching and learning. To summarize, the terministic screen is a sort of cluster of
terms that collectively afford as well as constrain the object of their description. In
chapters three, four, and ﬁve, terministic screens constitute the cornerstone of analysis.
In these chapters, I use the pentad to describe the cluster of terms for agent, act, scene,
purpose, and agency. These descriptions constitute the terministic screen because they
deﬁne the ways in which the drama of mathematics and its teaching and learning is
permitted to unfold. And by virtue of being deﬁnitive, they are also exclusive.

The dialectic of ambiguity and clarity: We deﬁne things because we want to be
understood clearly. Yet, the story of conﬂict, tragedy, and even comedy is one based on
misunderstanding, or understanding differently, the terms which are otherwise assumed
to be stated clearly. From the math wars we know that even the term mathematics, when
caught in two different discourses—that of mathematics and education—is rendered
ambiguous. The associated ambiguity is reﬂected in the following observation on the

mathematics education reform texts by the Berkeley mathematician Hsi-Wu:

The shock comes from the discovery that what passes for mathematics in these
publications bears scant resemblance to the subject of our collective professional

life. Mathematics has undergone a re-deﬁnition, and the ongoing process of

29

promoting the transformed version in the mathematics classrooms of K-14 (i.e.,
from kindergarten to the ﬁrst two years of college) constitutes the current

mathematics education reform movement (Wu, 1998, p. 1).

Viewed in the light of the above discussion on the relations between deﬁning as a
mode of action and conﬂict, Wu’s expression of dismay on not recognizing his
professional discipline in the deﬁnitions of reform texts is also a refusal of the system of
relations within the reform texts, which, according to him, shapes and deﬁnes what
mathematics could be. That mathematicians and mathematics educators do not talk about
the same thing when they refer to mathematics is also observed by some prominent
mathematics educators (see, for example, Sfard, 1998). When mathematics educators
deﬁned mathematics without including the perspective of folks like Hsi-Wu, ambiguity
ruled, and debates followed. Ambiguity, thus, has the power to move debates in action.
As my analysis in Chapters 3 and 4 will show, the term mathematics and the notions
about its learning and teaching mean different things, depending on who speaks it, to
which audience, in what context, and for what purposes.

So same terms may have different meaning because of being situated in two different
dramas. As Burke puts it: “Since no two things or acts or situations are exactly alike, you
cannot apply the same term to both of them without thereby introducing a certain margin
of ambiguity, an ambiguity as great as the difference between the two subjects that are
given the identical title” (Burke, 1945, p. xix). Below, I change this quote slightly to
illustrate the ways in which the assertions about ambiguity are relevant to my study:
“Since no two situations in which the term mathematics is spoken are exactly alike, you

cannot apply the term mathematics to both of them without thereby introducing a certain

30

margin of ambiguity, an ambiguity as great as the difference between the two subjects
that are given the identical title.” The utterance mathematics is ambiguous, and when
ambiguous it is conﬂictual. Action, drama, and conﬂict are brought into a relation
because of the contingent ambiguity that accompanies our efforts to deﬁne and clarify
concepts.

Based on the discussion above, I understand ambiguity as a suggestion that words
cannot shift from one discursive neighborhood to another without losing or changing
their meaning. Burke speaks quite dramatically of these one-to-many relationships
between the words and their particular investments by asserting that we do not even know
what was said when we hear someone uttering a word as simple as ‘yes’: “Let us suppose
that I ask you: “What did the man say?” And that you answer: “He said ‘yes.’” You still
do not know what the man said. You would not know unless you knew more about the
situation and about the remarks that preceded his answer... For there is a difference in
style or strategy, if one says “yes” in tonalities that imply “thank God” or .. in tonalities

that imply “alas!”” (Burke, 1966, p. 77).

Thus, Dramatism assumes, privileges, looks for, and describes ambiguities associated

with speciﬁc terms:

A perfectionist might seek to evolve terms free of ambiguity and in consistency
(as with the terministic ideals of symbolic logic and logical positivism). . .We take
it for granted that, insofar as men cannot themselves create the universe, there
must remain something essentially enigmatic about the problem of motives, and
that this underlying enigma will manifest itself in inevitable ambiguities and

inconsistencies among the terms for motives. Accordingly, what we want is not

31

terms that avoid ambiguity, but terms that clearly reveal the strategic spots at

which ambiguities necessarily arise (Burke, 1945, pp. xviii, italics mine).

If Conﬂict then Drama

So how does drama follow from the conﬂict? Drama happens on a stage, and
involves actors who appear to be acting purposefully. These actors, love, hate, kill, die,
laugh, cry, and do many other things when playing their part. Production of drama
requires presence of motives that must remain essentially ambiguous. For example, in
Shakespeare’s plays ambiguity is said to play an important role when the playwright
makes his characters say something whose signiﬁcance cannot be grasped with reference
only to the time of their utterance. “For what they say might have two meanings. The
one meaning which the speaker has in mind refers to the momentary situation, but the
other meaning may point beyond this moment to other issues of the play” (McDonald,
2004,p.51)

This theme about the function of ambiguity in conﬂict and drama will be revisited in
Chapter 6, where I will dwell upon the ways in which the utterances of the reform texts
respond to a momentary scene as well as to other moments in the historical scene. When
reformers uttered the term standards they responded to a current scene which called for
measurable standards as a large part of the educational reforms. However, they ﬁlled the
term standards with a meaning that went beyond the contemporary historical context,
thus making the term standards ambiguous. Conﬂict and drama followed, when different

people used the term standards but meant different things.

32

Ambiguity characterizes drama and sets it apart from what Burke terms scientistic
explanations. Scientistic explanations work to eliminate ambiguity. To the contrary,
drama thrives on ambiguity.

The dramatistic/scientiﬁc distinction is complemented by action/motion distinction.
In the spirit of deﬁnition by placement, Burke sets up Dramatism as a means of analysis
of human action, while leaving the analysis of motion to science. The concept of action
works to delineate motivated and purposeful human acts from mechanical responses to
Situations. As Burke puts it: “As for "act," any verb, no matter how speciﬁc or how
general, that has connotations of consciousness or purpose falls under this category.”
(Burke, 1945, p. 14). This Burkean distinction between motion and action, and
identiﬁcation of action with language and rhetoric is stressed in contemporary writings on
Burke. Robert Wess elaborates this distinction by linking motion to life and death as
biological: “Rhetoric is linguistic action, but language-users live and die in the biological
realm of motion” (Wess, 1996, p. 112).

In this section, I have argued that Dramatism becomes thinkable against a background
of assumptions about language as action, about the paradoxical nature of framing
deﬁnitions of ideas and ambiguity of terms and their clusters. As Burke puts it,
Dramatism as “a method of analysis and a corresponding critique of terminology is
designed to show that the most direct route to the study of human relations and human
motives is via a methodical inquiry into cycles or clusters of terms and their functions”
(Burke, 1968, p. 445).

To summarize, in this section, I described ways in which assuming language as a

mode of action helps us see conﬂict and debate as arising from attendant ambiguities.

33

To analyze the terms in mathematics education debate, I have used the ﬁve part
framework of Dramatism—also called the Pentad—which is outlined in Burke’s

Grammar of Motives (Burke, 1945). I will turn to its description in the next section.

Elements of Dramatism

At its simplest, a dramatistic analysis is built around a response to the following
broad question and its speciﬁc variations: “What is involved, when we say what people
are doing and why they are doing it?” (Burke, 1945, p. xv). By way of response, Burke

offers what he argues to be:

...basic forms of thought which, in accordance with the nature of the world as all
men necessarily experience it, are exempliﬁed in the attributing of motives. These
forms of thought can be embodied profoundly or trivially, truthfully or falsely.
They are equally present in systematically elaborated metaphysical structures, in
legal judgments, in poetry and ﬁction, in political and scientiﬁc works, in news

and in bits of gossip offered at random (Burke, 1945, p. xv).

The ﬁve elements of Dramatism are referred to as forms because of their emptiness,
generative principles of investigation because of their capacity to generate dramatistic
analysis, and sometime simply the elements of elements of Dramatism. For any particular

narrative these elements are signiﬁed by:

some word that names the act (names what took place, in thought or deed), and
another that names the scene (the background of the act, the situation in which it
occurred); also, you must indicate what person or kind of person (agent)

performed the act, what means or instruments he used (agency), and the purpose.

34

Men may violently disagree about the purposes behind a given act, or about the
character of the person who did it, or how he did it, or in what kind of situation he
acted; or they may even insist upon totally different words to name the act itself.
But be that as it may, any complete statement about motives will offer some kind
of answers to these ﬁve questions: what was done (act), when or where it was
done (scene), who did it (agent), how he did it (agency), and why (purpose).

(Burke, 1945, p. xv)

Act, Scene, Agent, Agency, and Purpose, then are the forms that form the system of
reasoning that Dramatism offers us. The next important point to explicate is that these
terms do not refer to ﬁxed categories. For example, an agent does not have to always be
a person, and agency does not always refer to some intrinsic ability to act freely in this
world. A somewhat longer quote ﬁ'om Burke provides us with a glimpse of the
malleability of these terms:

A portrait painter may treat the body as a property of the agent (an expression of
personality), whereas materialistic medicine would treat it as "scenic," a purely
"objective material"; and from another point of view it could be classed as an
agency, a means by which one gets reports of the world at large. Machines are
obviously instruments (that is, Agencies); yet in their vast accumulation they
constitute the industrial scene, with its own peculiar set of motivational
properties. War may be treated as an Agency, insofar as it is a means to an end; as
a collective Act, sub divisible into many individual acts; as a Purpose, in schemes

proclaiming a cult of war. For the man inducted into the army, war is a Scene, a

35

situation that motivates the nature of his training; and in mythologies war is an
Agent, or perhaps better a super-agent, in the ﬁgure of the war god.
(Burke, 1945, p. xx)
So although the analysis works with the ﬁve elements of drama mentioned above,
they are not used as ‘unvarying, frozen, literal categories. . .but as ﬂuid reagents,

”'z—to be explained shortly—for different problems. What

applicable in different “ratios
is one agent’s action is another agent’s scene. A given agent can be of someone else’s
agency—a tool to other ends—or he can be, again, a part of someone’s scene ”(Booth,
1974, p. 11). Also notice that I have used this malleability in the characterization of the
elements of pentad to organize this chapter. For example, to explain Dramatism--which
you may simply deﬁne as a method or technique of analysis—I have thought of it as an
agency. The assumptions about language as a mode of action, then, form the scene against
which the agency of Dramatism makes sense.
Dramatistic use of these terms is further exempliﬁed by the following quote in which
Burke is examining an excerpt from John Dewey’s Intelligence and Modern World.
...though Dewey stresses the value of "intelligence" as an instrument (agency,
embodied in "scientiﬁc method"), the other key terms in his casuistry,
"experience" and "nature," would be the equivalents of act and scene respectively.

We must add, however, that Dewey is given to stressing the overlap of these two

terms, rather than the respects in which they are distinct, as he proposes to

 

'2 I should acknowledge that Burke’s use of the term ratio will sound very unconventional to the readers
familiar with the term in mathematics. Ratio is not used here in its usual mathematical sense, but in the
sense of logic of the terms. While in mathematics ratio appears as a relation between two similar
magnitudes in respect of quantity, determined by the number of times one contains the other (integrally or
fractionally), Oxford English Dictionary also deﬁnes ratio as reason. Dramatism’s meaning of the term
ratio is closer to this latter sense.

36

"replace the traditional separation of nature and experience with the idea of
continuity. (Burke, 1945, pp. xxi-xx)
The sense in which Dewey might have deployed experience in relation to nature, then,

inserts them as act and scene respectively in the space of Dramatism.

The Analysis

A scholar using Dramatism to examine a text does more than simply decompose the
texts in terms of pentadic categories; he also uses Dramatism to develop insights about
relations between the various pentadic terms. I will elaborate these points through

examples of some texts.

Example of texts with stress on one or the other Dramatistic Element

Almost a year ago, as part of my many efforts to learn about the history of American
public education, I stumbled on an obscure text, written by a prominent name in
American educational history. I am speaking of William Bagley's book 'A century of
universal school' (Bagley, 1937). Bagley’s book was an appraisal of universal schooling
at the turn of the century, not just in the US, but also in other countries of the world. I
was particularly struck by a passage on Iraq. Bagley’s observations on Iraq pointed
toward a facet of Iraqi society that resonates so with current accounts of the tribal
conﬁgurations in Iraq:

Another serious educational problem in Iraq illustrates very clearly how
dangerous it is to assume that generalizations regarding human affairs can be
applied to all peoples. We hear a great deal today about the importance of

reducing the sentiments that attach to the idea of nationalism. But Iraq needs a

37

healthy growth of the feeling of nationalism as much as she needs literacy. The
loyalties of the people are tribal loyalties. Intertribal warfare has always been
taken for granted among the nomads. It has persisted over the ages--and for a
reason that is biologically sound. It is a means of keeping the population within
the limits of a very precarious food-supply. When the tribes settle, however, a
different situation develops; and yet, just as the powerful mores against the use of
water persist, so the tribal loyalties and enmities persist, making very difﬁcult the
enforcement of law and order—making impossible, indeed, an effective national
government on a democratic basis (Bagley, 193 7, p. 50).

On this view, the tribal motivations for intemecine warfare were explained in terms
that may be regarded as ‘natural.’ If we accept this account as a valid explanation of the
Iraqi situation, then the Iraqi people would come across as not really engaged in a tribal
warfare, but as merely moved to perform a natural function, that of keeping the
population in check in the face of a precarious food supply. This naturalist account,
then, does not stress Iraqi people as agents. Rather it stresses the ‘precarious food
supply’ as an element of scene that determines their motivation to kill. Furthermore, with
intemecine strife as part of the scene marked by precarious food supply, Bagley declares
a democratic government in Iraq as impossible. Thus, in this account, the precarious
supplies of food, the intemecine strife, and the impossibility of democracy are clustered
together in such a way as to stress the scene and construe the acts of as completely
determined by a scene characterized by a precarious food supply. The tribes do not act to
kill but follow a law of nature that explains conﬂict as a means to regain a balance

between the availability of food and size of population.

38

Contrast the above mentioned account of Iraqis with another account borrowed from
Kenneth Burke in Grammar of Motives (Burke, 1945) '3 :

Many people in the Great Britain and the United States think of these nations as
"vessels" of democracy. And democracy is felt to reside in us, intrinsically,
because we are "a democratic people”. Democratic acts are, in this mode of

. thought, derived from democratic agents, agents who would remain democratic in
character even though conditions required the temporary curtailment or
abrogation of basic democratic rights (Burke, 1945, p. 17).

Notice that, in contrast to Bagley’s text, this construes the scene [of democracy] as
emanating from the nature of the agents.

Burke extended this analysis to all major philosophical narratives of his time,
proposing that materialism, idealism, pragmatism, mysticism, and realism all feature one
or the other terms of the dramatistic pentad: The discourse of materialism—which
explains the universe sufﬁciently on the assumption of body and matter is aligned with
scene; Idealism—which accounts for universe as the work of reason and mind—features
agent”; Pragmatism—with its concern for instruments to act—features agency's;

Mysticism—with its concern for the valid ends for human existence—is aligned with the

 

‘3 The quoted passage does not present a theoretical position that Burke necessarily subscribes to. Rather,
he has used this passage as an example to clarify a distinction between the discourses that privilege agent
compared with the discourses that feature scene.

‘4 For details, see Burke (1945, p. 131 & 171).

‘5 Burke’s association of Pragmatism with agency seems due to pragmatists’ theory of knowledge as means
to desirable ends. He says: “In accordance with our thesis, we here seize upon the reference to means,
since we hold that Pragmatist philosophies are generated by the featuring of the term, Agency. We can
discern this genius most readily in the very title, Instrumentalism, which John Dewey chooses to
characterize his variant of the pragmatist doctrine. Similarly William James explicitly asserts that
Pragmatism is “a method only”” (Burke, 1945, p. 275).

39

element of purpose“; Finally, act is featured in realism.l7 This idea of prominence of any
one term from dramatistic pentad in major philosophical themes is important to my

subsequent analysis.

Concluding thoughts: Analyzing the Relations between Terms of

the Pentad

When a text stresses scene or agent—as in the case of Bagley’s texts from which I
quoted above, it also shapes the relations between various terms of the pentad. The
analytic stress, then, is not just on the terms of pentad but also on the relations between
them. Burke calls these relationships “ratios”. A ratio, according to Burke, is a formula
indicating a relation of precedence from one term to another. For example, a scene-act or
a scene-agent ratio gives priority to the scene, or background, in relation to the act or
agent. That is to say, the scene sets the parameters of what agents and acts can be.

So we cannot speak of an act or an agent, without also speaking of the scene.
Speaking in terms of ratios such as scene-act or scene-agent appears useful when
assigning relative weights to the elements of the pentad. The use of these ratios in
dramatistic analysis is exempliﬁed by the explanations for existence of democracy in

some countries in the above quote from Burke in the last section. Bagley’s text stressed

 

‘6 According to Burke, mystical philosophies appear in times of social chaos or confusion about human
purpose. “They are a mark of transition, ﬂourishing when one set of public presuppositions about the ends
of life has become weakened or disorganized, and no new public structure, of sufﬁcient depth and scope to
be satisfying, have yet taken its place. Thus, precisely at such times of general hesitancy, the mystic can
compensate for his own particular doubts about human purpose by submerging himself in some vision of a
universal, or absolute or transcendent purpose, with which he would identify himself” (Burke, 1945, p.
288).

'7 This association, according to Burke, is rooted in realism’s concern with actual embodying a sort of Act-
Actual ratio. Another example: if one replaces Act and Actual by God and Universe respectively in this
formulation, the real universe would appear to follow from a supreme act of creation—so it does in the
discourse of creationists.

40

scene more as compared with the act. By featuring the scene it set up a scene-agent and
scene-act ratio which worked to explain Iraqi people and their actions in terms of
contextual factors such as scarcity of food. On the other hand, the example that Burke
gives [which, I emphasize, is not his own theoretical stance] derives democratic acts from
the characteristics of an inherently democratic people. Notice that the text which stresses
scene deﬁnes acts in relation to the context [deﬁnition by placement] and the text which
stresses acts describes scene as derived from acts [deﬁnition by derivation].

In an important way, then, a dramatistic ratio signiﬁes a move from potential to
actual. As Burke puts it: “. . .a mode of thought in keeping with the scene-agent ratio
would situate in the scene certain potentialities that were said to be actualized in the
agent. And conversely, the agent-scene ratio would situate in the agent potentialities
actualized in the scene” (Burke, 1945, p. 262).

Insofar as men's actions are to be interpreted in terms of the circumstances in
which they are acting, their behavior would fall under the heading of a “scene-act
ratio.” But insofar as their acts reveal their different characters, their behavior
would fall under the heading of an “agent-act ratio.” For instance, in a time of
great crisis, such as a shipwreck, the conduct of all persons involved in that crisis
could be expected to manifest in some way the motivating inﬂuence of the crisis.
Yet, within such a “scene-act ratio,” there would be a range of “agent-act ratios,”
insofar as one man was “proved” to be cowardly, another bold, another
resourceful, and so on (Burke, 1978, pp. 332-333).

It is important to recognize that a ratio, in bringing together a potential-actual

relation, is not necessarily a cause-effect relation. As Burke put it: “It is not easy to know

41

just when one is deriving potentialities from actualities and the reverse” (Burke, 1945, p.
260)

Let me now brieﬂy allude to how these insights are put to use - and also complicated
- in the analysis of mathematics education texts. In Chapter 3, the texts that I examine
feature mathematics as scenic thus setting up a scene-agent and scene-act ratios. That is
to say, mathematicians and their mathematical acts are completely determined by the
scene. When mathematics is construed to be a priori, all the agents can do is discover it.
However, if we reverse these ratios, as it happens in the texts examined in Chapter 4, we
will also need to reconﬁgure the nature of mathematics to keep it coherent with the act-
scene ratio. Mathematics, in such a text, will need to be conceived as being constructed

instead of being out there.

Throughout this dissertation, the analysis involves exploring the consequences of
shifting the values of the dramatistic place holders, i.e. the scene, agent, act, agency, and
purpose. What changes in the relations between agent, acts, agencies, and purposes come
into play as the scene shifts? The analysis will sort the texts in terms of the dramatistic
categories ﬁrst, examining the ways in which these categories relate to each other in
particular texts, and answering the question about what insights can be developed about

mathematics and mathematics education based on these relations.

42

Chapter III: Mathematics, God, and Pedagogy

 

 

 

 

 

 

 

 

 

 

 

 

Dramatistic Classical Texts Reform Texts Counter-Reform
Pentad Texts
Scene Mathematics Mathematics Standards
Education—Based
on NCTM
Standards
Act Following, Constructing Following,
Reading, Mathematics Reading,
Discovering demonstrating
mathematics ﬂuency
Agent Mathematicians Children, Teachers Children
Agency Mathematical Mathematical Mathematical
Powers Powers proﬁciency
Purpose Illumination by Empowered Numerate Citizenry
natural light of citizenry
reason

 

Table l: Pentad at a Glance—Focus on Classical Texts

 

Reuben Hersh (1999) in his book What is Mathematics, Really? identiﬁes two

parallel descriptions of mathematics, labeling them Mainstream and Humanist:

For the Mainstream, mathematics is superhuman—abstract, ideal, infallible,

eternal. So many great names: Pythagoras, Plato, Descartes, Spinoza, Leibniz,

Kant, Frege, Russell, Camap. . .Humanists see mathematics as a human activity, a

human creation. Aristotle was a humanist in that sense, as were Locke, Hume, and

Mill. Modern philosophers outside the Russell tradition—maverickS—include

Peirce, Dewey, Roy Sellars, Wittgenstein, Popper, Lakatos, Wang, Tymoczko,

and Kitcher (Hersh, 1999. p. 92).

43

Among the humanists, Hersh also includes psychologist Jean Piaget, and mathematics
educators Paul Ernest and Anna Sfard. Some mathematicians have also made it onto
Hersh’s list of humanists, prominent among them Henry Poincaré and George Polya.

From these lists we may make two important observations. First, the mainstream
description of mathematics extends all the way from Pythagoras to our times (Davis &
Hersh, 1981; Sfard, 1998). Second, that the mainstream/humanist distinction also
suggests a schism between most mathematicians and most mathematics educators.

Hersh says that these descriptions of mathematics are not just diachronic. That is to
say, we cannot assume a linear progress from a mainstream view of mathematics to a
humanist view’of mathematics. Rather, the mainstream and humanist views exist side by
side. As someone who has moved past all of these discourses, while changing careers
from physics to mathematics education, I agree! Mathematics has changed its meaning
for me several times in a lifetime. As a student in elementary and high school, I
experienced mathematics as an abstract and meaningless but strictly rule-driven play of
symbols. Formalists, I learned later, have something similar to say about the nature of
mathematics. As a university student and later a teacher of physics, I experienced
mathematics as a collection of mysterious objects, which, if applied appropriately and
wisely, spoke, somehow, nothing but truth about the physical universe at scales
unimaginably large and small. As a mathematics educator, I came to see mathematics as
a product of human activity and socio-culturally constructed.

The mainstream and humanist approaches to mathematics practices are usually so
insulated that each appears distant to the other. That is, as a physicist, I had no clue of, or

even felt the need for a socio-cultural dimension of mathematics. But when working as a

44

teacher and mathematics educator, I came to work within a perspective on mathematics
grounded in the context of schooling. While the classical perspective on mathematics—
mainstream, to use Hersh’s phrase—projected mathematics as a bastion of absolute and
objective certainty, in the socio-cultural perspective—humanist in Hersh’s terms— as
Sfard puts it, there was “no absolute truth any longer” (Sfard, 1998, p. 491). Ultimately
on the educational landscape the distinction between the humanist and mainstream plays
out, as I will discuss in more detail in Chapters 4 and 5, in terms of mathematics as
invention and mathematics as discovery, with the former inscribed in the reform and the
latter in the counter-reform texts.

The difference between the mathematicians’ and mathematics educators’ descriptions
of the nature of mathematics has assumed a renewed importance in the wake of recent
mathematics education debates, and hence are the focus of this dissertation.

This chapter explores the descriptions of the nature of mathematics in some selected
mainstream texts, organizing these descriptions in terms of the ﬁve elements of Burke’s
(1945) dramatism, each deﬁned by the following questions:

1. Scene: How is the context of mathematical activity described?

2. Act: How are mathematical acts described?

3. Agent: Who is described as entitled to act, that is, to do mathematics?

4. Agency: What means are available to act, that is, to do mathematics?

5. Purpose: What is described as the purpose of mathematical activity?

The Texts: To explore the answers to these questions, I do not survey all the texts

produced by all the authors in Hersh’s list, and I include one mathematician—the 19th

45

century Harvard mathematician Benjamin Peirce's—who, curiously enough, has not
made it to Hersh mainstream listings. SpeciﬁCally, I explore the writings of Rene'
Descartes (1596-1650) because of a widely held perception about him as the father of
modernity, Benjamin Peirce (1809-1880), and Bertrand Russell (1872-1970). While
Descartes may not need much of an introduction here, Benjamin Peirce is not a name that
one frequently encounters, as is evident from his absence Hersh’s list of mainstream
mathematical thinkers. However, given his half a century long tenure at Harvard, Peirce
was uniquely positioned to inﬂuence American mathematics. Peirce was Hollis Professor
of Mathematics at Harvard University from 1833 until his death in 1880. He also served
as president of the American Association for Advancement of Science for seven years
from 1847 to 1954. Historian Daniel Cohen sums up Peirce’s inﬂuence on the American
mathematics community as follows:

Holding a position at Harvard for a pivotal half-century in which the university
shiﬁed its character away from clerical training toward a new research model, he
trained and guided the work of two generations of mathematicians who would go
on to train and guide countless others. Personally Peirce was responsible for
numerous theories and applications, and his linear Associative Algebra (1870)
was a landmark both for its new form of algebra and for its statements regarding
the nature of mathematics itself (Cohen, 2007, p. 42).

Bertrand Russell is usually known to the world of philosophy of mathematics as the
logicist par excellence. His monumental work with Alfred North Whitehead (1861-1047)
Principia Mathematica (Whitehead & Russell, 1997) is usually known as an attempt to
ﬁnd logical foundations to all mathematical truths, i.e. to reduce mathematics to logic.

However, for the purpose of analysis in this chapter, I was interested in Russell’s

 

'8 Benjamin Peirce was father of pragmatist philosopher Charles Sanders Peirce.

46

statements about the nature of mathematics. I have used what I found in Selected Letters
of Bertrand Russell (Grifﬁn, 2002).

The particular texts in which I searched for statements about the nature of
mathematics, mathematical practices, and mathematical purposes are Descartes’ Fifth
Meditation as well as the documented objections to it. For Descartes’ writings, I have
consulted Cottingham’s compilation of Descartes’ philosophical works (Descartes, 1984)
and translations of Descartes by Wollaston (Descartes, 1960). I access Benjamin Peirce’s
ideas through his book Ideality in Physical Sciences (Peirce, 1881) and his textbook
Linear Associative Algebra (Peirce, 1882). I have also looked at the writings of Peirce’s
colleagues and students, and drawn upon a recent historical survey of the connections
between religious faith and mathematics in 19’h century (Cohen, 2007), looking for clues
to answer the questions I posed above.

