PAUL ERNEST’S SOCIAL CONSTRUCTIVIST PHILOSOPHY OF MATHEMATICS EDUCATION

Erin Cecilia Wilding-Martin

Parkland College, Champaign, Illinois, USAEMartin@parkland.edu

A DISSERTATIONSubmitted in partial fulfillment of the requirements for the degree of Doctor of

Philosophy in Educational Policy Studies in the Graduate College of the University of Illinois at Urbana-Champaign, 2009. Urbana, Illinois

Doctoral Committee: Professor Emeritus Walter Feinberg, Chair; Professor Debra D. Bragg; Professor Timothy G. McCarthy; Assistant Professor Michele D. Crockett

© 2009 Erin Cecilia Wilding-Martin

ABSTRACT

This dissertation analyzes the theoretical underpinnings and implications of Paul Ernest’s social constructivist philosophy of mathematics education. Ernest sees learning as the social construction of knowledge through conversation. Therefore, he believes that mathematics education should foster knowledge construction through active engagement and student interaction. In addition, he claims that mathematics education should contribute to the development of democratic citizens who are able to critically evaluate political and social claims that are based on mathematical arguments. Ernest also makes recommendations for curricula and pedagogy, calling for a differentiated mathematics curriculum at the secondary level. While future mathematicians need to be prepared for advanced study in mathematics, he believes that mathematics education for other students should provide inquiry-based activities that encourage critical thinking, empower learners, and encourage students to be aware of and involved in social issues. This dual approach to mathematics education raises questions about Ernest’s underlying ideas on the nature of mathematical knowledge and the nature of democratic values.

The main focus of this project is to critically analyze Ernest’s philosophy of mathematics education and his conceptions of mathematical knowledge and democratic education. Specifically, I use textual analysis to interpret what he intends for each group of students and to critically analyze the conceptions of mathematical knowledge and democratic citizenship implicit in each set of recommendations. I examine whether these conceptions are consistent with each other and with his philosophy of mathematics education. I argue that while Ernest’s general conceptions of mathematical knowledge and democratic citizenship are similar for these two groups, he views mathematicians as

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producers of mathematical knowledge and the general public as users of this knowledge. This characterization seems to place research mathematicians in a privileged position. Finally, I explore the implications of Ernest’s framework for developmental mathematics at the community college level. I discuss what his view would contribute to conversations about educational opportunity, and I use specific examples to illustrate how this framework might be used to evaluate policy and practice in this context.

ACKNOWLEDGMENTS

Many people have helped to make this project possible through their support and encouragement. I would especially like to thank my advisor, Walter Feinberg, for reading numerous drafts and offering so much thoughtful advice and constructive feedback. I am also grateful to my committee members, Debra Bragg, Timothy McCarthy, and Michele Crockett, who offered guidance and support in their respective areas of expertise. It was extremely rewarding to consider my work from these different perspectives.

I would also like to express my appreciation to my family. Thank you to my parents, Don Wilding and Rachel Wilding, for teaching me to be inquisitive, instilling in me a desire to learn and to understand. I am grateful to my sister, Elizabeth Wilding, for commenting on early drafts and for always being there. Finally, I must thank my husband, Vance Martin, who has been infinitely patient and supportive through this entire process. I could not have finished without his encouragement and understanding.

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TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION.........................................................................................1Background..............................................................................................................3Overview of the Study.............................................................................................6

CHAPTER 2 CONSTRUCTIVISM..................................................................................11Debates in the Philosophy of Mathematics...........................................................12Constructivist Philosophies of Individual Knowledge..........................................17Constructivist Pedagogy........................................................................................21Summary and Conclusion......................................................................................26

CHAPTER 3 LAKATOS AND OTHER INFLUENCES.................................................27Vygotsky’s Social Theory of Mind.......................................................................28Wittgenstein’s Philosophy of Language................................................................31Lakatos’s Quasi-Empiricist Philosophy of Mathematics......................................35Summary and Conclusion......................................................................................61

CHAPTER 4 ERNEST’S SOCIAL CONSTRUCTIVISM...............................................63Philosophy of Mathematics...................................................................................64Philosophy of Mathematics Education..................................................................79Summary and Conclusion....................................................................................110

CHAPTER 5 ERNEST’S DIFFERENTIATED CURRICULUM..................................112General Mathematics Education..........................................................................114Education for Future Mathematicians.................................................................125Critical Comparison.............................................................................................137Tracking Issues....................................................................................................139Summary and Conclusion....................................................................................146

CHAPTER 6 IMPLICATIONS IN A DEVELOPMENTAL MATHEMATICS CONTEXT........................................................................................148

Community Colleges...........................................................................................151Developmental Mathematics...............................................................................156AMATYC’s Developmental Mathematics Standards.........................................160Developmental Mathematics Textbooks.............................................................175Developmental Mathematics Classroom Pedagogy............................................181Obstacles to the Application of Ernest’s Framework..........................................188Summary and Conclusion....................................................................................196

WORKS CITED..............................................................................................................201

AUTHOR’S BIOGRAPHY.............................................................................................211

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CHAPTER 1

INTRODUCTION

Mathematics education, it seems, has always been in the public eye. Are we

teaching students enough? Do students need all the mathematics we are teaching them?

From the recommendations of the National Council of Teachers of Mathematics,1 to the

standards adopted by each state, to the textbooks chosen by school boards, to teaching

decisions in the classroom, there is no shortage of opinions about what mathematical

topics should be taught, and to whom, and how. There are many difficult issues that arise

in discussions about school mathematics. What content should be covered? What

teaching methods should be used? Should students be tracked by ability, career goals, or

not at all? These questions must be answered within a philosophical framework that

defines the aims of mathematics education and outlines a theory of learning. One might

envision the role of mathematics education in society as training mathematically literate

citizens, educating a productive workforce, or enriching the individual mind. It might be

believed that students master content through repeated exposure and practice, or that they

must form a deeper understanding through a variety of experiences. These ideas, whether

clearly articulated or not, shape decisions that are made about policy and practice.

Though philosophical issues are often overlooked in conversations about practice,

they have an impact on issues in the classroom. Last year I overheard an argument

between two mathematics instructors about the proper way to teach students to factor

quadratic polynomials. The first instructor explained that he requires his students to use

trial and error, guided by some reasoning about the relationship between polynomials and

1 National Council of Teachers of Mathematics, Principles and Standards.

1

their factors. The second instructor responded that she prefers to teach students an

algorithm that in some cases takes more time, but allows the students to approach

factoring more systematically. The first instructor insisted that reasoned trial and error is

the method mathematicians actually use, that it better reveals the mathematical concepts

involved in factoring, and that it is more efficient for students in advanced mathematics

courses. The second instructor responded that most of her students will not become

mathematicians, and they need an easier approach to factoring, even if it takes longer.

These instructors were each trying to convince the other that theirs is the “best” method

for teaching factoring. What they did not realize was that the fundamental disagreement

was not about the best teaching method. Rather it was about their underlying visions and

goals for mathematics education and their assumptions about student learning. The first

instructor sees the role of mathematics education as helping students to understand

mathematical structure and preparing them for advanced mathematical study. He

believes students learn mathematics by engaging in the types of activities that are seen in

the professional mathematical community. For the second instructor, the main goal is to

help students who are not inclined to pursue advanced mathematics to learn the algebraic

skills they need to pass required courses. She believes that students need to practice

organized, step-by-step procedures to learn mathematics. Neither instructor will be able

to see the other’s point of view unless they recognize how those views have been

influenced by underlying philosophical differences. This may not help them to agree, but

it will allow their conversations to address the real issues, and hopefully to be more

productive. This is just one example of how philosophical views can shape our

understanding of educational practice and its related issues. The study of a particular

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view can highlight key issues or conflicts and may offer “radically different possibilities

that can stretch the imagination and expand the spirit.”2

A philosophy of mathematics education has at its core a set of aims and purposes

for mathematics education, a theory of mathematical learning, and a theory of teaching

which implements the learning theory within the stated aims. Other aspects of the

philosophy are generally informed by these core elements. Taken as a whole, the work of

Paul Ernest provides a comprehensive exposition of his social constructivist philosophy

of mathematics education, including an extensive description of the three core elements.

He draws on this philosophy to make recommendations about elementary and secondary

mathematics curricula and pedagogy. In general terms, the goal of my study is to analyze

the consistency of these recommendations with Ernest’s philosophy and to explore

possible applications in the community college developmental mathematics context.

Background

Constructivism has garnered much attention in the educational literature.

According to D. C. Phillips, “Across the broad fields of educational theory and research,

constructivism has become something akin to a secular religion.”3 Constructivist

philosophy addresses issues about the origins of human knowledge as well as the

development of individual understanding. In terms of the latter, constructivism refers to

a group of views which hold that students must construct meaning for themselves by

using new information to build on previous understandings.4 Constructivists such as

Ernst von Glasersfeld see this as an individual process, while others like Ernest believe it

2 Burbules and Warnick, "Philosophical Inquiry," 501.3 Phillips, "Good, Bad, and Ugly," 5.4 Ernest, "One and the Many," 461.

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requires social interaction and negotiation of meaning. In mathematics education, von

Glasersfeld’s radical constructivism has had wide influence. Based on Jean Piaget’s

theory of cognitive development, von Glasersfeld’s view is that learning happens through

a process of reflective abstraction. This process allows us to apply our existing

conceptual structures to new situations, and reorganize those structures if there is a

conflict.5 Von Glasersfeld’s work raises concerns for some about the nature of external

reality and social influences.6 Social constructivists hope to avoid this problem by

incorporating a view that human cognition is formed through social interaction.

Ernest’s social constructivism includes a philosophy of mathematics education

that incorporates a philosophy of mathematics, a theory of teaching and learning, and a

particular set of democratic aims for mathematics education. My focus will be on the

latter two elements, as his philosophy of mathematics addresses issues that are outside the

scope of this project. Ernest sees learning as the social construction of knowledge

through conversation. Therefore, he believes that mathematics education should

encourage the process of knowledge construction through active engagement and student

interaction. In addition, he claims that mathematics education should play a role in the

development of democratic citizens who are able to evaluate political and social claims

based on mathematical arguments.7 Ernest also makes recommendations for curricula

and pedagogy. He calls for a differentiated mathematics curriculum which takes into

account different talents and career goals. For future mathematicians, he believes that

school mathematics should be an enculturation into the mathematical community and its

conventions of conjecture, proof, refutation, and revision. However, Ernest asserts that

5 von Glasersfeld, Radical Constructivism, 104-5.6 Phillips, "Good, Bad, and Ugly," 8.7 Ernest, Philosophy of Mathematics Education, 207-8.

4

most students do not need this type of preparation.8 Instead he feels that students who are

not preparing to be mathematicians should have the opportunity to experience

mathematics as an interesting and relevant discipline, and should learn to question

mathematical arguments in the media. Thus he recommends a pedagogy that is focused

on investigation and problem solving. He feels that this will encourage critical thinking,

will empower learners, and will serve as an emancipatory force to counter social

reproduction “through a critical awareness of the role of mathematics in society.”9 While

it may seem obvious to some that students with different talents and career goals need to

learn different mathematics content, this dual approach to mathematics education raises

deeper issues about Ernest’s underlying ideas on the nature of mathematical knowledge

and the nature of democratic values.

In the existing literature, there appear to be no substantive analyses of Ernest’s

philosophy. While Ernest is cited often by philosophers and empirical researchers, no

one has conducted an extensive analysis of his theory or probed his work in any

significant way. Limited critiques of his view sometimes appear as part of larger

projects. These tend to focus on the adequacy of Ernest’s philosophy of mathematics or

his account of intersubjective agreement.10 There do not appear to be any in-depth

analyses of his approach to education or its internal consistency.

The main focus of this project is to critically analyze Ernest’s philosophy of

mathematics education. I do not attempt to make any determinations about the adequacy

of Ernest’s mathematical ontology and epistemology. Rather, I examine the consistency

of his arguments about mathematical knowledge, democratic aims, and mathematics 8 Ernest, Social Constructivism, 273.9 Ernest, Philosophy of Mathematics Education, 291-92.10 For example, see Macnab, "Epistemology, Normativity"; Matthews, "Appraising

Constructivism"; Stemhagen, "Beyond Absolutism and Constructivism."

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education. Then I explore how Ernest’s framework might be applied to a new setting,

developmental mathematics education at the community college level.

Overview of the Study

This project is a critical analysis of Ernest’s philosophy of mathematics education.

Ernest’s view provides an approach to mathematics education based on a social

constructivist account of mathematical knowledge and a particular set of democratic

ideals. However, his recommendations for the training of future mathematicians differ

from his recommendations for the mathematics education of other students. In this study,

I analyze the strengths and weaknesses of Ernest’s philosophy of mathematics education

and then examine his bifurcated approach to mathematics education. Specifically, I use

textual analysis to interpret what he intends for each group of students and to critically

analyze his conceptions of mathematical knowledge and democratic citizenship. Then I

examine whether the conceptions implicit in each area of mathematics education are

consistent with each other and with his philosophy of mathematics education. For

example, I consider whether Ernest sees mathematics playing the same role in the

education of all students or if there are different purposes for different groups. This

reveals underlying assumptions about mathematics education that are analyzed for

consistency with his educational aims and theory of learning. Finally, I demonstrate the

applicability of such an analysis through a consideration of developmental mathematics

at community colleges, showing how the issues I have highlighted impact discussions

and debates about teaching in this context.

In chapter two, I provide an overview of constructivism as a theoretical

background for Ernest’s work. A distinction is made between constructivism as a

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philosophy of mathematics and as a philosophy of mathematics education, and I explain

why this project need not consider the debates in mathematical philosophy. Then I

provide an introduction to constructivist philosophy as it relates to mathematics

education. I describe von Glasersfeld’s radical constructivism and contrast it with its

rival position, behaviorism. This provides a context for the development of Ernest’s

social constructivism and its departure from other constructivist views.

Chapter three examines the major theoretical influences that Ernest draws on to

distinguish his social constructivism from the radical constructivism of von Glasersfeld.

First I summarize Lev Vygotsky’s social view of mind and the important role he ascribes

to language in cognitive development. Then I describe Ludwig Wittgenstein’s theory

that language is based on social agreement through our participation in language games

embedded in human forms of life. Forms of life refer to the human purposes, rules, and

behaviors that give language meaning. Wittgenstein sees mathematics as a language

game with its own set of rules embedded in mathematical forms of life.11 The main focus

of this chapter is the quasi-empiricism of Imre Lakatos. Breaking from the work of Karl

Popper, Lakatos emphasizes that discovery is intertwined with justification in

mathematics, in a cycle he calls the “logic of mathematical discovery.”12 I analyze the

classroom dialogues in Lakatos’s Proofs and Refutations, which form the basis for

Ernest’s philosophy of mathematics education. Though these dialogues are set in a

classroom, they were intended to model the mathematical community rather than

mathematics education. I analyze the philosophy embedded in the dialogues and their

implications as a model for mathematics education.

11 Wittgenstein, Remarks, 381.12 Ernest, Social Constructivism, 99-100.

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In the fourth chapter, I critically analyze Ernest’s philosophy of mathematics

education. First I explain how he embeds Lakatos’s logic of mathematical discovery in

Wittgenstein’s forms of life to develop his social constructivist philosophy of

mathematics. I examine how these views have influenced Ernest’s account of the nature

of mathematics and mathematical knowledge, but I do not take a position on whether this

account is accurate or true. Then I outline Ernest’s philosophy of mathematics education,

which comprises a set of democratic aims and values for mathematics education, a theory

of learning that incorporates Vygotsky’s social view of mind, and a theory of teaching

based on mathematical investigation. I argue that while Ernest’s philosophy of

mathematics influences his philosophy of mathematics education, the connection is not

logically necessary. Finally, I examine the role Lakatos’s work plays in Ernest’s

approach to education and analyze its strengths and weaknesses.

Chapter five turns to Ernest’s recommendations for a differentiated curriculum. I

describe his vision for the general mathematics education of most students and the

preparation of future mathematicians. Ernest believes most students should be exposed

to mathematical topics that reveal the intrinsic value of mathematics, as well as its

connection to other disciplines and its presence in everyday life. He also wants students

to develop a critical understanding of mathematics in society.13 For future

mathematicians, Ernest envisions an education that enculturates them into the

mathematical community and prepares them for mathematical research. I use Ernest’s

vision for each group of students to analyze his implicit conceptions of mathematical

knowledge and democratic values. Based on this analysis, I argue that Ernest’s

conception of mathematical knowledge is fairly consistent from one group to the other,

13 Ernest, "Why Teach Mathematics?" under "Capability versus Appreciation."

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but that he places non-mathematicians outside the mathematical community. In addition,

his conception of democratic citizenship differs for the two groups. He sees social

activism as the democratic obligation of most citizens, but in the case of mathematicians

he calls for self-reflection. I close with a discussion of equity issues that places Ernest’s

curriculum recommendations in the context of debates about tracking.

In the sixth and final chapter, I illustrate the implications of my analysis by

exploring the issues that arise if Ernest’s view is applied to developmental mathematics at

the community college level. I describe the community college developmental

mathematics context and the issues it raises about educational opportunity. Ernest

conceives of educational opportunity in terms of access to meaningful mathematical

content, instruction that incorporates social construction, and activities that encourage

critical analysis of mathematical claims. To demonstrate how this can be used to explore

issues in developmental mathematics, I use Ernest’s view as a framework to analyze three

aspects of practice. First I look at the standards for developmental mathematics designed

by the American Mathematical Association of Two-Year Colleges, which influence

curricular development in community college mathematics departments.14 These

recommendations seem pedagogically similar to Ernest’s approach, but they do not rest

on the same theoretical foundation. Second, I consider the findings from a study by

Vernon Kays about the pedagogical intent of developmental algebra textbooks used by

community colleges in Illinois.15 Almost all of these textbooks were categorized as

providing detailed exposition and examples with skill-based problem sets. They do not

encourage a pedagogical approach that aligns with Ernest’s vision of investigation,

14 American Mathematical Association of Two-Year Colleges, Crossroads in Mathematics.15 Kays, "National Standards, Foundation Mathematics."