Below, I foreshadow the argument that I develop in the rest of the chapter.

Scene: The nature of mathematics in both Descartes and Peirce can be read as

grounded in a peculiar mix of theology and intellect, which renders mathematics as

existing in a heavenly realm in the mind of God. These mathematicians’ descriptions
of mathematics stress scene—the existence of a preexisting mathematical landscape.

Acts like discovering, reading, following: When scene is a priori mathematics, acts

are described by the dramatistic scene-act ratio. That is to say, if the scene is already

given, then acts cannot construct it. Thus the mathematician is positioned in a math
land in whose construction he does not play any role. Contrast this with the Hersh’s
humanists whose description of mathematics as a human activity will need a reversal

of this ratio.

47

Agent: The mathematician comes across as a seeker in the math land, or a prophet

receptive to the mind of the God.

Agency: Powers to practice mathematics lie beyond normal human faculties. If

mathematics exists in the mind of God, the powers to do mathematics are like spiritual

powers needed to provide access to the mind of God.

Purpose: To know the mind of God

Examining the implications of this description for teaching and learning of
mathematics, I argue that mainstream mathematics is incompatible with the humanist
mathematics practiced by math educators. That is, if we accept the arguments of
Descartes, Peirce, and Russell, mathematics learning and teaching cannot be thought
about in the ways in which we educators think about it in the public school contexts.
When mathematical truths exist in the mind of God as priori truths, the capacity of
individuals to do mathematics, the so-called mathematical powers must also be
understood as gifts of God.

This point may be illustrated metaphorically by considering a mountain range as a
scene, climbers as agents, and climbing as an act. Climbers do not invent or construct
the peaks that they scale. We believe that the job of climbers is to ﬁnd the best routes
and use them to reach the summits. Once a particular summit has been scaled, the beaten
paths come on record, and become available for future generations of climbers. These
climbers may also discover new, more efﬁcient, and safer paths to reach the summit.
Yet, they, and those before or after them, cannot change the shape of the mountain.

Like the mountain range and its many summits, mathematics comes across in the

classical texts as a priori. Just as the mountain range as a scene contains the summits

48

and the climbers, as well as their attempt to scale the summits, mathematics also works as
a scene. The texts examined in this chapter are those that assume the existence of a
priori math world—a universe populated with mathematical forms. On this view, the
mathematicians act—that is, they do mathematics—to gain access to the pre-existing
world of mathematics. Since the world of mathematical forms pre-exists us, it constrains
what can or cannot be done as mathematics. In these texts the preexisting world of
mathematics is described in two ways. First, in the Cartesian Meditations mathematics is
conﬂated with the mind of God thus commingling religion and mathematics. Second, in
the Platonist texts (Russell) it is construed as existing in a metaphysical realm
independent of mental and physical reality.

The idea of mathematics as conveyed in these texts can be analyzed according to
Burke’s pentad as follows: The mathematician [agent] reads the mind of God or
discovers mathematical objects, or follows the paths already tread by other
mathematicians, in a preexisting mathematical landscape [scene]. Since mathematics is
either in the mind of God or existing in a metaphysical math world, the powers to do
mathematics come across as abilities to read, comprehend, and follow the mind of God.
Those with the gift are the elect who can gain to access the math world. Such
speciﬁcation of the nature of mathematics implies an elitist approach to knowledge
focused on identifying the gifted. I make no claims that such is the perspective of all
contemporary mathematicians. I also make no claims about whether such a perspective is
inherently good or bad. Rather, I perform this rhetorical analysis in order to help identify
a cluster of terms—that is, a terministic screen—that is just as internally coherent as the

one that motivates us mathematics educators, teacher educators, and reformers.

49

However, the mainstream conception operates with a set of truths about mathematics that
is different from that of the humanists, and by implication their respective assumptions

about teaching and learning are also different from one another.

The Mountain Range: Mathematics as the Scene in Classical
Texts
There are striking similarities between the perspectives of René Descartes (1596-

1650) and Benjamin Peirce (1809-1880) inasmuch as both preserve the nature of
mathematical reason as the God-given light of the mind and the connection of this form of
rationality with the heavenly and divine. Both construe mathematics as a priori and
divine. By virtue of being a priori and divine this perspective frames mathematics not
just as preexisting but also as containing the totality of mathematical objects that can be
accessed by the mathematicians. It is this description of mathematics that, I argue,
provides it with the connotation of scene in the sense in which Burke uses the term, as
containing both the agent and the act, and when mathematics is the scene—the given—
then learning mathematics becomes a discovery process, and the role of the teacher
becomes that of forebear.

Recall from Chapter 2 that the scene contains the agents and their acts. Furthermore,
there are narratives that stress the scene, while there are others that stress agents. In a
particular narrative on the nature of mathematics that stresses mathematics as a scene, we
will expect the terms for mathematicians and mathematical practice to exhibit scene-
agent and scene-act relationships. What a mathematician can or cannot do will depend

on what kinds of agents and acts can be contained in the scene without destroying the

50

coherence of the narrative.19 For example, if the narrative describes mathematical objects
as existing independently of the human acts, then mathematics must be discovered;
mathematics can never be invented in such a narrative.

In the texts of Descartes and Peirce mathematics comes across as belonging to a
divine realm. This may be as surprising to others as it was for me. The name of
Descartes is usually spoken in relation to rationalism. Rationalism, rationality, and
reason have come to be understood as in opposition to religious dogma. When Descartes
is read as the father of modern rationalism, the relation of the divine and religious with
Cartesian reason is not remembered. Reading mathematical rationality as at the roots of
the modern scientiﬁc enterprise, which boasts to be secular, obscures the cordial
relationship that mathematics and religion enjoyed until the late nineteenth century. It
was signiﬁcant that I could recall nothing from my early encounters with mathematics
that suggested a deep relation between mathematics and divine. Yet, the relationship
between mathematics and divinity remained alive and well until very recently (Cohen,
2007). Professionalization is a kind of secularization, and at the end of the nineteenth
century, the professionalization of mathematics served to sever the link between religion
and mathematics (Cohen, 2007).

Yet, as expressed through the claims earlier, Platonism reemerges from this severing
of the link between religion and mathematics. Theologizing mathematics had only
located it in the mind of God. Modern Platonism released it from God’s house only to
relocate it (at least for some mathematicians) in some a metaphysical realm. As Hersh

observes, “the trouble with today's Platonism is that it gives up God, but wants to keep

 

'9 Burke uses the term narrative here where others might use the terms philosophy or epistemology.

51

mathematics a thought in the mind of God” (Hersh, 1999, p. 135). A dramatistic analysis
is useful for educators because it helps us see that substituting the theological basis for
mathematics with a Platonist base is not likely to change the scene-act or the scene-agent
ratios. That is to say, whether the mathematical forms are creations of God or they are
assumed to exist in some a priori realm, the key mathematical act remains discovery.

Let me begin with an analysis of selections from Descartes’ Meditations (Descartes,
1960). Throughout this analysis, I focus on the ways in which, from a dramatistic
standpoint, the Cartesian Meditations talk as not as if mathematics were an act, but rather
as if mathematics were the scene in which actors might take a position.

René Descartes and mathematics as scene. I read Descartes’ perspective on
mathematics through a reading of his Fifth Meditation.20 Descartes’ famous argument
comes across in four moves. First, he isolates the properties of the sensible world that are
susceptible to proportional reasoning, the most signiﬁcant being the discreteness (lending
itself to counting), extension (lending itself to the measure of degree of extent), and
duration (lending itself to the measure of change), and calls them the ‘quantiﬁable
dimensions.’ Second, he invokes the idea of reason, which he also calls natural light of
mind, as capable of illuminating the quantiﬁable dimensions. Third, after identifying the
particular geometrical forms in the sensible world he decides that he did not invent this
form, and that it existed a priori in the deepest reaches of his soul or Cogito, as he calls it
Cogito is mind or spirit, which contains ideas as forms; and the cogito is separate from

the body and sensations. Cogito, and not the world of sensations, becomes the spring

 

20 In the F ifth Meditation Descartes attempted a mathematical proof of the existence of God. He based this
truth on the geometry of the triangle. The certainty of triangle was used in this argument to imply the
certainty of religion.

52

from which the dimensions emanate. Fourth, he decides that since the form was already
there and it could not have been invented by him, that there must be a perfect being who
knows, and that Cogito is but the extension of that perfect being. This is the meaning of
“cogito ergo sum.” Let me retrace these moves in some more detail.

The dimensions (sometimes called essences) were written in Descartes’ text as
quantitative, or quantiﬁable, properties of matter, the magnitudes and dimensions that
could ultimately be isolated and subject to analysis by mathematical reason. Number
could be assigned to counting parts, ﬁxing a degree of extent, and assigning duration. As
Descartes put it:

Now, in the ﬁrst place, I have a distinct image of that quantity which philosophers
commonly call continuous quantity, the extension, that is to say, in length,
breadth, and depth, of that quantity, or rather of the thing to which quantity is
attributed. In addition, I can count its several parts, attributing to each various
sizes, ﬁgures, locations, and movements; and, ﬁnally, I can assign duration, in
varying degrees, to these movements (Descartes, 1960, pp. 144-145).

If continuous and discrete quantities are the dimensions, or quantiﬁable attributes of
things in the world, then what might their source be? Descartes locates the source of
these attributes of extension and of magnitude in the Cogito—the thinking substance:

And all this I know not only distinctly, when I consider it in general, but I have
only to apply my attention to become aware of a host of particulars regarding
numbers, ﬁgures, movements, and so on, of which the truth appears with so much
evidence, and seems so connatural with my mind, that it does not seem so much

that 1 am learning something new as that I am recalling what I knew before, or

53

perceiving what I already had in my mind, although I had not yet turned my
thoughts in that direction (Descartes, 1960, p. 145).

The a priori nature of the mathematical truth is stated clearly in this passage. The
truth of, say, a geometrical ﬁgure such as a triangle, must preexist its demonstration; the
triangle exists before it can appear in theworld. It is already there, much before it is
mapped onto the materiality of a physical shape. A shape such as a triangle possesses its
own true and unalterable nature—immutable and eternal like the metaphorical mountain I
alluded to earlier. However, even if the truth of triangle precedes its mapping on a
sensible triangle, where did it come from? Who invented it? According to Descartes, the
triangle could not possibly be invented by him. Humans can only conceive of a triangle,
more or less distinctly. So even when it was possible to demonstrate various properties
of the triangle, such as its three angles and their sum as equivalent to the sum of two right
angles, and so on, Descartes regarded these properties to be contained in the true and
immutable form of triangle. As he put it: “when I ﬁrst imagined a triangle, I had no
thought of these properties, which cannot therefore have been invented by me”
(Descartes, 1960, p. 145).

In sum (l) then, the properties of which the ideas of physical things are composed,
namely extension, shape, position, and movement, cannot exist formally in us since we
are merely a thinking being. Those ideas are the forms, “the garments,” as he puts it, “in
which substance is clad.” The form, existing independently of us, is supplied with
content from facts of experience to account both of a priority and applicability of

mathematics (Shabel, 2007).

54

Descartes’ account of mathematics then assigns a certain ideality and a priority to
mathematics. But there is another factor involved here. Descartes was a practicing
Jesuit, and he worked to make his philosophy rationally coherent with his religious
beliefs (see, e.g., Toulmin, 1990). This ideal and a priori nature of mathematics is
consistent with his religious beliefs. The apriority of mathematical truths follows from
their being “connatural” with his mind. That is to say, ideas belong to the realm of
Cogito (the spiritual world), and not the realm of the world, the body, or the sensations.
The expression and application of mathematical truths involves discovering and eliciting
the natural light of mind.

However, Descartes adds to the complexity of Cogito as he extends his argument to
prove the existence of God. Cogito in this sense may be read as relating God and Man,
and by the same token, the inﬁnite and perfect with the ﬁnite and imperfect, the same
way that scene is related to what it contains. The imperfect and the ﬁnite are contained in
the perfect and inﬁnite. God as inﬁnite and perfect contains what is ﬁnite and imperfect.
God becomes the scene of all scenes:

the idea of God is particularly clear and distinct, and contains [italics mine] in
itself more objective reality than any other, there is none that is more true in itself,
and less open to the suspicion that it is false. This idea, I say, of a sovereignly
perfect and inﬁnite being is wholly true. A pretense can be made that no such
being exists; there can be no pretence that its idea represents nothing real to me,
as I have said was the case with the idea of cold. And the idea of God, being

particularly clear and distinct, contains wholly within itself all that I perceive to be

55

real and true and as having some degree of perfection (Descartes, 1960, p. 128,
emphasis mine).

Finally, Descartes’ views on the relation between mathematics and God come out
even more explicitly in exchanges with those objecting to the claims in his Meditations,
chief among them the mathematician and French theologian, philosopher, priest, and
astronomer Pierre Gassendi (1592-1655). Gassendi was not happy with the idea of
granting to the triangle an immutable existence prior to its perception in the physical
world. Gassendi’s objection to the Fifth Meditation argues:

It seems very hard to propose that there is any ‘immutable and eternal nature’
apart from almighty God. You will say that all that you are proposing is the
scholastic point that the natures or essences of things are eternal, and that
eternally true propositions can be asserted of them. But this is just as hard to
accept. (quoted in, Descartes, 1984, pp. 221-222)

Gassendi then objects explicitly to the triangle as “out there” and argues that the
essence of a triangle is elicited from the experience of a triangle, an approach that
Descartes—and as I will show in the next section, also Benjamin Peirce——obviously
reject in forming a relation between mathematics and God. For Gassendi, the term
triangle is a mental construct which is put to use to sort triangles from other shapes in the
physical world, and as one it has been induced from material experience with triangles.
Gassendi explained his objection this way:

The Triangle is a kind of mental rule that you use to ﬁnd out whether something
deserves to be called a triangle. But we should not therefore say that such a

triangle is something real. For it is the intellect alone which, after seeing material

56

triangles, has formed this nature and made it a common nature. . .It follows that we
should not suppose that properties demonstrated of material triangles belong to
them because they derive them from the ideal triangle. Rather, they themselves
possess these properties in their own right, and it is the ideal triangle which does
not possess except in so far as the intellect, after inspecting the material triangles,
has attributed such properties to it, only to give them back to the material triangles
to again in the course of the demonstration. (quoted in, Descartes, 1984, p. 223)
Descartes, defends the immutability of mathematical truths in response to this

objection by claiming that the objection would have been worthy if he was claiming the

immutability of mathematical truths as apart from God:
Just as the poets suppose that fates were originally established by Jupiter, but that
after they were established he bound himself to abide by them, so I do not think
that the essences of things, and the mathematical truths which we can know
concerning them, are independent of God. Nevertheless I do think they are
immutable and eternal, since the will and decree of God willed and decreed that
they should be so. Whether you think this is hard or easy to accept, it is enough
for me that it is true (Descartes, 1984, p. 261).

Descartes reinforced this opinion in a letter to one of his contemporary French

mathematicians, theologian Marin Mersenne (1588-1648):
The mathematical truths which you call eternal have been laid down by God and
depend on Him entirely no less than the rest of his creatures. Indeed, to say that
these truths are independent of God is to talk of Him as if He were Jupiter or

Saturn and to subject Him to the Styx and the Fates. Please do not hesitate to

57

assert and proclaim everywhere that it is God who has laid down these laws in
nature just as a king lays down laws in his kingdom. (Descartes, Quoted in
Kenny, 1970, p. 693)

To summarize—using the dramatistic image of a stage, agent, act, purpose and
agency—Cartesian descriptions depict mathematics as a scene on which Descartes the
mathematician also appears as a worshipper, who does mathematics to simultaneously
connect with the mind of God through his notion of ego, and illuminate the workings of
the world by shining on it the light of mathematical reason.

Benjamin Peirce and mathematics as scene. For half a century (from 1833 until his
death in 1888) Peirce held the Hollis Professorship of Mathematics at Harvard
University. He also served as the President of the American Association for
Advancement of Sciences from 1852 to 1854. He was often called the “Father of pure
mathematics in America” (Cohen, 2007, p. 42).

Peirce was an impressive, inﬂuential, ﬁrebrand, and hard to follow. A witty New
York Times correspondent covering his presidential address at the sixth annual meeting of
AAAS in Washington, DC. in 1854, had this to say:

...the professor’s sentences were long and ﬁrll of metaphors, similes, allusions,
and the like, which no mortal man might hope to set down. They could be no
more sketched, than the shapes of a puff of smoke. They came out regularly and
beautifully, but they widened and enlarged, and they went up, and if you looked
too long at one you lost the next dozen. It was magnifying of geometry—a

gloriﬁcation of pure mathematics (Anonymous, 1854).

58

For Peirce’s views on mathematics, I consulted his books Ideality in Physical
Sciences (1881) and Associative Linear Algebra (Peirce, 1881, 1882) and also searched
for what was written about him by his contemporaries. However, while in the middle of
writing this chapter, I serendipitously came across a recent historical study of Victorian
mathematics by historian Daniel Cohen, which he aptly titled Equations from God
(Cohen, 2007). Given the analytical resonance between what I was trying to accomplish
in this chapter and Cohen’s work, I also draw heavily on his work to support my
argument.

To begin a discussion on Peirce’s ideas on the nature of mathematics, I ﬁrst draw on
his Harvard colleague, Andrew Peabody (1811-1893), a mathematician and Professor of
Christian Morals at Harvard. The following quote from Peabody, whose fundamental
premises are repeated in the writings of Peirce, stresses the a priori nature of
mathematical thought and its applicability to the physical universe. Peabody’s text is as
much an eulogy of Peirce, as it is a statement of belief about mathematics as the language
in which the universe seems to have been written. He says:

Two years ago, my colleague, Professor Peirce, who in his own department has no
superior among living men, delivered from this platform a course of Lectures, in
which be constructed a theoretical universe. He took his stand outside of the
visible creation, assumed merely the existence of brute matter and certain
fundamental mathematical laws, and determined by a masterly line of a priori
reasoning what the proportions and relations of a universe constructed in

accordance with those laws must have been. The result was the coincidence,

59

point for point, of this universe of theory with the actually existing universe
(Peabody, 1864, p. 196).

According to Peabody, proceeding deductively and step by step from a few ﬁrst
principles, mathematics recreates the universe, seemingly mimicking the work of the
Creator. What else would this coincidence of mathematically deduced universe with
“actually existing universe” signify, if not the privileging of a connection between the
realms of pure thought: between the Cartesian Cogito, and the divine?21

Peirce’s perspective, given in the following quotation from Ideality in Physical
Sciences is similarly poised toward the connection between mathematics as a priori:

Ideality is pre-eminently the foundation of the mathematics. Observation supplies
fact. Induction ascends from fact to law. Deduction, applying the pure logic of
mathematics, reverses the process and descends from law to fact [Italics mine].
The facts of observation are liable to the uncertainties and inaccuracies of the
human senses; and the ﬁrst inductions of law are rough approximations to the
truth. The law is freed from the defects of observation and converted by the
speculations of the geometer into exact form. But it has ceased to be pure
induction, and has become an ideal hypothesis. Deductions are made from it with
syllogistic precision, and consequent facts are logically evolved without

immediate reference to the actual events of Nature (Peirce, 1881, p. 165).

 

2' However, as I will argue later, the a priority of mathematics is not disturbed after the exclusion of God
from this perspective. Not surprisingly, the dualism that separates mathematics from the ‘real’ universe
continues to dominate ﬁelds such as theoretical physics. The effectiveness of mathematics in predicting
“real phenomena” is acknowledged and assumed by theoretical physicists, but also termed “unreasonable”
and mysterious. See, for example, (Eugene, 1960).

60

By accounting for the coincidence of thought with actual facts of creation, it may be
argued that Peirce believed his mathematical work to be an engagement with the
heavenly and divine:

“The loftiest conceptions of transcendental mathematics have been outwardly
formed, in their complete expression and manifestation, in some region or other of
the physical world. . .They are the reﬂections of the divine image of man’s spirit
from the clear surface of the eternal fountain of truth.” (Peirce (1854), quoted by
Cohen, 2007, p. 55)

This perspective on mathematics as the reﬂection of divine image was reinforced in
the nineteenth century by the successes of classical celestial mechanics: How could
mathematics be anywhere but existing in the mind of God if it could be used to predict
the existence of a planet in the farthest reaches of the solar system, starting from some
basic mathematical laws? Just to give an idea of the wide extent of the belief in the
connection between mathematics and God, I will quote brieﬂy Thomas Hill (1818-1891)
22, a Unitarian clergyman who served as president of Harvard University from 1862-
1868.

But in the pursuit of mathematical knowledge men began, at an early age, to
invent and investigate a priori laws, laws of which they had not received any
suggestion from nature. And the intellectual origin of the forms of nature was
made still more manifest when these a priori laws, of man’s invention, were, in

many cases, afterwards discovered to have been truly embodied in the universe

 

22 Thomas Hill is known to have had a lifelong friendship with Benjamin Peirce and shared his view of
the relationship between mathematics and divinity. Daniel Cohen speaks about Thomas Hill as a great
popularizer of Peirce’s work and an “exemplar of affinity between pure mathematics and idealist
Unitarianism in nineteenth century.” (See Cohen, 2007, p. 57.)

61

from the beginning; as, for example, Plato’s conic sections in the forms and orbits
of the heavenly bodies, and Euclid's division in extreme and mean ratio (Hill,
1874, pp. 4-5).

According to Thomas Hill, Newton and Descartes were led by nature’s own guidance

to their discoveries of calculus and algebra respectively, delightful discoveries indeed:
When we remember how intense the delight which man feels in the discovery of
mathematical truths. . .we may surely own, with gratitude, the marks of divine
wisdom and love, in this gift to man, of the power to penetrate space, and apply to
it the laws of time. (Hill, 1874, p. 21, emphasis added)

These mathematicians have inﬂuenced two generations of mathematicians directly,
and countless generations indirectly. I take the perspectives of Benjamin Peirce and
others like Thomas Hill and Andrew Peabody—all at Harvard University—on
mathematics to be stressing scene. Mathematics, under this perspective came across as a
preexisting expression of divine. It is transcendent and potentially capable of recreating
an entire universe. Only few, which have the God’s gift to man, or as Thomas Hill put it
in the above quote, the power to penetrate, could access mathematical truths and, of
course, take delight in this spiritual journey across the divine math world.

Russell’s math world. Bertrand Russell (1872-1970) is credited along with Gottlob
F rege and Alfred North Whitehead as a pioneering proponent of logicism—a story that
reduces mathematics to the terms of logic. Logicism also resonates with the works of

Descartes and Peirce on the nature of mathematics.

62

In a letter written to Helen Thomas23 in 1901, Russell expressed his out-of—this-world

conception of mathematics:

In the same passage Russell also betrays a longing for engagement with pure reason
and a feeling of shame in engagement with the transient things:

I have come to feel a certain shame in thinking of transient things, and to regard a
year spent, as this year has been, in human sympathy, as something weak and
slightly contemptible. But the life of pure reason remains a remote aspiration,
which (fortunately, you probably think) I do not ﬁnd myself attaining (Grifﬁn,
2002,p.218)

Russell’s description of the mathematical world as a cold, passionless, and eternal
realm of pure reason apparently puts him in the mainstream camp. However, unlike
Descartes, who is in conversation with seventeenth century Jesuits, Russell does not
connect mathematics to God. The common denominators between Descartes, Russell,
and Peirce are a priority and immutability of mathematical truths. The relations between
the elements of mathematical drama—i.e. the scene-act ratio discussed above—are the
same for all of these mathematicians. For all three of these mathematicians, mathematics
is a scene that can be discovered by actors. Mathematics as a scene cannot be invented,
and cannot be affected by actors.

This common denominator is important because it connects, as Hersh argues, all the

various shades of the mainstream in mathematics. So let us dwell on the ways in which

 

23 Helen Thomas was Carey Thomas’s sister, and was at Bryn Mawr College around the same time when
Carey Thomas was president of the college. Both sisters were a cousin to Bertrand Russell’s ﬁrst wife Alys
Pearsall Smith.

63

Descartes, Peirce and Russell are similar. Russell is working in a setting in which the
theological strand which connected mathematics and divinity, and which was prominent
in the works of Descartes and Peirce, had lost out to professionalization in the
mathematics departments (See, Cohen, 2007). However, as Russell’s description
suggests, this shift apparently only involved a disengagement of Descartes’ Cogito from
the divine24 and its re-engagement with Plato’s ideal world of mathematics that preceded
the conception of a monotheistic God. An important aspect of modern Platonism is that
unlike the reading-the-mind-of—God that characterizes Descartes’ and Peirce’s projects,
the modern Platonist is identiﬁed as being simultaneously God and human. On Russell’s
description, humans become Godlike when we think about mathematics, yet we must also
enter the perfect math world that other mathematicians have created. The earlier version
of Plato seems to have been separated from the modern version of Plato by two
interregna: Christianity (the belief that Christ is the coming of God into the world of
man) and the Age of Reason inaugurated by democratic revolutions in the West.

Whether it wells up from the Cartesian Cogito representing the natural light of mind,
or accessed from the independent abstract Platonist math world, mathematics is not

invented but can only be discovered in both of these views.

 

2" This disengagement is inscribed as a forgetting of the religious motivation of Cartesian rationalism. If
we see Descartes’ Cogito as the origin of modern science, then from our present perspective we do not
entertain the possibility that Descartes’ Cogito was conceived in response to questions about the existence
of God.

64

Climbing the Peaks: Mathematician (Agents) and Mathematical Practice
(Acts)

What do the conceptions of Descartes and later Peirce and Russell imply for the act
of doing mathematics?

First, the a priori nature of mathematical truths is a common thread in these texts. As
a physicist, I was initiated into such a conception of mathematics. All great equations in
physics were spoken about as discoveries. The list of these discoveries was long:
Newton discovered laws of motion; Hamilton discovered quatemions; Maxwell
discovered laws of electromagnetism; Einstein discovered relativity, Erwin Schrddinger
discovered wave mechanics, etc. Discover was the verb that described the act in this
discourse.

From the perspective of the dramatistic analytical scheme (and as I have said before)
the verb discover suggests a strict containment of the act within the scene, the so-called
scene-act ratio. The act of doing mathematics is completely contained in a mathematical
universe. Of course, one cannot discover a mathematical object that does not exist, even
though we only ﬁnd out after the fact that it did indeed exist.

Second, mathematics as a relation to divine, leads to the notion of acting as reading
the book of God. An exercise in pure thought, as proceeding from ﬁrst principles, and
deductively creating the universe, it signiﬁes that accessing mathematical language is

tantamount to reading the mind of God. Since mathematical conceptions are thoughts

65

from God’s mind, engaging in mathematics is tantamount to “adoration, praise, or
prayer.”25

Adoration, prayer, and praise are not the verbs that found their way in twentieth
century discourse of mathematical practices; as I have discussed earlier, the reference to
God was eliminated. The mathematician replaced God, indeed as suggested by Russell’s
quotation. For instance, when Russell enthusiastically assigned a combination of beauty,
an inhuman coldness, purity, sublimity, dignity, timelessness, and privileged access to
mathematics, he also placed the privileged mathematicians in a realm that had previously
belonged properly to God alone.