9

conversation, and open-ended problems. Finally, I examine an excerpt from a classroom

observation that is part of a case study by Donald Blais.16 It describes an algebra lesson

presented by a developmental mathematics instructor at a community college. She uses

what would traditionally be considered effective teaching methods: organized

presentation of the material and necessary formulas, detailed examples, and clear

explanations of the procedures used. However, this type of lesson does not employ the

conversation, investigation, and critical analysis that Ernest emphasizes. After analyzing

these three aspects of practice, I discuss potential obstacles to the implementation of

Ernest’s framework in this context. I close with a summary of what has been learned

from this analysis of Ernest’s philosophy and its applications, and I make suggestions for

future research.

While it focuses on Ernest’s view, this project also contributes to a general

understanding of the relationship between aims and purposes for mathematics education,

theories of learning, and educational practice. Such an understanding might influence the

way in which educators evaluate different forms of constructivism, a philosophy that is

significant to theorists and practitioners alike. My analysis could inform future

philosophical and empirical research on constructivist learning theory, mathematics

education, or even tracking. I also expect that this study will be useful to mathematics

educators in their discussions about the nature and justification of developmental

mathematics education.

16 Blais, "Constructivism Applied to Algebra."

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CHAPTER 2

CONSTRUCTIVISM

This chapter will provide some context and background for the discussion and

analysis of Ernest’s social constructivism. Constructivist philosophies of education,

while not new, have garnered much attention over the past decade. There has been much

controversy, earning constructivism a place as the topic of the National Society for the

Study of Education’s 2000 yearbook.17 It is within this conversation that Ernest’s social

constructivism has emerged.

Modern constructivism comes in many forms, all of which incorporate some

theory of mental structures built up from previous knowledge through conflict and

conceptual change.18 These different forms can be loosely categorized as psychological

and social. Psychological constructivism, such as von Glasersfeld’s radical

constructivism, sees knowledge as a personal construction. Knowledge cannot be

transmitted from one person to another—it is constructed in the human mind, either

individually or in response to interaction and discussion with others.19 Social

constructivism, on the other hand, is the view that the mind is socially formed.

Knowledge is not constructed in response to discussion with others. Rather knowledge is

constructed through socially situated conversations.20 Ernest’s social constructivism

departs from von Glasersfeld’s radical constructivism in important ways, but they share

common foundations. Ernest cites the work of von Glasersfeld as a key influence in his

17 Phillips, Constructivism in Education.18 Ernest, "One and the Many," 461. 19 Phillips, "Opinionated Account," 7. 20 Ernest, Social Constructivism, 220.

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own account of individual mathematical knowledge.21 For this reason, my focus in this

overview will be on von Glasersfeld as a precursor to Ernest.

In this chapter, I look at the ideas that have influenced von Glasersfeld’s work and

give a brief introduction to his view of mathematics education. I also clarify the

distinction between epistemology and pedagogy, because they are sometimes confused.

Both will be important, but they must be considered separately. To begin with, however,

I attempt to set aside certain debates in the philosophy of mathematics which stand

outside the scope of this paper. These issues are addressed in the work of both von

Glasersfeld and Ernest, but do not impact the current analysis.

Debates in the Philosophy of Mathematics

One cannot analyze mathematics instruction without considering the underlying

philosophy of mathematics and its set of assumptions about the nature of mathematics

and mathematical knowledge. However, there are some longstanding disputes that I will

want to avoid. Two of the great debates in the history of mathematical philosophy center

around the nature of mathematical objects and truth, and the nature of mathematical

justification. These debates are not critical to my project and would only serve to distract

attention from the main concern. Thus I will attempt to separate myself from them, and

focus instead on questions about individual mathematical knowledge and mathematics

education. Here I will give a brief outline of the ontological and epistemological debates,

including von Glasersfeld’s and Ernest’s positions. I will explain how their positions in

these debates are not relevant to the current project, and thus do not require more

extensive discussion. A brief overview of these debates will help to dispel certain

21 Ernest, Philosophy of Mathematics Education, 91.

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criticisms that might otherwise be offered against von Glasersfeld’s and Ernest’s work,

and will help to avoid distractions in my own analysis.

Ontological questions are concerned with the nature and status of mathematical

objects. There are many ontological positions, but the most commonly contrasted are

Platonism, a form of realism, and intuitionism, a form of constructivism. The Platonist

view is that mathematical objects exist independently of mathematical systems, and that

mathematical truths are statements about these objects.22 Mathematical statements are

true or false, independent of our knowledge of this truth status. Hilary Putnam suggests a

realist view that is not Platonist. In his view, mathematical truth exists independently of

the human mind, but mathematical objects do not. Instead, he sees mathematical objects

as abstract possibilities.23 In contrast, intuitionism holds that there is no independent

reality that contains mathematical objects and truths. Mathematics is constructed in the

human mind, and mathematical objects are dependent on human thought.24 In the current

study, this issue is largely irrelevant. Von Glasersfeld makes no ontological claims about

the existence of an external reality. He argues that we have no way of accessing any such

reality outside of our experience, so we have no way of knowing if it exists or not.25

Ernest does believe in an external reality, but not in an absolute knowledge of it. Thus he

believes that mathematical objects and truth are human constructions. However, his

social constructivist account of individual, or subjective, knowledge does not depend on

this ontological view. One could plausibly believe that mathematical objects and truth

exist independently of the knower, but hold a constructivist view of how the learner

comes to know and understand this truth. Thus I argue that the nature of mathematical 22 Steiner, Mathematical Knowledge, 109.23 Putnam, "What is Mathematical Truth?" 60.24 Heyting, "Intuitionist Foundation of Mathematics," 52-53.25 von Glasersfeld, Radical Constructivism, 4.

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objects and truth has little bearing on how we understand mathematical learning. As

Putnam asserts, “the investigation of mathematics must presuppose and not seek to

account for the truth of mathematics.”26 In general, accounts of individual knowledge

acquisition do not logically depend on a particular ontology. Therefore, I will not take an

ontological position in this paper.

Epistemological questions address issues of mathematical justification and

knowledge. How do we come to know mathematical truth? In the mathematical

community, the primary means of justification is proof. A proof begins with a set of

axioms, and generates a sequence of statements, through inference, that ends in the given

proposition.27 The debate that arises surrounds the nature of the knowledge that results.

The traditional philosophy of mathematics has been one which asserts that mathematics

offers absolute certainty.28 Proof generates knowledge of absolute truth. But the skeptic

asks, how can we be sure that what we have come to know is true? Epistemological

Platonism holds that this knowledge is acquired through a sort of intuitive perception.29

Accounts of this perception are generally considered inadequate, however. Putnam

describes a realist view in which we aren’t sure that we are correct. But we can be

reasonably sure through quasi-empirical inference, based in some cases on the necessity

of mathematical statements for scientific applications.30 Intuitionists believe that we

mentally construct the system of mathematics through intuition, and make logical

inferences to construct mathematical statements. While intuitionists believe mathematics

is constructed by humans, there is still a sense of certainty about it because mathematical

26 Putnam, "Truth and Necessity," 11. Italics in the original.27 Ernest, Social Constructivism, 6.28 Stemhagen, "Beyond Absolutism and Constructivism," 25.29 Steiner, Mathematical Knowledge, 110.30 Putnam, "What is Mathematical Truth?" In fact, Putnam argues that in some cases, quasi-

empirical methods should be allowed to replace proof as justification.

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statements are constructed through logical inferences. In this view, truth is provability—

there is no external truth, and thus no gap between what is true and our knowledge of it.31

For von Glasersfeld, mathematical knowledge is not some external, objective

truth: it is a system of concepts and symbols that have been given meaning through

human construction. It is not possible to haphazardly construct any mathematical ideas

you want, however, because they are subject to corroboration from others in the

mathematical community. For example, 2 + 2 = 4 is a fact that is certain because we

have come to construct units in a particular way and have agreed on how they are to be

counted.32 We can be sure of facts that logically follow from this construction. For

someone’s own construction to be considered correct, it would have to be compatible and

produce these facts. Therefore, even though mathematics is a human construction, von

Glasersfeld believes that it is still objective and true. The influence of von Glasersfeld’s

account of mathematical knowledge can be seen in that of Ernest.

Ernest’s epistemology incorporates features of intuitionism. Unlike intuitionists,

however, Ernest holds the view that mathematical judgments are not guaranteed any

certainty through logical inference. This is a fallibilist constructivist view, rather than the

absolutist view of intuitionist constructivism. Ernest argues that mathematics is fallible

because its foundation is a set of axioms that cannot be proved, and because its system of

deduction is uncertain. Mathematics is built on a set of assumptions, and is deduced from

those assumptions using rules of inference that we must assume are sound.33 Because we

are working under agreed-upon assumptions, mathematics is actually determined by

social agreement. Objective mathematical knowledge is intersubjective and shared, that

31 Dummett, "Intuitionist Logic," 108-9.32 von Glasersfeld, "Aspects of Radical Constructivism," 313.33 von Glasersfeld, "Exposition of Constructivism," 28.

15

which has come to be accepted by the mathematical community as rigorously proven.34

To contrast it from objective knowledge, Ernest refers to an individual’s knowledge as

subjective knowledge. While the phrase subjective knowledge seems contradictory,

Ernest has chosen this terminology to reflect the idea that when we learn, we internalize

the body of objective knowledge and recreate it in our own minds. This means that we

each will have “unique subjective representations of mathematical knowledge.”35 It is

this process of creating subjective knowledge that pertains to the current project. The

process of creating objective knowledge through social agreement is irrelevant in this

analysis, if we begin by taking the body of objective mathematical knowledge as given.

With this restriction, Ernest’s view of mathematical learning could be consistent with

more traditional accounts of objective knowledge, such as realism. Making this

separation will allow me to investigate the implications of his account of individual, or

subjective, mathematical knowledge, without committing to a particular view on the

origins and certainty of the accepted knowledge base.

With this study now separated from the debates in mathematical philosophy, I can

turn to the development of constructivist philosophies of individual knowledge. I will

provide a brief background in constructivist epistemology, with a focus on the

foundations of von Glasersfeld’s radical constructivism in mathematics education.

Constructivist Philosophies of Individual Knowledge

Constructivism must be considered in contrast to its rival position, behaviorism.

Associated with the work of B. F. Skinner and others, behaviorism is based in behavioral

34 Ernest, Philosophy of Mathematics Education, 43-44; Ernest, Social Constructivism, 144.35 Ernest, Philosophy of Mathematics Education, 43.

16

psychology. Behaviorism holds that learning can only be evaluated through observable

behaviors. Therefore, it focuses on the transference of knowledge and behavioral

reinforcement. Behaviorists believe that learning is sequential, and that students are only

ready to learn a particular topic after mastering all prerequisite concepts. Learning

happens when students are induced to make proper responses to questions, which

“primes” their behavior. This behavior is reinforced when they are praised for a correct

answer.36 While a sequential account of learning seems to make sense, it is somewhat

unappealing to describe human learning in a way akin to training a dog. Skinner’s theory

of behavior reinforcement reduces learning to acquiring the habit of giving right answers,

rather than a complex process of developing and understanding concepts. In his review

of Skinner’s Verbal Behavior, Noam Chomsky provides a detailed criticism along these

lines. He argues that Skinner’s theory of learning is hopelessly incomplete because it

does not consider how the learner processes information, and cannot account for how we

generalize what we learn.37 In contrast to behaviorism, constructivism seeks to explain

how the mind develops and learns, focusing on mental construction rather than

observable behavior.

Constructivist epistemology is a descendant of Kantian thought. Kant believed

that intuition provides a mental framework for the possibilities of experience, such as

spatial, temporal, and causal relations. He envisioned knowledge as a mental

construction created through the interaction of intuition and experience.38 Von

Glasersfeld’s radical constructivism is a form of psychological constructivism, based on

the Kantian idea that the status of a knowledge claim cannot be determined by comparing

36 Skinner, "Shame of American Education," 951.37 Chomsky, review of Skinner's Verbal Behavior.38 Bredo, "Reconsidering Social Constructivism," 129.

17

it to some external reality. Any comparison made by us happens within our thoughts and

experience, and would thus be tainted. Instead, reality is constructed in the individual

mind. Von Glasersfeld’s account of this construction draws on Piaget’s theory of

cognitive development.

Piaget’s work describes a genetic epistemology, in which children progress

through cognitive stages. This progression happens through a process of adaptation, by

means of assimilation and accommodation. Assimilation happens when a person takes a

new experience and fits it into her existing conceptual structure.39 She recognizes some

feature of the experience which causes her to categorize it with other experiences she has

had that shared this feature. For example, she encounters a small, brown animal about

the size of a cat. Because it fits the requirements in her conceptual scheme, she

assimilates the experience as seeing a cat. Every new experience is processed in this

way, at all stages of cognitive development. The assimilation of a new experience leads

to a certain set of expectations about what would result from particular actions. Piaget

calls this a scheme: a perceived situation, a possible activity, and an expected result.40

Because the girl has assimilated her experience as seeing a cat, she expects that petting

the animal would cause it to purr. Accommodation occurs when new information causes

a perturbation—it conflicts with the existing scheme. The girl pets the animal, and

instead of purring, it barks. She then reviews the situation, recognizing previously

unacknowledged differences between it and what was perceived.41 Accommodation

occurs as she realizes that this is actually not a cat, but rather some new animal, which

she later learns is called a dog. This all happens through a process called reflective

39 von Glasersfeld, Radical Constructivism, 62.40 Ibid., 65.41 Ibid.

18

abstraction, which involves a mental reorganization.42 To von Glasersfeld, it is through

our experiences, by means of assimilation and accommodation, that reality is constructed.

Cognition is adaptive toward a fit or viability with our experience. Reflective abstraction

allows us to apply existing constructions to new situations, to reflect on what we know,

and to reconstruct and reorganize our conceptual structures as needed.43 Skinner would

argue that this is all well and good, but that cognitive psychology has yet to teach us

anything new about teaching and learning. For example, he claims that psychologists

were already aware that students connect new information to what they already know.44

Constructivists would answer that von Glasersfeld’s account gives a much more

comprehensive idea of how the learner’s mind operates, which makes it possible to create

richer learning experiences.

A common criticism of radical constructivism is that it does not allow for real

communication or shared meaning. Because knowledge and truth are human

constructions and not reflections of an independent reality, reality is invented by each of

us.45 This relativism is what makes constructivism difficult for many philosophers to

accept. Kenneth Howe and Jason Berv argue that humans must be able to share some

meanings, even if it is just references to objects. This much is necessary to be able to

communicate coherently with other people.46 Eric Bredo suggests that constructing

meaning does not mean we must construct the existence of objects for ourselves. Instead,

we construct ideas of how different objects are to be used.47 For example, a child learns

which things are called tables from her mother pointing to many different tables and

42 Piaget, Genetic Epistemology, 18.43 von Glasersfeld, Radical Constructivism, 104-5.44 Skinner, "Shame of American Education," 949.45 Bredo, "Reconsidering Social Constructivism," 131.46 Howe and Berv, "Constructing Constructivism," 34-35.47 Bredo, "Reconsidering Social Constructivism," 144.

19

saying, “Table.” The child must construct an idea of the common features that come to

define a table, what a table in general is used for, and the uses for specific types of tables.

Before the child understands their uses, tables have no meaning. This is a more

commonly accepted version of psychological constructivism: that we share definitions

and a common reality, but we must construct underlying meanings for ourselves.

In mathematics education, constructivists such as Paul Cobb have incorporated

sociocultural perspectives into radical constructivism. Cobb sees mathematical learning

both as individual construction and as enculturation.48 Through classroom interactions,

the teacher and students negotiate taken-as-shared meanings and interpretations which

allow communication and interaction. “Students, in the course of their individual

cognitive development, actively participate in the classroom community’s negotiation

and institutionalization of mathematical meanings and practices.”49 Because this theory

does incorporate notions of social interaction and negotiated meanings, Phillips warns

that one must be careful with terminology in the literature. Psychological constructivists

who believe knowledge is constructed individually in response to social influences are

sometimes referred to as social constructivists.50 Because Cobb’s theory does not

incorporate a social view of mind formation, it will not be considered a social

constructivist philosophy in this paper. Therefore, it should not be confused with

Ernest’s social constructivism.

Constructivism’s focus on the construction of knowledge rather than observable

behavior suggests the use of teaching methods different from those generally seen in a

traditional, behaviorist mathematics classroom. While adhering to a constructivist

48 Cobb, "Where is the Mind?" 13.49 Cobb et al., "Mathematics Project," 6.50 Phillips, "Opinionated Account," 11.

20

philosophy does not limit one to a particular set of activities, it will most likely influence

a teacher’s pedagogical decisions.

Constructivist Pedagogy

Pedagogical behaviorism and constructivism must be considered separately from

their philosophical counterparts, because each may or may not be accompanied by a

particular philosophical perspective. For example, it is possible to believe in the value of

a constructivist pedagogy without committing oneself to the idea that reality is a

construction. In fact, most classroom applications assume that students’ constructions

should be consistent with some set of acceptable results, which suggests more traditional

views on reality and truth.51 Behaviorist and constructivist pedagogies make use of the

psychological principles in their corresponding philosophies, but not necessarily the

ontological or epistemological stances.

Behaviorist pedagogy views the curriculum as a set of measurable objectives,

defined by observable student behaviors.52 Behaviorists believe that the curriculum

should be broken down into its component parts and presented in a systematic way.

Lessons should move the student through an organized sequence of increasingly complex

concepts, with frequent assessment to see if objectives have been achieved.53 One

implementation of this is Skinner’s concept of programmed instruction. This involves

individualized instruction, moving the student through carefully designed activities at her

own pace, with frequent testing to assess skill mastery.54 Behaviorism has been very

influential in mathematics education, encouraging rote learning of formulas and 51 Noddings, "Constructivism in Mathematics Education," 17.52 Fey and Graeber, "New Math to Agenda," 540.53 Ediger, "Philosophies of Education," 180.54 Skinner, "Shame of American Education," 951.

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procedures. Repetition is used to teach basic skills.55 However, behaviorism has also

come under great scrutiny in the mathematics education community. While it provides

students with clear expectations and a clearly-defined learning plan, it has been accused

of encouraging memorization without understanding.

In contrast to behaviorist pedagogy, constructivist pedagogy focuses on creating

situations in which students can construct meaning. While this is consistent with the

principles of the constructivist philosophy of education, it need not rely on them.

Constructivist pedagogy encourages personal understanding, with no commitment to a

particular view of truth. It is quite possible to have a realist view of objective

mathematical truth, but a constructivist idea of how we come to know that truth. I will

briefly outline what von Glasersfeld has to say about this, and then look at what is

generally referred to as a constructivist pedagogy.

Von Glasersfeld states that to be consistent with its own philosophy, radical

constructivism must hold that there is no one right way to teach. He is purposefully

vague on specific teaching methods and learning activities that should be used. Radical

constructivism can, however, provide a framework within which a teacher can try

different ideas, and von Glasersfeld gives some suggestions within this framework. Von

Glasersfeld’s work with physics students convinced him that students need a deep

conceptual understanding of the material in order to solve unfamiliar problems.56 “To

know mathematics is to know how and why one operates in specific ways and not in

others, how and why the results one obtains are derived from the operations one carries

out.”57 In an effective constructivist classroom, the teacher must provide opportunities

55 Handal, "Philosophies and Pedagogies," 6.56 von Glasersfeld, "Constructivist Approach to Teaching," 5.57 von Glasersfeld, introduction to Constructivism in Mathematics Education, xvi.

22

for students to actively construct meaning, and be alert to student interpretations that are

incompatible with those of the mathematical community. To provide opportunities for

active construction, teachers should keep students actively engaged at the proper

cognitive level. Von Glasersfeld explains that children need concrete representations at

first. The transition to abstract ideas and symbols can gradually be made when the

children are ready, as symbols become more than meaningless marks on the page. In

order for learning to happen, however, students must encounter conflict. For this

purpose, von Glasersfeld encourages group work. Working actively in groups can be

productive because students must organize their thoughts to explain their solutions to

others, and because disagreements can uncover errors in their understanding. 58

Consistent with the suggestions of von Glasersfeld, “constructivist pedagogy”

generally refers to one in which students engage in exploratory projects, problem solving,

and group work. The teacher begins with what the student brings: her pre-existing

knowledge, conceptions, attitudes, and interests. By reflecting on her experiences, the

student is able to construct understanding.59 In mathematics, the hope is that students will

learn to think more deeply—conceptual understanding is emphasized with the intention

that procedural knowledge will follow. According to Jere Confrey, the goal of

constructivist instruction is for an instructor to “promote and encourage the development

for each individual within his/her class of a repertoire for powerful mathematical

constructions for posing, constructing, exploring, solving and justifying mathematical

problems and concepts.”60 Each student finds relationships between mathematical ideas

58 von Glasersfeld, Radical Constructivism, 188.59 Howe and Berv, "Constructing Constructivism," 30-31.60 Confrey, "What Constructivism Implies," 112.

23

in a way that makes sense to her. Students are encouraged to make comparisons, find

patterns, look for problems, and construct solutions.61

Deborah Loewenberg Ball and Hymn Bass have written several articles about

constructivist research in Ball’s third grade classroom. For her, the process of fostering a

constructivist classroom consists of establishing a public base of shared knowledge,

developing mathematical language, and providing opportunities for the construction of

new mathematical ideas through reasoning. The base of publicly shared knowledge is a

collection of facts, concepts, definitions, and procedures that have been accepted and

understood by all in the class.62 Reasoning takes place within the community of the

classroom, much like it would among members in the mathematics field. Students work

in groups, making conjectures and debating their validity. All reasoning must be justified

based on publicly shared knowledge and a series of logical steps that can be understood

and articulated by the student.63 This is the core of mathematical knowledge

construction. Sometimes reasoning is used to see patterns and develop new ideas. For

example, Ball writes that one of her students wants to call the number 6 odd because it

can be broken up into three groups of two.64 Through reasoning, the class determines that

this does not fit the definition of odd. However, the teacher leads them to consider the

concept in its own right. The class determines that there are other such numbers, so they

decide to give them a name. They call them “Sean numbers,” after the student who

discovered them.

Skinner would accuse constructivist pedagogy of not teaching enough, fast

enough. He would claim that the instructional objectives are unclear, and that the 61 McCarty and Schwandt, "Seductive Illusions," 49-50.62 Ball and Bass, "Making Believe," 204.63 Ibid., 203.64 Ibid., 214.

24

activities do not organize the material sequentially for efficient learning. In addition,

Skinner would worry that constructivist education will frustrate children, because it does

not teach them the necessary basic skills before advancing to more complex projects.65

Educators who endorse a constructivist pedagogy would respond that open-ended

problem solving activities better prepare students to be critical thinkers and to solve

problems that require creativity and reasoning. Children will learn the basic skills, and

will understand and remember them because they have experienced the need for them

firsthand, in context.

The behaviorist-constructivist dichotomy has been the source of much

controversy in mathematics education, and the argument will not be settled here. It is

sufficient to present an outline of each approach. While the use of one of these

pedagogies does not necessitate a belief in its corresponding theory of knowledge, there

is a natural connection. Von Glasersfeld does not endorse one particular set of teaching

methods, but he does suggest that the constructivist pedagogy described here is consistent

with a constructivist theory of individual knowledge formation.

Summary and Conclusion

In this chapter, I have attempted to clarify the philosophical scope of this study.

Ontological and epistemological issues in the philosophy of mathematics are highly

contested, and are not immediately relevant to my analysis of Ernest’s theory of

individual knowledge. Therefore, I will not take a position on the nature of mathematical

objects and truth. I will accept the body of mathematical knowledge as given, and

consider how the individual obtains this knowledge.

65 Skinner, "Shame of American Education," 950-51.

25

I have also provided an introduction to constructivist philosophy as it relates to

mathematics education. Constructivism has emerged in reaction to behaviorism, moving

the focus from observable behavior to mental construction. Building on Piaget’s work, it

describes a process in which a learner must construct his or her own understanding to

create knowledge. As a pedagogy, constructivism advocates active learning and critical

thinking. Von Glasersfeld is an important constructivist figure in mathematics education,

because he has specifically applied his radical constructivism to the learning of

mathematics. His work has also formed the basis of Ernest’s theory of individual

knowledge.

While Ernest’s work is based on the constructivist ideas outlined here, his social

constructivism incorporates a theory of knowledge construction through socially situated

conversation. This departure from von Glasersfeld is influenced by Vygotsky’s social

view of the mind, Wittgenstein’s philosophy of language, and Lakatos’s quasi-empiricist

philosophy of mathematics. In the next chapter, I will examine the work of these figures.

Later, I will examine how Ernest has integrated these views with those of von Glasersfeld

to create his social constructivist philosophy of mathematics and mathematics education.

26

CHAPTER 3

LAKATOS AND OTHER INFLUENCES

While Ernest’s social constructivism draws on the radical constructivism of von

Glasersfeld, it diverts from his work in some important ways. Where von Glasersfeld

bases his theory of cognitive development on the work of Piaget, Ernest turns instead to

the work of Vygotsky. Vygotsky’s theory contributes the idea that language and social

interaction are integral to the construction of knowledge. Ernest then uses Wittgenstein’s

philosophy to elaborate on the role of language in mathematics, and borrows from

Lakatos’s quasi-empiricism to describe the role of social interaction.

It is the work of Lakatos that will require the most consideration in this study.

While Ernest borrows concepts from Vygotsky and Wittgenstein, significant portions of

both his philosophy of mathematics and his philosophy of mathematics education are

taken from Lakatos. He reframes Lakatos’s philosophy of mathematics as a learning

tool, which forms the basis for his philosophy of mathematics education for future

mathematicians. He uses Lakatos’s logic of mathematical discovery to describe both the

generation of individual mathematical knowledge and the enculturation of new

mathematicians. These concepts form the framework for Ernest’s view. Therefore, I will

only give a brief overview of Vygotsky’s social theory of mind and Wittgenstein’s

philosophy of language, to show how they have contributed to Ernest’s work. This will

be followed by a more complete analysis of Lakatos’s quasi-empiricism.

27

Vygotsky’s Social Theory of Mind

Unlike Piaget and von Glasersfeld, who see knowledge as the mental organization

of individual experience, Vygotsky regards knowledge as cultural. It is only within a

system of social behavior that activities gain meaning.66 The prevailing system of

culturally-defined meanings must be learned as part of a child’s cognitive development.

Vygotsky explains that biological and sociocultural forces come together to influence

human development, which involves the internalization of this “socially rooted and

historically developed” system of behavior.67 It is within this framework that Vygotsky

forms his theory of cognitive development. He disagrees with Piaget about the nature of

knowledge construction, and therefore the relationship between instruction and

development. Vygotsky feels that Piaget’s description of assimilation and

accommodation does not account for confrontation with reality and its effect on logical

thinking.68 Instead, he argues that concept formation happens socially, in interaction with

others. Because of their influence on the work of Ernest, two key features of Vygotsky’s

model will be considered here: the role of language and the role of social interaction.

Language plays an integral role in Vygotsky’s theory of cognitive development.

Vygotsky believes that children first learn language as a communicative tool in the form

of external speech, which is talking to others.69 Through external communication, they

learn the rules of social behavior. As a child develops, external speech is internalized,

which contributes to logical thinking. Egocentric speech, or speaking vocally to oneself,

is a transitional form between external and internal speech, in which the child has

66 Vygotsky, Mind in Society, 30.67 Ibid., 57.68 Vygotsky, Thought and Language, 52.69 Ibid., 35.

28

internalized social behavior and begins using it to organize his own activities.70 Given a

problem-solving task, he may talk to himself to devise a plan. Language is connected to

both behavior and concept formation. It provides the means for social interaction, which

teaches children how they are supposed to act. But it is also the vehicle for learning,

providing them with the ability to formulate thoughts and to construct their own

conceptual understandings.

In his theory of cognitive development, Vygotsky distinguishes between two

groups of concepts: spontaneous and scientific. Spontaneous concepts are those that are

constructed by children in response to their everyday experiences.71 For example, this

would include concepts such as “family” or “home,” as well as the contexts and

assumptions attached to them. Family may be understood as a mother, a father, and two

children, because that is what the child’s family looks like. Scientific concepts are more

logical and are constructed in response to formal instruction.72 Students learn about the

associated topics in school, but do not have any personal experience to attach to them.

Vygotsky explains that this is analogous to learning a foreign language, the learning of

which, like scientific concepts, is approached consciously and intentionally. The native

language, learned in childhood through contact with others, serves the role of

spontaneous concepts.73 In order to learn a scientific concept, one must build on related

spontaneous concepts—a student must have enough experience to understand and

contextualize the new scientific concept. In turn, these scientific concepts provide a new

70 Vygotsky, Mind in Society, 27.71 Kozulin, "Vygotsky in Context," xxxiii-xxxiv.72 Ibid., xxxiii.73 Vygotsky, Thought and Language, 195.

29

context for the examination of spontaneous concepts, which are analyzed and developed

as the student matures.74 Vygotsky uses the language analogy to illustrate:

Success in learning a foreign language is contingent on a certain degree of maturity in the native language. The child can transfer to the new language the system of meanings he already possesses in his own. The reverse is also true—a foreign language facilitates mastering the higher forms of the native language. The child learns to see his language as one particular system among many, to view its phenomena under more general categories, and this leads to awareness of his linguistic operations.75

The development of spontaneous concepts prepares students to encounter scientific

concepts in school. However, Vygotsky cautions against focusing instruction on those

concepts and tasks the child is ready to master independently. Each child can be pushed

beyond his or her current level of preparation with some prompting or assistance.

Vygotsky refers to the difference between a child’s independent ability and his ability

with assistance as the zone of his proximal development.76 This is the level at which the

most efficient learning happens. It is also here that the role of social interaction is critical

—learning is inherently social, as the student works in the zone of proximal development

in cooperation with others.77

Vygotsky’s theory is different from the others we have examined in important

ways. It is quite different from the behaviorism of Skinner. Skinner’s programmed

instruction requires students to master all prerequisite concepts before moving to a new

topic, and then prompts them for the responses they are ready to give independently.

Vygotsky claims that students should be deliberately pushed beyond that level, to their

zone of proximal development. But his view also differs from the constructivist work of

Piaget and von Glasersfeld in how it sees the role of communication in learning. For 74 Ibid., 194.75 Ibid., 195-96.76 Ibid., 187.77 Vygotsky, Mind in Society, 90.

30

them, instruction and interaction with others serve as sources of perturbation which

promote accommodation and development. For Vygotsky, instruction works with the

child’s current conceptual structure, building on it rather than creating a perturbation that

conflicts with it. Learning happens through interaction and cooperation with the teacher

and peers.78 As we will see in the next chapter, Ernest incorporates these ideas to

distinguish his social constructivism from the radical constructivism of von Glasersfeld.

Vygotsky sees language and social interaction as key elements to cognitive

development. Ernest finds a link between this and the philosophy of Wittgenstein,

though the focus is quite different. Rather than looking at cognitive development,

Wittgenstein considers the role of language and social interaction in the creation of a

social body of knowledge.

Wittgenstein’s Philosophy of Language

Wittgenstein’s philosophy links language to mathematics. While Vygotsky

connects language to the formation of the mind, Wittgenstein connects language to the

formation of a body of knowledge. Ernest draws heavily from his own interpretation of

Wittgenstein’s later philosophy of language. A proper treatment of Wittgenstein’s

philosophies of language and mathematics would be very long and outside the scope of

this paper. Instead, I will briefly summarize the role that his philosophy of language

plays in his philosophy of mathematics, focusing on the features that are pertinent to

Ernest’s work.

78 Vygotsky, Thought and Language, 148.

31

For Wittgenstein, language is based on social agreement—we have to agree on

the meanings of words for communication to be possible.79 These meanings and the rules

for their use are negotiated in context, through social interaction. Wittgenstein refers to

the linguistic patterns we follow and the rules that govern them as language games, which

are embedded in a larger social structure and its forms of life.80 Forms of life include the

human purposes, rules, behaviors, and language games that provide the context in which

language acquires meaning. The meanings of words emerge through social patterns of

use, and can change over time or between contexts. For example, the term “queer” can

mean strange, it can be an inclusive term referring to a person who has a non-traditional

sexual identity or orientation, or it can be a derogatory term for gay men. The meaning

shifts based on the context and the intentions of the person using the word.

The negotiation of meaning through language games extends to mathematical

terms and concepts as well. Wittgenstein states that mathematics itself is a language

game.81 Therefore, its terms also gain meaning through patterns of social use. It is these

two aspects of Wittgenstein’s philosophy that are most important to Ernest’s own

philosophy of mathematics: that mathematics is a language, and that it is constructed

socially. Mathematical language has developed over time as the mathematical

community has come to agree on the meanings of terms, symbols, syntax, and proof,

along with rules for their use.82 There is, for the most part, agreement about

mathematical notation and vocabulary. But there is also agreement about laws of

inference and standards of proof, which Wittgenstein claims are determined by

mathematical custom. “‘So you are saying that human agreement decides what is true 79 Wittgenstein, Remarks, 342.80 Ernest, Social Constructivism, 69.81 Wittgenstein, Remarks, 381.82 Ernest, Social Constructivism, 79.

32

and what is false?’—It is what human beings say that is true and false; and they agree in

the language they use. That is not agreement in opinions but in form of life.”83

Wittgenstein believes that we accept laws of inference not because we must due to some

inexorable truth, but because they are an underlying part of our language games. “[T]he

laws of inference do not compel him to say or to write such and such like rails

compelling a locomotive.... [I]t is for us an essential part of ‘thinking’ that—in talking,

writing, etc.—he makes this sort of transition.”84 This differs from the traditional view of

the role of proof, changing it into a more communicative process. Rather than justifying

statements of mathematical truth, proof is meant to convince. “Proof must be a procedure

of which I say: Yes, this is how it has to be; this must come out if I proceed according to

this rule.”85 This actually creates new concepts and introduces them into mathematical

language: “the proof changes the grammar of our language, changes our concepts. It

makes new connexions, and it creates the concept of these connexions.”86

This is a controversial part of Wittgenstein’s philosophy. Reuben Hersh, for

example, interprets this to mean that mathematics is completely arbitrary, and that people

can define the rules however they want.87 Ernest disagrees. In the interpretation that

Ernest has incorporated into his own work, Wittgenstein sees mathematics as constrained

by the rules of its language games. It is an internally consistent system, based on a set of

linguistic rules and rules of proof. Mathematical truth has certainty within this system of

rules.88 For example, the truth of the statement is not arbitrarily decided. It is

certain, given our conventions of naming numerals and the way we have defined

83 Wittgenstein, Philosophical Investigations, 88. Italics in the original.84 Wittgenstein, Remarks, 80. Italics in the original.85 Ibid., 160.86 Ibid., 166.87 Hersh, What is Mathematics, Really? 202.88 Ernest, Social Constructivism, 79-80.

33

addition. The same is true for inferential proofs. A proof shows that within our

mathematical system, a proposition is true and it could be no other way.