Russell the mathematician—one with the access rights to this territory—acts by
traveling around in this [pre-existing] mathematical universe. Like Descartes and Peirce,
doing mathematics on the stage so set is like ﬁnding one’s feet, and ultimately, one’s way
on this mathematical landscape. Russell, or any other mathematician for that matter, has
the license to be there, only to ﬁnd his way around and tell others what they found. The
agent does not act to create, but it is the scene—the timeless world of mathematics—that
determines what the agent’s act can be.

Russell’s description, like that of Descartes and Peirce, both affords, and limits, what
a mathematician could be, as someone endowed with powers—mathematical powers—to
follow the contours of the paradisiacal place called mathematics. Once again, discover,

not invent or construct, is the verb that describes his act. While the mathematician may

be seen as representing mathematics, the act seems to involve appropriately describing

 

2’ The quotation refers to a statement made by a friend of lchabol Nichols (1784—1859), a pioneering liberal
theologian who was also a mathematician, and a formative inﬂuence on Benjamin Peirce. For a detailed
exposition see Cohen, (2007, p. 51).

66

the maps he forms of a pre-existing math-world, thus conﬂating production with
discovery.

These aspects of mathematical acts are also exempliﬁed in the perspectives that
support theoretical physics and other mathematical sciences. Induction and empiricism
reign in the experimental sciences with their reliance on experiential; however,
mathematical physicists continue to preserve a belief in a preexisting mathematical
language in which the universe is written (See, for example, Tegmark, 2007).

This does not mean, however, that there are no mathematical texts other than the one
that belongs to the divine (or Platonist) realm. Of course, mathematical treatises are
made available to the students to understand and learn about the existing maps of the
math world. However, acting here also means reading and following the text.

I exemplify this aspect of mathematical acts by taking an example from a nineteenth
century English economist William Stanley Jevons (1835-1888). In an attempt to
translate a book written by his French contemporary Augustin Coumot, J evons reported
that he was unable to follow all aspects of Coumot’s analysis, and attributed this situation
to his own lack of mathematical powers (Fisher, 1898). If by power we mean an ability
to act, then a mathematical ability to act, in J evons’s text, is an ability to comprehend and
follow the mathematical reason. I read Jevons’ purpose—the desire to follow Cournot—
as akin to an urge to occupy the same vantage point as occupied by Coumot, from where
the latter illuminated the nature of economics with mathematical reason.

In sum, then, the mathematician and his mathematical practice is constrained by a
perspective shaped by the a priori nature of mathematics. Under Descartes, the a priori

nature of mathematics is elicited from Cogito, which is an extension of the divine.

67

Likewise, under Peirce, engaging with mathematics is engaging with the divine.
However, under the contemporary Platonists—as described by Davis and Hersh—the
engagement is with a mysteriously, but still a priori, math world. In all of these
perspectives, reading, following, discovering, comprehending are the terms that describe

mathematical acts.

The Power to attain Natural Light of Reason: The Agency and
the Purpose

Mathematical power is a term that I will discuss in more detail in the following
chapter as related to mathematics educators’ perspectives on mathematics. In the
classical texts, power seldom comes across as a term, but is, arguably, implied. In the
dramatistic pentad outlined above, agency is interpreted as the means that an agent uses
to act purposeﬁrlly. When mathematical objects appear as natural light of mind or as
inhabiting a Platonist world, existing a priori in both cases, the agency can be construed
as power to navigate this pro-existing math world, the privilege that Russell speaks about,
and the mathematical powers that Jevons bemoans as lacking.

The idea of power is also found in Descartes’ Meditations. Descartes’ rationalism
does not imply that the ultimate source of the “natural light of reason” is inherent in self-
contained human subjects. Cogito, as he puts it, is always up for growth in its quest to
approach the divine perfection. Power is what is needed to keep the Cogito in motion
toward the divine:

Perhaps I am something greater than I take myself to be; perhaps all these
perfections I attribute to the nature of a God, are in me potentially, even though

they have not yet been realized or shown themselves in act. For my experience

68

teaches me that my knowledge increases and grows more perfect little by little,
and I see no reason why it should not grow to inﬁnity and, being thus augmented
and made perfect, acquire for me all the remaining perfections that belong to the
Divine nature; and if I have this power, the power to acquire these attributes, that
should be sufficient for my own mind to produce the ideas of them (Descartes,
1960,p.128)

So Cogito becomes a vehicle to potentially access the inﬁnite wisdom of God only if
sufﬁciently powered.

Dramatistically speaking, then, this narrative speaks of doing mathematics as an
engagement with divine or with an independent and pre-existing (but unexplained)
ontological reality. On this view, when an agent (mathematician) acts (does
mathematics), it is an exercise of mathematical powers through the use of right methods
(Agencies) that lets him be at one with the 'paradisiacal' (as Russell would have us
believe) world of mathematics by discovering, reading, and following what is already

given.

The Relationship between Mathematics and its Teaching and
Learning in Classical Texts

At the risk of repeating, I have argued that in none of the descriptions given above—
Descartes, Russell, Peirce, Jevons, and Wigner—does the practice of mathematics come
across as an act of creation. Classical mathematicians do mathematics when they
discover, read, and follow either as yet unknown mathematical truths, or, as students,
those that are already known. Not enough data are available to reﬂect on the implications

of foregoing analysis for the teaching of mathematics. We do not know much about

69

Descartes’ or Russell’s teaching philosophies. Yet, dramatistic ratios can help us
complete the picture. The question is if mathematics is believed to be a preexisting
scene, then what could we say about teaching by imagining it in terms of a dramatistic
scene-act ratio?

Since the motivations for a speciﬁc kind of mathematical pedagogy are well
documented for Peirce and his colleagues, one way to answer this question would be to
see if Peirce’s teaching conforms to a scene-act logic with scene deﬁned by mathematics.
Peirce was well known as an eminent educational reformer (See, Cohen, 2007, p. 42).
SR. Peterson (1955) in an essay titled Benjamin Peirce: Mathematician and Philosopher
has documented Peirce’s disposition toward teaching. I will consider what has been said
about Peirce by his students. Through these views, teaching comes across as not about
making all students learn mathematics but about ﬁltering away those who cannot-—
perhaps reforms urged by Peirce at Harvard may be some of the ﬁrst sorting practices
based on elective courses in American education; learning appears as an exercise in
silently following the great master’s work; students appear in this perspective as either
gifted or not

Peirce was contemptuous of teaching any but a gifted few students. The reforms he
introduced at Harvard University were designed to offer mathematics to a select few. His
move to have mathematics made into an elective discipline for all the four years of
college at Harvard University is widely interpreted as a step to ward off the unworthy
from his classes:

One compelling motive for this action may have been his intense dislike of

teaching any but the most gifted students. When mathematics was made an

70

elective, the students stayed away in droves, and the mathematics department
became known as small, difﬁcult, and unpopular. Generations of unhappy
students have recorded what they have suffered at Peirce’s hands, a combination
of respect for his enthusiasm and genius with a total befuddlement as to what he
was trying to say (Peterson, 1955, p. 93)26.

Charles Eliot, the president of Harvard University from 1869 to 1909, and member of
the famous Committee of Ten, was Peirce’s student from 1849 to 1853. Eliot reminisced
about Peirce’s teaching in the following words:

He was no teacher in the ordinary sense of that word. His method was that of the
lecture or monologue, his students never being invited to become active
themselves in the lecture room. He would stand on a platform raised two steps
above the ﬂoor of the room, and chalk in hand cover the slates which ﬁlled the
whole side of the room with ﬁgures, as he slowly passed along the platform; but
his scanty talk was hardly addressed to the students who sat below trying to take
notes of what he said and wrote on the slates. No question ever went out to the
class, the majority of whom apprehended imperfectly what professor Peirce was
saying (Eliot, Lowell, Byerly, Chace, & Archibald, 1925, p. 2).

Peirce’s conduct as a teacher is consistent with the demands of the scene-act ratio.
He may be looked upon as a bad teacher when viewed from our perspective as educators.
But when mathematics is the scene, then it makes sense to narrate the accounts from
one’s adventures and discoveries in the math world, then to target individual leamer’s

capacities to learn mathematics. Peirce was not there to teach mathematics, but—as a

 

26 Also see Cohen (2007) for a similar description of justiﬁcation for declaring mathematics an elective
subject.

71

forbear—to provide images of the math world to those who were adequately gifted. As
Eliot wrote:
If a question [asked by a student] interested him, he would praise the 'questioner,
and answer it in a way, giving his own interpretation to the question. If he did not
like the form of the student’s question, or the manner in which it was asked, he
would not answer it at all, but sometimes would address an admonition to the
student himself which went home (Eliot, et al., 1925, p. 2).

The acts of the ‘unrepentant elitist’ as a teacher are consistent with a description of
mathematics as scenic. For the true believer in the connection between mathematics and
divine, reforming education meant getting rid of those without mathematical powers, not
inculcating them in those who did not possess them. Perhaps Peirce, even Descartes, and
Russell would be horriﬁed if we brought them into a room and posed to them the problem
of teaching mathematics to all children in the primary school. Slogans like
“Mathematical power for all,” might appear to them as incomprehensible gibberish.

If access to natural light of reason is a gift of God as Descartes would have us believe,
and if mathematics is indeed out there, existing in a divine realm which can be accessed
only by few, then mathematical powers have a less psychological and more spiritual
connotation. The questions, such as “how do humans learn to reason?” did not arise in a
setting deﬁned by mathematics as a divine and heavenly habitat open only to a privileged
few. Pedagogy, in the terms in which educators and educational psychologists might
render it, did not seem to have a place on the registers of Cartesian and Platonist

mathematics.

72

Chapter IV: Revisiting Mathematical Power for all

To some extent, everybody is a mathematician. ..School mathematics must endow
all students with a realization that doing mathematics is a common human
activity.

(NCTM, 1989, p. 8)

Truth and beauty, utility and application frame the study of mathematics like the
muses of Greek theater. Together, they deﬁne mathematical power, the objective
of mathematics education.

(NRC, 1989, p. 43)

 

 

 

 

 

 

 

 

 

 

Dramatistic Classical Texts Reform Texts Counter-Reform
Pentad Texts
Scene Mathematics Mathematics Standards
Education—Based
on NC T M
Standards
Act Following, Constructing Following,
Reading, Mathematics Reading,
Discovering demonstrating
mathematics ﬂuency
Agent Mathematicians Children, Teachers Children
Agency Mathematical Mathematical Mathematical
Powers Powers proﬁciency
Purpose Illumination by Empowered Numerate Citizenry
natural light of citizenry
reason

 

 

Table 2: Pentad at a Glance—Focus on Reform-based Mathematics Education Texts

I would like to remind the reader at the beginning of this chapter what I wrote in the
introduction to this dissertation project. As an outsider to mathematics education
discourse in the United States, and a mathematics educator, I was introduced to a set of
terms—like mathematical power, constructivism, child-centered pedagogy, and so on—

which were part of progressive mathematics educators’ repertoire in the United States. I

73

encountered those terms as part of ‘traveling reforms’.27 When I arrived in the United
States, the only major change—albeit very signiﬁcant one—inasmuch as these terms
were concerned was the recognition that they were part of a progressive reform
package—something that did not travel with them to the distant lands. This move did not
change the terms that I used. I just saw them as part of a different scene. Later, I
recognized this scene to be ﬁercely contested and in a state of conﬂict with its detractors.
This chapter is about the scene of mathematics education and the drama it entails.

The Drama of Reform Texts

The drama of reform texts unfolds at two different levels. At one level, the scene is
the ﬁeld of mathematics education itself with several independent but interrelated
discourses reverberating within it, and constituting the descriptions of its actors, acts,
agencies, and purposes. At the second level, the scene of mathematics education also
permits an act-scene ratio. That is to say, it construes mathematics as produced due to
human acts.

In this drama, the key term is mathematical power that table—2above indicates as
occupying the place of agency in the dramatistic pentad. It occupies the same place, as
the table shows, in the case of classical texts as well. However, as the analysis in this
chapter will suggest, the dramatistic ratio associated with the term mathematical power is
different in these two cases. In the reform texts mathematical power follows the logic of
act-scene ratio. That is to say agents (children) are described as exercising their

mathematical powers to engage in mathematical practice. Unlike the classical texts,

 

‘7 Traveling reforms is the term used by Gita Steiner-Khamsi, a scholar of comparative and international
education, to describe the import and export of educational discourses across cultures (Popkewitz &
Steiner-Khamsi, 2004).

74

mathematics does not preexist to be discovered and thus discovered in the due course, but
is actively constructed through mathematical practices (thus the act-scene ratio).

However, these mathematical practices (with act-scene ratio) are themselves enabled
by the scene of mathematics education. This enabling is, in an important sense, a certain
kind of democratizing of mathematics. Mathematics education works as a scene for this
democratizing impulse much in the same way as the modern liberal democratic state
works as a scene for democratic processes in the society at large.

A study of this scene of mathematics education reveals a contingent coming together
of independent but related discourses in the wake of the calls for reforms in the 19803.
Three distinct streams—pedagogical progressivism deriving from as Wilson puts it,
“Dewey-ian conception of the child and curriculum as two sides of the same coin”
(Wilson, 2003, p. 18), the constructivist learning theories, and the quasi-empiricist
philosophy of mathematics attributed to Imre Lakatos—come together and spread into a
policy space under the banner of NCTM standards—to be described shortly—and other
related reform texts. The task this chapter does is to map this scene of mathematics
education and the ways in which it enables agents with mathematical powers as their
agencies.

To do this I will recycle the questions that were applied to the classical texts in the
last chapter.

Scene: How is the context of mathematical activity described?

Act: How are mathematical acts described?

Agent: Who is described as entitled to act, that is, to do mathematics?

Agency: What means are available to act, that is, to do mathematics?

75

Purpose: What is described as the purpose of mathematical activity?

In what follows, I will ﬁrst describe the reform texts that I have used in this analysis.

Apart from those texts, the analysis also draws heavily on academic writing in

mathematics education to map the coordinates of the scene of mathematics education and

other elements of the dramatistic pentad.

The Reform Texts

The research reported in this chapter has involved chasing similar terms in the reform

documents produced in the late 19805 and early 19905, and then collecting and sorting

the texts according to the dramatistic pentad. Around 1990 there was a heavy outpouring

of reform documents from many different directions, by many different organizations,

and at different levels. The notable documents are:

National Research Council’s Everybody Counts (NRC, 1989)

NCTM’s Curriculum and Evaluation Standards (N CTM, 1989)

The Mathematical Sciences Education Board’s Reshaping School Mathematicqu
Philosophy and Framework for Curriculum (MSEB, 1990)

Professional Standards, and (NCTM, 1991)

California’s Department of Education’s Mathematics Framework for California
Schools (CDE, 1992) (henceforth called the Framework).

Measuring What Counts (NRC, 1993)

Measuring Up: Prototypes for Mathematics Assessment (MSEB, 1993)

Assessment Standards (NCTM, 1995)

76

Of these documents, I draw primarily on NCTM Curriculum and Evaluation
Standards28 and Mathematics Framework for California Public Schools. I chose these
documents because these texts have been at the center of the drama of the so called
reforms and counter reforms in mathematics education. These documents are also
amenable to a dramatistic analysis because together they make up the dramatistic
structure of motivations about mathematical practice, teaching and learning. I describe
them in some detail below.

NCTM Standards

NCTM projects the standards as “one facet of the mathematics education
community ’s response to the calls for reform in the teaching and learning of
mathematics” (NCTM, 1989, p. 1, emphasis added). That is to say, mathematics
education in these texts does not signify just a field of practice, but also a professional
community. I will discuss the implications of this professional posture in detail later in
this chapter. Here it sufﬁces to note that this set of documents is projected as
representing the professional consensus of the mathematics education community.

The Curriculum and Evaluation Standards [to be called NCTM Standards from this point
onward in this chapter] published in 1989 make recommendations for improving and
updating the mathematics curriculum and the evaluation of students’ achievement. The
NCTM standards were part of, and in a sense a catalyst for, a broader standards-based
reform movement called into existence by the report A Nation at Risk, which made an

urgent call for reforms for higher and more rigorous standards (N CCE, 1983). The Bush

 

28 The term standards is used by both reform texts and counter-reform texts. In both sets of texts, however,
like other terms, standards are dramatistically different. In this chapter, when the term standards is used, it
refers only to the NCTM standards. In other chapters, when 1 use the term standards, I will call the
attention of the reader to the text in which it is located.

77

administration (1989—1993) used NCTM’s standards “as a free-market model for
educational reform” (Ravitch, 1995, p. 28). Following their example, the US.
Department of Education provided grants for creation of similar “voluntary national
standards in science, history, geography, foreign languages, the arts, English, and civics”
(Ravitch, 1995, pp. 28-29). Yet, the standards, as well as the policy documents and
curricula based on them, were contested by critics who accused them of being devoid of
mathematical content. (For a detailed documentation of mathematics education debates,
see, Wilson, 2003; also, see, Wu, 1998; Wu, 2000).29
Mathematics Framework for California Public Schools

Publication of Mathematics Framework for California Public Schools answers a

call from ...the National Council of Teachers of Mathematics, the Mathematical

Sciences Education Board, and the teachers and mathematics educators who

served on the Framework Committee. That call is to change what mathematics we

teach, how we teach it, and to whom (CDE, 1992, p. vii).

The 1992 Framework is an important document because of its proximity with NCTM
standards. By proximity here, I mean that many of the writers who are part of the NCTM
were also members of the team of authors who wrote 1992 Framework (NRC, 1989).
Deﬁning Mathematical Power in terms similar to those of the NCTM, the 1992
Framework devotes an entire chapter—one of its four main chapters—to describing and
explaining this notion. The remaining four chapters are entitled Developing

Mathematical Power in the Classrooms, Structure and Content of the Mathematics

 

29 Eventually, NCTM published a revised document entitled Principles and Standards of School
Mathematics (PSSM) in the year 2000. PSSM no longer used the phrase mathematical power.

78

Program, Mathematical Content in Kindergarten Through Grade Eight and
Mathematical Content and Course Structure in Grades Nine Through Twelve.

The 1992 Framework works with the terms strands and unijying ideas. Strands refer
to the mathematical content that is covered at each level and unifying ideas refer to the
terms such as patterns, proportional reasoning and so on. The unifying ideas are assumed
to cut across strands. There are eight strands namely number, measurement, geometry,
functions, statistics and probability, logic and language, algebra, and discrete
mathematics and several different unifying ideas at different levels. An example of

strands and unifying ideas is given in table 3 below

 

 

 

Strands Unifying Ideas
Elementary Middle Grades High School
Functions How many? How Proportional Mathematical
Algebra much? relationships modeling
Geometry Finding, making, and Multiple Variation
Statistics and describing patterns representations Algorithmic thinking
probability Representing quantities Patterns and Mathematical
Discrete and shapes generalization argumentation
mathematics Multiple
Measurement representations
Number
Logic and
language

 

 

 

 

 

 

' Table 3: Strands and Uniﬂing Ideas at various levels in the 1992 Framework

The 1992 Framework requires curriculum units to be created in ways that bring
together the strands and unifying ideas through experiences that would result in the
development of mathematical power. The framework urges that these curriculum units
be investigations that enable the exercise of mathematical power.

Both NCTM Standards and the 1992 Framework appeal to the notion of empowering

all students mathematically and justify this emphasis on the basis of multiple objectives

79

of ensuring equity, meeting the demand for a mathematically literate work force, and the

need for a numerate electorate.

The Scene of Mathematics Education

In dramatistic analysis, the scene is a container of the agent and his/her acts. The
scene, as discussed in Chapter 2, is not a physical landscape. In particular narratives it
can be anything that could be construed as containing the agent and the act.

I will map the scene of reform mathematics ﬁrst by contrasting it with the scene of
the classical texts. Examining the classical texts in the last chapter, I argued that some
notable texts produced about mathematics between the sixteenth and nineteenth centuries
described mathematics as existing in the mind of God or in a metaphysical Platonist
realm. I also argued that from the classical perspectives on mathematics as a priori,
teaching made sense as a project to identify the gifted few. This may sound elitist, and
surely different from what we, as mathematics and teacher educators, have come to
believe as the purpose of teaching these days. Yet, identifying the mathematically
powerful—i.e., giﬁed——students was wholly consistent with a mathematical practice that
was based on epistemological discovery because mathematical truths were assumed to
have been derived from a divine or Platonist realm. In those classical narratives, then, it
made sense to aim for identifying the gifted few.

So, in the classical texts mathematics came across as scenic in the dramatistic sense
because, by being out there, it framed the mathematical practice as well as its purpose.
With mathematics as the scene, doing mathematics was described by verbs such as
discovering, reading, and following. In the mathematics education reform-texts, as I will

discuss in detail in this chapter, it is the ﬁeld of mathematics education which works as

80

the scene. Dramatistically, I take mathematics education to mean a frame—not a
collection of individuals or a phrase signifying the teaching of mathematics—which
legitimates the meaning of the terms such as mathematics and mathematical power.

As the frame—pr the scene—mathematics education deﬁnes what mathematics is.
The reform texts deﬁne the nature of mathematics as being produced as a result of
purposive human action with the ‘real’ world.30 When the scene shifts from classical to
recent reform texts, the wanderer in the math world becomes a creator of mathematical
landscapes. While the term mathematical power in the classical texts comes across as the
ability to discover, read, and follow, in the reform texts, under a different scene, it
becomes capacity to construct, conjecture, communicate, and validate.

In what follows I will ﬁrst map two critical features of the scene of mathematics
education. First, the scene of mathematics education is constituted by an emergent
professionalization of the ﬁeld in the early 19805. Second, I present, borrowing heavily
from Batarce and Lerman (Batarce & Lerman, 2008), an argument about an inherent
issue with the term mathematics education that invites intervention from mathematicians
to reclaim the mathematics part of mathematics education.

Mathematics Education and the Rhetoric of Professionalism

With the publication of NCTM’s Curriculum and Evaluation Standards (N CTM,
1989), the air seemed ﬁlled with the talk of the professional status of a mathematics
education community. The professionalization of mathematics education can be
described as a scene through an analogy with the profession of medicine.

Professionalization creates particular roles for the physician, the patient, and the relation

 

30 This view of mathematics as being inductively elicited from the ‘world’ sounds much like one embodied
in Gassendi’s objections to Descartes’ Fifth Meditation. See previous chapter for details.

81

between these entities with respect to the sickness. Put differently, the acts of a physician
as well as a patient are circumscribed by professionalism. Likewise, mathematics
education, in the spate of reforms in the last two decades, reached out to all aspects of
professional practice like never before. Professionalization involved deﬁning
mathematics, mathematical practice, teaching, learning, and the roles and responsibilities
of teachers and learners, and curricula in particular ways. It is in this sense that I take
mathematics education discourse as assuming a scenic connotation, namely that of
professionalism.

Mathematics education was also seen by its participants as a young ﬁeld still in
search of a professional identity (Sfard, 1998; Sierpinska & Kilpatrick, 1998). Indeed, at
the turn of the 20‘h century, it may have been difﬁcult to imagine a separation between
mathematicians and mathematics educators. Concerns regarding the learning of
arithmetic had belonged traditionally to educational psychology, a ﬁeld that already had a
status as a professional community.

Mathematics education—as a ﬁeld of research and practice as an academic group—
had never led a huge reform effort until the mid 19805. For instance, the New Math
reforms of the 19605 that erupted in the wake of the launching of Sputnik were almost
wholly organized by mathematicians. The pronouncements from NCTM ofﬁcials in the
wake of their publication of the Curriculum and Evaluation Standards (NCTM, 1989)
lent support to the above claim about the emerging professional status of mathematics
education in the 19805. In an essay outlining the vision for implementation of these
standards, F. Joe Crosswhite, one the past presidents of NCTM, described the standards

project as “the most ambitious, and certainly the most expensive, program ever”

82

undertaken by the council (Crosswhite, Dossey, & Frye, 1989, p. 513). The standards,
said Crosswhite, would also be “remembered as a prototype for the development of a
professional consensus. . .And none has ever had the prepublication endorsement of so
many professional groups” (Crosswhite, et al., 1989, p. 513, emphasis mine). The image
of a professional identity for mathematics education was further reinforced by then
president of NCTM, Shirley M. Frye. She wrote:
In this next decade our profession has an unparalleled opportunity to revitalize
mathematics education and to make that mathematics education effective for all
students. Our initial step in developing the Standards was unique! No other effort
of this kind had ever been undertaken in any discipline by a professional
organization (Crosswhite, et al., 1989, p. 518).

Thus, Frye’s rhetoric positioned NCTM within the rank of professional organizations
in a uniquely triumphant manner. Mathematics educators, speaking through their
professional organization NCTM, seemed to have a solid grasp of what needed to be
done to ﬁx mathematics education in the United States. And they seemed poised to do it.
Sociologist of professions Andrew Abbott has deﬁned the term jurisdiction as a relation
that a group of professionals establishes with the task it sets for itself (Abbott, 1988). For '
mathematics educators, this relationship was embedded in the standards document that
they would strive hard to implement in the next decade or 50. Through the standards, the
mathematics education community spelled out what mathematics and its learning and
teaching could be. It was now only to be implemented through policy shifts.
Signiﬁcantly enough, Frye spelled out ways in which the standards construed

mathematics by calling attention to a set of verbs:

83

Our challenge is to translate the Standards into action by raising expectations and
by bringing vitality and vigor into our classrooms. The verbs used in the
Standards vividly describe the behavior our instruction should aim to achieve.
Words like explore, communicate, construct, use, and represent stress the
involvement of students in the active "doing" of mathematics. Words like
collaborate, question, express, value, share, and enjoy bring a new ﬂavor to the
work of the students. Words like reﬂect, appreciate, connect, apply, and extend
build a new attitude toward mathematics and its uses. . .Mathematics itself is
changing, and with it, the entire school mathematics curriculum is entering a
period of unprecedented change (Crosswhite, et al., 1989, pp. 519-520).

I should also observe that professionalism among educators, like educational reform,
was the order of the day in the 19805. Mathematics educators were not the only group
attempting to cross professional thresholds in the mid and late eighties. The efforts to
turn teaching “from an occupation into a genuine profession” were also underway (See,
for details, Holmes Group, 1986).

Mathematics under erasure

Second, the term mathematics and education was not a simple conjunction of two
words. Rather it came across as an agonistic pairing during the math wars. The term
education—here my focus is on the meanings associated with education—brings a set of
concerns to mathematics that may be different from the concerns that shape professional
mathematics communities. For example, the professional—that is university—
mathematicians do not have as their top priority the education of all students.

Mathematicians’ conceptions of mathematics arise from a different set of disciplinary

84

concerns, namely academic research and the preparation of the next generation of
professional mathematicians. In contrast, mathematics education’s concerns arise from
its domain that takes into account the concerns of teachers and school classrooms as well
as mathematics. The calls for a diﬂerent mathematics in the reform texts (NCTM, 1989,
p. 1) can be seen as an example of this recontextualization.

The rearrangement of the meaning of mathematics and mathematical practice within
the folds of mathematics education, as I will discuss in the subsequent sections, reﬂects
some aspects of mathematics and deﬂects its other aspects, thus working as a sort of
terministic screen (See Chapter 2). This redeﬁnition of mathematics circulates through
other terms of mathematics education reforms—such as mathematical power—both
affording and constraining what mathematics and mathematical practice could be.

A simple comparison with the drama of classical texts described in the previous
chapter tells us that mathematics education involves knocking mathematics off from the
top row in the table that describes pentad at a glance (See table 1). The displacement of
mathematics by mathematics education entails a rearrangement of the table. This
rearrangement of the table is signiﬁed by the emergence of a new professional
jurisdiction as I have discussed in the previous section.

By displacing mathematics from its place, mathematics education works to efface the
possibility of any role of mathematicians in this professional enterprise. Mathematicians
must ﬁrst become mathematics educators, before they can be insiders, if we go strictly by
the logic of the table. Thus mathematics education decides both what mathematics is and

what mathematics education can be.

85

Batarce and Lerman (2008), while interpreting the conjunction of terms mathematics
and education, propose to read mathematics education as mathematics education (Batarce
& Lerman, 2008), with the strike-through being read as putting mathematics under
erasure. They argue that the term mathematics education cannot be imagined prior to
both mathematics and education. Mathematics Education must contain the traces of both
of its constituent terms. However when the identity of mathematics education is given
precedence, the hierarchy is disrupted in a way that undermines the mathematics (Batarce
& Lerman, 2008, p. 45). This is expressed in mathematics education through a
simultaneous mistrust as well as a dependence on mathematical knowledge. As Batarce
and Lerman put it:

On the one hand, mathematics education, at some point, must mistrust the nature
of mathematical knowledge (for instance, for not being adequate for the
mathematics teacher education); on the other hand, mathematics education, when
it attempts to delimit itself and institute its episteme (for instance, by its notion of
methods of research), must ask for help from the nature of mathematical
knowledge (Batarce & Lerman, 2008, pp. 46-47).

Thus, the term mathematics education is in perpetual debate—it is, as Burke (1945)
would term it, agonistic. The terms for mathematics and education need each other, yet
are agonistic. It “effaces the ‘mathematics’ word written within the term ‘mathematics
education” (Batarce & Lerman, 2008, p. 47). The studies in the psychology of
mathematics education did not draw such debate because psychology of mathematics
education did not concern itself with, or raise epistemological questions about, the nature

of mathematical knowledge. Reading mathematics under erasure in mathematics

86

education is signiﬁcant, and expresses a democratizing impulse. When mathematics is
recontextualized in a way that deemphasizes authoritative knowledge and places in the
hands of a child the authority to create mathematical knowledge, mathematics is
democratized. Whether it is a good or a bad thing is not a question that I answer.
However, I only suggest that when the scene shifts from mathematics to mathematics
education, it also inserts a democratic impulse into the dramatistic conﬁguration.

The Elements of the Scene of Mathematics Education

The professional stance of mathematics education shapes a scene which is supported
by a contingent merger of three distinct strands with remarkable dramatistic similarity.
The ﬁrst emanates from Piagetian advances in mathematics education, namely
constructivism. The second approach can be described as proceeding from John Dewey’s
notion of experience. The third may be loosely described as Imre Lakatos’ articulation of
Popper’s philosophy of science into the ﬁeld of mathematics (Lakatos, 1979), which sees
mathematics advancing as a dialectic of proofs and refutations.

Constructivism: Constructivism was a theory of learning that spun off from the neo-
Piagetian work in mathematics education (Steffe & Kieren, 1994). Not everyone agrees
on what constructivism is.3 I I am not as much concerned with the range of meaning
associated with the term constructivism as with its dramatistic implications. So, while
there is considerable variation in meaning, all versions of constructivism regard students
as active constructors of knowledge rather than as passive recipients thereof (Wilson,

2003,p.40)

 

3' For a range of thinking about constructivism in mathematics education, see, Confrey, 1990; Davis, 1990;
Glaserfeld, I990; Goldin, 1990; Noddings, 1990; Phillips, 1995; Simon, 1995.

87

This theory reverberates strongly through mathematics education texts. As I
mentioned earlier, when I had ﬁrst encountered the term constructivism, it came across to
me as a learning theory. Yet, in the reform texts, constructivism crisscrosses, the
pedagogical and epistemological domains. Overall the reform texts appealed to
constructivism as their perspective on learning but insisted this theoretical recognition be
expressed in one’s teaching. For instance, the NCTM standards articulated an
unambiguous commitment to constructivism while insisting: “constructive, active view of
the learning process must be reﬂected in the way much of mathematics is taught”

(N CTM, 1989). The text perceived children as “active individuals who construct,
modify, and integrate ideas by interacting with the physical world, materials, and other
children” (N CTM, 1989). The 1992 Framework also echoed this emphasis in
assumptions about children as active creators of knowledge rather than its passive
absorbers. The reign of constructivism was extended from the domain of learning
theory—as I will discuss in more detail in the next section—to epistemology with the
admission of Imre Lakatos’s view of mathematics as fallible (Ernest, 1991; Lakatos,
1979). Not surprisingly, as I will discuss in more detail in the next chapter, the counter-
reform texts contested this amalgamation of constructivism and epistemology by
reestablishing a clear distinction between content and pedagogy.

Pedagogical Progressivism: I see this strand as proceeding from John Dewey’s
lamentations about educational sectarianism in America as emerging from a bogus
separation between the child and the curriculum (Dewey, 1902). Dewey had talked about
two “sects” of education who were ﬁghting over the curriculum. One sought to

“subdivide each topic into studies; each study into lessons; each lesson into speciﬁc facts

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and formulae. Let the child proceed step by step to master each one of these separate
parts, and at last he will have covered the entire ground.” For the other sect, the “child is
the starting point, the center, and the end” (Dewey 1902). Dewey sought to collapse this
binary distinction by claiming both child and curriculum as the two sides of the same
coin. Dewey’s larnentation and his campaign to collapse the binary distinction between
the child and curriculum may be seen as constituting what has been described as
progressivism by some and pedagogical progressivism by others.

Pedagogical progressivism appears as inseparable from constructivism in the reform
texts in mathematics education, as well as in progressive reform texts of the last two
decades in all subject areas. As Labaree (2005) puts it, pedagogical progressivism
means:

Basing instruction on the needs, interests and developmental stage of the child; it
means teaching students the skills they need in order to learn any subject, instead
of focusing on transmitting a particular subject; it means promoting discovery and
self-directed learning by the student through active engagement; it means having
students work on projects that express student purposes and that integrate the
disciplines around socially relevant themes; and it means promoting values of
community, cooperation, tolerance, justice and democratic equality. . .this adds up
to ‘child-centered instruction’, ‘discovery learning’ and ‘learning how to learn’.
And in the current language of American education schools there is a single label

that captures this entire approach to education: constructivism (Labaree, 2005, p.

277).

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In mathematics education pedagogical progressivism seeks to be faithful to both
mathematics and the child. Thus, it seeks a sort of mathematical authenticity as an
element of classroom practices. As Ball puts it, in her attempts to build a teaching
practice that responds to this challenge, “With my ears to the ground, listening to my
students, my eyes are focused on the mathematical horizon” (Ball, 1993, p. 376).

The point I am trying to make is that constructivism and pedagogical progressivism
are distant cousins brought together by historical happenstance to form a scene in which
students are seen as constructors of authentic mathematical knowledge. It is also a happy
happenstance that these strands are met midway with a particular streak in epistemology
of mathematics, the so-called quasi-empirical tradition of Imre Lakatos, which I describe
below.

Imre Lakatos and mathematics education

In Proofs and Refutations, Lakatos (1979) depicted the history of evolution of a
famous conjecture—known as Euler-Descartes F ormula3 2—as a conversation in a
classroom between ﬁctional characters. The book is narrated as a story of a conversation
between a ﬁctional teacher and his students. The teacher presents a particular proof of
the Euler-Descartes formula to his students. The students begin to offer counter
examples—which are drawn from the writings of the actual mathematicians—to dislodge
the proof offered by the teacher. The result is a conversation—spread over not a single
lesson but over many centuries of mathematicians’ struggles with the proof of the Euler-
Descartes formula—that helps shape the form of the proof, constantly modifying it

through a process of proofs and refutations. Thus, the proposals and counter proposals

 

’2 The formula is V — E + F = 2; in this formula, E = number of edges of the polyhedra, V = number of
vertices of the polyhedral, and F = number of faces of the polyhedra.

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spread over many centuries appear as a classroom conversation moderated by a teacher.
Lakatos’ is not just a story of a particular proof, but a conception of mathematics as
fallible and progressing through a dialectic of proofs and refutations similar to the logic
of scientiﬁc discovery proposed by Popper. Lakatos also makes a distinction between the
informal and formal mathematics. That is, from the coded mathematics resulting after
the dialectic of proofs and refutations is exhausted, one cannot tell the intense back and
forth and zigzagging which allowed a proof to grow in maturity and strength before it
was ﬁnally formalized and validated as mathematical knowledge.

This zigzagging behavior of mathematical progress resonates with the ways in which
the reform texts construct mathematics. For example, conjecturing, refuting, and
validating are the verbs that deﬁne the mathematical power of students in these texts and
not their abilities to read, and follow the already existing and coded mathematics—as was
suggested in classical texts. Lakatos’ epistemology, which constructs mathematics as
fallible, incomplete and uncertain, therefore, could be inserted comfortably into the
repertoire of reform texts.

Let me emphasize this point about the discursive proximity between reform texts and
Lakatos’ epistemology. By construing the object of its investigation as fallible, and,
therefore, in process, Lakatos’ view of informal mathematical discourse support the
conception of a classroom environment in which knowledge could be seen as being in the
process of construction through application of higher order thinking skills. Interestingly,
the higher order skills are not catalogued in response to ways of knowing mathematics
proposed by Lakatosian epistemology. The psychological imperatives underlying higher

order thinking skills do not necessarily make any reference to Lakatosian epistemology.

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The reform texts’ emphasis on the students’ construction of their own mathematical
knowledge, and their use of higher order thinking skills to do this does not directly follow
from a philosophy of mathematics. Rather, Lakatosian epistemology is comfortably
absorbed into this structure because it supports the tenets of pedagogical preferences of
reform texts. In other words, the reform texts pick and chose the nature of mathematics
as a perfect ﬁt within an existing terministic screen.33

Thus, in the reform texts the philosophy of mathematics of Lakatos became the
privileged vantage point from which the traditional classroom practices—such as
memorizing, practicing, and giving the one right answer—appear as undesirable aspects
of traditional mathematics teaching and learning that reform texts wish to exclude from
their approach to teaching mathematics.

Below, I provide another extended quote from Lampert to reinforce two points. First,
reform texts, through a sort of synecdochical move34 identify mathematics with
Lakatosian epistemology. Second, through this identiﬁcation reform texts stake a claim

as enacting practices of the discipline of mathematics in the classroom.35 In the words of

Lampert (1990), then:

 

33 In the 19605, the discourse of new math reforms exhibited a search for an educational psychology to ﬁt
with the preferences of the new math. As Steffe and Kieren document this: “In a sense, the mathematicians
who have guided the recent curriculum reforms have been waiting to be shown that psychological theories
of learning and intelligence have something relevant to say about how mathematics shall be taught in the
schools. These reformers (and I speak now not only of SMSG) have been so successful in teaching
relatively complex ideas to young children, and thus doing considerable violence to some old notions about
readiness, that they have become highly optimistic about what mathematics can and should be taught in the
early grades” (Steffe and Kieren, 1994, p. 712).

3" The term synecdoche implies taking the part for the whole. The point I am making here is that while
Lakatos’ epistemology illuminates some aspects of mathematics, it obscures other aspects of it.

35 lronically——-and as the reaction of mathematicians during the math wars suggest—mathematicians did not
view the reform texts as included in this dispensation.

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At every level of schooling, and for all students, reform documents recommend
that mathematics students should be making conjectures, abstracting
mathematical properties, explaining their reasoning, validating their assertions,
and discussing and questioning their own thinking and the thinking of others.
These activities do not ﬁt within the tasks that currently deﬁne mathematics
lessons. Moreover, they require both teachers and students to think differently
about the nature of mathematical knowledge. Little research has examined what
the intellectually generative sort of mathematical activities espoused by NCTM or
MSEB might look like in classrooms or the role that the classroom culture plays
in the social construction of a view of mathematical knowledge; studies of this
sort are needed if we are to understand what it will take to transform discipline-
derived standards into school practice... (Lampert, 1990, pp. 32-33, emphasis
added)

Thus positioned, reform texts construed the ideals of Lakatos about mathematical
practice as contrasting “sharply with the way in which knowing mathematics is viewed in
popular culture and in most classrooms” (Lampert, 1990, pp. 31-32).

However, what does not come across in the above quotation is that the ideas of
Lakatos also contrasted sharply with those of a signiﬁcant number of mathematicians.
Lakatos was not directly debated, but mathematicians’ emphasis on axioms and
mathematical truths as the starting line for a deductive approach toward proofs did not
suggest Lakatos’ epistemology as the cornerstone of the totality of mathematical

practices. Hung-Hsi Wu’s quotation below suggests this to be the case:

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On the one hand, logical deduction—proof—is the backbone of mathematics. . .On
the other hand, it would be a grave mistake to insist that every statement in
elementary mathematics, up to and including calculus, be given a proof. There is
no reason to impose the kind of training designed for future professional
mathematicians on the average student. . .What is important, however, is to give
students adequate training in making logical deductions. . .A reasonable

mathematics education should aim for at least this much (Wu, 1996, p. 4).

To summarize, constructivism, pedagogical progressivism, and Lakatosian quasi-
empiricist mathematical philosophy are brought together as allies under the banner of
standards-based reforms in mathematics education. In Chapter 6, I will discuss the ways
in which this threesome was both made possible as well as undermined by the language
of standards. Here, I want to speak of it as part of the scene of mathematics education,
constituting a drama in which the agents (children) are provided with mathematical
powers, which they can use to construct authentic mathematics.

To wrap up this conversation, in this section I have made the case that mathematics
education from the mid 19805 worked as a scene with two overarching characteristics.
First, the texts of mathematics education loudly pronounced a professional jurisdiction
over school mathematics, and second, mathematics education was projected as an
agonistic term in which mathematics was reshaped, reconstituted and rearranged. This
reconstitution, I have argued, happened through a contingent coming together of three
independent but, ultimately, interrelated discourses of constructivism, pedagogical

progressivism, and Lakatos’ philosophy of mathematics.

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This reconstitution of mathematics is coalesced in the term mathematical power in the

reform texts, which I will discuss in the following section.

Mathematical Power: Examining Act and Agency in the Reform

Texts

The development of mathematical power in all students was mobilized as the
overarching goal of school mathematics in the mathematics education reform texts in the
last two decades of the twentieth century (See, MSEB, 1990; NCTM, 1989, 1991, 1995;
NRC, 1993) until its near replacement recently by Mathematical Proﬁciency for all (Ball,
2003; California Department of Education, 1999, 2006; Kilpatrick, Swafford, & Bradford
Findell, 2001).

When in use, mathematical power worked as an umbrella term for a (reform-based)
school mathematics that signaled creative construction of mathematical knowledge
through engagement with the problems in “real world” involving investigational work,
discussion and argumentation in the classrooms. (See, California Department of
Education, 1985, 1992; NCTM, 1989, 1991, 1995; NRC, 1989). In these documents the
need to empower all students mathematically was also seen as part of the attempts to
foster creation of mathematical communities in K-12 classrooms. The image of
mathematical power, as one commentator noted, went way beyond “the usual minimalist
prescriptions. . .by including performance criteria more associated with professional

mathematicians in the mathematical sciences” (Bishop, 1990, p. 362).36

 

3" After remaining the focus of reforms proposed by the NCTM Standards (NCTM, 1989, 1991, 1995) for a
little over a decade, the term mathematical power completely disappeared from the more recent NCTM
publications (N CTM, 2000, 2006). Likewise, the mathematics framework for K-12 education published by
Califomia’s Department of Education in 1985 and 1992 articulated mathematical power for all as a central
goal of school mathematics. However, the 1999 Framework, which I will examine in more detail in the

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What do the reform texts mean by “doing mathematics”? From the terms (mainly
verbs) that deﬁne this activity, I infer the parameters within which mathematics is
conceived in these texts. In what follows, I will describe the ways in which reform texts
deﬁne mathematical power. Then, isolating the terms that describe mathematical acts
and, thus, expressions of mathematical power, I will, suggest the ways in which these
texts both afford and constrain what can be viewed as mathematics and ways of knowing
it.

Mathematical Power in Standards and Framework:

Standards articulate mathematical power in terms of ﬁve goals for all students:
...K-12 standards articulate ﬁve general goals for all students: (1) that they learn
to value mathematics, (2) that they become conﬁdent in their ability to do
mathematics, (3) that they become mathematical problem solvers, (4) that they
learn to communicate mathematically, and (5) that they learn to reason

mathematically. (NCTM, 1989, p. 6).

The NCTM Standards take these goals to imply educational experiences for the
students in which these goals could be met. Such experiences are expected to encourage
students to “value the mathematical enterprise, to develop mathematical habits of mind,
and to understand and appreciate the role of mathematics in human affairs” (N CTM,
1989, p. 6). The standards also use verbs such as explore, guess, conjecture, test and
build arguments to describe student actions as they undergo the experiences prepared for

them. With a voice sounding a professional consensus, the NCTM Standards exclaim,

 

next chapter, purged this term, establishing mathematical proﬁciency for all students as the new consensus.
I will examine the texts that take the idea of mathematical proﬁciency for all as the goal in more detail in
the next chapter.

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“We are convinced that if students are exposed to the kinds of experiences outlined in the
Standards, they will gain mathematical power” (N CTM, 1989, p. 6).

The 1992 Framework deﬁnes mathematical power in terms that are similar to NCTM
Standards. Mathematically powerful students, according to the 1992 Framework, are
those who think and communicate drawing on mathematical ideas and using
mathematical tools and techniques. It, then, deﬁnes thinking and communicating in
terms of other acts that could be used to indicate the doing of mathematics, and ideas,
tools, and techniques in terms of what the programs make available to students as
mathematical knowledge, technological tools such as computers and calculators, as well
as algorithms. As stated in the Framework:

0 Thinking refers to intellectual activity and includes analyzing, classifying,
planning, comparing, investigating, designing, inferring and deducing, making
hypotheses and mathematical models, and testing and verifying them.

0 Communication refers to coherent expression of one's mathematical processes
and results.

0 Ideas refer to content: mathematical concepts such as addition, proportional

relationships, geometry, counting, and limits.

Tools and techniques extend from literal tools such as calculators and compasses and
their effective use to ﬁgurative tools such as computational algorithms and making visual
representations of data.

Expressions of mathematical powers: What does a student do when expressing her
mathematical powers? To answer this question, I have created lists of verbs that reform

texts mobilize as expressions of mathematical powers by the individuals.

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The various ways mathematical power is described in the reform texts are restricted
to the terminological cluster of verbs mentioned in the table below. The 1999
Framework describes mathematically powerful students as those who do all of the things
mentioned in the table above. That is, “think and communicate, drawing mathematical
ideas and using mathematical tools and techniques. ” The verbs in the table are also
sorted in terms of thinking and communicative verbs. In addition some verbs also appear

to show students’ attitude toward mathematics.

 

Type of Verb Verbs
Thinking Verbs Analyze
Classify
Compare
Deduce
Design
Estimate
Hypothesize
Infer
Investigate
Model (construct
patterns)
Plan
Validate
Construct
Communicative Verbs Communicate
Present
Share
Attitudinal Verbs Value mathematics
Appreciate its role in
human aﬂairs
Table 4: Verbs of Mathematical Power

 

 

 

 

 

 

 

Furthermore, the thinking skills mentioned in the table above are also nominated as
higher-order thinking skills which are marked off from the other kind of thinking skills by
the following characteristics:

0 Higher-order thinking is non-algorithmic. That is, the path of action is not

fully speciﬁed in advance.

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o Higher-order thinking tends to be complex. The total path is not visible
(mentally speaking) from any single vantage point.

0 Higher-order thinking often yields multiple solutions, each with costs and
beneﬁts, rather than unique solutions.

0 Hi gher-order thinking involves nuanced judgment and interpretation.

0 Higher-order thinking involves the application of multiple criteria, which
sometimes conﬂict with one another.

- Higher-order thinking often involves uncertainty. Not everything that bears on
the task at hand is known.

0 Higher-order thinking involves self-regulation of the thinking process. We do
not recognize higher order thinking in an individual when someone else “calls
the plays” at every step.

0 Higher-order thinking involves imposing meaning, ﬁnding structure in
apparent disorder.

0 Higher-order thinking is effortful. There is considerable mental work involved

in the kinds of elaborations and judgments required (CDE, 1992, p. 21).

If by doing mathematics the students must exercise higher order thinking skills, then
mathematics ought to be envisaged as the site of this exercise. That is to say,
mathematical activity, to be consistent with the characteristics of higher order skills, must
also appear as non-algorithmic and complex; it should appear to yield multiple solutions
with multiple criteria for judgment; should be uncertain; it should involve self-regulation
with no one else ‘calling the play’; it should require effortful investigative work. Thus,

the reform texts framed the nature of mathematics in ways that made it consistent with

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the deﬁnitions of higher order thinking skills, which must be exercised in the classroom

settings to make the whole enterprise of reforms internally coherent. As one of the

reform texts put it:
Real mathematics is rarely prestructured or marked with key words. Real
situations seldom look like recipes; more often, they are complex and ambiguous.
A single task can encompass many problems, often not clearly deﬁned. There
may be many ways to go about ﬁnding a solution or even deciding what
constitutes a solution. Completing a task may take hours, weeks, or even years of
sustained, persistent work (California Department of Education, 1992, p. 16).

The 1992 Framework, for example, regarded the student work in mathematics as
complete only if it demonstrated all of the four dimensions of mathematical power:
“when a student successfully ﬁnishes a large-scale project that demonstrates all the
dimensions of mathematical power, that accomplishment will be-referred to as complete
mathematical wor ” (CDE, 1992, p. 5). While, the dimensions of mathematical power
involved reasoning and use of higher order thinking skills to investigate, conjecture,
justify, validate, so on and so forth, they de—emphasize other skills such as memorization,
automatic recall of number facts, and so on.

In order to make school mathematics compatible with a child engaged in practices
deﬁned by the above-mentioned terms, mathematics would have to be conceptualized in
such ways as to offer possibilities for such higher-order thinking acts to exist. This
conceptualization would also be such as to exclude the use of other schools. The extreme
manifestation of this posture toward mathematics is shown poignantly in the following

passage from Wilson’s documentation of mathematics education debates in California.

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As a participant-observer in one professional development seminar, when asked
to solve a math problem, I used the quadratic formula. When asked to present my
results, I did, surprised to discover the hostility that erupted among my
colleagues: “You can ’t use formulas. You have to think” (Wilson, 2003, p. 49).

As I have noted above, it is happy coincidence that the reform texts eventually found
support for their notion of “real mathematics” in a particular epistemology of
mathematics, which, like reform texts, viewed mathematics as progressing when
mathematicians made and refuted conjectures. Armed with Lakatos, mathematics
education could now see mathematics classrooms as images of mathematical
communities and children as little mathematicians.

To wrap up the conversation in this section, the reform texts construe mathematics as
an object of higher order thinking skills expressed through action verbs given in table 3.
Agents (students) do mathematics as they apply higher order thinking skills purposefully
to solve ‘real world’ problems. Learning mathematics is the same as doing mathematics,
which is the same as using higher thinking skills. Equating learning with doing
mathematics enables the reform texts to project the learning communities as
mathematical communities engaged in mathematical practice.

This image is consistent with the broader tenets of pedagogical progressivism and
constructivism Pedagogical progressivism’s concern with the separation of child and
curriculum ﬁnds a resolution, when on one hand constructivism empowers the child to
construct mathematics and on the other hand Lakatos provides support to this

constructability of mathematics from an epistemological standpoint.

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At this point, I would like to call the attention of the reader to the contrasts between
the drama of classical texts and the reform texts that I have examined in this chapter. The
coincidence of pedagogical progressivism, constructivism, and Lakatos’ philosophy may
be a narrow terministic screen highlighting some aspects of mathematics and
mathematical practice. Yet, it works to displace mathematics from the mind of God or in
a mysterious heavenly realm, and places it in the hands of a human agent. Its imagery of
mathematical power is quintessentially democratic inasmuch as it is shifting the locus of
authority from the heavens to earth. Just as democratic governance displaced the
sovereign in the society, the conjunction of pedagogical progressivism, constructivism

and Lakatosian philosophy works to create a similar effect in the classroom.

Purpose: the Mathematically Empowered, Democratic Citizen

Before going any further let me digress to emphasize that the reform-based texts
make individual responsibility central to empowerment. This emphasis on individual
responsibility lets us read empowerment as a technology for governing conduct. The
California Mathematics Framework, for example, which declares mathematical power for
all students as the central goal of K-12 school mathematics, makes a loud pronouncement
about what it requires from the students in terms of taking responsibility for their own
learning:

Students are also members of the school community; they must change as well.
Like their teachers, students will learn to work in different ways and to play
different roles. And like their teachers, they will need “staff” development in
these new ways of working and studying. . .To be ready [read mathematically

powerful], California students need to develop—and everyone needs to nurture—

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a mathematical work ethic that requires self-discipline and efforts to meet a high-
quality standard (CDE, 1992, pp. 12-13. Italics mine).

At this point, I would like to draw attention to an analogy with the emergence of
constitutional democracies. In this emergence, the inﬂuence of sovereign monarchies
was curtailed in developments that diffused the power throughout the social body. This
is expressed as an emphasis on individual responsibility (See, for example, CDE, 1992).
As I mentioned earlier, while the reform texts permit an act-scene ratio—i.e., children as
active constructors of knowledge--this enabling is itself circumscribed, almost enforced,
by the scene of mathematics education. The act-scene relation, then, exists within a
scene-act relation in which the scene, as argued earlier, is mathematics education. The
counter reforms targeted this scene. When the scene is replaced, as it happened in the
case of Califomia’s 1999 F ramework—a complete set of relations, signiﬁed by the term
mathematical power, disappeared with it.

Agency: Mathematical Power as an Exercise of Power

In the beginning of my discussion on mathematics education as a scene I used
Batarce and Lerman’s (2008) argument to suggest the undermining of the term
mathematics in mathematics education. In a discussion of mathematical power as an
agency, I use a similar argument to argue that even though mathematical power works as
agency in the reform texts, it does so in a way that may undermine both mathematics and
the intention to empower students.