Ernest believes that Wittgenstein’s philosophy of mathematics as language can be

applied to individual knowledge formation. Wittgenstein holds that we learn language

(and by extension, mathematics) through practice.89 For Wittgenstein, a word’s meaning

is defined by its use. “When I think in language, there aren’t ‘meanings’ going through

my mind in addition to the verbal expressions: the language is itself the vehicle of

thought.”90 This means that the learning of language cannot be separated from its

application. A child must learn by watching others use language, and by practicing it

with them. This is somewhat like Vygotsky’s internalization of social behavior, in that

children learn language by using external speech in interaction with others. As with

Vygotsky, it is this external speech that makes internal speech, or thinking, possible.

Wittgenstein explains that one cannot even have non-verbalized thoughts and intentions

without language. “If the technique of the game of chess did not exist, I could not intend

to play a game of chess. In so far as I do intend the construction of a sentence in

advance, that is made possible by the fact that I can speak the language in question.”91

Ernest feels that these ideas lead to an implicit social view of mathematics learning,

which would require an immersion in the mathematical community’s shared forms of life

and an enculturation into its language games.92 Ernest borrows this concept for his

philosophy of mathematics education.

Ernest sees Wittgenstein’s work as a precursor to a social constructivist account

of mathematics learning. The role of language in Wittgenstein’s philosophy only hints at 89 Wittgenstein, Remarks, 41.90 Wittgenstein, Philosophical Investigations, 107.91 Ibid., 108.92 Ernest, Social Constructivism, 93.

34

the social nature of mathematics, but Ernest sees this as a useful background for his

application of Lakatos’s theory. He takes Lakatos’s philosophy of mathematics and

embeds it in Wittgenstein’s language games and forms of life to draw out its social

aspects.

Lakatos’s Quasi-Empiricist Philosophy of Mathematics

Lakatos draws heavily on Popper’s philosophy of science. Popper believes that

scientific theories are speculations which have been tested empirically with the

possibility of falsification.93 Similarly, Lakatos puts forth a philosophy of mathematics in

which mathematical conjectures and their proofs are subject not to falsification in any

final sense, but rather to refutation through counterexamples. All conjectures must be

subjected to this process, which he bases on the Principle of Retransmission of Falsity.

Lakatos elaborates on this idea in several of his essays, explaining that classical

epistemology relies on a downward transmission of truth. One begins with a finite set of

true axioms, and truth flows down through the proof, “through the safe truth-preserving

channels of valid inferences.”94 In science and mathematics, however, rather than having

certain truth at the top, there can be certain falsity at the bottom. If a counterexample is

presented that shows the conjecture is false, then there must be an error somewhere in the

conjecture or proposed proof. Thus falsity “flows upward from the deductive channels

(explanations) and inundates the whole system.”95 This creates a somewhat combative

vision of the mathematical community, in which mathematicians continually subject their

work to the criticism and rebuttal of others. Refuted conjectures are not abandoned,

93 Hersh, What is Mathematics, Really? 209.94 Lakatos, "Renaissance of Empiricism," 28.95 Lakatos, "Infinite Regress," 5. Italics in the original.

35

however. Theories, or revised versions of them, may eventually be saved through a

variety of means. Some of the methods are rational in the traditional sense, while others

are not.

Lakatos calls his philosophy quasi-empirical because counterexamples are

generated abstractly, not observed in the spatio-temporal world.96 Both he and Popper

incorporate into their philosophies a cycle of conjecture, refutation, and new conjecture.

Lakatos diverges from Popper, however, in an important way. Popper clearly separates

the context of justification and the context of discovery. The context of justification

deals with the objective, logical, and rational aspects of knowledge. The context of

discovery, on the other hand, has to do with human or historical invention.97 Popper does

not see the context of discovery as part of the philosophy of science, so his cycle of

conjectures and refutations does not account for the origin of new conjectures.98 In

contrast, Lakatos incorporates the formation of new conjectures into his cycle of proofs

and refutations, which he refers to as “the logic of mathematical discovery.” Conjectures

are critiqued by others, who offer counterexamples. The counterexamples are then

analyzed to find key features. This information is used to revise the old conjectures,

creating new ones. The cycle of proofs and refutations intertwines discovery and

justification by accounting for the element of human invention in proof.99 The element of

conjecture generation is important, because it provides a description of where

mathematical ideas come from and how they are refined. This contrasts with the usual

emphasis solely on mathematics as justification through proof. It validates the creation of

new conjectures as a valuable part of mathematics, worth considering and studying in its 96 Hersh, What is Mathematics, Really? 213.97 Ernest, Social Constructivism, 44.98 Ibid., 99.99 Ibid., 99-100.

36

own right. As a result, this also makes the creation of new conjectures an important part

of mathematics education.

The joining of discovery and justification is key to Lakatos’s philosophy of

mathematics, and it strongly influences the way in which Ernest describes mathematical

learning. By giving mathematical discovery a place in the creation of mathematical

knowledge, Lakatos moves away from the predominant view of what it is to do

mathematics. Rather than giving the traditional picture of a person in a quiet room

creating linear, deductive proofs, Lakatos depicts mathematics as a lively, interactive

enterprise. The logic of mathematical discovery introduces conversation, conflict, and

argument into the mathematical process. Doing mathematics is not just about formalized

logic—it is about mentally experimenting with mathematical ideas in conversation (or

argument) with others, in order to discover new relationships. Ernest incorporates this

view of mathematical practice into his vision of mathematics education. While Lakatos’s

work is not intended as a philosophy of mathematics education, Ernest sees potential for

extending it in this direction. Where Lakatos means to model the generation of the

collective body of mathematical knowledge, Ernest describes the development of

individual mathematical knowledge. He believes that the classroom dialogue in Proofs

and Refutations suggests a pedagogical approach different from “the traditional formal

pattern of textbook presentations (definition, lemma, theorem, proof).”100 This new

approach arrives at mathematical knowledge developmentally, through an emphasis on

informal mathematics. In order to examine the logic of mathematical discovery as a

philosophy of mathematics education and a pedagogical method, I will examine this

dialogue in more detail. I will first summarize the method of proof-analysis presented in

100 Ibid., 113.

37

Proofs and Refutations, and then I will explore its potential and limitations as an

educational model.

Proof-Analysis and the Creation of Mathematical Knowledge

Lakatos wrote Proofs and Refutations as an attack on formalism, and to elucidate

his philosophy that mathematics grows through quasi-empirical “speculation and

criticism” rather than the development of formal theorems.101 Mathematical knowledge is

generated through a creative process of speculating, finding counterexamples, and

making appropriate adjustments to one’s reasoning. The text is a fictional dialogue set in

a mathematics classroom, while the footnotes trace its parallels to real events in the

mathematical community. The dialogue presents what Lakatos calls a “rationally

reconstructed” account of the history of the Euler conjecture.102

The dialogue opens with the teacher reminding the students what occurred during

the previous class. They had discovered that for all regular polyhedra, ,

where V is the number of vertices, E is the number of edges, and F is the number of faces.

A student offered the conjecture that this is true for all polyhedra.103 Several were tested,

and it held for all of them. Now, the teacher begins class by offering a proof of the

conjecture, which he refers to as a thought-experiment.104 The proof consists of a hollow

polyhedron made of rubber. One face is removed, and what remains is stretched flat.

Assuming for the original polyhedron, now, with a face removed,

. Each face that is not already triangular is made so by drawing diagonals,

which will not change the relationship between V, E, and F. Triangles are removed one

101 Lakatos, Proofs and Refutations, 5.102 Ibid.103 Ibid., 7. Lakatos notes that this is Euler’s conjecture.104 Ibid.

38

by one, preserving the relationship , until only one is left. For this triangle

holds, which the teacher claims proves the conjecture.105 In the subsequent

discussion, a polyhedron for which is called Eulerian.

With a conjecture and proposed proof to consider, the students argue about the

validity of both. This is where Lakatos sees the real mathematical work happening.

Three students speak up immediately to question the proof’s implicit assumptions, or

hidden lemmas. Pupil Alpha asks whether it can be assumed “that any polyhedron, after

having a face removed, can be stretched flat on the blackboard.”106 Pupil Beta is not sure

that drawing diagonals will preserve the relationship , and Pupil Gamma

questions the certainty of the triangle-removing procedure. Rather than being

discouraged, the teacher is excited by these doubts: “This decomposition of the

conjecture suggested by the proof opens new vistas for testing.”107 In fact, this forms the

basis for Lakatos’s logic of mathematical discovery. A proof is analyzed and criticized,

thereby leading to new discoveries as well as the strengthening of the proof.

As the dialogue continues, the class discusses not only the validity of the

conjecture and proof, but also the appropriate methods to use in their own inquiry. Alpha

gives a counterexample that is both global, refuting the main conjecture, and local,

refuting the lemma he has questioned about stretching a polyhedron flat. He points out

that a pair of nested cubes cannot be stretched flat if a face is removed, and that

does not hold for this polyhedron.108 This is followed by more

counterexamples, and an argument over the definition of a polyhedron. Pupil Delta

105 Ibid., 7-8.106 Ibid., 8. Italics in the original.107 Ibid., 10.108 Ibid., 13.

39

makes several attempts to nullify these counterexamples by changing the definition of a

polyhedron to exclude them: “You should really find a more appropriate name for your

non-Eulerian pests and not mislead us all by calling them ‘polyhedra’.”109 Alpha feels

that this continuous re-definition is counterproductive. He calls it “the latest linguistic

trick, the latest contraction of the concept of ‘polyhedron’! Delta dissolves real

problems, instead of solving them.”110 The teacher agrees, admonishing Delta for

“monster-barring,” or avoiding global counterexamples by redefining key terms. Beta

suggests that it would be more productive to engage in “exception-barring,” an extension

of monster-barring that restricts the domain of both the conjecture and the refuted lemma

to avoid known counterexamples. Instead of changing the definition of a polyhedron, he

limits the conjecture to convex polyhedra, which so far have appeared to be Eulerian.111

The teacher criticizes this method as well, warning that a somewhat arbitrary domain

restriction could be more extreme than necessary, and still adds no certainty to the proof.

He calls for a more careful proof-analysis:

When you restrict your conjecture to a ‘safe’ domain, you do not examine the proof properly, and, in fact, you do not need to for your purpose…. I build the very same lemma which was refuted by the counterexample into the conjecture, so that I have to spot it and formulate it as precisely as possible, on the basis of a careful analysis of the proof…. Your method does not force you to give a painstaking elaboration of the proof, since the proof does not appear in your improved conjecture, as it does in mine.112

The teacher points out that the counterexamples, such as the nested cubes, refute Alpha’s

lemma because they cannot be stretched flat. Thus, the conjecture only needs to be

restricted to simple polyhedra, or polyhedra that can be stretched flat, to preserve the

109 Ibid., 19.110 Ibid., 20.111 Ibid., 28.112 Ibid., 36. Italics in the original.

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proof.113 The teacher feels this is a better approach, because it links the counterexample

to the proof, both of which are used to improve the conjecture. To Lakatos, improving

the conjecture (or the proof, in some cases) means examining a counterexample,

discovering something new about it, and revising the conjecture (or proof) based on this

information.

Lakatos is trying to illustrate that the value of proof-analysis lies in its

contribution to a deeper understanding of the related concepts. Rather than restricting the

domain of a refuted lemma, the mathematician should analyze the proof to learn why the

lemma does not hold. What is learned about this lemma can be incorporated into the

conjecture, thus improving the conjecture while leaving the proof intact. “Our method

improves by proving. This intrinsic unity between the ‘logic of discovery’ and the ‘logic

of justification’ is the most important aspect of the method of lemma-incorporation.”114

This is an important point in Lakatos’s work—that the process of proofs and refutations

actually fuels mathematical discovery, generating new knowledge.

Lakatos formalizes his proof-analysis method as a heuristic later in the dialogue.

It is based on the Principle of Retransmission of Falsity, requiring that any global

counterexample also be a local counterexample.115 In other words, for a proof to be

correct, any global counterexample must also be found to violate one of the lemmas.

This principle drives the cycle of proofs and refutations, which is now made explicit:

Rule 1. If you have a conjecture, set out to prove it and to refute it. Inspect the proof carefully to prepare a list of non-trivial lemmas (proof-analysis); find counterexamples both to the conjecture (global counterexamples) and to the suspect lemmas (local counterexamples).

113 Ibid., 34.114 Ibid., 37. Italics in the original.115 Ibid., 47.

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Rule 2. If you have a global counterexample discard your conjecture, add to your proof-analysis a suitable lemma that will be refuted by the counterexample, and replace the discarded conjecture by an improved one that incorporates that lemma as a condition. Do not allow a refutation to be dismissed as a monster. Try to make all ‘hidden lemmas’ explicit.Rule 3. If you have a local counterexample, check to see whether it is not also a global counterexample. If it is, you can easily apply Rule 2.116

In this heuristic, the Principle of Retransmission of Falsity allows counterexamples to be

used to improve the proof. A proof is outlined, and a global counterexample is noticed

(Rule 1). The Principle states that this must also be a local counterexample. However,

the lemma that it refutes may not be articulated yet, so a hidden lemma may need to be

found. The refuted lemma can then be analyzed to see why it does not hold, and this

information can be incorporated into a reformulation of the conjecture (Rule 2).

Alternately, one may review an explicit lemma in the proof, and notice a local

counterexample in the process. This should be checked to see if it is a global

counterexample, and handled as before (Rule 3).

But what if the local counterexample is not also global? This indicates a problem

not with the conjecture, but with the proof. No restriction is necessary—instead, the

proof should be analyzed and the faulty lemma replaced with a carefully chosen new one.

This adds a new rule to the heuristic:

Rule 4. If you have a counterexample which is local but not global, try to improve your proof-analysis by replacing the refuted lemma by an unfalsified one.117

The value of this situation is that it creates knowledge by increasing the content of the

theorem instead of decreasing it.118 For example, Gamma finds a counterexample in

which the triangle-removing procedure (after the polyhedron is stretched flat) does not

116 Ibid., 50. Italics in the original.117 Ibid., 58. Italics in the original.118 Ibid.

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preserve the relationship between V, E, and F. The removal of an interior triangle rather

than one from the boundary would only remove a face, leaving .119

However, holds for the original polyhedron. Instead of restricting the

domain of the conjecture to the domain of this faulty lemma, the lemma itself is replaced

with one that specifies the order of removal. This approach can be extended to include

global counterexamples, as well:

Rule 5. If you have counterexamples of any type, try to find, by deductive guessing, a deeper theorem to which they are counterexamples no longer.120

Instead of restricting the domain of the conjecture, one can analyze a counterexample to

create new conjectures. In the dialogue, Pupil Zeta performs a thought-experiment in

which he pastes together Eulerian polyhedra. This allows him to explore why some of

the newly created polyhedra are not Eulerian. Zeta uses this analysis of global

counterexamples to form new conjectures about different types of configurations.

Through this investigation, he and Pupil Sigma are able to devise the conjecture that for

n-spheroid polyhedra, .121 Pupil Rho achieves the same result, but

approaches the thought-experiment from the opposite direction. He starts with one of the

polyhedra that were offered as counterexamples to the original conjecture, and breaks it

into pieces that are Eulerian. This is another approach that uses counterexamples to find

deeper patterns and conjectures. Both of these methods generate new knowledge by

analyzing and extending the concepts that arise in a proof.

Lakatos has provided several examples of proof-analysis methods, which fall into

two main categories: restricting content or increasing content. A global counterexample,

119 Ibid., 10.120 Ibid., 76. Italics in the original.121 Ibid., 77.

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or a local one that is also global, may lead you to restrict your conjecture. But a strictly

local counterexample will lead to a new lemma, and a global counterexample could result

in a deeper conjecture. Both of these moves increase content by exploring in a new

direction. Whether restricting or increasing content, the method of proofs and refutations

generates new knowledge. We either refine our existing knowledge or create new

concepts altogether.

A possible criticism of the dialogue is that rather than uncovering errors in the

proof, the class generates counterexamples that are not in the spirit of the conjecture. In

other words, as Delta tries to maintain through his monster-barring, the counterexamples

are not the types of polyhedra that were intended when the conjecture was made. Instead

of refuting the theorem, the counterexamples pick apart the definition, looking for

loopholes and causing unnecessary conflict. Lakatos acknowledges this, but counters

that it is necessary for the growth of mathematical knowledge. The mathematical

community needs the “stimulus of counterexamples” to stretch its thinking, and to avoid

the false sense of security which is “a symptom of lack of imagination, of conceptual

poverty.”122 By stretching a concept beyond its intended domain, mathematicians find

counterexamples that trigger a subsequent expansion of the surrounding theory. This

produces a more powerful theory which can explain the counterexample.123

Lakatos’s philosophy is somewhat controversial. Its basis runs counter to

mathematical Platonism, as it begins with the assumption that any mathematical system

must be based on a set of axioms whose certainty we cannot prove. Attempting to prove

these axioms will result in infinite regress. Lakatos believes that we must accept the fact

122 Ibid., 87.123 Ibid., 94.

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that we cannot establish foundations of mathematical knowledge or mathematical

certainty. Instead, we can only make guesses, and then critique and improve those

guesses.124 Many criticisms have been offered of Lakatos’s philosophy of mathematics.

Reuben Hersh, while not a Platonist, complains that Lakatos does not provide an

ontology. Hersh wants an account of what mathematics is about, and why it is

important.125 Lakatos also prompts complaints because he discounts the emphasis

traditionally put on formal mathematics. He sees informal mathematics as the primary

mathematical activity, which produces formal mathematical theories.126 Lakatos believes

that this mirrors the history of mathematics and its development as a discipline. D. A.

Anapolitanos comments on the weaknesses of this explanation. He claims that it does not

account for the sudden shifts in conceptual frameworks that can result from foundational

crises or unexpected reconceptualizations.127 Criticisms of Lakatos’s philosophy of

mathematics are somewhat moot in the context of this paper, however. The current

project is more interested in analyzing its potential as a philosophy of mathematics

education. As addressed in the last chapter, this allows us to avoid many of the issues

that pertain to its adequacy as a philosophy of mathematics. Instead, I will analyze this

philosophy’s potential as an account of individual mathematical knowledge.

Educational Implications

While Lakatos wrote Proofs and Refutations as an analogy for the mathematical

community, this work can also be interpreted more literally, as an educational example.

It can be read as an account of individual knowledge creation in a classroom setting.

124 Lakatos, "Infinite Regress," 9-10.125 Hersh, What is Mathematics, Really? 212.126 Ernest, Social Constructivism, 117.127 Anapolitanos, "Proofs and Refutations Reassessment," 338-40.

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Here I will consider what this account would look like in order to form a Lakatosian

philosophy of mathematics education.

A Lakatosian Philosophy of Mathematics Education

To make the shift from the generation of mathematical knowledge to individual

knowledge more clear, we can substitute the word understanding for knowledge. Rather

than an account of the creation of new mathematical knowledge, we can see in the

dialogue a description of the development of individual understanding through the logic

of mathematical discovery. Read in this way, we see that the process of finding and

analyzing counterexamples helps the students to develop a conceptual understanding of

the geometric principles that are involved. As the class argues about the definition of a

polyhedron, some of the students begin to understand that polyhedra can be much more

complex than they originally thought, and they attempt to refine their understanding of

what, exactly, constitutes a polyhedron. The class experiments with different restrictions

on the conjecture to see if they can exclude the counterexamples. The teacher guides the

students in finding the most productive ways to revise the conjecture and the proof, by

uncovering hidden assumptions and exploring why the counterexamples contradict those

assumptions. This prompts them to consider which features are necessary for a

polyhedron to be Eulerian. Through this exploration, the students refine and deepen their

understanding of geometric relationships. Some of them find polyhedra that contradict

one of the hidden assumptions in the proof, but are still Eulerian. They try to determine

what purpose the contradicted assumption was serving, and how it could be rephrased or

altered. Other students begin experimenting by combining Eulerian polyhedra to create

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new examples for testing. They apply what they have learned to these new examples,

creating new conjectures and expanding their understanding.

A Lakatosian description of individual mathematical knowledge formation would

look very much like Lakatos’s account of the historical development of mathematics. In

a sense, the class is a miniature version of the mathematical community, recreating parts

of mathematics as if they were not already known. Through a cycle of conjecture, proof,

counterexample, and revised conjecture, students can continuously develop and expand

their understanding of mathematical concepts. This is similar to what Ball envisions for

her third grade classroom, as described in the previous chapter. Her class works with a

base of publicly shared knowledge, making conjectures and debating their validity. An

important difference, however, is that Ball emphasizes deductive, “top-down” reasoning,

while Lakatos focuses on counterexamples. Ball’s students criticize each other’s line of

reasoning without necessarily presenting counterexamples to disprove the original

conjecture. Lakatos’s fictional students do question aspects of the explanation, but then

turn to counterexamples to test those aspects.

While this explains how concepts are introduced in the classroom, we are still

missing an account of how the logic of mathematical discovery generates individual

mathematical knowledge. Lakatos provides an explanation of how an abstract body of

knowledge grows and evolves within a community, which can be modified to refer to

concept generation within a classroom. But when speaking of education, one must also

consider how the individuals come to learn about, or internalize, the concepts they are

generating and discussing. The logic of mathematical discovery can be used to explain

how students recreate mathematical content through discussion and argument, but it does

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not describe how students fit the various pieces of content together to form some sort of

mathematical understanding. Because Lakatos does not provide an account of this, it

must be filled in with an explanation that is consistent with his logic of mathematical

discovery.

In many ways, the dialogue is consistent with a radical constructivist philosophy.

As in the work of von Glasersfeld, a student assimilates a new experience (such as

noticing a pattern and forming a conjecture). When counterexamples are presented, this

creates a perturbation, and accommodation happens as the conjecture is revised. This

makes the counterexamples a critical part of the learning process. They force students to

reconsider what was previously held to be true, pushing the boundaries of their current

knowledge base. New individual knowledge is generated as a student revises or expands

her previous understanding. The radical constructivist account, however, does not seem

to give an adequate role to the dialogue itself. It sets up an opposition between each

student and the others, describing the learning process as an individual reaction to the

perturbations created by the critiques of the other students. On examining Lakatos’s

dialogue, however, there seems to be more importance placed on interaction than

reactions. Perturbations affect the whole group, and attempts to resolve them are

verbalized for group consideration. Each student’s thoughts build on the comments of

others, and together they construct a common understanding of the concepts at hand. We

must turn to a social version of constructivism if we want to give dialogue this more

fundamental role in the learning process.

We can take a more social view by incorporating elements from Wittgenstein and

Vygotsky into the philosophy. Ernest sees the logic of mathematical discovery as the

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“dialectical logic of human conversation and interaction,” which he feels is based in

Wittgenstein’s concepts of language games and forms of life.128 Thus human language

and conversation allow mathematical discovery to happen. Drawing on Vygotsky, this is

also the medium through which learning occurs. Students use language and interaction

with each other to formulate their ideas and internalize concepts. One could even say that

encountering unexpected counterexamples pushes students into their zone of proximal

development, where new understanding can be developed quite rapidly.

The pedagogy suggested by a Lakatosian philosophy is the one modeled in Proofs

and Refutations, which makes use of certain constructivist pedagogical methods.

Constructivist instruction is problem-centered, which means students are “engaged in

attempting to resolve problematic situations for themselves.”129 This usually includes

projects and group work. The communication that occurs in these situations allows

students to learn by verbalizing their thinking, reconceptualizing a problem, and

analyzing erroneous solution methods.130 In Lakatos’s dialogue, the class is presented

with a problem: to determine if a conjecture and proposed proof are correct. They

verbalize their thinking, analyze erroneous lemmas, and reconceptualize the concept of a

polyhedron. These activities allow the student to gradually construct an increasingly

complex understanding of the related concepts.

Analysis

The perceived strengths and weaknesses in a Lakatosian philosophy of

mathematics education depend on one’s perspective. One must work within a stated set

of goals to judge its educational value. Some may say that a strength is how it presents a

128 Ernest, Social Constructivism, 125.129 Yackel, Cobb, and Wood, "Mathematical Communication," 34.130 Ibid., 36.

49

view of mathematics education that is modeled on mathematical practice, and reflects the

manner in which mathematical knowledge has actually been generated in many cases. To

learn mathematics, students “do” mathematics. The method of proofs and refutations

reflects the fact that mathematics is an evolving discipline, with new branches emerging

that defy previous assumptions.131

From a pedagogical perspective, the method of proofs and refutations encourages

logical inquiry and argument. Students are taught not to take things at face value, but to

examine them carefully. Counterexamples are offered not as the final word, but as

learning opportunities. Because the students are required to revise either the conjecture

or the proof in a suitable manner, they must learn to analyze the proof to see where it

breaks down. This teaches students to draw out relevant features of a counterexample in

order to uncover hidden assumptions in a proof, and to deduce what revision is necessary

to resolve the conflict. There are also opportunities to synthesize pieces of information

into a new conjecture. Throughout the process, students must be able to articulate their

thinking to others. All of these skills are important in the mathematical community, and

the hope is that they provide students with a deeper conceptual understanding while

forcing them to be active and engaged in the learning process.

There are some concerns that could be posed, however. Some of these are merely

questions of implementation, but should be addressed. Other concerns are more

substantive, revealing important weaknesses in this approach. The issues related to

implementation center around teacher preparation and classroom norms.

131 For example, non-Euclidean geometry is not subject to many of the geometric principles we take for granted: parallel lines may intersect, etc.

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Teacher preparation. In terms of teachers, the method of proofs and refutations

would be subject to the same criticism as other constructivist methods, such as discovery

learning and problem-solving approaches: it is difficult to implement well. The teacher

needs a stronger background in the subject matter to be able to handle the unexpected

directions that the conversation may go. In most situations, teachers must balance

freedom of exploration against a required curriculum. To make sure the class achieves

prescribed outcomes, the teacher must carefully plan a conjecture or topic that is

accessible, conveys the intended material, and yet is interesting enough to generate

debate. Then she must provide the right level of guidance to encourage mathematical

creativity while making sure the students are acculturated into standard mathematical

practice.132 To execute this type of lesson effectively takes planning and practice.

Classroom norms. The other implementation issue has to do with student

personalities and classroom norms. Some students adjust better than others to a

classroom that does not follow the familiar patterns. In traditional mathematics

classrooms, most of the talking is done by the teacher. When the teacher does ask for a

response from the class, she is looking for a particular answer. Students who offer

alternate methods are downplayed or ignored.133 This pattern becomes familiar over the

years, and it can make it very uncomfortable for a student who encounters a teacher with

different expectations. Terry Wood, in his work with Paul Cobb and Erna Yackel, has

found that this makes “the mutual construction of classroom norms … crucial to

establishing the setting for learning.”134 A teacher using a Lakatosian approach must be

alert to this and begin orienting students to her classroom norms from the first day. There

132 Simon, "Mathematical Learning in Classrooms," 103.133 Wood, Cobb, and Yackel, "Whole-Class Discussion," 58.134 Wood, "Creating an Environment," 19.

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will still be obstacles, however. Shy students may be hesitant to share their ideas.

Students who lack self-confidence in the area of mathematics may be unprepared to

handle the criticism that is inherent in the process of proofs and refutations. In Lakatos’s

dialogue, students go so far as to heckle each other’s ideas and to call each other names.

Delta tells Alpha, “I turn in disgust from your lamentable ‘polyhedra’, for which Euler’s

beautiful theorem doesn’t hold.” Alpha responds, “You are a real old-fashioned Tory!

You blame the wickedness of anarchists for the spoiling of your ‘order’ and ‘harmony’,

and you ‘solve’ the difficulties by verbal recommendations.”135 While Lakatos is

exaggerating the conflict for dramatic effect, it is not so hard to imagine students

becoming frustrated and lashing out at each other when their ideas are criticized. This

will create a challenge for the teacher, who will need somehow to help the less confident

students engage in classroom discussion. Magdalene Lampert describes a fifth-grade

classroom in which she used a Lakatosian approach with some success. She taught her

students to use phrases like, “I want to question so-and-so’s hypotheses,” and to present

their reasons, so that their comments would take the form of logical refutations rather

than judgments.136 Students were also encouraged to use the phrase, “I want to revise my

thinking,” to convey that revising a conjecture was not like admitting a mistake.137 This

provides an approach for reducing the conflict that can arise in a Lakatosian classroom.

It is something that the teacher must address early, modeling the expected behavior. She

will need to work very hard to create a classroom in which students see thoughtful

critique as a normal part of the learning process.

135 Lakatos, Proofs and Refutations, 19.136 Lampert, "Problem Not the Question," 40.137 Ibid., 52.

52

The more substantive concerns cannot be addressed through teacher preparation

and training. They may have to be dealt with by adjusting the approach, or by using it in

conjunction with other methods.

Absence of formal proof. Mathematicians may be concerned about the de-

emphasis of formal proof and deductive logic. There is a sense in which proof can be too

formal for an educational setting. Hersh explains that in the classroom, proof should be

more informal and presented in natural language. Rather than providing an airtight

mathematical justification, proof in this context serves “to give insight why a theorem’s

true.”138 It should increase understanding, or else be omitted. However, it seems that

Lakatos goes beyond what Hersh suggests. Lakatos’s Principle of Retransmission of

Falsity puts the focus on counterexamples rather than proof, even in the informal sense.

The priority is disproving rather than proving, showing that something is false rather than

explaining deductively why something is true. I do not believe, though, that Lakatos

completely abandons proof and deductive logic. Instead, he uses informal proof and

deduction in conjunction with refutations. First, deduction is used to devise the original

conjecture. While it may seem that the conjecture is arrived at inductively by observing

examples, Lakatos emphatically argues that this is not the case. He claims that as

examples are observed, we move through a series of early conjectures that are quickly

refuted. Through a process of deductive guessing, we arrive at the main conjecture.139

Second, the original proof is an informal expression of deductive logic. Third, a

counterexample is presented, and deduction must be used to figure out how to improve

the conjecture or proof. Recall that in the dialogue, the teacher urges the class not to

138 Hersh, What is Mathematics, Really? 59.139 Lakatos, Proofs and Refutations, 73.

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restrict the conjecture haphazardly. Instead, one should uncover a hidden lemma, and

examine why it is refuted. This information is used to revise the conjecture. This

employs a process of deduction, though not in the formal sense. A student must run

through the proof, establish what assumption is being used but is not expressed, and

determine how it is operating in the string of deductive logic. Then he can deduce in

what way the counterexample refutes the lemma, and why this contradiction occurs. The

conjecture can be revised accordingly. A similar process occurs if a strictly local

counterexample is found. The student must examine the place the lemma holds in the

thread of logic, and determine which aspects of the lemma are necessary to that

argument. The lemma can then be replaced with one that retains the necessary elements

but is not refuted by the counterexample. In these ways, I see informal deduction

occurring throughout the proof-analysis.

I do acknowledge, however, that this will not satisfy mathematicians who believe

that the backbone of mathematics is formal proof, and that this must be reflected in

mathematics education. They will say that the method of proofs and refutations does

indeed use reasoning, but not deductive proof. There is some validity to this argument.

While I maintain that the logic of mathematical discovery incorporates deductive

reasoning, it does not give students the opportunity to practice the logical steps involved

in constructing a proof. After the initial “rough-draft” of the proof, the focus shifts to

finding counterexamples. The students may notice suspect assumptions, but they do not

examine the proof for logical errors. This sends the message that errors can only be

found through counterexamples. Students are thoroughly schooled in the retransmission

of falsity from the bottom up, but receive little experience in the transmission of truth

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from the top down. It seems there should be a place for classical deduction, even within

Lakatos’s quasi-empiricist scheme. In mathematics practice, both the transmission of

truth and the retransmission of falsity are important. Counterexamples fuel many

mathematical discoveries, but at the end of the day, mathematicians need to be able to

turn what they have learned into a formal proof. Students can learn about proof even at

elementary levels, in a setting like Ball’s third grade class. The retransmission of falsity

can be taught by incorporating more counterexamples into this environment. By

integrating both aspects into a classroom, students can be given a better sense of what

“doing mathematics” really means.

Absence of applications. Mathematics educators may be more concerned about

the lack of applications in a Lakatosian approach than the lack of formal proof. The

method of proofs and refutations is designed to generate new knowledge. But many

educators and mathematicians alike would argue that “doing mathematics” is also about

applying that knowledge in order to solve problems. In fact, there has been a growing

emphasis on problem solving in mathematics education research over the past twenty

years. In a study of a high school geometry class, Alan Schoenfeld found that the lack of

complex problem solving opportunities left students unable to see connections between

different areas of geometry, such as proofs and constructions.140 The National Council of

Teachers of Mathematics (NCTM) also sees connections to real-world applications as an

important part of the curriculum, to bring significance and relevance to the material.141

These concerns point to a need for more than Lakatos’s mathematical discovery.

140 Schoenfeld, "Good Teaching, Bad Results," 150.141 National Council of Teachers of Mathematics, Principles and Standards, 66.

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Students need to see how the mathematical facts and principles they have discovered

connect to other aspects of the discipline and to practical applications.

As with the lack of formal proof, I believe this weakness in the Lakatosian

approach can be remedied by combining it with other strategies. Specifically, it can be

incorporated into a problem-based curriculum. Sarah Lubienski explains that while some

teachers add applications at the end of a unit, teaching through problem solving is a

distinct pedagogical approach in which problem solving is actually used to introduce and

develop mathematical concepts.142 This seems like a natural complement to proofs and

refutations. One day, the problem could be exploring and revising a proposed conjecture

and proof through counterexamples. Then the next day, students could work on problems

that apply the concepts they explored the day before. This allows students to experience

the more theoretical side of doing mathematics, as well as the practical applications and

problem solving.

Lack of relevance for non-mathematicians. Kurt Stemhagen raises the concern

that is perhaps most pertinent to the current project. He points out that the mathematics

discussed in Proofs and Refutations is abstract and theoretical, far removed from

everyday practice.143 Indeed, Lakatos intended the dialogue to model the generation of

mathematical knowledge within the mathematical community. Does this make a

Lakatosian philosophy of education relevant only to future mathematicians? What about

students who are either not prepared, or do not desire, to pursue advanced mathematics?

There are two questions we can ask about the education of students who are not future

mathematicians. Could the method of proofs and refutations be used with these students?

142 Lubienski, "Problem-Centered Mathematics Teaching," 254. 143 Stemhagen, "Beyond Absolutism and Constructivism," 134.

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And, would this method be a benefit to these students, or should efforts be focused

elsewhere?

To answer the first question, yes, I believe the method of proofs and refutations

can be used in classes not necessarily intended for future mathematicians. As long as the

original conjectures are appropriate to the age of the students and to the course

curriculum, students should be able to look for counterexamples, and make revisions to

the conjecture or form new conjectures of their own. Lampert’s use of a Lakatosian

approach in her fifth-grade classroom illustrates this. Each lesson began with the teacher

posing a problem. Rather than focusing on the solution to the problem, she required

students to make and test mathematical hypotheses about a solution strategy.144 The

students engaged in class discussions about these hypotheses, refuting and revising

conjectures. Lampert found that students were able to participate fully, and that these

lessons succeeded in teaching students to engage in “disciplinary discourse” in a way that

reflects what it means to know and do mathematics in the Lakatosian sense.145

The second question, however, is harder to answer. We cannot assess how

beneficial this method is without specifying our values and goals. If we believe that all

students should be taught to think like mathematicians, then we might think that the

method of proofs and refutations is beneficial to all students. On the other hand, maybe

we believe that students going into fields other than mathematics need more emphasis on

applications, or that underprepared students need an emphasis on basic skills. If these are

our goals, then we may prefer less abstract classroom activities. The teacher could still

have students explore why things are true, without using proofs and refutations. If she

144 Lampert, "Problem Not the Question," 39-40.145 Ibid., 58-59.

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wants the students to experience mathematical discovery, she could use some well-

designed problem solving activities instead. In these cases, the method of proofs and

refutations could be seen either as not necessary, or not appropriate.