Here it is useful to refer to the analysis of classical texts in the previous chapters.
Mathematical power was situated in a different structure of motivations in the classical

texts. People were either gifted with mathematical powers or not. Those who were thus

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gifted were able to glimpse a divine order. However, in the reform texts mathematical
power is an expression of teachable higher order thinking skills. That is to say, it is a site
of pedagogical intervention. It is assumed that by creating appropriate educational
experiences, teachers will enable their students to be mathematically empowered.

However, the purpose of empowerment here is not just mathematical empowerment,
but about the possibility of introducing a democratic attitude toward mathematics. By
reversing the dramatistic ratio, the reform texts are mapping the drama of democracy in
the mathematics classrooms. This is happening, at the cost of undermining the notion of
mathematics as a bastion of certainty, de-authorizing it from its traditional sanctioning
authorities, and re-authorizing the children as authentic denizens of a math world. The
point I am making is that empowerment is loaded with larger societal and
unmathematical concerns and can be construed as preparation for a democratic
citizenship a lot more than merely mathematical preparation for social efﬁciency.

The mathematically powerful mathematician did not have to be a great citizen just
because s/he was mathematically empowered. The mathematicians like Benjamin Peirce
were not interested in initiating all students into mathematics. The schoolteachers of the
nineteenth century perhaps did not think they were creating a community of
mathematicians in their classrooms. Unlike classical texts, however, the reform texts
suggested the possibility of creating ‘genuine’ mathematical communities in the
classrooms. The texts sought to do so with reference to the language of empowerment.
Thus was created a scene in which the teachers had to come across as the repositories of
power. This scene made possible both the powerful and the to-be-empowered. The

drama engaged the powerful in acts of empowering those who were less powerful. The

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power in such a scene comes across as the capacity of one actor to alter the behavior of
another actor. However, the empowerment itself becomes an exercise of power. That is
to be empowered, the students must do some things and not do others. That is to say, it is
assumed that being mathematically powerful must mean being able to investigate,
conjecture, refute, validate, communicate, and all the other things that I discussed in the
previous sections. Yet, the skill to form and communicate a conjecture may not always
be needed in real life situations. Some situations may need it, but others may need
automaticity with computational skills. When found in such situations, the students who
used to associate mathematical power with a well deﬁned and narrow set of skills will not

necessarily feel powerful.

Concluding Thoughts

This chapter took the dramatistic pentad and applied it to the reform texts. The drama
of reform through this analysis appeared at two levels. At the ﬁrst level, it is deﬁned by a
dramatistic act—scene ratio. A contrast with classical texts will help me summarize this.
In the classical texts, mathematical powers were described as a gift of God, and enabled
one who possesses it entry to the preexisting math world. In contrast, in the mathematics
education reform texts the idea of mathematical power conjures up the image of a
democratized math world in which a human can think and produce mathematics,
revalidate it. That is to say, in contrast with the classical texts, the mathematical power
in reform texts becomes a capacity to construct, not discover, mathematics. Thus, on this
stage agents (children) appear as working—like mathematicians—in mathematical

communities, actively constructing their mathematical knowledge in this process.

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At the second level, empowering itself appears as an exercise of power. That is to
say, the enabling of mathematical power appears as an act that is sanctioned by the scene
of mathematics education. The legitimating discourses of mathematical power, which
contingently come together in this scene, are pedagogical progressivism, constructivism,
and Lakatos’ philosophy of mathematics. I have explicated this discursive merger'by
using the arguments of Batarce and Lennan (2008), showing that it undermines the term
mathematics in mathematics education, and mathematical in mathematical power.

Thus, the registers on which mathematical power makes its appearance are vastly
different in classical texts and the texts of reform mathematics. That is, the notion of
mathematical power responds to a different structure of motivations, is mobilized by a

different set of actors, and is directed toward different purposes.

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Chapter V: The Drama of Counter Reforms: Implication
for Mathematics Education

[W]hat passes for mathematics in these publications bears scant resemblance to the
subject of our collective professional life. Mathematics has undergone a re-deﬁnition, and
the ongoing process of promoting the transformed version in the mathematics classrooms

of K-14 (i.e., from kindergarten to the ﬁrst two years of college)
constitutes the current mathematics education reform movement.
(Wu, 1998, p. 1)

If there was ever a time in the United States when no one cared about mathematics
education, it certainly has not been the past couple years. Mathematics education has
been written about in the local, regional, and most important national newspapers and

magazine. Reports have also appeared on radio and national television. The focus of
attention has been the so-called “Math Wars” that center on reform in school

mathematics curriculum and its teaching.

(Becker & Jacob, 1998)

 

 

 

 

 

 

 

 

 

 

 

Dramatistic Classical Texts Reform Texts Counter-Reform
Pentad Texts
Scene Mathematics Mathematics Standards
Education—Based
on NCTM
Standards
Act Following, Constructing Following,
Reading, Mathematics Reading,
Discovering demonstrating
mathematics ﬂuency
Agent Mathematicians Children, Teachers Children
Agency Mathematical Mathematical Mathematical
Powers Powers proﬁciency
Purpose Illumination by Empowered Numerate C itizenry
natural light of citizenry
reason

 

Table 5: Pentad at a Glance—Focus on Counter-reform Texts

The landscape of mathematics education may be in ﬂux in the middle of the ongoing

mathematics education debates. I argue this based on my observations of the shifts in the

107

scene and the relations between various elements—agents, acts, agencies, and
purposes—of the drama of mathematics education. When the scene shifts, the deﬁnitions
of mathematics also shift together with the descriptions of what constitutes the act of
learning mathematics as well as the learners. This chapter, through an analysis of the
counter reform texts, reveals shifts in the relations between the mathematics and its
learning and teaching.

Before proceeding I would like to call attention to how this dissertation is advancing
its argument by recalling the central concerns of the last two chapters.

In those chapters, the ultimate purpose of dramatistic analysis has been to answer the
questions adapted from the dramatistic pentad and use those answers to develop insights
about the relations between the subject matter of mathematics and its learning as they
appear in particular texts. 37

In Chapter 3, the historical mathematical texts that I examined projected mathematics
as a scene. That is to say, mathematics was projected in those texts as a priori and
existing independently of human cognition. The mathematical practice of
mathematicians could be described, in metaphorical terms, as ﬁnding their way in a
sprawling terra incognita. In dramatistic terms, the mathematical practice was described
in terms of a strict scene-act ratio. Once discovered, the mathematical truths no longer
remained in terra incognita but available for others to read, comprehend, follow, and use.

The guiding terms for mathematical acts, therefore, were discovering, comprehending,

 

37 The speciﬁc questions guiding the dramatistic inquiry are:

Scene: How is the context of mathematical activity described?

Act: How are mathematical acts described?

Agent: Who is described as entitled to act, that is, do mathematics?
Agency: What means are available to act?

Purpose: What is described as the purpose of mathematical activity?

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and following the preexisting mathematical forms. Mathematical powers became
thinkable in these texts as abilities to discover, follow, and comprehend.

Chapter 4, in which I examined the mathematics education reform texts, suggested a
shift in the scene. While the classical texts posited mathematics itself as scenic, in reform
texts the scene was mathematics education. But this scene of mathematics education also
contained within it the act-scene ratios expressed through a redeﬁnition of mathematics
as a human construction—the act-scene ratio here implied that it was human acts that
preceded mathematics and not vice-versa. Consistent with this view of mathematics, the
reform texts construed mathematical powers as abilities to construct, and also as an object
of pedagogical intervention: teaching mathematics well meant expanding students’ power
to do—construct, invent—mathematics.

In classical texts, pedagogy was not important because access to mathematics was
described as a gift of God. In reform texts, pedagogy was important because
mathematical power was conceptualized as an object of intervention and development,
hence the slogan “mathematical power for all.”

In this chapter, I examine one of the counter reform texts, the Mathematics
Framework for California Public Schools (CDE, 1999) and the California content
standards (which were published together with the 1999 Framework). I have selected
these documents, not because they are the only example of counter reform texts, but
because they replaced a reform-based framework (CDE, 1992). From this point on, I will
refer to both the 1999 Framework and the content standards published with it as the 1999

Framework.

109

Through this analysis, I will show that the 1999 Framework constitutes the scene for
the learners in a similar way as mathematics was scenic for mathematicians in the
classical texts (as discussed in Chapter 3). This is not to suggest that the 1999
Framework was derived from a Platonist perspective on mathematics. Rather I mean to
indicate that classical texts and counter-reform texts share a dramatistic similarity; both
the classical texts and the counter reform texts appear as dramas in which the agents
(mathematicians in the former and children in the latter) and their mathematical practices
are constrained to wander about in a preexisting math land. This contrasts sharply with
the drama of reforms in which the scene of mathematics education redeﬁned mathematics
as a product of human activity and spoke of children as constructing mathematics.

The classical texts described the mathematician as a discoverer in the preexisting
math land. The content standards associated with the 1999 Framework work to place
students on a narrow set of pathways and benchmarks with mathematical proﬁciency
signaling an individual’s incremental progress along those pathways. Curriculum,
instruction, and assessment are called forth to ensure that all students stay on course, that
is to say, no child is left behind.

I further argue that the dramatistic arrangement of the 1999 Framework, by reversing
the relations between mathematics and pedagogy described by the reform texts, also
tended to map onto content and pedagogy a separation that exists between the ﬁelds of
mathematics and education. Since the reform texts were premised on merging content
and pedagogy, taking them apart works to contest the claims of professional mathematics
educators and to raise questions about what mathematics education is as it emerges from

the debris of the math wars.

110

What follows is a description of the 1999 Framework, followed by its dramatistic

analysis.

Counter Reforms

The Framework and the associated content standards that this chapter examines were
published in 1999 after the counter-reformers in California were able to reverse the 1992
mathematics education policy in California that had been based on the Mathematics
Framework for California Public Schools (CDE, 1992) and Curriculum and Evaluation
Standards (NCTM, 1989).

The 1992 Framework, the 1989 NCTM Standards, and the curricula based on these
documents, had all became targets of intense criticism by different groups for different
reasons. The heat generated by these ‘debates’ earned them the title of “math wars” (for
a detailed description of math wars in California and nationally, see, Wilson, 2003).

When the curtain ﬁnally dropped in California in 1999, the framework had changed
hands. The 1999 Framework, while ‘accurately’ laying down what students needed to
know at each level, raised the ﬂag of Mathematical Proﬁciency for All as its goal. The
notion of Mathematical Power—which was central to earlier framework and NCTM
standards—was expunged from the framework and proﬁciency inserted in its place.
Beginning with the 1999 Framework, the calls for mathematical proficiency for all have
been pouring into the system nationally and regularly. Notable among the publications
deﬁning and building on this notion are Adding it Up (Kilpatrick, et al., 2001 ),
Mathematical Proﬁciency for All Students (Ball, 2003), and Foundations for Success:
The Final Report of the National Mathematics Advisory Panel (U SDOE, 2008). Notable

NCTM publications such as Principles and Standards of School Mathematics and

Ill

Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics (N CTM,
2000, 2006) also dropped the language of mathematical power.

When I ﬁrst read the 1999 Framework, I did not read it with a focus on terms such as
mathematical proﬁciency. In fact, in my ﬁrst reading of both the reform and counter
reform texts, I was simply looking for ways in which these texts constructed
mathematics, its teachers and learners, without reference to terms such as power or
proficiency. However, the shifts in vocabulary felt irresistible, almost seducing me to
focus on the texts which stirred the controversy and those that eventually replaced power
by proﬁciency.

This shift seemed signiﬁcant, and, perhaps, a key to understanding the terms of
debates. Part of the reason this seemed so was, as I discussed in the previous chapter,
that the language of mathematical power was part of the professional claims of the ﬁeld
of mathematics education. But the claim of reform texts, were also perceived as
undermining the mathematicians’ perspective on mathematics. Some mathematicians
sprang to action. As mathematician Hung-Hsi Wu,38 who was an inﬂuential critic of the
reform texts, put it: “One positive outcome of the current mathematics education reform
may very well be the revival of the idea that mathematics is important in discussions of

mathematics education. The battle over the standards is a stunning illustration of this

 

38 1 often cite Hung-Hsi Wu because of his passionate involvement in mathematics education debates, and
possible inﬂuence on both 1999 Framework and several reports produced nationally after the publication of
1999 Framework. Wu was a critic reforms initiated by the 1992 framework in California and one of the
authors of the 1999 Framework. After 1999, he has been part of the NRC committee along with Jeremy
Kilpatrick, Deborah Ball, Hyman Bass and others. This committee published the report Adding it Up
which brings the notion of mathematical proﬁciency for all in the national conversation about mathematics
education. More recently, he has also been part of the National Mathematics Advisory Panel (NMAP)
commissioned by President George W. Bush along with other notable mathematics educators. The ﬁnal
report of NMAP published in March 2008 captures many of the themes initiated by the 1999 Framework,
including the ideas of mathematical proficiency and of all school mathematics as a preparation for a
culminating experience in Algebra I and Algebra II.

112

fact” (Wu, 2000, p. 27). For Wu, the battle over the standards was a battle over correct

representation of mathematics in school mathematics.

1999 Framework

The 1999 Framework is a 351-page document, nearly half of which (Chapters 2 and
3) is taken up by a description of mathematics content standards; 28% of the document is
ﬁve appendices consisting of samples of instructional proﬁles, three lesson plans, and
math problems; and the remaining 22% contains small chapters on Instructional
Strategies; Assessment; Universal Access; Responsibilities of Teachers, Parents,
Students, and Administrators; Professional Development; Use of Technology; and
Criteria for Evaluating Instructional Resources.

Standards in this document are deﬁned as essential content for all students,
identifying what all students should know and be able to do at each grade level. For
kindergarten through grade seven, the statements describing the standards are distributed
vertically across the grade levels and horizontally across ﬁve content strands, Number
Sense; Algebra and Functions; Measurement and Geometry; Statistics, Data Analysis,
and Probability; and Mathematical Reasoning. A particular strand keeps becoming
increasingly dense as the grade level advances. The standards for grades eight through
twelve are not organized in strands. Referring to them as ‘discipline names, the
framework rearranges them as Algebra 1 & 2, Geometry, Trigonometry, Mathematical
Analysis, Linear Algebra, Calculus, and Advanced Placement Probability and Statistics.
Each standard is followed up by its component parts giving the appearance of a
hierarchical results framework. The 1999 Framework also cites research to support its

emphases. However, the citations show a clear preference for experimental research.

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Wilson observes this preference when describing the 1999 Framework, “When research
is cited (in the 1999 Framework), it is research that ﬁts-for the most part-a narrower
paradigm of experimental and quasi-experimental research. Research is presented as
deﬁnitive and authoritative. No mention is made of disputes over certain practices (like
tracking). No mention is made of highly regarded research (among mathematics
educators) that draws on other research traditions that are more qualitative” (Wilson,
2003,p.173)

The framework does not specify how the curriculum should be delivered explicitly
leaving pedagogical choices to teachers: “Teachers may use direct instruction, explicit
teaching, or knowledge-based discovery learning; investigatory, inquiry-based, problem-
solving-based, guided discovery, set-theory-based, traditional, or progressive methods; or
other ways in which to teach students the subject matter set forth in these standards”
(CDE, 1999, p. 19).

Yet, the 1999 Framework also contains three sample lesson plans as appendices.
The lesson plans provide precise statements of the goals of the lesson, a list of materials
to be used, the plan of activities, and precise steps to be taken by the teachers, sometimes
also suggesting the exact phrases that the teacher may use while addressing students.

Whereas the 1992 Framework appealed to higher order thinking skills and
constructivism as a theory of learning, I did not ﬁnd any appeal to developmental
appropriateness or any other psychological considerations for justiﬁcation of
mathematical content. Rather, the 1999 Framework appears to be building up K-7

mathematics as an increasingly complex set of knowledge and skills along a clearly laid

ll4

out roadmap with all strands leading to the culminating experiences in Algebra 1,

Geometry, and Algebra II in grade 8-12.

Standards as Pathways and Milestones: The Scene of
Mathematical Proficiency

In Chapters 3 and 4, I have discussed narratives in which the stress on the scene
varied. The classical texts presented a scene-act ratio stressing mathematics as scenic—
being out there to be discovered—and mathematicians as wandering around in this
preexisting math land. The reform texts presented an act-scene ratio, stressing the agent
as having the power to construct mathematics. The 1999 Framework, as I argue in detail
below, presents a scene-act ratio, stressing the mathematics content standards embedded
in it as scenic.

The 1999 Framework sets development of mathematical proﬁciency in all students as
the ultimate goal of mathematics education.39 Mathematical Proﬁciency is said to be
gained when all students “can solve meaningful, challenging problems; demonstrate both
a depth and breadth of mathematical understanding; and perform both simple and
complex computations and mathematical procedures quickly and accurately, with and
without the aid of computational tools” (CDE, 1999, p. vi). What solving, demonstrating

understanding, and performing computations might mean is stipulated in a set of

 

’9 I would also like to stress that my analysis of the term mathematical proﬁciency in this chapter is limited
to the text of the 1999 Framework. 1 should observe that the language that seems to have emerged from
California has an interesting trajectory. For instance, the language of mathematical power was ﬁrst used in
the 1985 framework. It then reappeared as part of a more emphatic professional consensus in the 1989
NCTM standards, and was later re—emphasized in the 1992 California Framework. Likewise, mathematical
proﬁciency, after appearing in the 1999 Framework has found its way into many inﬂuential documents.
Notable are NRC’s Adding it Up (Kilpatrick, et al., 2001), Deborah Ball’s monograph Mathematical
Proﬁciency for All (Ball, 2003), and more recently published report Foundations for Success: The Final
Report of the National Mathematical Advisory Panel (USDOE, 2008). These documents have reﬁned this
idea after its ﬁrst appearance in the 1999 Framework.

115

California mathematics content standards that replace NCTM’s standards as the basis for
the framework.

In this section, my central thrust is to direct your attention to the work that standards
do as a scene. As scenic, standards embedded in the 1999 Framework work to describe
the entire mathematical landscape for school mathematics with its pathways and
benchmarks in a way that is akin to the mountain range metaphor I used in Chapter 3 to
describe mathematics as a scene in the classical texts. As such, mathematical proﬁciency
is contained by this scene as the ability of the student to reach the benchmarks prescribed

by the standards.

Standards as Pathways and Benchmarks
Let me revisit the mountain range metaphor I used in Chapter 3 to talk about the

relationships between the mathematics and mathematical acts suggested by the classical
texts. The classical texts presented mathematics, like the mountain range, as existing a
priori. The mathematician, like the mountain trekker/climber, could not create
mathematics, but only discover it. Just as the job of climbers was to ﬁnd the best routes to
zaswe34 Platonists at the turn of the 20th century. In the case of both Cartesians and
Platonists, a large body of mathematics existed in terra incognita which became gradually
accessible to those with mathematical powers. The high priests alone—the
mathematicians who also happened at times to be the clergymen—could access this terra
incognita and draw its maps for others to understand, read, and follow.

In what ways does this description contribute to an understanding of the 1999
Framework and the standards as scenic? By foregrounding the mountain range metaphor,

I am certainly not implying an exact parallel between the classical and counter reform

116

texts. I am not claiming that perspectives of those contemporary mathematicians who
agitated against the reform texts, and who authored or validated the I 999 Framework, are
similar to nineteenth century mathematicians. The link between these two very different
kinds of texts becomes obvious only when one resorts to metaphorical thinking to
understand the scenic stress in both documents. That is to say, the classical texts and the
contemporary counter reform texts such as the 1999 Framework are similar only
inasmuch as both stress mathematics as existing externally, in the heavenly or mysterious
domains in the case of the former, and in the mathematically rigorous standards in the
case of the latter. Let me elaborate.

The 1999 Framework speaks of the standards as a “context for a coordinated effort to
enable all California students to achieve rigorous, high levels of mathematics
proﬁciency” (CDE, 1999, p. vi, emphasis added). It is the description of the nature of
standards as a strict frame in this text, which stresses them as scenic from a dramatist
perspective. And standards as a scene, I argue here, are similar to the mountain range
metaphor inasmuch as both conjure the image of an externally existing landscape. But
unlike the classical math texts, the counter-reform mountain range has been converted
into a national park with well deﬁned pathways, trail markers, and milestones. Indeed,
the authors of the 1999 Framework pride themselves on their clarity and explicitness.

To understand this claim, consider a student entering a kindergarten classroom
modeled on the 1999 Framework. Much of what is contained in the standards is
embedded in the 1999 Framework is terra incognita for the entering student. Yet, there

is a preexisting map of mathematical knowledge and skills spread out in time and space

117

from kindergarten to grade 12—a sort of math world—prepared for him by
mathematicians.40

Consider now the relationship between the acts of the students and standards and
compare it with the one that I argued to have existed between a mathematician and a
Platonist or a divine world of mathematics. Along the contours of the metaphorical
mountain range, all the pathways and benchmarks are already in place in the standards.
The student is placed on these pathways upon entering kindergarten is required to cover
the distance marked out by every path. In fact, once placed on the path, the student must
ﬁnd herself guided along the pathways, nudged forward if left behind, by the external
forces such as those wielded by the curriculum, instruction, and assessment. In much the
same way that the individual members of a climbing team are roped in so that no one
falls in a crevasse, there are forces and regimens—such as the legislation No Child Left
Behind—put in place to ensure that students stay on course. That is, this map does not
merely preexist but has strictly regulated pathways on which succeeding benchmarks
cannot be reached by any other means except ﬁrst reaching the preceding benchmarks.
As the authors of the 1999 Framework put it, “Mastery of almost all the material at each
level depends on mastery of all the basic material at all previous levels” (CDE, 1999, p.
198). As mathematics educator Sinclair points out, “this image, by its very nature,

discourages connective paths, circuitous paths, short cuts... because such things would

 

4° 1 stress mathematicians among a great variety of actors who may have contributed to the 1994
Framework and Standards, primarily because the standards is what mathematicians targeted for their
involvement and ‘jurisdiction’. Although, the NCTM standards had the endorsement of mathematical
organizations such as MAA and AMS. However, many mathematicians considered that endorsement to be
a mistake (see, Klein, 1997) and sought direct involvement in the creation and sanction of content
standards, and the sanction of professional community of mathematicians. As Wu put it: “It is often
forgotten in the war of words that mathematics education has a substantive component: mathematics” (Wu,
2000, p. 27).

118

make it almost impossible to judge who is left behind, and also makes the presences of
crevasses harder to see” (Sinclair, N., personal correspondence, Aug 14, 2008).

The strands—number sense, algebra and functions, measurement and geometry,
statistics and probability, and mathematical reasoning—may be imagined as the major
pathways on which the student must remain as s/he advances to the next level. On each
strand, within each grade level, standards are stated as benchmarks to be achieved. With
the scene set up in this manner, mathematical proﬁciency can be imagined as the
achievement of what is stated in the benchmarks. The following table shows the ways in
which standards are stated as increasingly complex but precise statements about what
students must know and be able to do for the ﬁrst standard (labeled 1.0 in each standard,
followed by its component standards) in the ‘number sense’ strand for the kindergarten to
grade 3.

The students advance along the pathways by displaying proﬁciency in each one of
these standards. A cursory glimpse on this table shows the path for developing the
number sense as laid out careﬁrlly. Students begin by simply comparing sets of objects to
determine their comparative size. These comparisons are replaced by ordering numbers
and using the symbolic notations of less than (<), equal to (=), or greater than (>) as they
advance to the next level. As they keep advancing, the place value of the numbers they

must be able to count is pushed up by one place.

119

 

Kindergarten

 

1.0 Students understand the relationship between numbers and quantities (i.e., that a set of
objects has the same number of objects in different situations regardless of its position
or arrangement):

1.1 Compare two or more sets of objects (up to ten objects in each group) and identify
which set is equal to, more than, or less than the other
1.2 Count, recognize, represent, name, and order a number of objects (up to 30)

 

Grade I

 

1.0 Students understand and count up to 100

1.1 Count, read, and write whole numbers to 100

1.2 Compare and order whole numbers to 100 by using symbols for less than, equal to,
or greater than (<, =, >)

1.3 Represent equivalent forms of the same number through the use of physical
models, diagrams, and number expressions

1.4 Count and group objects in ones and tens

1.5 Identify and know the value of coins and show different combinations of coins that
equal the same value

 

Grade 2

 

1.0 Students understand the relationship between numbers, quantities, and place value in
whole numbers up to 1,000.
1.1 Count, read, and write whole numbers to 1000 and identify the place value for each
digit
1.2 Order and compare whole numbers to 1000 by using symbols <, =, >

 

Grade 3

 

1.0 Students understand the place value of whole numbers
1.1 Count, read, and write whole numbers to 10,000
1.2 Compare and order whole numbers to 10,000
1.3 Identify the place value for each digit in numbers to 10,000
1.4 Round off numbers to 10,000, to the nearest ten, and use expanded notation to
represent numbers

 

 

 

Table 6: Standard for the Number Sense Strand in grades K-3

1 have tabulated the number strands standards above to illustrate the way they
describe the signposts on a path deﬁned by a particular strand. There are 391 standards

statements for K-7 in all, distributed across strands as follows:

120

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Strands Number of Standards
Number Sense 122
Algebra and Functions 60
Statistics, Data Analysis and Probability 55
Mathematical Reasoning 95
K-8 through 12

Algebra 1 and 11 59
Geometry 22
Trigonometry 21
Mathematical Analysis 10
Linear Algebra 12
Advance Placement Probability and 8
Statistics

Calculus 34
Total 391

 

 

 

Table 7: Number of Standards associated with 1999 F rﬂewogt

With these signposts prescribed by the standards crossed, and the content speciﬁed at
each signpost mastered, the goal for mathematical proﬁciency in grades K-12 stands
achieved.

Thus, from a dramatistic perspective, the standards in the 1999 Framework are scenic
in largely—but not exactly—the same ways that mathematics was for the mathematicians
in the classical texts. That is to say, that they are scenic similarly inasmuch as both refer
to an externally existing a priori entity. However, it is important to point out that
mathematics, as it is projected in classical texts as a creation of God or mysteriously in a
Platonist realm, is qualitatively different than standards constructed by humans for the
purpose of providing a purposive educational experience at the K—12 level. It is
important to realize, however, that similarity is dramatistic. That is to say, both Platonist
mathematics and Standards-based mathematics is experienced as preexisting. While the

scenes are different, the scene-act ratios are similar.

121

They are also different since students—unlike the early mathematicians of classical
texts—are always already in an environment in which they must do things as prescribed
(not just preexisting): commit facts to memory, learn and apply procedures, solve
problems according to well deﬁned heuristics, and so forth, and do all of this under the

gaze of a teacher.

Standardized Tests as Aspects of the Scene

With the scene laid out in terms of strictly regulated pathways and milestones, the
assessment comes across as an aspect of the scene as well and following from the path-
dependent nature of mathematical progress. The 1999 Framework construes progress in
achieving the goal of mathematical proﬁciency as almost entirely history- and path-
dependent. As it is put in the 1999 Framework:

A feature unique to mathematics instruction is that new skills are almost entirely
built on previously learned skills. If students’ understanding of the emphasis
topics from previous years or courses is faulty, then it will generally be
impossible for students to understand adequately any new topic that depends on
those skills. For example, problems with the concept of large numbers as
introduced in kindergarten and the ﬁrst grade may well go unnoticed until the
ﬁfth grade, when they could cause students severe difﬁculty in understanding
fractions. The biggest problem facing mathematics assessment is, therefore, how
to devise comprehensive methods to detect the mastery of these basic learned
skills (CDE, 1999, p. 197).