One framework that may be used to analyze the benefit of the method of proofs

and refutations for non-mathematicians is that of NCTM’s Principles and Standards for

School Mathematics. The organization has been widely influential with this standards

document, emphasizing inquiry, problem solving, and reasoning in mathematics

education for all students at the pre-kindergarten through twelfth-grade levels. These

standards are relevant to this discussion because they are not aimed at future

mathematicians, but are intended for all students. While NCTM certainly does not

represent every stakeholder, it does express the opinion of many mathematics educators.

I will consider the method of proofs and refutations within the framework of these

standards, using them not as benchmarks for evaluation, but rather as a tool to highlight

issues that may be important in this setting. The recommendations that most directly

influence pedagogy are the Process Standards, which describe processes, or methods, that

students should be able to use to acquire and work with content knowledge. Teachers are

encouraged to model these methods in their teaching. They include Problem Solving,

Reasoning and Proof, Communication, Connections, and Representation.146

The method of proofs and refutations at least partially satisfies the standards of

Problem Solving, Reasoning and Proof, and Communication. The Problem Solving

standard states that students should be able to build mathematical knowledge through

problem solving; solve problems in context; develop, choose, apply, and adapt

appropriate strategies; and monitor their own progress and adjust their approach as

146 National Council of Teachers of Mathematics, Principles and Standards, 29.

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necessary.147 A Lakatosian approach satisfies these criteria very well in terms of abstract

problem solving. However, what NCTM has in mind are application problems. As

discussed above, the method of proofs and refutations is weak in this area, but can be

strengthened by incorporating it into a problem solving curriculum. The method also

satisfies parts of the Reasoning and Proof standard. This standard intends for students to

see how integral reasoning is to mathematics, and to develop conjectures along with

arguments and proofs. NCTM wants students to understand that proof goes beyond

seeing a pattern—they must be able to explain why something will always be true.148 I

have already described how reasoning, argument, and proof are present in the Lakatosian

approach. While most of the time is spent finding counterexamples, this still requires

quite a bit of reasoning, some of it deductive, and a certain amount of argument and

proof-adjustment. However, the method of proofs and refutations is mostly concerned

with figuring out when something is not true. Beyond the initial “rough sketch” of a

proof, this would not encourage students to explain why a particular conjecture is always

true. This is a potential weakness of the Lakatosian approach. Because students

engaging in the logic of mathematical discovery do their reasoning and argument as a

class discussion, the Communication standard applies. NCTM expects students to

discuss their ideas and solutions with other students. This will help them to learn the

importance of clearly defining a concept, how to formalize thoughts with conventional

terminology, and how to make a clear and convincing argument.149 Here the method of

proofs and refutations excels. Students will argue about definitions and terminology,

which will illustrate why clear definitions are so important. When making revisions to

147 Ibid., 52-54.148 Ibid., 56-59.149 Ibid., 60-63.

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conjectures and proofs, students who do not present a clear and convincing argument will

be refuted by other students in the class. Having others point out the weaknesses in an

argument will help students learn logical principles and mathematical argument styles.

From NCTM’s standpoint, the main weaknesses in the method of proofs and

refutations appear in reference to the Connections and Representation standards. NCTM

wants students to see connections between conceptual understanding and procedures,

between different areas of mathematics, and between mathematics and other

disciplines.150 With its emphasis on abstract problems, Lakatos’s logic of mathematical

discovery does not emphasize these elements. It is possible that the search for

counterexamples could reveal links to other branches of mathematics, but it will not draw

attention to the other types of connections mentioned. The Representation standard

encourages students to develop and use their own representations for mathematical

concepts before learning conventional forms. They should be able to select the best

representation for a given problem, and use it to model mathematical situations.151

Because proofs and refutations is primarily a verbal method, it does not address visual

representations in any conscious way. It could be said that it encourages alternate proofs

which make use of different mental representations, but this only begins to do what

NCTM intends. A proofs and refutations approach could be strengthened in the areas of

Connections and Representation if it were combined with a problem solving curriculum.

This would allow the teacher to use problems which highlight the connections between

concepts and procedures and between mathematics and other disciplines. She would also

150 Ibid., 64-66.151 Ibid., 67-69.

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be able to encourage the use of written and visual representations as students work on

problems and explain their solutions.

NCTM’s Process Standards provide a framework for examining the strengths and

weaknesses of the method of proofs and refutations for students are not future

mathematicians. Through this analysis, we see that this approach makes use of abstract

problem solving, reasoning, and communication. However, some educators may feel that

it focuses too much on abstract mathematical principles, and not enough on applications

and connections. In addition, it does not address the need for written mathematical

representations. These weaknesses do not mean that the method is not useful, but they

suggest implementation issues that should weighed against one’s particular goals.

Teachers may want to combine the approach with a problem solving curriculum that will

incorporate these missing elements. We will see in the next chapter that for students who

are not future mathematicians, Ernest suggests a pedagogical method that is influenced

by Lakatos but emphasizes problem solving rather than proofs and refutations.

Summary and Conclusion

Ernest’s philosophy of mathematics education draws its constructivist roots from

von Glasersfeld’s radical constructivism. However, Ernest accounts for intersubjective

agreement by incorporating Vygotsky’s social view of cognitive development and

Wittgenstein’s philosophy of language as a social convention. Lakatos’s logic of

mathematical discovery is adapted for classroom application to explain the generation of

mathematical knowledge and to form the backbone of Ernest’s philosophy. In his

pedagogical recommendations, Ernest distinguishes between mathematics education for

future mathematicians and for everyone else. While his recommendations for general

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mathematics education resemble NCTM’s standards in many ways, his vision for the

training of future mathematicians is based on Lakatos’s proofs and refutations. In the

next chapter, I will explore how Ernest incorporates the work of von Glasersfeld,

Vygotsky, and Wittgenstein into a Lakatosian framework to form his philosophies of

mathematics and mathematics education. Then in chapter five, I will analyze the

distinction he makes between the training of future mathematicians and mathematics

education for everyone else.

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CHAPTER 4

ERNEST’S SOCIAL CONSTRUCTIVISM

Ernest’s social constructivism consists of two philosophies: a philosophy of

mathematics and a philosophy of mathematics education. The philosophy of

mathematics includes his ontology and epistemology, while the philosophy of

mathematics education addresses issues of teaching and learning. Each has been

influenced to some degree by the ideas of von Glasersfeld, Wittgenstein, Lakatos, and

Vygotsky. Ernest believes that his philosophy of mathematics forms a critical part of his

philosophy of mathematics education, but the former is bound to meet with severe

resistance. His philosophy of mathematics challenges traditional realist views that have

been pervasive in their influence on mathematicians and teachers alike. If the realist

philosophy of mathematics is correct, and there is a necessary link between Ernest’s two

philosophies, then his philosophy of mathematics education falls with his philosophy of

mathematics. As I indicated in chapter two, I would like to avoid such ontological and

epistemological debates in the philosophy of mathematics. Therefore, in this chapter I

will explore whether his philosophy of mathematics education can be supported

independently of his philosophy of mathematics.

This chapter will also provide the theoretical background for a later analysis of

Ernest’s curriculum recommendations. Ernest has developed his ideas over many years,

and has articulated his views in two books and many articles. During that time, he has

revised and elaborated some aspects of his theories. In addition, his two books each

focus on a different aspect of his philosophy. In his earlier book, The Philosophy of

Mathematics Education, Ernest briefly outlines his philosophy of mathematics in order to

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frame a more extensive conversation about mathematics education. His second book,

Social Constructivism as a Philosophy of Mathematics, focuses instead on the nature of

mathematical knowledge. In it, he discusses the learning of mathematics as it pertains to

his epistemology and the transmission of knowledge from one generation to the next. I

will attempt to present as accurate a summary of Ernest’s views as possible by

considering his work as a whole and using the most recent writing where ideas have been

revised. In this chapter, I will first outline Ernest’s philosophy of mathematics to provide

context. After discussing his philosophy of mathematics education, I will explore its

connection to his philosophy of mathematics to determine if the former depends on the

latter. Then, I will analyze and evaluate Ernest’s philosophy of mathematics education

and its incorporation into the classroom.

Philosophy of Mathematics

Ernest’s social constructivist philosophy has been influenced in many ways by

von Glasersfeld’s radical constructivism: mathematical knowledge is constructed, not

discovered. However, Ernest emphasizes interpersonal communication in a way not

present in the work of von Glasersfeld. He does this through the use of conversation, a

unifying concept that runs through his philosophy. Conversation is a metaphor used to

describe an interchange between people, both verbal and written. It can be private,

between two or more people, but it can also occur as a published exchange in the public

sphere. Conversation includes any “sequence of linguistic utterances or texts in a

common language (or languages) made by a number of speakers or authors, who take it

in turn to ‘speak’ (contribute) and who respond with further relevant contributions to the

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conversation.”152 Ernest borrows from Wittgenstein and Lakatos to form a version of

constructivism that recognizes the social and linguistic dimensions of mathematical

knowledge, and that accounts for its genesis in addition to its justification. He uses

conversation to provide an account of the genesis of mathematical knowledge and to

explain how human agreement is possible. I will summarize Ernest’s philosophy of

mathematics and offer a limited analysis in order to provide background for his

philosophy of mathematics education. I will not, however, take a position on its accuracy

or truth.

Ernest’s philosophy of mathematics is based on the fallibilist assumption that

mathematical truth is never absolutely certain. He rejects, therefore, what he calls the

absolutist position: that mathematical truth is infallible and objective.153 Like von

Glasersfeld, Ernest sees the traditional, absolutist view of mathematics as a futile search

for an objective, independent truth. Even if it did exist, we could not know for sure.

Mathematics is therefore a fallible human institution, created through human relations.

Drawing from Wittgenstein, Ernest believes that mathematics is a set of language games,

based in shared forms of life. This provides socially situated conversation as the basis for

Ernest’s philosophy. Mathematical objects are just the social constructs of mathematical

discourse, signified by certain symbols or notations.154 Mathematical truth is determined

by the rules of its language games. Ernest explains that the necessity of logical and

mathematical truth is found in the linguistic conventions, rules, and social practices that

are required for participation in language games.155 If these conventions are not accepted,

then one is no longer participating in the same language game. One must operate under 152 Ernest, Social Constructivism, 163.153 Ibid., 9.154 Ibid., 193.155 Ibid., 146.

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the accepted logical and mathematical rules, or else sacrifice shared understanding with

the rest of the mathematical community. In other words, these rules are necessary

because if they are not followed, it is not the language of mathematics. Thus Ernest feels

that this theory of language supplies a justificatory basis for mathematics, providing an

epistemological foundation for mathematical knowledge.156

While Wittgenstein’s framework provides the foundation for mathematical

knowledge in Ernest’s philosophy, it does not describe its genesis. This foundation does

not account for mathematical invention or the actual process of warranting mathematical

claims. Ernest uses the ideas of Lakatos to describe how mathematical knowledge is

created. With socially situated conversation as its foundation, Lakatos’s logic of

mathematical discovery is reinterpreted not as an example of abstract ideal logic, but

rather as “alternating voices in dialogue.”157 This dialogue provides a model for the

formation of the body of objective mathematical knowledge.

Objective Knowledge

Because of Ernest’s belief that mathematics has no claim to absolute truth, he uses

the phrase objective knowledge in a non-traditional way. He defines objective

mathematical knowledge as that which is intersubjective and shared among the

mathematical community. This includes mathematical theories, conjectures, axioms, and

proofs, as well as shared conventions and rules of language use.158 While this kind of

objective knowledge loses the sense of certain truth that Platonism can provide, Ernest

feels that it maintains the “external thing-like character” that is associated with a body of

disciplinary knowledge.159 One can imagine that as a body of knowledge is handed down 156 Ibid., 135.157 Ibid., 125.158 Ibid., 144-45.159 Ibid., 145.

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throughout history, with its own set of rules and standards, it seems to have an existence

of its own that is separate from those who have contributed to it. The body cannot be

altered on the whim of an individual, but rather only by widespread agreement. In

mathematics, issues of terminology and notation are decided through the social

negotiation of meaning, while statements and propositions must come to be accepted by

the mathematical community as rigorously proven.

With socially situated conversation as a basis, Ernest builds on the work of

Lakatos to provide a social, dialectical account of the creation of objective mathematical

knowledge. In response to the concerns discussed previously that Lakatos’s philosophy

does not account for sudden shifts in conceptual frameworks in the history of

mathematics, Ernest extends and generalizes the account of mathematical knowledge

generation into a “generalized logic of mathematical discovery.”160 Set within the

background context of the mathematical community, the cycle begins when a new

conjecture, proof, problem solution, or theory is presented publicly to the community. As

if in conversation, a dialectical response is generated by “a subsection of the

mathematical community,” either accepting or critiquing the proposal.161 Ernest expands

Lakatos’s list of critical responses that can be offered if the proposal is not accepted. In

addition to counterexamples and refutations, members of the mathematical community

may offer counterarguments or other criticisms of the proposal.162 As a result of the

critique, the proposal may undergo either a local or global restructuring. A local

restructuring would be a modification of the proposal. Ernest does not go into detail

about the possible modifications, but one would assume these would include the potential

160 Ibid., 149.161 Ibid., 150.162 Ibid., 151.

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revisions described in Lakatos’s Proofs and Refutations, a reworking of the argument, or

other modifications appropriate to the type of proposal and the criticism received. A

global restructuring would involve a major change to the background context in which

the mathematical community operates. This could include a change in mathematical

concepts, methods, language, or standards of proof, as well as an alteration of accepted

meta-mathematical views about the scope and structure of mathematics.163 For example,

Ernest notes that proofs generated using computers raised suspicion in the past but are

coming to be accepted.164 It is this aspect of the generalized logic of mathematical

discovery that allows for major shifts in conceptual frameworks within the mathematical

community. Whether resulting from a local or global restructuring, the cycle continues

as the new proposal or mathematical context is presented publicly and subjected to

further critique. If any iteration of the cycle results in an acceptance of the proposal by

the mathematical community, Ernest explains that this would result in the proposal being

added to the body of objective mathematical knowledge.

For Ernest, the most important factor in the acceptance of a proposed item of

mathematical knowledge is its proof. By increasing the emphasis on formal deduction,

Ernest avoids some of the criticism directed toward Lakatos. While for Lakatos it is the

counterexamples that drive mathematical discovery, for Ernest the process is centered on

proof:

Proof serves as more than just a warrant for propositional knowledge in mathematics. It also provides such things as the central test of the fruitfulness of new concepts and definitions, the means to elicit the consequences of informal and axiomatic theories, the means to test proposed solutions to problems, and the way to establish the consequences of hypotheses and conjectures. Consequently, the public construction, scrutiny, and testing of proofs is undoubtedly the central

163 Ibid., 152.164 Ibid., 157.

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activity around which [the generalized logic of mathematical discovery] revolves.165

For Ernest, proof is a narrative text. He explains that “as a proof is an argument intended

to convince, a listener/reader is presupposed.”166 But proof is also part of a continuing

dialogue or conversation, because it assumes a response. The author expects to receive

either acceptance or critique from other mathematicians.167

The acceptability of a proof is determined by its structure and properties. Certain

assumptions and inferences are sanctioned by the mathematical community, and their

presence is expected in a legitimate proof. For example, mathematical propositions are

built on past knowledge, so previously established definitions, rules, and theorems may

be assumed.168 In addition, certain informal elements are acceptable. A proof may

contain informal assumptions that appeal to common knowledge or intuition. Arguments

can be made by analogy to another proof, or with the claim that a skipped step is easily

verified.169 All assumptions, but these informal elements especially, must be considered

carefully by those evaluating the proof. In some cases, there will be disagreement in the

mathematical community over the proof’s validity. There may be differences of opinion

about what constitutes a complete proof, or about whether a particular argument is sound.

When this happens, Ernest suggests that social forces will play a role in determining the

final acceptance or rejection. This involves factors such as the status, role, or

relationships of a particular member of the mathematical community.170

165 Ibid., 153.166 Ibid., 169.167 Ernest, "Dialogical Nature of Mathematics," 43.168 Ernest, Social Constructivism, 184-85.169 Ibid., 185-86.170 Ibid., 189.

69

A recent example of the social forces involved in proof acceptance is the proof of

the Poincaré conjecture, a famous problem concerning three-dimensional manifolds that

was posed by Henri Poincaré in 1904. In 2002 and 2003, Grigory Perelman posted a

proof of this conjecture in three installments on the internet. According to an article in

The New Yorker, the proof was quite short and contained many compressed arguments.171

Nonetheless, it convinced most mathematicians who had the background necessary to

evaluate it, and the consensus was that the proof was correct. Mathematician Shing-Tung

Yau, however, believed that the compressed arguments revealed mathematical gaps. Yau

felt that credit should instead be given to his students Xi-Ping Zhu and Huai-Dong Cao,

whose proof of the same conjecture was published in 2006. They reconstructed much of

Perelman’s proof, substituting new arguments where they felt Perelman had skipped

steps.172 Yau felt that Perelman’s proof was incomplete and therefore invalid, allowing

Zhu and Cao’s proof to be considered the first. The official decision of the mathematical

community went in Perelman’s favor, however, and he was offered the prestigious Fields

Medal in 2006. This vignette illustrates how in cases of disagreement, a proof’s

acceptance is determined both by its ability to convince and by social dynamics within

the mathematical community.173

The generalized logic of mathematical discovery and standards for proof

acceptance explain the fallibilist, human nature of mathematics. Because of this, Ernest’s

view holds that the cultural values and preferences of the mathematical community are

involved in objective knowledge formation.174 This incorporates a social responsibility,

171 Nasar and Gruber, "Manifold Destiny."172 Ibid.173 The article in The New Yorker gives a more elaborate account of the issues in question and the

resulting debate in the mathematical community. It also reveals issues surrounding mathematical collaboration, building on the work of others, and assigning credit for the final product.

174 Ernest, Social Constructivism, 270.

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with ethical consequences. Even a Platonist could acknowledge the social responsibility

associated with human choices about avenues of research or application. But Ernest goes

further, stating that because knowledge is constructed by humans, cultural values and

interests shape and form that knowledge, becoming part of it.175 Traditionally,

mathematics is characterized by logic and reasoning and is thus seen as objective and

value-free. Any values or preferences are subjective human distortions, which are kept

separate from the body of knowledge.176 But Ernest claims that cultural values have

influenced the mathematical norms themselves. The mathematical community has come

to favor the formal, abstract, and objective, and through the process of social construction

has implicitly chosen to define the discipline as logical and rational. This has had a great

impact on the direction and nature of mathematical knowledge and its creation. While

this knowledge is usually seen as absolute and value neutral, Ernest’s view is that

the cultural values, preferences, and interests of the social groups involved in the formation, elaboration, and validation of mathematical knowledge cannot be so easily factored out and discounted. The values that shape mathematics are neither subjective nor necessary consequences of the subject. Thus at the heart of the absolutist neutral view of mathematics, fallibilism claims to locate a set of values and a cultural perspective, as well as a set of rules which renders them invisible and undiscussable.177

Mathematical thought is characterized as abstract and objective, which makes it a tool for

the dehumanization of social issues and decisions. This “stripping away of the softer

human and moral aspects” allows decisions to be made without regard for human factors,

revealing the ethical consequences of mathematics.178

Ernest believes that the values of the mathematical community have also led to

the omission of valid mathematical ideas from the body of objective knowledge. In 175 Ibid., 269.176 Ibid., 270.177 Ibid., 270-71.178 Ernest, "Social and Political Values," 201.

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mathematics, preferences and values of the mathematical community have influenced

what kind of knowledge is most esteemed, and the process by which knowledge is

validated. Some mathematical concepts and procedures are not acknowledged because

they are not seen as legitimate parts of the discipline. For example, studies in

ethnomathematics reveal that cultural practices of indigenous people in Fiji and other

locations are often based on concepts of a mathematical nature.179 Systems of record

keeping, methods of measurement, and patterns in craft work, for example, make use of

mathematical concepts that are not part of mainstream mathematics. These often

complex ideas are not accepted as objective knowledge because they are undervalued by

the mathematical community. Ernest sees this as a social injustice:

In absolute terms, there is no basis for asserting that the system of values of one culture or society is superior to all others. It cannot be asserted, therefore, that Western mathematics is superior to any other forms because of its greater power over nature. This would be to commit the fallacy of assuming that the values of Western culture and mathematics are universal.180

Ernest is not saying that what is considered true in one culture may be considered false in

another—for example, that bringing together two groups of three creates a group of six.

Rather he is making the claim that different cultures may have different uses for

mathematics, shaped by their unique sets of cultural values and goals. This creates

systems of measurement or mathematical applications that are not valued by the

mathematical community, and thus have not been acknowledged in its western-

dominated body of objective knowledge. Ernest feels that this lack of recognition

disrespects and undermines the integrity of these other cultures.181 Michael Matthews

cautions that Ernest’s view conflates ethical and epistemological arguments. He claims

179 Stillman and Balatti, "Contribution of Ethnomathematics."180 Ernest, Philosophy of Mathematics Education, 264.181 Ibid.

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that the moral imperative to respect all cultures and an epistemological decision about the

validity of a system of values or mathematics do not in any way depend on each other.182

Ernest would respond that all of our epistemological decisions about validity are based on

our own value system, and that respect for other cultures requires that we acknowledge

other value systems as valid alternatives. We must allow ourselves to see the complexity

and power of mathematical systems different from our own. Matthews has a point,

however. Respecting another culture may ethically require us to consider the possibility

that its different mathematical value system is equally valid. It does not, however,

require that we come to an affirmative conclusion.

A strong criticism often leveled against social constructivism is that of relativism.

If mathematics is determined by social agreement, then it is arbitrary. Ernest

acknowledges that his view is relativist to some extent, but to a limit. Because of certain

standards imposed on the construction of mathematical knowledge, any alternate system

would have to be similar in certain ways. For example, mathematical applications are

used to model certain aspects of the physical and social world, and thus are subject to

empirical verification or refutation.183 Any alternate system of mathematics would also

have to accurately describe these phenomena. It might look different, but it would most

likely have a similar structure. In addition, Ernest feels that the standard of proof

prevents the body of mathematical knowledge from being arbitrary. A person cannot

make up whatever mathematical system he pleases—to be recognized as contributing to

mathematical knowledge, he must work within the accepted system of linguistic rules and

its assumptions and rules of reasoning. The system itself is subject to empirical

182 Matthews, "Appraising Constructivism," 185.183 Ernest, Philosophy of Mathematics Education, 61.

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constraints, and within its set of rules many propositions are determined by logical

necessity. In Ernest’s view, mathematical certainty is provided by the rigor of the

mathematical community in reviewing these knowledge claims.184 But Gila Hanna, who

has also written on the social process of proof acceptance, points out that most proofs are

not rigorously scrutinized. Only theorems deemed significant warrant attention, and even

then the proofs are not examined closely if the results seem plausible.185 Even Ernest

indicates that sometimes status of the mathematician, rather than professional rigor,

determines which proofs are accepted. And if a proof is accepted, it still may not garner

attention and thus not be disseminated widely.186 Ernest would probably respond,

however, that this is all part of the social construction of knowledge. Bias is one of the

social forces that shape the objective knowledge base. But Ernest and Hanna both note

that accepted proofs are often flawed.187 Sometimes a crucial error is discovered many

years later, and the proof is ultimately rejected or revised. Ernest’s response on this point

is that even with the potential for error, certainty is present at the time. There is no

absolute truth, just certainty until convinced otherwise. Still, his comments on bias and

error do not seem to put much faith in the rigor of the mathematical community on which

his account of certainty, even by his definition, depends.

Despite these doubts, Ernest holds that the body of objective mathematical

knowledge has been, and continues to be, rigorously constructed. He believes that its

basis in language games and forms of life gives it a solid foundation, and the generalized

logic of mathematical discovery describes how objective knowledge is constructed

184 Ernest, Social Constructivism, 159.185 Hanna, "More than Formal Proof," 22.186 Ernest, Social Constructivism, 159.187 Ibid., 29; Hanna, "More than Formal Proof," 22.

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through social agreement. The nature of objective knowledge in Ernest’s view has

helped to shape how he sees individual knowledge and the formation of the mind.

Subjective Knowledge

A comprehensive discussion of individual knowledge is fairly unique to social

constructivism. It is usually not emphasized in the philosophy of mathematics, often

being viewed as a topic for the philosophy of education or educational psychology.

Ernest feels that the learning of mathematics is important here because it accounts for

mathematics as it is passed from one generation to the next, and allows for the creativity

of individual mathematicians.188 Therefore, he includes it as part of his philosophy of

mathematics, though it is also part of his philosophy of mathematics education. Here I

will discuss Ernest’s general theory of subjective knowledge development, and in the

next section I will elaborate on the role of schooling in this process.

For Ernest, mathematical knowledge includes propositional knowledge as well as

know-how.189 Thus when he describes the nature and genesis of individual knowledge,

he is referring to propositions and statements, but also symbolic and conceptual

procedures. “To know one of these items is to be able to offer a valid justification for it,

if it is explicit, or to be able to demonstrate that knowledge through appropriate

behaviors.”190 As discussed in chapter two, Ernest refers to individual knowledge as

subjective knowledge. This reflects the fact that each individual constructs his or her

own subjective representation of the body of objective mathematical knowledge.

Knowledge cannot be learned through transfer from teacher to student. Each individual

188 Ernest, Social Constructivism, 265.189 Ibid., 136.190 Ibid., 160.

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must actively construct her own meaning based on her experiences, through a process of

internalizing and reconstructing objective knowledge.191

Ernest bases his view of subjective knowledge on von Glasersfeld’s radical

constructivist theory that learning is the construction of meaning through experience.

However, he is concerned that von Glasersfeld does not allow for communication and

agreement between individuals. In The Philosophy of Mathematics Education, Ernest

attempts to remedy this by emphasizing that an individual interacts with the physical and

social worlds, which improves the fit of knowledge with experience. Linguistic rules

constrain communication so as to allow a fit between different individuals’ meanings.192

In Social Constructivism as a Philosophy of Mathematics, he greatly expands on the role

of the social world and conversation in the formation of subjective knowledge. Unlike

von Glasersfeld, who bases his theory of development on Piaget’s cognitive view of

mind, Ernest models his on Vygotsky’s social view of mind. For Ernest, socially situated

conversation is instrumental in the formation of the mind, and thus also in its use.193 He

borrows from Wittgenstein the idea that thought is not possible without language, which

is learned socially.194 Language is the medium for learning and concept formation, and

thus for the genesis of higher level thought. Therefore, the learning process involves

conversation and interaction with others. This is different from von Glasersfeld’s view,

in which interaction with others is merely a source of perturbation. As I interpret this

difference, for von Glasersfeld learning is an individual process that is informed by

interaction with others, while for Ernest it is a social process that actually happens

through the interaction with others. In von Glasersfeld’s theory, a person first constructs 191 Ernest, Philosophy of Mathematics Education, 44.192 Ibid., 72-73.193 Ernest, Social Constructivism, 211.194 See chapter three for an elaboration on this idea.

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her understanding individually. Then conversations with others create perturbations, or

reveal mistakes or inconsistencies in her construction. She then goes back and

individually revises her construction to accommodate this experience. While

conversation contributes to her understanding, the construction itself happens

individually. In Ernest’s theory, the construction of a concept is a social rather than

individual process. He believes this “because (1) individual thinking of any complexity

originates with and is formed by internalized conversation, (2) all subsequent individual

thinking is structured and constitutively shaped by this origin, and (3) some mental

functioning is collective (e.g., group problem solving).”195 I interpret this to mean that

the human mind is inherently social. Thought itself is based in language, and thus the

very process of thinking and learning is shaped by social experience. Even when we are

alone and thinking about something, we use mental functions that have been socially

formed. Therefore, we learn through socially situated conversation: articulating thoughts

to ourselves, listening and talking to others, and reading and responding to texts. This

interaction is not only the means for constructing subjective knowledge, but it also

ensures that an individual’s knowledge is consistent with that of others and with the body

of objective knowledge.

Subjective mathematical knowledge is gained through “prolonged participation

in many socially situated conversations in different contexts with different persons.”196

Ernest holds that mathematical learning begins very early. Influenced by Vygotsky and

others, he believes children engage in socially situated conversation through play. When

children play, they impose meaning on objects and actions, constructing and interpreting

195 Ernest, Social Constructivism, 212.196 Ibid., 220.

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alternate realities. This helps children learn two concepts that are central to mathematical

thought: the use of symbol or signifier, and the creation of imaginary realities.197 These

concepts prepare children for the later construction of subjective knowledge. In his first

book, Ernest elaborates on the actual mechanisms through which he believes subjective

mathematical knowledge is constructed. There is first a vertical process that generates

subjective knowledge through generalization, abstraction, and reification. New concepts

are created as properties are abstracted from lower level concepts, or through the

reification of concrete experiences.198 For example, one is presented with examples of

various continuous functions and, over time, constructs the abstract concept of continuity.

Second, and more importantly here, there is a horizontal process that refines and

elaborates concepts. This process, according to Ernest, is Lakatos’s logic of

mathematical discovery, as described in Proofs and Refutations.199 Learners push the

boundaries of concepts, and through a cycle of conjectures and refutations learn more

about those concepts and refine their personal constructions to achieve consistency and

compatibility with others. This is a simplified account, describing only how learning

occurs when it is successful. It does not take into account obstacles to learning and how

they might be overcome. For example, it does not explain what happens when a learner

becomes confused and cannot seem to make any progress. It is possible in these cases

that a point has been reached at which the learner requires another level of abstraction

that she has not yet achieved.

Ernest’s theory of subjective knowledge outlines the social nature of the mind and

the central role of conversation in mathematical learning. This will be an important part

197 Ibid., 217-18.198 Ernest, Philosophy of Mathematics Education, 77-78.199 Ibid., 78.

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of his philosophy of mathematics education, providing the basis for his theories of

learning and teaching.

Philosophy of Mathematics Education

Ernest’s philosophy of mathematics education incorporates what he calls the

“ideology of public educators.” He explains that social constructivism by itself does not

constitute an approach to education, but must be combined with a set of values and an

educational ideology.200 Ernest defines an ideology as a larger view that includes general

philosophies and aims, such as an epistemology, a philosophy of mathematics,

psychological and social theories, and general aims for education, as well as more

specific ideas like theories about teaching and learning mathematics and specific aims for

mathematics education.201 Social constructivism holds that mathematics is value-laden,

but it does not specify which values should be promoted. It needs a value system and an

ideological framework to give it a stance on issues of social responsibility. Ernest

identifies five ideologies that have emerged in the British context and adopts the ideology

of public educators, which he describes as “a radical reforming tradition, concerned with

democracy and social equity.”202 This tradition emerged in the late 1800’s in response to

a perceived need to empower the working classes. According to Ernest, it is defined by

its egalitarian values and the desire for social justice. The public educators see social

inequality as a widespread problem, and see “the masses as disempowered, without the

knowledge to assert their rights as citizens in a democratic society.”203 They consider this

to be the defining feature of social injustice, and so they aim to empower the individual 200 Ibid., 264.201 Ibid., 131-33.202 Ibid., 199.203 Ibid., 198.

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through education. The ideology includes a fallibilist epistemology and a social view of

mind, incorporating a social constructivist theory of teaching and learning.

Ernest describes the public educator ideology in mathematics education as the

only one based on the general aims of democracy and social equity, to be achieved

through the development of critical thinking. The goal is not just to create citizens who

can participate in a democratic society, but to empower learners and encourage

autonomy.204 Ernest asserts that the public educator ideology has only been possible in

mathematics education since the emergence of Lakatos’s fallibilist view of mathematics,

because before this time mathematics was not recognized as value-laden and thus was

seen as unrelated to issues of social justice.205 Ernest believes that a social constructivist

philosophy of mathematics is necessary to the public educator ideology, and that it is the

only ideology which incorporates these views.206 However, the public educator ideology

is a collection of theories and values that do not necessarily depend on each other. One

does not need a particular philosophy of mathematics in order to hold a certain learning

theory or set of educational values. I will argue this more fully as I describe Ernest’s

philosophy of mathematics education. As for the claim that only this ideology includes

social constructivist views, this may be true of the five ideologies identified by Ernest.

However, these are drawn only from the British context, and are quite generalized. It is

possible that different ideologies, or more nuanced combinations of those listed, exist

elsewhere and share a social constructivist philosophy.

Ernest adopts the ideology of public educators and builds on it to form his

philosophy of mathematics education, so the ideology will be further analyzed as part of

204 Ibid., 206.205 Ibid., 205.206 Ibid., 210.

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this view. Ernest addresses three main areas of interest: aims and values for mathematics

education, a theory of learning, and a theory of teaching.

Aims and Values

As Ernest places an element of social responsibility on the mathematics

discipline, he also lays an ethical obligation on mathematics education. He explains that

education is an intentional activity, and different social groups have different educational

aims.207 The values behind these aims shape theories of teaching and learning, and the

activities that take place in the classroom. Borrowing the general framework of the

public educator ideology, Ernest believes that the aims for mathematics education should

include the promotion of social justice through the development of democratic citizens

who can think critically in mathematics. He believes that mathematics educators have an

obligation to challenge the perception that mathematical arguments in the media and

other aspects of public life are beyond scrutiny.208 Students should develop the skills they

need to analyze these arguments, and the awareness and confidence to challenge unjust or

unsubstantiated claims.

Ernest feels that mathematics is presented as the product of white western males,

with no acknowledgement of the racism and sexism that he believes is inherent in its

creation.209 He explains that women and minorities are disadvantaged by the

dehumanization of mathematics through its portrayal as neutral, objective, and

disconnected from other disciplines and from humanity. The mathematical community

values reason over feeling, representing the “aggressive masculine half of human

nature.”210 While Ernest may have a point about whose contributions are emphasized in 207 Ibid., 123.208 Ernest, Social Constructivism, 274.209 Ernest, "Social and Political Values," 201.210 Ernest, Philosophy of Mathematics Education, 279.

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the history of mathematics, the portrayal of aspects of knowledge as masculine or

feminine draws on gender stereotypes. He does state, however, that these classifications

are culturally constructed and should be worked against in the classroom. Not only

should the contributions of women and other cultures be recognized, but stereotypes

about the mathematical capabilities of women and minorities should be revealed and

combated.211 To this end, he believes that the mathematics taught in schools should

reflect the social nature of the discipline. Mathematics education should value discovery

and creativity as much as justification, and it should reflect the contributions of non-

western and non-traditional mathematics. To promote social equity, the curriculum must

reflect the “diverse historical, cultural and geographical locations and sources” of

mathematics as well as its embeddedness in our social institution.212

According to Ernest, mathematics educators have the responsibility to work for

social justice by presenting mathematics in a way that provides a “representative

selection from the discipline of mathematics,”213 but also counters socially reproductive

forces in the system. I take this to mean that the mathematical content itself should be

rigorous and complete, not altered in some way to push an agenda. But Ernest also wants

teachers to encourage critical awareness by explicitly discussing both how bias appears in

mathematics and how it can be revealed by mathematical analysis. They should discuss

multicultural contributions to mathematics, misleading uses of statistics, and how

statistics can be used to study social situations.214 Ernest’s hope is that this will empower

students to use mathematics critically and to work toward social change when this

analysis reveals a source of social inequity.211 Ibid., 272, 278.212 Ibid., 210.213 Ibid., 237.214 Ibid., 272-73.