The framework proposes this detection of mastery (or its absence) to take place

before, during and after the instructional intervention. That is to say, the assessment is

122

performed to establish baseline data, monitor progress, and evaluate mathematical
proﬁciency developed after the intervention—exposure to curriculum and instruction in
this case-——has been completed. Taken together, these forms of assessment “provide a
road map that leads students to mastery of the essential mathematical skills and
knowledge described in the Mathematics Content Standards” (CDE, 1999, p. 196,
emphasis added).

Another aspect of assessment as scenic is the emphasis on assessing and grouping
students according to proﬁciency levels for comparisons, as well as for assigning them to
different diagnostic groups for treatment. If proﬁciency is to be interpreted in terms of a
measure of whether or not students have reached a speciﬁc benchmark (characterized by
speciﬁc knowledge and skills), then assessment becomes thinkable as a mechanism to
discover the extent to which the knowledge and skills required at the benchmark, are
present or not. Furthermore, if the framework is to guide all students to gain proﬁciency
in standards, then the assessment must appear as part of one or the other risk management
strategy that identiﬁes those who are behind so that an appropriately adjusted
instructional treatment is provided. The framework also recommends use of assessment
to identify those who are above the norm.

It may be useful to contrast brieﬂy the description of assessment in the 1999
Framework with the one given in the1992 framework In the 1992 framework, the sites
of student assessment were focused on assessing expressions of mathematical power—
construed more as acts than facts. For example, the 1992 framework emphasized the acts
of investigating, conj ecturing, and communicating more than automatic recall of number

facts on a timed multiple-choice test. In order to collect information about acts, the

123

sources of assessment information in this case moved away from the traditional timed
tests to more open-ended questions (tasks that could take a few minutes or be spread over
several weeks): observations of students at work; and Student Portfolios (see, CDE, 1992,
pp. 66-72). This recourse to different sources of information for assessment data was
also supported by NCTM Curriculum Standards:
An assessment method that stresses only one kind of task or mode of response
does not give an accurate indication of performance, nor does it allow students to
show their individual capabilities. For example, a timed multiple-choice test that
rewards the speedy recognition of a correct option can hamper the more
thoughtful, reﬂective student, whereas unstructured problems can be difﬁcult for
students who have had little experience in exploring or generating ideas (N CTM,
1989,p.202)

While the 1999 Framework acknowledges the existence of other ways of collecting
assessment data—as the one mentioned in the 1992 Framework and cited above—it
nevertheless accords timed tests a privileged place in measuring mathematics proﬁciency.
Furthermore, a preference for timed tests is justiﬁed with appeals to the nature of
mathematics.

All of these techniques can provide the teacher and the student with valuable
information about their knowledge of the subject. However, they also represent a
serious misunderstanding of what mathematics is and what it means to understand
mathematical concepts. Assessment methods such as timed tests play an essential
role in measuring understanding—especially for the basic topics, the ones that

must be emphasized (CDE, 1999, p. 198, emphasis original).

124

The timed tests are justiﬁed with appeals to automaticity as a requirement for
mathematical understanding. The argument rests on the assumption that if students are
not able to quickly recall knowledge and answer questions, then their knowledge is
superﬁcial. Students need to master all standards at each level in order to meet the
standards at the next level. And since they cannot move to the next level if their
knowledge is superﬁcial, it is important to detect and eliminate superﬁciality. And,
according to the 1999 Frameworks, the only way quick recall and automaticity (or its
absence) can be detected is through the standardized timed tests (CDE, 1999).

In 1999 Framework, then, the other sources of assessment information—such as
open-ended questions, portfolios, and student observations—privileged by 1992
framework—are deemphasized. The timed tests are reemphasized and justiﬁed on the
basis of an appeal to the nature of mathematics and its learning. This may be understood
better by extending the metaphor of strands as pathways and standards as benchmarks to
include assessment as a recording mechanism (assessment sources) placed on various
points along the path with the task of activating triggers. The triggers are activated when
the recording mechanism records a score. This sounds akin to the ways in which a
prescription is triggered by the medical discourse that follows the diagnosis of an ailment.
Diagnosis is a sort of interpretation of the body’s score on various scales. Likewise the
score on timed tests is also interpreted in terms of a diagnostic regimen.

This brings us to the path-dependent nature of mathematical instruction and learning
suggested in the framework. As teachers and teacher educators we know that education
is complex, uncertain, and dynamic. However, assessment on timed tests simpliﬁes the

management of this complexity, uncertainty, and dynamism by simply locating the

125

coordinates of students on narrow pathways. These coordinates at a given time place
students as either behind, on the benchmark, or as gone past it. A trigger is activated
when an individual is not recorded as doing all s/he must to reach the benchmark, or if an
individual reaches the benchmark too fast. This metaphorical trigger is meant to reroute
the slow (or fast) individuals to a processing facility.

In sum, then, assessment, as it is set up in the 1999 Framework, can be construed as
setting up recording devices (the timed tests), which activate triggers for treatment
according to ﬁxed scoring algorithms. That is to say, it activates a remedial program for
those performing two standard deviations below the mean score and an accelerating
program for those performing two standard deviations above the norm (the gifted or
advanced learners). Nothing needs to be done for those students whose score hovers
around the mean.

The 1999 Framework identiﬁes three such groupings depending on the individual’s
test score on entry level and formative assessments. These are: Benchmark group
(containing children whose score is around the mean score); Strategic group (one or two
standard deviations below the mean score); and Intensive group (more than two standard
deviations below the mean). The treatments, which the 1999 Framework suggests after
the students have been labeled as Benchmark Strategic, or Intensive are broadly
described in the document. They are also not important to my argument. My purpose in
this conversation has been to highlight those aspects of standards that render them scenic
from a dramatistic standpoint. I have argued that in the 1999 Framework standards work
to structure the space in terms of strands (pathways) and benchmarks for each strand.

Mathematical proﬁciency comes across in this scene as the ability to reach and get past

126

the benchmarks in each strand, without getting caught in the strategic or intensive net.
The recording devices (mostly timed tests) are positioned at various points (at the entry
level, throughout the pathway, and at the end) to generate data aboutwhether the students
are meeting standards (gaining proﬁciency). The scores on these devices are associated
with triggers that, once activated, must send students into different streams for
differential treatments aimed at ensuring that all move on to the next level.

From a dramatistic standpoint, the discussion above suggests a scene-act ratio, in
which standards completely contain the pathways (strands) as well as the benchmarks in
relation to which the notion of mathematical proﬁciency becomes operative. Once
placed on these pathways, children are sorted into falling behind, going in the middle, or
leading, depending on the diagnosis of their proﬁciency levels.

The scene-act ratio of the 1999 Framework is a reversal of act-scene ratio expressed
in 1992 Framework. That is to say, in the 1999 Framework the content of mathematics is
laid out in advance and learning, knowing, or doing school mathematics is contained in
this layout, whereas in the 1992 Framework, existing mathematics was viewed as a tool

with which students could engage in constructing their (new) mathematical knowledge

(CDE, 1992).

Rigor and Scene
So far, I have argued that the 1999 Framework stresses a scene which is composed of

1) content standards (containing the pathways as well as the milestones), 2) the
mechanisms for documenting the individual’s proﬁciencies in terms of their placement on
the pathways, and 3) the triggers to sort students according to such placement. Before

advancing the discussion on the ways in which the element of rigor is inserted into this

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scene, I would like to digress to a contrast between classical and reform texts once again.
In the classical texts the scene is mathematics. In the counter-reform texts, however, the
scene is a construction, a layout. In the classical texts the mathematical landscape is
beyond and independent of humans. In the 1999 Framework, however, the scene is a
selected representation of mathematics neatly laid out by humans. Thus, even the notion
of rigor works differently in these two domains. The classical drama, for example,
permits Fermat’s theorem whose rigorous proof can wait for two centuries. Fermat’s
theorem—~much like a revelation—does not have to be deﬁned by existing deﬁnitions. In
the drama of the 1999 Framework the notion of rigor, as I will explain below, spells out a
technology of control.

The 1999 Framework frequently appeals to this qualiﬁer to bolster its exhortations
about standards, curriculum, instruction, and assessment. The 1999 Framework, unlike
NCTM standards and the 1992 Framework, does not appeal as much to the psychological
arguments—such as constructivism and ideas about higher order thinking skills—as to
what it terms as the rigorous mathematical content standards containing essential
mathematical content that must be learned by all students. At the same time, the term
rigorous does not just refer to mathematics but spills over to other domains of
mathematics education. It works as an additional qualiﬁer for mathematical proﬁciency,
curriculum, and instruction. For example, calls are made for rigorous levels of
mathematical proﬁciency, rigorous mathematics standards, rigorous and challenging
mathematics program, rigorous and demanding curriculum and instruction (pp. 1-2);
engagement with rigorous mathematics is offered as a recipe to solve classroom

management issues (p. 210).

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As a qualiﬁer for standards, rigor is linked with the need to be explicit. The need for
being rigorous and explicit is projected as an aid to learning and motivations: since,
“students are not biologically prepared to learn much of this material on their own, nor
will all of them be inherently motivated to learn it. . .explicit and rigorous standards,
effective teaching, and well-developed instructional materials are so important” (p. 185,
emphasis original).

Rigor, then, permeates the scene of the 1999 Framework as its qualiﬁer and as its
appeal and justiﬁcation. Rigor is not just about mathematics. As I have shown above,
rigor is ubiquitous and works as a qualiﬁer for curriculum, instruction, and assessment.
Rigor, as I will discuss at length in the next chapter, was used in the rhetoric of counter
reforms as a symbolic weapon which aimed at associating the counter reform texts with
the powerful cultural values of science, while at the same time dissociating reform texts
from these values. Rigor as an element of the scene is extended into the domain of
teaching, assessment and research. By corollary, then, in the drama of counter reforms,
the rigor is projected on acts [of students and teachers] as well as agency [the

mathematical proﬁciency].

Separating Content and Pedagogy: Contesting the Claims of
Mathematics Education
In this discussion so far, I have called your attention to a scenic stress in 1999

Framework, implying that students and teachers must move along prescribed pathways
and milestones—what the 1999 Framework calls rigorous and essential mathematical
content. In this section, I suggest that this arrangement works to separate content from
pedagogy and that that separation of content and pedagogy works to undermine the

professional claims of mathematics educators contained in the reform texts.

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In the 1999 Framework, separation of content from pedagogy is stated explicitly and
is achieved by regarding the content as essential. Mathematical proﬁciency is stated in
terms of mastery of essential mathematical content deﬁned by standards. With
proﬁciency so deﬁned, the zone of pedagogical intervention is separated from deﬁnition
of content. Teachers are called upon to use any methods that they think would work to
achieve the standards. As the framework puts it:

The standards do not specify how the curriculum should be delivered. Teachers
may use direct instruction, explicit teaching, or knowledge-based discovery
learning; investigatory, inquiry-based, problem-solving-based, guided discovery,
set-theory-based, traditional, or progressive methods; or other ways in which to
teach students the subject matter set forth in these standards (CDE, 1999, p. 19).

The NCTM Curriculum and Evaluation Standards and 1992 Framework had blurred
the distinction between content and pedagogy by regarding knowing mathematics as
doing it (see, NCTM, 1989, p. 7), conceptualizing mathematical power in terms of
aspects of doing, and ultimately focusing pedagogical attention on development of
mathematical power. Therefore, once enacted this separation between content and
pedagogy contests and undermines the professional claims of mathematics educators
established through the reform texts signaling standards as a zone of inﬂuence for
mathematicians. It signals standards as formal space whose contents need professional
mathematician’s involvement and sanction.

I am not arguing that the shifts signaling separation of content and pedagogy are good
or bad. Rather, I am indicating, as I suggested at the beginning of this dissertation, that

policy is an extension of the ways in which debates are concluded. To give you an

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extreme example, the manner in which World War II concluded shaped the geography of
Europe for the next 50 years. Likewise, the manner in which the recent round of
mathematics education debates appear to be concluding may be shaping the geography of
mathematics education. This constitution, as I have argued in this section, may not
require educators to think hard about what mathematics is before deciding how to teach
it. It wants them to be competent in the discipline as it is deﬁned by the professional
community of mathematicians and turn their efforts on how best to teach it. This
proposal to reshape the geography of relations between mathematics and mathematics
education is repeated in several documents (CDE, 1999; USDOE, 2008; Wu, 1998,
2007). Berkeley mathematician Hung-Hsi Wu proposes that mathematics education be
considered akin to ‘mathematical engineering.’ This phrase, according to Wu, is not
offered as a metaphor for mathematics education. As he puts it, “I am not making an
analogy. I am not using “engineering” as a metaphor. Rather, I am giving a precise
description of what mathematics education really is” (Wu, 2007). Wu appeals to the
notion of Engineering as involving customization of abstract scientiﬁc principles to
satisfy human needs. Like the chemical engineer’s work with the science of chemistry to
develop products such as “the plexi-glass tanks in aquariums, the gas you pump into your
car, shampoo, Lysol...,” mathematical engineers [read educators] have to develop from
abstract mathematics a “mathematics that meets the needs of students and teachers in the
k-12 classrooms” (Wu, 1998).

This proposal is premised on a separation of content and education. At this point, I

would like to recall Nell Noddings’ reﬂections about mathematics education (Noddings,

1990). Although Noddings is thinking about the methodological and pedagogical aspects

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of constructivism, her comments also speak to the analysis I am offering in this section.
Noddings suggests that the assumption that all knowledge is constructed did not resolve
the matter of locus of power in mathematics, “pr is a mathematical statement, we are
more likely to accept it if George Polya or John von Neumann is its source than if, say,
Ronald Reagan or a local high school student came up with it. The mathematicians have
an authority that the other two do not have” (N oddings, 1990, p. 11). In other words,
Noddings is suggesting what this dramatistic analysis identiﬁes as a vulnerability in the
reform discourse. Noddings was pointing toward the cultural reality about
mathematicians’ power over what deﬁning what counts as correct irrespective of what the
mathematics educators deﬁne as correct.

Nonetheless, in reform discourse, mathematics educators redeﬁned mathematics by
combining content and pedagogy. By becoming part of the counter reform discourse the
mathematicians contributed toward redeﬁning mathematics education by separating

content and pedagogy again.

Concluding Thoughts

The school mathematics conceptualized in the 1999 Framework, I have argued in this
chapter, can be described by a dramatistic scene-act ratio. The content standards
contained in the 1999 Framework deﬁne the entire landscape of school mathematics as
divided in pathways—the strands of number sense, algebra and functions, measurement
and geometry, statistics and probability, and mathematical reasoning—and milestones to
be reached on each one of these pathways. A student’s journey through these pathways is
supposed to be strictly regulated by the milestones. Standardized timed tests work to

determine the coordinates of students in relation to milestones (criterion reference) and in

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relation to the mean scores. Students are distributed into benchmark, strategic, or
intensive categories depending on the standard deviation of their scores from the mean.
This analysis suggests an irony in thinking about the possible relations between
mathematics and school mathematics. Mathematics educators claimed to create authentic
mathematical communities in the classrooms (Chapter 4). The professional claims of
mathematics educators redeﬁned mathematics and mathematical powers as objects of
pedagogical intervention. However, these efforts, the reforms, constituted a terministic
screen with important inclusions and exclusions. The irony is contained in the
observation that the strongest challenge to attempts to construct authentic mathematical
communities came from the mathematicians. Ultimately, when mathematicians turned to
school mathematics, they rendered it a highly rationalized system of pathways and
benchmarks in which the progress in mathematics is deﬁned strictly and linearly. While
the reformers talked in terms of authentic practice, and counter-reformers in terms of
mathematical content, both suffer from the paradox of deﬁnitions that Burke had
suggested in his term terministic screen. Although both reform and counter-reform texts
seek vocabularies that they hope to be faithful reﬂections of mathematical reality, to this
end they must develop vocabularies that are selections of reality. “And any selection of
reality must, in certain circumstances, function as a deﬂection of reality” (Burke, 1945, p.

59).

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Chapter VI: The American Jeremiad and Reforms: The
Dramatistic Vulnerabilities of Reform Texts

[l]n a time of great crisis, such as a shipwreck, the conduct of all persons
involved in that crisis could be expected to manifest in some way the motivating
inﬂuence of the crisis. Yet, within such a "scene-act ratio," there would be a
range of "agent-act ratios," insofar as one man was "proved" to be cowardly,
another bold, another resourceful, and so on.

(Burke, 1978, pp. 332-333)

A Nation at Risk: The Imperative for Educational Reform

(NCEE, 1983)

In his penetrating analysis of early religious and political sermons in New England,
Sacvan Bercovitch identiﬁes a particular rhetorical trope that worked to generate in its
audience a deep anxiety by showing them the images of an impending damnation, while
at the same time promising redemption through proper acts (Bercovitch, 1978).
Bercovitch called it the American Jeremiad to highlight the peculiarity of this trope as
New England Puritans’ unique variation on the religious sermons of the Hebrew prophet
Jeremiah.

I ﬁnd the trope of the American J eremiad together with the elements of dramatism
particularly generative as a way of understanding the drama of mathematics education
reforms and counter reforms as acts within a scene set up by jeremiads. From the
standpoint of dramatism, as I will discuss in detail subsequently, jeremiads come across

as narratives that stress scene, albeit one that is always characterized by decline and

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crisis. That is to say, the jeremiads declare their audience to be in a state of deep crisis
and doomed if they did not act to transform the scene containing the crisis. As such, the
jeremiads constitute possibilities for reforms as acts seeking to transform the crisis ridden
scene. In an important way, then, jeremiads appear to be the generators of reforms.4| In
this chapter, I mobilize this rhetorical relation—between the jeremiads as scenes
describing decline and potential damnation and acts [of reforms] promising salvation—to
think about both reforms and counter reforms as acts within a scene set up by a jeremiad.

The jeremiad in question here is A Nation at Risk (N CEE, 1983), the report was
produced by the National Commission for Excellence in Education (N CEE). The report
is famous for its phrase, “. . .the educational foundations of our society are presently being
eroded by a rising tide of mediocrity that threatens our very future as a Nation and a
people” (NCEE, 1983, p. 113). From a dramatistic standpoint, A Nation at Risk, as I will
explain at length in this chapter, describes the scene of a crisis in terms of declining test
scores, and thus circumscribe possible reforms as efforts to establish higher and more
rigorous standards for the students to eventually reverse the decline. I will use Burke’s
pentad, in particular the notion of the scene—act ratio to account for the vulnerability of
the rhetorical strategies employed by mathematics education reform texts to respond to A
Nation at Risk.

With standards becoming the terms in which the possibilities of reforms were stated,
the scene-act logic demanded production of standards as the proper reform acts. And
indeed, in the decade following A Nation at Risk, the air was ﬁlled with the talk of

standards. In school mathematics—as well as in other subjects—the reform proposal for

 

4' The ubiquity of jeremiads taken together with ubiquity of reforms in America may also explain the
absence of these terms in the cultures that do not possess jeremiads as a dominating rhetorical trope.

135

higher standards was justiﬁed with reference to call for reforms by A Nation at Risk and
other reports that reinforced the jeremiad (CDE, 1985, 1992; NCTM, 1989; NRC, 1989).

This chapter consists of two main sections. In the ﬁrst section, I survey the notion of
American jeremiad as a rhetorical trope that stresses a scene of crisis. I will provide
examples of earlier jeremiads to show the ways in which they, as descriptions of scene,
circumscribe the acts of reforms. I will then elucidate how A Nation at Risk works as an
instance of American jeremiad, and as such, constitutes the scene for both mathematics
education reforms and counter reforms. This discussion creates the analytical space for
consideration of both reform and counter-reform texts in mathematics education as
competing responses to the same calls for action constituted by A Nation at Risk.

In the second section, I examine the rhetorical strategies of the reform-texts, chieﬂy
those produced by the NCTM, as vulnerable afﬁrmations of the scene-act ratio. I argue
that the reform texts in mathematics education used the term “standards” but infused the
term with a meaning that was shown by counter reformers to be incompatible with the
acts called for by the scene that the jeremiad A Nation at Risk had described. That is, if
the scene is ajeremiad, then reform texts could not be heard on the stage because of their '

simultaneous presence in a different scene, that of progressive discourse.

The American Jeremiad

The term jeremiad is derived from the name of the Hebrew Prophet Jeremiah who
lived circa seventh century BCE. Jeremiah is considered to be one of the Major Prophets
(with Isaiah and Ezekiel, and sometimes Daniel) in the books of the Bible that Christians
call “the old testament.” Jeremiah’s sermons were crafted to warn ancient Israel of its

impending doom, and urged repentance for its sins so that it could be redeemed and

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restored to its proper place. The Oxford English Dictionary shows the term jeremiad as
in use since the late eighteenth century. J arnes J asinski’s Sourcebook of Rhetoric also
corroborates this claim highlighting its use as a form of public discourse that echoed the

key themes of the religious sermons of prophet Jeremiah (J asinski, 2001).

Elements of Jeremiad

The eighteenth century American jeremiad was a ritualized evaluation of the state-of-
the-covenant ritual. It had three recognizable parts. First, it emphasized the covenant
between the Puritans and God, afﬁrming the belief that God had chosen Puritans as the
historic instrument of His will. This relationship between Puritans and God seemed to be
based on a balanced compact implying that “if they kept their part of the bargain, God
would keep His promise and grant the Puritans salvation” (Jasinski, 2001, p. 335).
Second, jeremiads routinely depicted the state-of-the-covenant as characterized by
Puritans’ deviation from the covenant. This deviation was used to account for the
calamities that had befallen them in the present and to warn them of a terrible future if the
earlier robust state-of-covenant was not restored. “What was happening, Puritans were
told over and over again, was a process of decline or a falling away from the terms of the
promise” (Jasinski, 2001, p. 336, emphasis in original). As such then, the jeremiads
worked to rhetorically alter people’s perceptions about tragedy, disease, and other
calamities, real or perceived. Through the announcements of jeremiad, they appeared as

expressions of the declining state of covenant between Puritans and God.42

 

’2 1n rhetorical studies Jeremiad is considered part of what is broadly called the epideictic genre. Booth
describes epideictic genre as consisting of rhetorical attempts to “reshape views of the present” (Booth,
2004, p. 17). This eﬂect seeking rhetorical genre includes a wide range of rhetorical activities including

137

The third element of the Puritan jeremiad was a prophetic reassurance to its audience
that God had not forsaken them: That even though God had let wolves loose on them,
they were still His sheep, and that redemption would ultimately follow if appropriate
reforms were undertaken. “The only way to avoid the shipwreck,” announced Winthrop,
“is to follow the counsel of Micah: to do justly, to love mercy, to walk humbly with our
God” (Quoted in, King, 2004, p. 206, emphasis added). The ultimate purpose of this
sermon was, as Bercovitch puts it, to “direct an imperiled people of God toward the
fulﬁllment of their destiny, to guide them individually toward salvation, and collectively
toward the American city of God” (Bercovitch, 1978, p. 9).

There are three important aspects of this description from the standpoint of
dramatistic grammar. First, there is a stress on the existence of an imperiled people
together with the conditions of their imperilment, which comes across as the description
of a scene. Second, the imperiled people are directed to escape this state of imperilment
to fulﬁll their destiny, which is an exhortation to act or reform. Finally, and more
importantly from the standpoint of the argument in this chapter, the jeremiads deﬁne the
terms of the reforms, in effect constituting what could be said or done as reforms—“do
justly, love mercy, walk humbly with our God” or “increase the church staff” etc.

This last element of the jeremiad, then, by asking for a rhetorical resonance between
the scene and the act, can also be seen as constituting the possibilities of debate about
what could be an appropriate response to the calls of reform. I imagine debates between
various reform positions as afﬁrmations of this scene-act ratio. Conceivably, the acts

that identify themselves with the calls for reform in a particular jeremiad are expected to

 

education (Sullivan, 1994), funeral oratory (Wills, 1992), even poetry. The notion of education being an
epideictic rhetoric is not, however, a universally accepted notion.

138

play on the terministic grounds prepared by the jeremiad. Below, I have shown the

alignment of these early J eremiads with elements of Dramatism.

 

 

 

 

 

 

Elements of Pentad The Early American Jeremiad
Scene State of the C ovennant as in Decline
Act Reforms

Agent Reformers

Agency Church

Purpose Restore the State of Covenant

 

 

 

 

Table 8: The Dramatistic Elements in American Jeremiad

To ﬂesh out this description of the rhetoric of American jeremiad and to further
highlight its relevance to an understanding of the terms of debate in education, I provide
below an example from a late seventeenth century New England’s Synod—a commission
of Puritan clergymen,43 who were called upon to inquire into the causes of the perceived
decline of New England and to suggest reforms to rid the Puritans .of maladies that
afﬂicted them.44 The Synod was appropriately titled The Reforming Synod (Walker,
1893,p.10) ‘

Two questions deﬁned the work of the synod:

1. What are the evils that have provoked the Lord to bring His judgments on

New England? (Walker, 1893, p. 426, Caps as in original text)

 

’3 Synod’s similarity with its more recent equivalents, the special and presidential commissions such as the
National Commission for Excellence that authored the report A Nation at Risk in 1983, is quite pronounced
4" The descriptions of the context of New England’s synod and quotes from its ﬁnal report have been
documented in several archival works. The descriptions that I use are compiled by church historian
Williston Walker (1860-1922). Williston Walker was Titus Professor of Ecclesiastical History at Yale
University from 1901 until his death in 1922.

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2. What is to be done so these evils may be reformed? (Walker, 1893, p. 433)
While the ﬁrst question suggests the scene populated with evil and, therefore, one that
needs to be cleansed, the second leads to a description of acts [reforms].
Scene: Past as a Place of Bliss, Present that of Crisis. The scene constructed by the
jeremiad is New England’s glorious past and current calamities. The synod also justiﬁes
its existence as a means to provide an explanation for the calamities, and to prescribe a
way out and back to the glory of the past. The explanations bolster the scene by adding
to it the elements whose presence is advanced as the reason for decline. The glorious
past—characterized by a healthy state-of-the-covenant between the Puritans and God—
appears to cast a nostalgic shadow on the present providing the motivational inﬂuence for
the restoration of a glory. The past is mobilized as a place of bliss, delight and peace and
characterized by a healthy state-of—the-covenant between the Puritans and God.
Describing this scene, its authors observe:
The good will of him that dwelt in the bush hath been upon the head of those
that were separated from their Brethren: and the Lord hath (by turning a
Wilderness into fruitful land) brought us into a wealthy place; he hath planted a
Vine, having cast out the Heathen, prepared Room for it, and caused it to take
deep rooting, and to ﬁll the land, which hath sent out its boughs unto the Sea, and
its branches to the River. If we ask of the dayes that are past, and look from the
one side of heaven to the other, where can we' ﬁnd the like to this great thing
which the Lord hath done? (Walker, 1893, pp. 423-424, spellings as in original)
But, then, as Walker puts it, “this course of prosperity was rudely interrupted...” by

wars and natural disasters. The J eremiad came into play to account for the wars and

140

 

 

disasters by rhetorically identifying these calamities as expressions of the ‘wrath of
God’ :45
But that we have in too many respects, been forgetting the Errand upon which

the Lord sent us hither. . .And therefore we may not wonder that God hath
changed the tenour of his Dispensations towards us, turning to doe us hurt, and
consuming us after that he hath done us good. If we had continued to be as once
we were, the Lord would have continued to doe for us, as once he did. (Walker,
1893, p. 424, spellings as in original)

Explanation of the Calamities. The second element of the scene, as explained
above, is a description of the ways in which New Englanders were seen as deviating from
the covenant. In the synod’s report, these descriptions are offered as responses to the ﬁrst
question above. Throughout these descriptions, the synod seeks to persuade its audience
by ﬁrst describing what it observed as sins of the Puritans and then specifying why they
are sufﬁcient reasons for God’s wrath and punishment. Below, I brieﬂy describe the sins
documented by the synod in order for you to see the manner in which they form a scene
for the proposed reforms. These are: “a visible decay of the power of Godliness among
many Professors” in the churches; neglect of church fellowship and other divine
institutions; Sabbath breaking; sinful indulgence of masters of servants and parents of
children; “sinful heats and hatreds” among the members of the church; the frequent

“heathenish and Idolatrous practice of” drinking; promise breaking and lying; inordinate

 

’5 In rhetorical studies, the narratives that seek to alter the sense of reality of their audience are part of what
is called the epideictic genre. Booth describes epideictic genre as consisting of rhetorical attempts to
“reshape views of the present” (Booth, 2004, p. 17). This effect-seeking rhetorical genre includes a wide
range of rhetorical activities including education (Sullivan, 1994), funeral oratory (Wills, 1992), even

poetry.

141

affection of the world; opposition to the work of reformation; an absence of public
spirit”; and ﬁnally sinning against the Gospel (Walker, 1893, pp. 427-432).

The Future Scene. The third element of Jeremiad’s scene is an afﬁrmation of
Puritan’s destiny. The ways in which the rhetoric of synod afﬁrms the status of Puritans
as God’s sheep is sprinkled all over in the jeremiad of the Synod. Nowhere in the report,
is even a hint of God’s intention to break the Covenant because of the Puritan’s sins. The
Puritans could be punished and chastised by God but they were not abandoned. “We
have therefore cause to fear that the Wolves which God in His holy Providence hath let
loose upon us, have been sent to chastise His Sheep for their dividings and strayings one
from another...” (Walker, 1893, p. 427, emphasis added). This afﬁrmation of a special
and unbreakable bonding between the God and Puritans together with existence of sins
forms a scene which rationalizes the acts of reforms.

Acts

Finally, the Synod responds to the question of what needs to be done to reform and
restore the covenant. With eradication of the sins as the telos, the Synod prescribed the
Church as a major site of reforms describing it as in need of more discipline, more staff
and more encouragement Interestingly, the Synod also mentioned schools of learning as

important sites for reform. “As an expedient for Reformation,” it says, “it is good that

 

’6 The synod does not seem to understand public in the same sense in which we understand it. Describing
the absence of public sentiment as an afﬂiction of New England, the synod says, “all seek their own, not
the things that are Jesus Christ’s. Matters pertaining to the Kingdome of God, are either not regarded, or
not in the ﬁrst place” (Walker, 1893, p. 432). In this report, the symbol of Jesus Christ and Kingdome of
God combine to assign to the term public its collective meaning. Interestingly, and in the next breath, the
synod also associates lack of public sentiment with the languishing state of the schools of learning. The
synod clearly recognizes the importance of school for a collective allegiance to the image of Jesus Christ.
Comparing these lines with the place of the symbol of America in the contemporary rhetoric, one may see
the symbol of America may be doing for the secular what Jesus did for the sacred.

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effectual care should be taken, respecting Schools of Learning. The interest of Religion
and good Literature have been wont to rise and fall together” (Walker, 1893, p. 43 7).

The parallel between the Reforming Synod ’s report with the modern day J eremiads
such as A Nation at Risk are unmistakable and striking. Replace, for instance, the
Covenant by the Nation and Church by School, and the similarity comes in full View.
This similarity and its implications, I will discuss later in this chapter.

The operation of scene-act logic in jeremiad

Here, I want to emphasize the dramatistic logic of scene-act ratio as applicable to any
act of reform that refers to jeremiad for its justiﬁcation. The reforms in the wake of the
synod’s report would be constrained as efforts to strengthen the church and the ‘schools
of learning.’ Imagine a hypothetical reform proposal devised as a response to the synod’s
report. Suppose that this hypothetical program sought to restore the covenant and
justiﬁed itself with reference to and as a response to the synod’s report, but instead of
being a program to strengthen the church or the schools of learning, as suggested by the
synod, it decided to strengthen the prison houses and lunatic asylums. Just because it
justiﬁed its existence with respect to the ﬁndings and prescriptions of the Synod, its acts
(reform prisons and lunatic asylums) would be perceived as out of synch with the
purposes (restore the covenant) they intended to serve. Alternatively, reforming prisons
will need to be defended as a better alternative than strengthening the Church in order to
achieve the same purpose (restore the covenant). However, if the terms of reforms did
not follow the prescription of the synod, or did not defend them with reference to the
purposes of reform, they would be seen as a rhetorical misﬁt due to the violation of

scene-act ratio.

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As I will argue in the next section, the scene-act ratio unites the earlier religious with

the contemporary secular jeremiads.

From Jeremiads to Jeremiads: Transformations from Sacred to
Secular

Although, I do not intend to summarize the whole history of the jeremiad as a form of
American political prose, it makes sense for the purposes of this argument to survey the
major shifts in jeremiad, while highlighting that they perpetuate the same dramatistic
scene-act ratio. One important shift is the secularization of jeremiads.

Horace Mann’s lectures delivered in the early nineteenth century can be seen as part
of this ongoing transformation. Like the earlier jeremiads, Mann’s rhetoric too directs
the attention of its audience toward a growing menace and decline. But the source of
impending doom is not God but (humankind’s) unschooled mind. In one of his lectures
delivered in 1837, he said:

The mobs, the riots, the burnings, the lynching, perpetrated by the men of the
present day, are perpetrated, because of their vicious or defective education, when
children. We see, and feel, the havoc and the ravage of their tiger-passions, now,
when they are full grown; but it was years ago that they were whelped and
suckled. And so, too, if we are derelict from our duty, in this matter, our children,
in their turn, will suffer. If we permit the vulture's eggs to be incubated and
hatched, it will then be too late to take care of the lambs (Mann, 1845, p. 13).

The theme of damnation and salvation is preserved in Mann’s rhetoric. However,
unlike the jeremiad of the Reforming Synod, the focus, and object, of reform is no longer
the church but the common school. Tyack and Cuban also read Horace Mann’s rhetoric

similarly, when they see him as taking his “audience to the edge of the precipice to see

144

the social hell that lay before them if they did not achieve salvation through the common
school” (Tyack & Cuban, 1995, p. 1). Mann, like the synod I discussed earlier, persuades
his audience by altering their perceptions about business as usual without reforms. When
his audience see themselves as teetering at “the edge of the precipice,” his calls for action
also assume urgency. Although common school replaced church in these rhetorical
substitutions common school replace church as the focus of attention, Horace Mann’s
case for common schools is similar to 17th century Puritan jeremiads inasmuch as it also
contains warnings of an impending doom. Furthermore, like its predecessors, it also
seeks salvation through reform. The agency of redemption, however, is not church but
common school.

The secularization of the American jeremiad over the past two centuries also entailed
other rearrangements. For example, jeremiads were transformed from the earlier state-of-
the-covenant sermons to the state-of-the-American Dream statements. This was also
exempliﬁed in the famous “I Have a Dream” jeremiad by Martin Luther King Jr. Like
the Synod’s report, Martin Luther King’s speech also mentioned a promise and depicted
it as at risk. Martin Luther King’s promise, however, was not to be traced back to the
scriptures but to the emancipation proclamation, a covenant not with God but with the
people. It then moved on to depict America’s aberration from this covenant, and ended
with a prophecy of what his American dream—a replacement for the term destiny in
earlier Jeremiads—would look like, if only we rededicated ourselves to the spirit of
emancipation proclamation. Emancipation proclamation then can be interpreted as the

secularized version of the Covenant.

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The rhetorical substitutions of religious with secular symbols entailed by the
secularization of American J eremiad (Jasinski, 2001) is succinctly summarized by
J asinski in the following quote from Sourcebook of Rhetoric:
First, the idea that Puritans were the chosen people was replaced by the idea
that the American people (or more abstractly, America) were somehow
special. . .Through this substitution or replacement, “America” becomes a
condensation symbol—a verbal cue that embodies our most basic beliefs and
values—and the promise contained in the Puritans’ covenant with God becomes
our collective promises to each other embodies in the idea of the “American
Dream.” Second, the Bible and the lives of the saints were replaced by new texts
and new heroes—the Declaration of Independence, the Constitution, Jefferson,
Washington, Lincoln, and so on—as the sources from which secular Jeremiads
could draw to instruct their audiences on the appropriate policies and practices for

realizing America’s special role in the world (Jasinski, 2001, p. 336).

The promise or destiny, frozen in the symbol of America, would be continually
rhetorically affirmed in the expressions of the fear of losing them and in the calls for
subsequent frenetic efforts—i.e. reforms—to restore to the symbol its earlier vitality.

The perpetual state of being at risk will come to characterize the state of American dream

and destiny and calls for reforms would surface as acts needed to restore it.

146

 

 

 

 

 

 

Elements of Pentad The Puritan Jeremiad Secularized Jeremiad

Scene State of the Covenant as State of the American Dream
in Decline as in Decline

Act Reforms Re orms

Agent Reformers Reformers

Agency Church Varied, Schools in the case of

Horace Mann

Purpose Restore the State of Restore the State of A merican

Covenant Dream

 

 

 

 

 

Table 9: Comparison between the Puritan and Secularized Jeremiads

Similarities between Theologized and Secularized Jeremiads

I have one more aspect of jeremiad to highlight before moving on to a discussion of
jeremiad as a scene for mathematics education debates in the last two decades. So far my
discussion of jeremiads has delineated them as a belonging to a rhetorical genre that aims
at shaping the perception of its audience about a present declining state of things before
grounding its calls for reforms in this perception. I would like to examine the dramatistic
implications of the jeremiadic reference to the term state (as in the phrase “state of
being”). Under a dramatistic translation, the term state assumes a scenic connotation.
But the state at any given time can also be imagined as an enactment. The term state in
most narratives is also indicated by a particular arrangement of things. Let me elaborate
this by considering the 17th Synod report again. When the Synod directed the attention of
its audience toward a bad state-of-the-covenant, it depicted the scene as characterized by
a proliferation of sins. Sinning, however, is an act. But in the synod’s report it
characterizes the state of Puritans, and hence has a scenic connotation.

Dramatistically speaking, acts and scenes can be interpreted to be in a symbiotic

relationship. Scenes can be seen as derived from acts, and vice versa. As descriptions of

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scenes, the 17‘h century J eremiads depicted the state of the covenant betWeen Puritans
and God. The point I am making is that it is possible to view J eremiads as acts. I could
do it if I wanted to, but since I am concerned more with reforms as acts, and since
jeremiads appeared to frame the reforms, I prefer to emphasize the scenic connotation of
jeremiads in my argument.

To sum up, the jeremiads work to mix lamentations about a present, and a declining,
state of the covenant (with God for 17th century jeremiads and with the Nation for the
secularized jeremiads) with prophecies about eventual salvation and fulﬁllment of a
covenant. The perpetuation of a similar scene-act ratio across religious and secular
jeremiads renders them dramatistically similar.

The acts sought by the jeremiads are expected to exorcise the scene by removing
from it the elements that it identiﬁes as at the root cause of anxiety. The purpose of
jeremiad, as Bercovitch puts it, is to “exorcise, revitalize, and consolidate” (Bercovitch,
1978,p.169)

Whether the acts appeal to the audience as serious and effective, or comic and
ineffective afﬁrms the logic of scene-act ratio. Recall from Chapter 2 that there is a
relation of consistency between the scene and the act:

It is a principle of drama that the nature of acts and agents should be
consistent with the nature of the scene. And whereas comic and grotesque works
may deliberately set these elements at odds with one another, audiences make
allowances for such liberty, which reafﬁrms the same principle of consistency in

its very violation (Burke, 1945, p. 3, emphasis added).

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Burke points toward the possibility of understanding the comic as a consequence of the
violation of the principle of consistency.

My interpretation of reform texts in mathematics education (as acts) is based on the
evaluation of this scene-act ratio. It is commonplace to think of the Math Wars as
between conservatives and progressives. As such, the overarching labels associated with

a set of binaries are given in the following table.

 

 

 

 

 

 

 

Conservative Progressive

Mathematical proﬁciency Mathematical power

Traditionalist Constructivist

Math as skills Math as conceptual
understanding

Direct instruction Child-centered instruction

Social efﬁciency Democratic equality

Positivist lnterpretivist

 

 

 

 

Table 10: Binaries of Math Wars

While this table highlights the terms of difference between the so-called progressive
and conservative responses to calls for reforms, it leaves out the terms that united both in
their claims as valid responses to those calls. As practicing mathematics educators we
know that progressive reforms were associated with a description of mathematics
education that articulated all the terms in the right hand column of the table 10 above as

various facets of reform. However, not much attention is paid to the implications of the

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language of standards that was common to both sides in the reforms, and the relation of
the language of standards with the other categories in this list. But when we see the
reforms as acts formulated in response to a jeremiad, the common denominator of
standards, as I will describe in more detail below, is sent into sharp relief. Thus the lens
of dramatism helps us see the rhetoric of reforms as arising from the requirements of a
scene-act ratio set up by the jeremiad.

As I will discuss below in detail, the term standards, qualiﬁed by the term
measurable, was foremost in the jeremiadic pronouncements that were used as
justiﬁcation by the reformers. The analysis in the next section will outline the ways in
which the reformers’ mobilization of the term standards together with other terms in the
right column of the table above worked as a double edged sword.

Even as it worked to qualify the reform discourse—as a proper act, it ultimately
worked to undermine its own rhetoric. Ultimately, the reform texts were made to appear
in the public discourse as comic by the traditionalist activists in a way that afﬁrmed the
consistency of a scene-act ratio by making the claims of reform-text appear as its

violation.

American Jeremiad and Mathematics Education Debates

The mathematics education reform and counter-reform texts that this dissertation is
concerned with are largely seen as part of the so called tidal wave of reforms generated in
the wake of the report A Nation at Risk by the National Commission for Excellence

(NCEE) in Education in 1983.

The NCEE was formed by US. Secretary of Education Terrel Bell in 1981, to study

perceived problems in education and prescribe ways of resolving them. Echoing the tone

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of earlier jeremiads, A Nation at Risk warned that US. society was losing its economic
preeminence to other competitors and that its security, prosperity, and democracy were at
grave risk due primarily to falling quality of American public schools. Brieﬂy, the report
described American schools as faltering in their mission due to the falling test scores,
linked this pattern to an eventual, urgently called for and then spawned a nation wide
movement for higher standards.

For the next decades or so, talk of standards ﬁlled the air. With the eruption of the
national standards in all subject areas in the hindsight, A Nation at Risk can be seen as
constituting the scene within which the drama of standards-based reforms would unfold
in the next two decades or so. This scene, then, framed both reform and counter-reform

4 . . . .
texts 7 m mathematics educatron as competrng acts. 48

A Nation at Risk had its share of detractors and supporters. The critics called it part
of a burgeoning “conservative restoration” that erupted a decade after the civil rights
movement (see, for example, Apple, 1982, 1993, 1996; Shor, 1986). The report has also
been critiqued for its ﬂawed use of evidence and accused of joining the wave of attacks
on public schools (Berliner & Biddle, 1995). My analysis of A Nation at Risk does not
pigeonhole it as either a progressive or conservative document. I am interested in

thinking about A Nation at Risk as an instance of American J eremiad. Jeremiads have not

 

47 1 am not defending the use of the terms reform and counter-reform texts on any essential grounds. From
a dramatistic perspective they are both regarded as competing acts within a scene that calls for them.
Reform and reformers is merely a place created by the scene and, as such, can be claimed by multiple
agents. 1, however, use the terms as they are used conventionally.

’8 I would like to emphasize that I do not make a simplistic assertion about A Nation at Risk as the only
motivating inﬂuence behind mathematics education reforms. Because of its formal similarity with the
jeremiads I discussed earlier, I am interested in thinking about what this scene-act relation means for
understanding the conﬂict between various competing responses to this report.

151

been a monopoly of conservatives alone (Bercovitch, 1978).49 In each case, however,

they have worked to constitute a context for subsequent reforms.

The relation between jeremiad and reform. Theorizing A Nation at Risk as an
instance of jeremiad permits an understanding of the mathematics education reforms (as
acts) in relation to a scene ﬁlled with crisis. Central to the analysis provided in the
preceding section is an understanding of jeremiads as constitutive of scenes in which
something of critical value to the society is described as eroding or declining. But
erosion, decline, and other such nouns presuppose a past state as a datum, which by
necessity should be free of these characteristics. That is to say, the descriptions of
erosion and decline in the present simultaneously evoke a past and a future state free of
them. The reforms, then, appear as acts sandwiched between an impending damnation—
a scene that calls for them—and a promised salvation—a scene that is expected to result

from them.

Dealing out the hand. Furthermore, a jeremiad is always announced from a high
pulpit.50 As such, then, the cards dealt out by the jeremiads are expected to be picked up
and played. The jeremiads thereby indicate general directions that reforms ought to
follow in order to appear as proportionate responses to them. For example, the report of
the reforming synod describes reforms as actions needed to strengthen the Church,
Horace Mann deﬁnes them as making common school accessible to all children, and

Martin Luther King Jr. calls for making good the promises contained in the emancipation

 

’9 For instance, even those who were talking of ‘conservative restoration’ in the United States in the 19805,
had their own jeremiad going on. The progressives have their own jeremiad. In fact, the particular battles
in the math wars may, then, be viewed as a polemic that pits jeremiad against jeremiads.

’0 I ﬁnd it signiﬁcant that jeremiads are not documented as the speech of the ordinary public. It is prophets,
saints, synods, political leaders, or the commissions—such as the one that authored A Nation at Risk—that
can speak the jeremiad.

152

proclamation. So the purposes and the natures of reform-acts are dealt out by the
jeremiad like hands in a card game. That is to say, the players—reformers in this case—
are constrained to play with the cards dealt out to them.

However, I am not arguing for a simplistic, necessary, and strict relation of
correspondence between terms in which calls for reforms and responses to such calls are
stated. Rather, I see such a scene-act relation as generating ambiguities, and, therefore,
also drama and conﬂict. As we will see in more detail, the jeremiad of A Nation at Risk
called for reform in terms of standards. That is to say, standards were dealt out by A
Nation at Risk as the hand in the game of reforms. To be of relevance on the scene
created by A Nation at Risk, both the so-called conservatives and progressives picked up
this card. As such, then, the conﬂict that followed in mathematics education in
particular—and to some extent in other school subjects—can also be seen as a battle over
the meaning of the term standards. As highlighted in Chapters 4 and 5, both the
mathematics education reforms and counter-reform texts were based on, supported by,
and repeated the term standards; the former associated the standards with the terms
mathematical power while the latter with mathematical proﬁciency for all students.

Revitalizing function. If we view A Nation at Risk as merely a politically
conservative document then we will not be able to make sense of the ways in which the
so called progressive reform documents, such as the NCTM Standards and the 1992
California Mathematics Framework, used it as their justiﬁcation. Alternatively, if we
adopt a rhetorical approach, we can see that a jeremiad such as A Nation at Risk had a

little something for all sides in the arena of education. In other words, it provided the

153

 

 

context for both the emergence of the so-called progressive as well as conservative

reform texts in mathematics education by providing rhetorical openings to both.

The Scene Described and Acts Called for by A Nation at Risk:
The Scene-Act Ratio for Mathematics Education Reforms

A Nation at Risk, like a characteristic jeremiad, emphasized a promise, declared it to
be at risk, called for the reforms for its restoration, and ended on a prophetic note
stressing an eventual success with past glory as the guiding light for future destiny. The
parallels between the synod’s report and A Nation at Risk are striking. Like the synod’s
report A Nation at Risk also set out to describe a dismal scene. While the synod had
described the scene in terms of the moral pitfalls of Puritans, the commission described it
in terms of falling test scores. While the synod held the church responsible for both the
past glories and the present predicament of Puritans and called for reforms to strengthen
it, A Nation at Risk held the schools responsible for both the American success and its
current risks and demanded for reforms that strengthened schools and made them
accountable for raising educational standards. While the ultimate purpose of the synod
was to rid the Puritans of the social evils and their consequences (sins against the
covenant with God), A Nation at Risk saw the telos of maintaining national preeminence
in terms of raising what appeared to be falling—the standards and the standardized test
scores. I provide this comparison between the two jeremiads to direct your attention to
the way role of A Nation at Risk, like all jeremiads, works to generate as well as to limit
the reforms by providing them with the terms in which they must be conceptualized.

The terms in which the scene of crisis was constituted by A Nation at Risk were

standards and standardized test scores. Any subsequent reforms, which were to refer to

154

A Nation at Risk as their justiﬁcation would then be constrained to defend them as
proposals to improve what A Nation at Risk had declared to be in decline—i.e. the
standards and standardized test scores. The report conceptualized both the problem of
equity as well as excellence in terms of declining standards. Test scores alone constituted
the evidence of the risk to the nation at large (N CEE, 1983, p. 113). The extended quote
below shows the emphasis on test scores in setting the stage on which the scene of crisis

was laid out by A Nation at Risk:

International comparisons of student achievement, completed a decade ago, reveal
that on 19 academic tests American students were never ﬁrst or second and, in
comparison with other industrialized nations, were last seven times.

. Some 23 million American adults are functionally illiterate by the simplest tests
of everyday reading, writing, and comprehension.

. About 13 percent of all 17-year-old5 in the United States can be considered
functionally illiterate. Functional illiteracy among minority youth may run as high
as 40 percent.

. Average achievement of high school students on most standardized tests is now
lower than 26 years ago when Sputnik was launched.

a The College Board's Scholastic Aptitude Tests (SAT) demonstrate a virtually
unbroken decline from 1963 to 1980. Average verbal scores fell over 50 points
and average mathematics scores dropped nearly 40 points.

. Both the number and proportion of students demonstrating superior achievement

on the SATs (i.e., those with scores of 650 or higher) have also dramatically

declined (NCEE, 1983, p. 115).

155

A Nation at Risk set up its descriptions of the risk in terms of declining test scores
and, in keeping with this description, called for higher and measurable standards. If the
scene of decline is characterized by results on standardized tests, and if the excellence is
articulated in terms of higher standards, then the calls to establish measurable standards
together with ways of meeting them appears to be a proportionate response.5 '

Another term that found its way into both reform and counter-reform documents is all
students. The term all obviously appealed to concerns about equity. In attempting to
trace the use of the term all students, I looked for similar usage in the previous waves of
reforms in the United States, but did not ﬁnd it stated as explicitly and directly as it was
in A Nation at Risk and the reforms texts spawned by it. The post Sputnik reforms, for
example, articulated a rhetoric of quality and excellence and not equity.52 However, A
Nation at Risk foregrounded the term all: “All, regardless of race or class or economic
status, are entitled to a fair chance and to the tools for developing their individual powers
of mind and spirit to the utmost” (N CEE, 1983, p. 115, emphasis added). This emphasis

on all was inscribed in both the competing responses in mathematics education as

mathematical power for all and mathematical proﬁciency for all.

 

5 ' Not everyone agreed with the conclusions and recommendations of A Nation at Risk. But even its critics
played on the terministic grounds prepared by this report If the report claimed that test scores had
declined, the critics disputed the evidence without disputing the measure itself (see, for example, Berliner
& Biddle, 1995). Thus even the critiques repeated the terms in which the scene of educational decline was
set up.

’2 To direct your attention to the inscriptions of the scene set up by A Nation at Risk in the subsequent acts,
I observe that the term all was not part of the vocabulary of post-sputnik wave of reforms. The post sputnik
reforms squarely focused on what Jerome Bruner worded as “a concern for quality and intellectual aims of
education—but without abandonment of the ideal that education should serve as a means for training well
balanced citizens for a democracy” (Brunet, 1977, p. I). The focus on all children that became a hallmark
of reforms constituted in response to A Nation at Risk points to scene-act logic.

156

Setting up excellence for all as the ultimate purpose of the reforms, the document
shaped a scene in which reforms would need to ultimately articulate the expectations and

goals—ultimately to be understood as standards:

Excellence characterizes a school or college that sets high expectations and goals

for all learners, then tries in every way possible to help students reach them.

Excellence characterizes a society that has adopted these policies, for it will then

be prepared through the education and skill of its people to respond to the

challenges of a rapidly changing world. Our Nation's people and its schools and

colleges must be committed to achieving excellence in all these senses.

(NCEE, 1983, emphasis added)
With excellence described in these terms, the scene prepared by A Nation at Risk

called for the acts [of reforms] to frame high expectations and goals [read standards], a
mechanism for achieving them, and ways of assessing progress toward them.
Furthermore, consistent with the terms in which the indicators for decline were stated, A

Nation at Risk qualiﬁed standards as measurable:

We recommend that schools. . .adopt more rigorous and measurable standards,
and higher expectations, for academic performance and student conduct. . .This
will help students do their best educationally with challenging materials in an

environment that supports learning and authentic accomplishment.
(NCEE, 1983, p. 125)

It also suggested regular conduct of standardized testing as a device to keep students

on course:

157

Standardized tests of achievement. . .should be administered at major transition
points from one level of schooling to another and particularly from high school to
college or work. The purposes of these tests would be to: (a) certify the student's
credentials; (b) identify the need for remedial intervention; and (0) identify the
opportunity for advanced or accelerated work. The tests should be administered as
part of a nationwide (but not Federal) system of State and local standardized tests.
This system should include other diagnostic procedures that assist teachers and

students to evaluate student progress. (NCEE, 1983, p. 125)

To summarize, A Nation at Risk used the rhetorical form of jeremiad to take its
audience—to use the terms used by Tyack and Cuban to describe Horace Mann’s
jeremiad—to the edge of the precipice and to show them the social and economic hell
that lay ahead if the decline in public education was not reversed. Like all jeremiads, it
also translated the terms of crisis—scene—declining scores and achievement gaps—in the
corresponding terms for reform-acts—higher and measurable standards and
accountability linked to improved achievement on such standards. From a dramatistic
standpoint it set up the expectations from the reforms in terms of a scene-act ratio.

In what follows, I will focus particularly on various responses [acts] to the calls for
standards, evaluating their conformity with the expected scene-act ratio. For each of these
responses, I would suggest to the reader to imagine them as acts in response to a
jeremiad. In this sense, the reform and counter-reform texts, the politicians, and the
critics all appear as reinforcing, the terms of reform set up by A Nation at Risk. The

scene-act ratio will remain the anchor of the analysis that follows. Through this

158

dramatistic ratio, 1 will also attempt to explain the ways in which the rhetoric of the

progressive reforms was rendered vulnerable.

Echoes of Jeremiad: The Expressions Scene-Act Ratio in
Reforms and Counter-Reforms

Politicians and the Discourse of Standards

The terms in which political leaders responded to the jeremiad of A Nation at Risk
can be seen as conforming to the scene-act ratio outlined in the previous section. My
evidence for this claim is based on data gathered from archives of the New York Times in
and the periodical Education Week I searched for and picked up pieces of news reports
that described the pronouncements of the political leaders in the decade following A
Nation at Risk, hoping to see the ways in which the terms of the jeremiad—measurable
standards, declining test scores, standardized tests as evidence of progress—were
repeated [or not] in the political pronouncements about education.

About the same time A Nation at Risk was released, another task force consisting
mainly of governors was reﬂecting on "Education for Economic Growth." The
jeremiadic tone of this task force echoed that of A Nation at Risk. It also spoke with a
similar urgency as A Nation at Risk in calling for a deep and lasting change in the
American education to put the country on a par with other economic competitors. “If we
are serious about economic growth in America—about improving productivity, about
recapturing competitiveness in our basic industries and maintaining it in our newer
industries, about guaranteeing to our children a decent standard of living and a rewarding
quality of life, then we must get serious about improving education” (Fiske, 1983). Like

A Nation at Risk, the governors also drew the motivation for educational reform by

159

identifying the imperative of economic growth with that of educational reform. One of
the governors was reported in the New York Times as saying, “The fate of the country is
in the balance. . .Education is the issue, and this country will rise or fall with it.”
(Hechinger, 1983).

A Nation at Risk's calls for higher and measurable standards were faithfully echoed by
the political actors. In particular, the media reports suggest a repetition and
reinforcement of the messages of A Nation at Risk in governors’ pronouncements in the
1980s The issue of educational standards appeared so frequently in pronouncements
from the governors that one of the New York Times reports found it apt to describe it as
“Gubernatorial activism” (Hechinger, 1983). The governors repeated A Nation at Risk’s
emphatic association of state-of—the—economy/Nation/People with state-of-education.
Also reverberating through their talk was the notion of higher and measurable standards
as instruments of public accountability.

The politicians’ call for more dollars for education was linked to demands for
accountability. With the measurable standards being seen as instruments of
accountability, the talk of reforms was transﬁxed by the discourse of measurable
standards. Just a year after the publication of A Nation at Risk, political and business
leaders were reported as pushing for evidence about whether or not increased ﬁnancing
of elementary and high schools was producing results. The interpretation of standards as
measuring sticks reverberated through this discourse, often expressed as calls to “measure
the condition of education across the country by developing a set of standards similar to
the gross national product, the Consumer Price Index and other indicators published by

the Government to describe the state of the economy” (Fiske, 1984).

160

To summarize, the discourse of political leaders in the decade after the publication of
A Nation at Risk was less ambiguous than, as I will show in the next section, that of
mathematics educators. They repeated and reinforced A Nation at Risk's call for
measurable standards and measuring instruments as the backbone of the much needed

reforms.

Standards in Mathematics Education Reform Texts: The scene-act ratio +

The Standards may be remembered as the ﬁrst attempt by any teachers' organization
to specify national, professional standards for school curricula in their discipline.
(Crosswhite, et al., 1989, p. 513)
The mathematics education community signaled a rhetorical identiﬁcation with the
scene-act ratio constituted by A Nation at Risk by fully employing the term Standards. 1
will refer to NCTM’s Curriculum and Evaluation Standards (N CTM, 1989) as CES to
distinguish them from other uses of the term standards.
The process leading to CES is described succinctly by Alan Schoenfeld as follows:
In 1986, the NCTM’s board of directors established the Commission on Standards
for School Mathematics, chaired by Thomas Romberg. The following year,
NCTM President John Dossey appointed a team of 24 writers to produce the
Standards. The group produced a draft in the summer of 1987, obtained feedback
on the draft during the 1987-1988 working year, and revised the draft in the
summer of 1988. NCTM published the Standards in the fall of 1989 and began a
major effort to bring the work to the attention of its membership. Copies of the

Standards were mailed to all members, and various aspects of the Standards

161

became the themes for regional and national NCTM meetings (Schoenfeld, 2004,
p.265).

CES were received in an environment in which A Nation at Risk ’s rhetoric of
restoration and revitalization of the nation through reforming schools, and reforming
schools through putting in place higher and measurable standards had already been
repeated, reinforced, and entrenched by a plethora of reports coming about the same time
as A Nation at Risk. “Between 1983 and 1984,” as documented by Wilson,
“approximately twenty national commissions reported on the ills of American schools”
(Wilson, 2003, pp. 121-122).

The CES were warmly received by a constituency already prepared for reception of
standards. As Diane Ravitch puts it:
The NCTM standards quickly became a dynamic force in changing staff
development, instructional practices, teacher education, textbooks, technology,
and assessment. Every new commercial mathematics textbook or instructional
program that entered the market since 1989 has claimed to incorporate the NCTM
standards. According to the NCTM, 30% to 40% of the nation's mathematics
teachers were using the new standards by 1992. Even the federally sponsored
National Assessment of Educational Progress (N AEP) began to change in
response to the NCTM standards. (Ravitch, 1993)
Some even spoke of CES as a national model for standards in other subject areas
(Winston & Royer, 2003, p. 193). The enthusiasm with which the CES were received

reinforces the point made earlier about the strength of scene-act ratio under the

162

jeremiads.53 The language of standards as a model of reform was a rhetorical ﬁt within
the scene-act ratio set up by the jeremiad of A Nation at Risk. Not everyone rejoiced, and
some scholars cautioned that suggesting a certain vagueness in the CES was kept there so
“that powerful groups or individuals who would otherwise disagree can ﬁt under the
umbrella” (Apple, 1992, p. 413).

However, the CES enunciated by the mathematics education reforms were also
contained simultaneously by multiple scenes. That is to say, while the rhetoric of
standards, particularly the CES, worked well within an air ﬁlled with the talk of
standards, it dropped the qualiﬁer measurable. The scene-act ratio set up by the
jeremiad had called for higher and measurable standards for all. Dropping the qualiﬁer
of measurable created the necessary space for the articulation of a conception of
mathematical power that was more consistent of a Dewey-ian conception of the relation
between content and pedagogy (I have discussed this in more detail in Chapter 4).

While the language of “standards” in the scene was constituted by the jeremiad, the
idea of mathematical power for all was circumscribed by a different historical scene, that
of pedagogical progressivism. 54 While the CES endorsed the scene constituted by A
Nation at Risk by using the language of standards for their proposals, it also incorporated
the elements of pedagogical progressivism. The CES did this by changing the deﬁnition
of what it meant to succeed in school mathematics, and, therefore, transforming the
objectives of reforms (Rothman, 1989). As a part of an explanation of the connection

between ambiguity and dramatism, I had referred to Russ McDonald’s observation about

 

5’ In fact the standards based reforms may be seen as the primary mode of reforms with other reforms being
spawned within the framework of standards.

5 The term pedagogical progressivism is used by David Labaree to distinguish the Dewey-ian tradition in
education from the administrative progressives (Labaree, 2005).

163

Shakespeare’s use of ambiguities to heighten the drama thereby bolstering the
explanation of connection between ambiguity and dramatism. Russ McDonald observes
that in critical moments of a Shakespearean play, his characters say utterances that cannot
be grasped with reference to the scene in which they take place. “For what they say
might have two meanings. The one meaning which the speaker has in mind refers to the
momentary situation, but the other meaning may point beyond this moment to other
issues of the play” (McDonald, 2004, p. 51). Likewise, I suggest that reformers’
rhetorical strategies involved using the language of standards to mean two things; as a
reference to momentary situation—i.e., as responses to calls for standards by a jeremiad:
as a site for reawakening the progressive discourses in American education, bolstered by,
as l have discussed in Chapter 4, by constructivism as a learning theory and availability
of Lakatos’ philosophy of mathematics as a fallible human activity”. While the reform
texts’ use of the term standards had the initial effect of strengthening their claims as
proper responses to A Nation at Risk, the meanings that reform texts infused into the term
standards diverged from the other competing references to the standards as measuring
sticks, i.e., as enablers of precise and measurable statements about students’ learning
outcomes.

Thus, the standards based on the conjunction of pedagogical progressivism,
constructivism, and Lakatos were a response to the call for standards, but one that used
the language of standards to challenge the scene of decline itself by changing the
deﬁnition of what it meant to succeed in mathematics. These transformed objectives

were stated in terms of developing mathematical power in all students.

 

55 For the sake of brevity, 1 will use Pedagogical Progressivism as an umbrella term for the conjunction of
progressivism, constructivism, and Lakatos ' philosophy of mathematics that l have described in Chapter 4.

l. 64

What it meant to succeed in mathematics, that is, to develop mathematical power, as I
argued in Chapter 4, was articulated largely as an open ended engagement with
mathematics in the classrooms. Such an engagement between the learner and subject
matter belonged to a scene which was constituted, not by A Nation at Risk, but by John
Dewey’s lamentations at the turn of the century about the educational sectarianism in
America, as resulting from a bogus separation between the child and the curriculum
(Dewey, 1902). Dewey had talked about two “sects” of education who were ﬁghting
over the curriculum. One, according to him, sought to “subdivide each topic into studies;
each study into lessons; each lesson into speciﬁc facts and formulae. Let the child
proceed step by step to master each one of these separate parts, and at last he will have
covered the entire ground” (Dewey 1902, p. 12). For the other sect, the “child is the
starting point, the center, and the end” (Dewey 1902, p. 13). Dewey sought to collapse
this binary distinction by claiming both child and curriculum as the two sides of the same
coin. Dewey’s lamentation and his suggestion to collapse the binary distinction between
the child and curriculum is seen by many as constituting the basis of pedagogical
progressivism (Labaree, 2005). However, it is important to note at this point that some
commentators observe that pedagogical progressivism lost out in the longer struggle for
American curriculum (Labaree, 2005; Lagemann, 2000).

Yet, just as the jeremiad of A Nation at Risk sought to restore and revitalize American
schools by calling for higher and measurable standards, the pedagogical progressivism
sought to respond to the challenge thrown by John Dewey. Indeed, some notable
reformers interpreted the NCTM standards and the ideas it contained, as creating

opportunities to imagine a resolution of the century old lamentations of John Dewey

165

about the separation of child and curriculum. As Ball observed: “Across school subjects,
current proposals for educational improvement are replete with notions of

99 ‘6

“understanding, authenticity,” and “community,” about building bridges between the
experience of the child and the knowledge of the expert” (Ball, 1993, p. 374). Ball’s
observation resonates with the terms of reform discourse—such as, Teaching for
Understanding and Authentic Assessment—that circulated in the educational discourse
through the 19905. In Chapter 4, I argued that reform texts’ way of building bridges
between the child and curriculum was expressed through a collapsing of content and
pedagogy. In making that claim, I am not as much referring to the construct of
Pedagogical Content Knowledge (Shulman, 1988)——a specialized knowledge base of
teaching—or its recent incarnation in mathematics education that goes by the name of
Mathematical Knowledge for Teaching (MKT). Rather, I wish to direct your attention to
statements that sought to narrowly redeﬁne mathematics (See Chapter4). This
redeﬁnition was expressed in the reform discourse through phrases such as mathematics
as reasoning, communication, problem solving, etc.

Reverberating through this scene, as Ball (1993) also suggests, are articulations of
Schwab’s idea about the possibility, and desirability, of a school curriculum that
approximates not just the knowledge but also the ways of knowing in a particular
discipline (Schwab, 1964), and Bruner’s hypothesis that any subject can be taught
effectively in some intellectually honest form to any child at any stage of development
(Bruner, 1977). Ball speaks of this tension succinctly in describing her stance toward

teaching: “With my ears to the ground, listening to my students, my eyes are focused on

the mathematical horizon” (Ball, 1993, p. 376). But the scene in which a practice which

166

 

requires teachers to be both grounded in their students with their eyes ﬁxed at the
disciplinary horizons at the same time does not entertain measurable standards in a
straight forward way.

This point can be understood with reference to Dewey’s, observations on the term
standards—though made in relation to the critical judgment of the works of arts—is
relevant to this discussion. Dewey deﬁned standards as precisely measurable attributes
that could be used for comparison of physical attributes of things. Using such a notion of
standards to compare works of art, thought Dewey, could be misleading:

When, therefore, the word “standard” is used with respect to judgment of works
of art, nothing but confusion results, unless the radical difference in the meaning
now given standard from that of standards of measurement is noted. The critic is
really judging, not measuring physical fact. He is concerned with something
individual, not comparative—as is all measurement. His subject matter is
qualitative, not quantitative. There is no external or public thing, deﬁned by the
law to be the same for all transactions, that can be physically applied (Dewey,
1959,p.307)

But it is individuals, and groups that measurable standards in education would
ultimately seek to compare. Yet, the progressive reformers, nonetheless, used the
language of standards to ﬁnd a place in the scene constituted by the talk of measurable
standards. They also simultaneously attempted to respond to Dewey’s challenge to
eliminate the bogus distinction between the child and curriculum. Within the scene-act

framework this could be done—as Dewey had suggested in the case of debates about

167

standards of judging the works of art—by rearticulating the meaning of the term
standards.

Indeed the meaning of the term standards multiplied in the reform discourse, with all
meanings carefully avoiding the connotation of standards as measures. Thomas
Romberg, for instance, interpreted them as a vision and a set of criteria for judging the
quality of curriculum and instruction (Romberg, 1992). Standards were also called the
rallying ﬂag for teachers (Ball, 1991; Romberg, 1992, 1993). The official interpretations
of the term standards offered in the reform texts also kept changing over time. NCTM’s
Curriculum and Evaluation Standards (CES) deﬁned them as a project to “guide the
revision of the school mathematics curriculum and its associated evaluation toward this
vision” (N CTM, 1989, p. 1) and also as “statements of criteria for excellence in order to
produce change...” (NCTM, 1989', p. 1). The CBS, under this View, were to be viewed as
facilitators of reform. They were also portrayed as based on a grand consensus of the
professional mathematics education community about the nature of mathematics and its
teaching and learning (Carl & Frye, 1991; Crosswhite, et al., 1989).

The Principle and Standards of School Mathematics (N CTM, 2000), discussed
standards as instruments of improvement in school mathematics by providing
opportunities for debates and improvement:

As with the previous NCTM Standards, Principles and Standards offers a
common language, examples, and recommendations to engage many groups of
people in productive dialogue. Although there will never be complete consensus
within the mathematics education profession or among the general public about

the ideas advanced in any standards document, the Standards provide a guide for

168

focused, sustained efforts to improve students' school mathematics education

(NCTM, 2000, p. 5).

Calling them a vision, criteria, professional consensus, rallying ﬂag for teachers of

mathematics, or a trigger for debates, these reform documents kept holding the language

of standards in a tight embrace until recently. The most recent document published by

the NCTM—The Curriculum Focal Points: A Quest for Coherence—replaced the

language of standards with that of the focal points (NCTM, 2006). The NCTM focal

points are statements about a small number of signiﬁcant mathematical “targets” for each

grade level. These statements attempt to distinguish them from the “commonly accepted

notions of goals, standards, objectives, or learning expectations” (NCTM, 2006, p. 1).

Rather, the document claims to be containing descriptions of the most signiﬁcant

mathematical concepts and skills that must be attained at each grade level.

I construe the shifts in the deﬁnitions of standards and ultimate departure from this

language as signaling a vulnerability that can be accounted for with reference to the

scene-act ratio constructed by a jeremiad.

 

 

 

 

 

 

Jeremiad of A Nation at NCT M Standards 1999 Framework
Risk
Scene Declining Test Scores Declining Test Scores Declining Test Scores
Act Reforms: Put in Place Reforms: Put in Place Reforms: Put in Place
Higher and Measurable Higher Standards Higher and Measurable
Standards Standards
Agent A wide range of Reformers Counter-Reformers
stakeholders, including
what ultimately became
reformers and counter
reformers
Purpose Excellence for all Mathematical Power for Mathematical
All Proﬁciency for All

 

 

 

 

Table 11: Comparison of Scene-Act ratio in A Nation at Risk, NCT M Standards, and

1999 F rgmework

169

 

Standards after the NCTM standards: The Afﬁrmation of scene-act Ratio

“What if these standards are wrong” asked Chester E. Firm (1993). Acknowledging
the initial success of standards Finn suggested what I interpret as violation of scene-act
ratio, indicating what he thought was the misunderstanding between the content standards
and student-performance standards:

The content standards describe what schools should teach and—presumably—
their pupils should learn. . .Content standards are about curriculum: its goals,
frameworks, scope and sequence, etc. They are intended mostly for
educators. . Student-performance standards are something else. They involve how
well youngsters must do in order to be said to have met the expectations of the
content standards (Finn, 1993).

Firm observed that while NCTM had provided us with Curriculum Standards, it did
not provide student performance standards needed so that “students (and parents) see how
well they—and their schools—are doing vis-a-vis what's expected of them. Student-
perforrnance standards are truly about results and outcomes” (F inn, 1993).

Details of the math debates are messy and, though not irrelevant, are not needed for
the analysis I am rendering here. Taking into consideration the fact that a progressive
agenda was reversed in California, I was interested in using dramatism to account for the
rhetorical vulnerabilities associated with using the term standards by the progressive
reformers, when the scene-act ratio was constituted by a jeremiad.

Arguably, the drama, its scenes acts and actors, its situation of comedy and tragedy
may be seen as constituted in affirmation and violation of dramatistic ratios. If the scene

calls for crying, and the actor laughs out loud, the latter can be interpreted as a violation

170

of scene-act ratio. The act of laughing when the scene calls for an expression of sadness
could make sense as insanity or, perhaps, a move to sweeten the air of a bitter scene by
doing something unexpected. If the actor is able to alter the scene the ratios may
reversed. If not, the act would not appear to be contained by the scene. In the case of
mathematics education reform texts, the standards were infused with a meaning that was
not anticipated, or called for, by the jeremiad of A Nation at Risk. Not so, however, with
the counter-reform texts.

In Chapter Five, I examined the 1999 Framework and the embedded California
Content Standards. I would like to repeat my ﬁndings from Chapter 5 brieﬂy to argue
that 1999 Framework appears perfectly aligned with the scene-act ratio set up by A
Nation at Risk and endorsed and repeated by politicians.

The 1999 Framework sets up standards as precise statements about strands and
benchmarks, specifying in precise and measurable terms what the students should know
and be able to do. The idea of mathematical proﬁciency for all is advanced as a
measurable construct, and a regimen of standardized testing is put in place to assess
progress, make diagnostic judgments, and place students in diagnostic groups
accordingly. The idea of rigor is introduced to bolster the rhetoric of measurable
standards. Students are supposed to learn ‘rigorous mathematics,’ are to be assessed
‘rigorously,’ and ‘rigorous’ research is needed to evaluate interventions that are more

likely to help students achieve the standards.

Reform Texts, Comedy and Tragedy: The Afﬁrmation of scene-
act Ratio

The violation of scene-act ratio by the reform texts created the opportunity for the

comic and the satire. Funny images of reformers and reform texts began to ﬁll the air,

171

already contaminated by the language of war. Nearly all of these depicted the reform
texts as working against the reforms. I am not interested in whether these renderings of
the reforms were right or wrong. Rather I am thinking about the possibility of comedy in
this drama in which the scene was set up by a very serious jeremiad stating the nation to
be in jeopardy due to falling test scores. Yet, the discourses that took the lead in
occupying the reform ﬁeld appeared to be working. This was akin to doing something
that the public believing in the jeremiad did not expect. For example, one column written
by Deborah Saunders“, a columnist writing for San Francisco Chronicle, suggested that
the reforms called for by A Nation at Risk had been hijacked by the ‘edu-crats’.57 Quoting
A Nation at Risk, she wrote:
More than a decade ago, a national commission issued a devastating report,
"A Nation at Risk." It warned, “If an unfriendly foreign power had attempted to
impose on America the mediocre educational performance that exists today, we
might well have viewed it as an act of war.” In the wake of the report, reforms
were promised. Some improvements resulted, but also some horriﬁc trends
accelerated full bore. No surprise: The very people [the reformers] behind the
“mediocre educational performance” were piloting the so-called reforms
(Saunders, 1995).

The detractors rhetorically identiﬁed reformers and reform-texts with the very ‘sin’

1 that the jeremiad of A Nation at Risk had identiﬁed for elimination—the falling standards

 

’6 Deborah Saunders wrote frequently, launching a ruthless tirade against the progressive reforms. Between
1995 and 2000, she wrote over 25 columns in the San Francisco Chronicle.

57 Wilson has documented what she terms as the labels assigned conservative journalists for the reformers.
The list included. “pedagogical imperialists,” “educrats,” “the educational establishment,” or “curriculum
Nazis” (Wilson, 2003, p. 145),

172

and the test scores. What began may be termed as another jeremiad in the public sphere
against the reforms and reformers. This jeremiad was played out by the inﬂuential
parents, conservative politicians, conservative journalists, and like all jeremiads, it did
not need accurate evidence for its claims.58 Where A Nation at Risk had called for
reforms, this jeremiad called for war against the progressives ‘masquerading’ as
reformers. This jeremiad set up a scene of its own with its own scene-act ratio. A scene
described as one in which an ‘evil’ power had come to dominate the policy space and
victimize children and parents. The scene called for its own heroes.

The ﬁgure of war was used as an act by the counter reformers to identify them as
rescuers and saviors of the ‘oppressed’ children. What would be more effective as a
rhetorical ﬁgure than Abraham Lincoln’s 271-word address given on the dedication
ceremony of the Gettysburg National Cemetery? I quote from the take off from the
Lincoln’s address put up as the dedication of the warriors on mathematically correct,59
one of the ﬁrst and best known of the websites maintained by the counter reformers:

Over four score and seven decades ago philosophers brought forth into this
world a new mathematics, conceived in correct computational formulae and
dedicated to the proposition that two plus two equals four.

Now we are engaged in a great educational war, testing whether Algebra I or
any form of mathematics so conceived and so dedicated can long endure. We are

met on a great virtual battleﬁeld of that war. We have come to dedicate a portion of

 

5" This rhetoric depended for its sayability, not on evidence but on mobilizing the ﬁgure of war and
repeatedly identifying reformers with denigrating titles (for details of this counter-reform campaign, see,
Wilson, 2003, pp. 132-165).

59 1 ﬁrst accessed mathematically correct in the beginning of 1996, a few days after setting my feet in the
United States. It is still online and can be accessed at http://www.mathematicallvcorrect.com

173

that ﬁeld to those who are giving up the quality of their education so that
California's Math Framework might live. It is altogether ﬁtting and proper that we
should do this.

But in a larger sense, we cannot dedicate, we cannot consecrate, we cannot
hallow their loss. The brave children who now must struggle to learn math outside
of the classroom have consecrated it far above our power to add or subtract. The
world will little note nor long remember the actions of a few irate parents, but it can
never forget what fate has befallen the children. It is for us, the mathematically
competent, rather to be dedicated here to the unﬁnished work, to the battle to save
basic math skills that has thus far been so nobly advanced. It is rather for us to be
here dedicated to the great task remaining before us—that from these honored
children we take increased devotion to the cause for which they gave up their
weekends and vacation time—that we highly resolve that these children shall not
have suffered in vain, that this state shall have a rebirth of computational skills, and
that a mathematics of Algebra 1, Geometry, and Algebra 11 shall not perish from our
schools (Clopton, 1995, italics mine).

This takeoff from the Gettysburg Address substitutes “a new nation, conceived in
Liberty, and dedicated to the proposition that all men are created equal” with “a new
mathematics, conceived in correct computational formulae and dedicated to the
proposition that two plus two equals four,”; the “great civil war” with “great educational
war”; and “the honored dead” with “these honored children”. It’s a rhetorical move that

sets up the counter reformers as rescuers and saviors of the mathematics.

174

The scene of math wars, so constructed in the rhetoric of counter-reforms, came
complete with innocent victims (the children and parents) victimized by villains
(reformers), and in need for heroic rescue by those who were competent and cared
enough to rise to the occasion. The reform texts were projected as harbingers of tragedy
and destruction in a scene that had called for reforms to suppress the ‘rising tide of
mediocrity’ (see, NCCE, 1983). In sum, the counter-reform criticism projects reformers
as mathematically incompetent villains whose victims, children and parents stood in need
of rescue, and the mathematically competent as heroes under obligation to rescue them.

Furthermore, the language used by the critics of reform texts often invoked the
important cultural notions of ‘fundamentals’ and ‘rigor,’ a rhetoric apparently well suited
to engage the sympathy of scientists and mathematicians. The reformers’ views were
projected as compromising the aspects of fundamentals and rigor and replacing them
with what critics termed as fuzzy notions of mathematics and mathematical practice.
Reform texts were satirized as Weapons of Math Destruction, reformer as educrats and
educational establishment.

All of this undermined what mathematics educators termed as the professional
consensus (see, Carl & Frye, 1991; Crosswhite, et al., 1989; Frye, 1990) by making their
public claims about the content and pedagogy of school mathematics appear as falling

short on such culturally important criteria as, say, rigor.

Concluding Thoughts

In the previous chapters, I had described the drama of reform and counter reform
texts mapping it out internally. That is to say, I presented to my reader the respective

scenes within which the terms of both reforms and counter reforms made sense. In this

175

chapter, I went at a level of dramatistic analysis that sets up both reforms and counter-
reforms as acts within a scene constituted by the jeremiad of A Nation at Risk.

I have argued that this was a drama in which both reforms and counter-reforms were
responses to calls for reform embedded in A Nation at Risk. Both, therefore, were
constrained to use the language of standards. Reform texts, however, were not a
straightforward response to the calls for higher and measurable standards. Beyond the
scene constituted by the jeremiad of A Nation at Risk they were also a response to another
inﬂuential text, namely Dewey’s lamentation about the bogus separation of the child and
curriculum. The math education debates, then, were battles that pitted one seminal
educational text against the other. The reformers were at a disadvantage primarily
because they set themselves an impossible task: to respond to both of these key
documents at the same time.

Since 1999, the descriptions of standards have seen some shifts that can be interpreted
as reassertion of the scene-act ratio constituted by the jeremiad of A Nation at Risk
Nearly all documents released by the NCTM and other panels and groups, including the
recently published report by National Mathematics Advisory Panel (N MAP) endorse the
notion of measurable standards (Ball, 2003; Kilpatrick, et al., 2001; USDOE, 2008). The
cumulative effect of the reforms and counter reforms, so far, has been to reinforce and

afﬁrm the drama set in motion by the Jeremiad.

176

 

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