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Ernest wants mathematics education to empower students, but there is no standard

definition for student empowerment in mathematics. For Alan Schoenfeld, mathematical

empowerment involves quantitative literacy. This means being able to think

mathematically in daily life. A quantitatively literate person can interpret and make

informed judgments about quantitative data, and can use mathematics in practical

applications. But such a person can also approach problems from multiple perspectives,

and think analytically about issues and arguments.215 Eric Gutstein offers another

perspective. In addition to fostering the development of mathematical power in students,

he hopes to promote sociopolitical awareness and a motivation to act for social change.216

I think Ernest would find that these accounts are both correct, addressing different aspects

of student empowerment. While Schoenfeld seems to focus on a sense of personal

empowerment, Gutstein is more interested in social empowerment, or empowerment to

act for social change. Ernest refers to empowerment most often in Schoenfeld’s sense, as

mathematical ability that empowers the individual “to take control of their life, and to

participate fully and critically in a democratic society.”217 In an article on the subject,

Ernest explains that this type of empowerment has three dimensions. Empowered

students have mathematical ability, the ability to use mathematics in their lives, and

confidence in their abilities.218 The ability of students to use mathematics in their lives

includes having the skills needed to secure higher education and a good job, as well as

the ability to make critical judgments about mathematics used in the media and politics.219

This sounds much like Schoenfeld’s definition of empowerment, but also includes

215 Schoenfeld, "Learning to Think Mathematically," 335.216 Gutstein, Reading and Writing the World, 24-28.217 Ernest, Philosophy of Mathematics Education, 199.218 Ernest, "Empowerment in Mathematics Education," under "What is Empowerment?"219 Ibid., under "Social Empowerment."

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Gutstein’s sociopolitical awareness. When Ernest talks about critical awareness in

students, he seems to have an end goal of inspiring social action in Gutstein’s sense.

Ernest does not elaborate on this much, but he does refer to “social engagement and

empowerment,” which seems to include both awareness and a desire to act, and he notes

that an aim of the public educators is the “empowerment and liberation of the individual

through education … to initiate and participate in social growth and change.”220 This

indicates that Ernest also intends for empowerment to encourage students to work toward

social justice by participating in movements to change social institutions that contribute

to social inequity.

Ernest’s aims for mathematics education as an agent of social change reflect an

underlying set of democratic values. Together, these aims and values help to shape

Ernest’s theories of learning and teaching.

Theory of Learning

Ernest’s theory of learning begins with his account of subjective knowledge. As

discussed above, subjective mathematical knowledge is formed by the social construction

of meaning through conversation and through Lakatos’s logic of mathematical discovery.

As part of a philosophy of mathematics education, this account must be extended to

include issues involved in formalized schooling. Ernest explains that learning

mathematics in a school setting is different from early childhood learning. In school,

subjective knowledge is acquired through the internalization and reconstruction of the

objective knowledge found in textbooks.221 This is much more artificial than early

development, “because school mathematical knowledge is not something that emerges

220 Ernest, Philosophy of Mathematics Education, 199.221 Ernest, Social Constructivism, 244.

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out of the shared meaning and purpose of a pregiven form of life. Instead it is a set of

artificially contrived symbolic practices, a significant part of the meaning of which is not

already given but deferred until the future.”222 Therefore, school mathematics involves

enculturation into a new form of life and language games, which will form the basis for

the learning that takes place there. This may raise the concern that, like traditional

mathematics education, those who are not already familiar with the academic culture or

form of life will be disadvantaged. For example, some students will come from families

that begin this enculturation in the years before formal schooling and continue to

reinforce it at home, while other students will not. Ernest hopes to address this issue by

making it explicit, discussing power relations and social disadvantages in the

classroom.223 This seems idealistic, however. An awareness of these issues will not

compensate for them. If Ernest is basing his aims on social justice, then he should

consider ways to help students offset or overcome the effects of these inequities.

Ernest’s theory of learning assumes that students need to actively engage with

mathematics in order to learn it. He rejects the behaviorist idea of sequential, or

hierarchical, mathematics learning, citing several studies which have found that

individual learners do not all follow a single hierarchy in their learning.224 He also argues

that mathematical concepts are often composite conceptual structures. One cannot master

a concept before moving on to the next sequential topic, because concepts interconnect

and are gradually understood at a deeper and deeper level in various contexts over many

years.225 Therefore, students need experiences that allow them to construct and refine

mathematical concepts by discovering connections and testing ideas in new contexts. 222 Ibid., 221.223 Ernest, Philosophy of Mathematics Education, 272.224 Ibid., 239-40.225 Ibid., 240.

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Conversation is critical in this process. Language is the vehicle of thought, so dialogue is

necessary to the construction of subjective knowledge. Conversation and interaction also

allow students to compare their ideas and test their validity. Each individual constructs

his or her own understanding of mathematical concepts, and Ernest claims that the

artificial nature of school mathematics makes it more likely that these constructions will

be different for each student. Interaction is necessary to negotiate a fit with the

knowledge of others. Through dialogue, students “generate, test, correct, and validate

mathematical performances” to ensure that their constructions are consistent with those

of others and with the system of objective knowledge.226 Thus students need to be

actively engaged in discussions about mathematics, challenging each others’ ideas and

confronting different perspectives.

Learning is also affected by the classroom environment. Ernest sees the social

context of the classroom as a social form of life with its own language games, and thus it

has an impact on the significance and meaning of the mathematical knowledge items that

are encountered there. Students are enculturated into the classroom form of life, and this

social context shapes how students think about themselves and mathematics. The

classroom context is determined by many components, including the classroom aims and

purposes, the people involved and their relationships, the classroom discourse, and the

available material resources.227 The classroom aims and purposes include the sometimes

competing aims of all influential parties: teachers, parents, school administrators, school

boards, politicians, etc. The teacher’s aims, and the pressure put on her to fulfill the aims

of others, affect how the teacher views her responsibilities, how she designs classroom

226 Ernest, Social Constructivism, 221.227 Ibid., 231-32.

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activities, and other aspects of the social context. The social regulation and norms of the

classroom are influenced by all people involved, through teacher-student relationships,

but also through the school procedures and structures that give the teacher power.228 In

this way, the relationship between the administration and the teacher impacts

relationships in the classroom. Ernest states that there is usually a power differential

between teacher and student, which constructs the mathematics learner in a particular

way.229 He does not think this power differential needs to be erased, but its impact should

be recognized. Another important part of the social context is the classroom discourse,

formed by the personal interactions in the classroom. These are determined by the

teacher’s classroom management style, communication styles of teacher and students, the

mathematical content, and the nature of written assignments.230 For example, an

emphasis on symbolic manipulation in written assignments contributes to the idea that the

symbols have an existence of their own, and thus encourages students to form a Platonist

ontological view.231 Material resources such as textbooks and technological tools also

impact the social context. The incorporation of calculators into the curriculum conveys

the message that mathematics is about concepts and applications that require calculation

as a step, but not as an end goal. In these ways and others, the social context of the

classroom affects learning and influences student perceptions.

Ernest’s theory of learning is accompanied by a theory of teaching that addresses

the social construction of subjective knowledge. It also takes into account the importance

of the social context, using it to further his aims for mathematics education.

228 Ibid., 232.229 Ibid., 233.230 Ibid., 232.231 Ibid., 227.

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Theory of Teaching

Ernest’s theory of teaching reflects both his theory of learning and the aims and

values of the public educator ideology. Thus he believes that the mathematics teacher

should work toward twin goals: to teach mathematics and to promote social justice. She

should design activities that will foster the construction of subjective knowledge through

conversation. But she should also provide as democratic a classroom context as possible,

to encourage critical thinking and social engagement.

One of the main goals of mathematics teaching is to transmit mathematical

knowledge. For Ernest learning entails the social construction of meaning, so students

must engage in genuine discussion about mathematical ideas. The role of the teacher is

to “direct and control mathematics learning conversations both (a) to present

mathematical knowledge to learners directly or indirectly (i.e., teach) and (b) to

participate in the dialectical process of criticism and warranting of others’ mathematical

knowledge claims (i.e., engage in assessment).”232 These conversations foster the

construction and refinement of knowledge, as well as the enculturation of students into

the forms of life of the mathematics classroom. In order to encourage social engagement

and empowerment, Ernest also wants teachers to create a democratic classroom context

in which students learn to think critically. Students should be presented with socially

relevant topics such as race and gender issues in mathematics.233 This will help them to

think critically about the social uses of mathematics and the values implicit in the

discipline. The curriculum should also reflect social diversity, acknowledging the real

contributions of women and non-European cultures.234

232 Ibid., 222.233 Ernest, Philosophy of Mathematics Education, 209.234 Ibid., 265.

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To provide opportunities for the social construction of mathematical concepts

within a democratic context, Ernest proposes an investigatory approach to teaching

mathematics. He explains that investigations are similar to problem solving approaches,

but are more open-ended. Ernest sees problem solving in the usual sense as an activity in

which the teacher poses a problem, and then leaves the students free to find their own

methods for solving it. In an investigatory approach, on the other hand, the teacher

merely provides a beginning situation for students to consider.235 This is only the starting

point for the investigation—if students are to be truly engaged and encouraged to think

critically, then they must be allowed some latitude to explore different aspects of the

topics presented. It is up to them to define a problem on which they would like to focus.

Ernest borrows from Paulo Freire the idea of “problem posing” pedagogy, which

encourages empowerment and social engagement by allowing students to question both

the curriculum and pedagogy in the classroom.236 The students investigate issues and

choose, or pose, the problems they will study. With a problem solving pedagogy,

students are exploring different ways to solve a particular problem. With an investigatory

pedagogy, the students are exploring in all different directions to uncover many different

potential problems. This approach makes inquiry and problem solving central to

mathematics education. They are not just added at the end of a unit—problem solving

and investigation should be used as a pedagogical approach to the whole curriculum,

drawing problems and topics from socially relevant contexts.237

Ernest sees investigations as essentially similar to problem solving, with the

crucial addition of a problem posing step.238 Therefore, he often refers to the 235 Ibid., 286.236 Ibid., 202.237 Ibid., 288.238 Ibid., 286.

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investigatory approach as “problem posing and solving.” Ernest recommends that this

problem posing and solving should be done both individually and in groups.239 Students

need to collaborate in groups to build confidence and engage in conversation with others.

They should also work alone at times, however, to encourage creativity and self-

direction. Ernest cautions that students should not be left completely unguided, though.

While students should be allowed freedom in their mathematical explorations, the teacher

should monitor and guide them when they have misconceptions or if their approach

seems unproductive. This is in contrast to progressive education, which Ernest feels does

not give the teacher a strong enough role.240 Students need someone more experienced to

make sure their ideas fit with the body of mathematical knowledge. This puts pressure on

the teacher to have a deep understanding of the subject in order to guide in-depth

investigations that may go in unexpected directions.

For Ernest, problem posing and solving are at the heart of mathematical learning,

incorporating the aims of conversation, mathematical discovery, and empowerment. The

use of investigations encourages both student-student and student-teacher mathematical

dialogue. Students working in groups can discuss their ideas and negotiate meaning

within the context of the situation they are exploring. They can also engage in

conversation with the teacher, to check their constructions for fit with hers. Ernest sees

this process as driving the cycle of Lakatos’s logic of mathematical discovery. Ideally,

students are investigating a situation, posing problems, making and testing conjectures,

discussing their ideas and those of others, debating and offering counterexamples, and

revising their conjectures. They are the creators of knowledge.241 This furthers

239 Ibid., 208-9.240 Ernest, "One and the Many," 464.241 Ernest, Philosophy of Mathematics Education, 291.

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democratic goals by engaging students in socially relevant discussions and empowering

them to make and defend conjectures. Ernest hopes that these experiences will

enculturate students into mathematical forms of life, while raising social awareness and

encouraging students to challenge social injustice.

Ernest’s philosophy of mathematics education combines his social justice aims

with a social constructivist learning theory and an investigatory pedagogy. He sees a

need for teachers to facilitate the social construction of subjective mathematical

knowledge while fostering critical awareness and a sense of empowerment.

Independence From Philosophy of Mathematics

An important concern to address is whether Ernest’s philosophy of mathematics

education can be considered independently of his philosophy of mathematics. As his

philosophy of mathematics is somewhat controversial, it would make his philosophy of

mathematics education seem plausible to a wider audience if the link between the two

were not necessary. Here I will consider whether Ernest’s philosophy of mathematics is

necessary to the three aspects of his philosophy of mathematics education: aims and

values, theory of learning, and theory of teaching.

Aims and Values

As I described above, Ernest does feel that his fallibilist philosophy of

mathematics has made the aims and values of the public educator ideology relevant to

mathematics education. However, these aims and values do not depend on that

philosophy of mathematics for their basis. One need not believe in the social

construction of mathematics in order to value social justice and mathematical literacy.

And one need not hold that mathematics itself is value-laden to worry that the current

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state of mathematics education may be privileging some and discriminating against

others through a “hidden curriculum.” In an absolutist view, there is still the possibility

of discrimination and bias in the way mathematics is used and in the way mathematics

education is structured. Thus democratic values for mathematics education and aims for

social justice are consistent with non-constructivist philosophies of mathematics.

Theory of Learning

Ernest’s theory of learning is an extension of his theory of subjective knowledge,

applying the social construction of individual knowledge in the classroom context.

Because he also views his theory of subjective knowledge as part of his philosophy of

mathematics, I will analyze whether this theory depends on the more controversial

ontological and epistemological aspects of that philosophy.

The first issue is whether one must agree with Ernest’s ontology in order to grant

the premise of his theory of learning, that subjective knowledge is constructed through

socially situated conversation. The traditional, realist view is that mathematical objects

exist independently of the human mind, and mathematical truth is found in true

statements describing these objects. Knowledge, then, is characterized by justified true

belief of these statements. For Ernest, mathematical objects do not exist independently of

the human mind. They are constructs, created through human discourse. Mathematical

truth is that which has been rigorously proven and accepted, and knowledge is the ability

to provide a justification. But as Volker Gadenne argues, it would not be logically

inconsistent to hold a realist ontology but believe that “cognition is a constructive

process.”242 He concludes that one can believe in an objective reality, but still assert that

242 Gadenne, "Construction of Realism," 154.

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our knowledge about this reality is uncertain. It follows that Ernest’s theory of learning

does not depend on his ontology.

The second issue is one of epistemology. Objective and subjective knowledge are

closely linked in Ernest’s social constructivism. The body of objective mathematical

knowledge is socially constructed through the acceptance of proposals by the

mathematical community. Through conversation, the individual constructs a subjective

version of this mathematical knowledge base. She draws on this subjective knowledge to

develop a new conjecture or claim for submission to the mathematical community as

potential objective knowledge, and she also participates in the evaluation of proposals

submitted by others.243 In this way, the individual contributes to the generation of

objective knowledge, creating a cycle. Given this cycle, does Ernest’s theory of

subjective knowledge, and thus his theory of learning, depend on his account of objective

knowledge? There are actually three cycles involved in Ernest’s theory. First, the

generalized logic of mathematical discovery which describes the formation of objective

mathematical knowledge. Second, Lakatos’s logic of mathematical discovery, which

accounts for the construction of subjective knowledge. Third, there is the cycle that links

the two together, in which objective knowledge is reconstructed as subjective knowledge,

and subjective knowledge generates proposals for consideration as objective knowledge.

The pertinent question here is whether the second cycle, of subjective knowledge, can

stand alone. Are the other two cycles necessary to Ernest’s theory of subjective

knowledge, and in turn to his theory of learning? I gave a brief argument in chapter two

that one’s ontology and philosophy of objective knowledge do not dictate a particular

243 Ernest, Social Constructivism, 148-49.

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theory of learning. Now that we have examined Ernest’s philosophy, we can consider

this question in more depth as it pertains to social constructivism, in particular.

Ernest’s theories of objective and subjective knowledge obviously share common

theoretical influences, as they follow similar models. But while they complement each

other, neither is logically necessary to the other. Once the body of mathematical

knowledge exists, the way it was formed does not have to impact the way it is learned by

individuals. For example, one could agree with Ernest’s social constructivist account of

the genesis and justification of objective knowledge by the mathematical community, but

hold a behaviorist view of learning. Ernest believes that learning mimics the historical

evolution of mathematical knowledge, but one could just as easily take the view that

learning is sequential and requires behavioral reinforcement. More importantly for this

study, one could have a Platonist philosophy of mathematics, but agree with Ernest’s

social constructivist view of subjective knowledge. Cobb suggests that “[i]n practice, the

issue of mathematical foundations is tangential to the processes by which a theory

becomes the way mathematical reality is until further notice.”244 One can hold a Platonist

ontology, but hold also that the process by which mathematicians create and pass on a

body of knowledge is social. But one need not grant even this much. A social

constructivist view of subjective knowledge can be held without Ernest’s account of

objective knowledge. This view would probably look something like this:

“Mathematical objects and the truth status of mathematical propositions exist

independently of the human mind. The body of mathematical knowledge of which

humans are aware has been discovered primarily through deductive proof, using the

objective rules of logic and reason. Truth may be absolute, but human understanding is

244 Cobb, "Experiential, Cognitive, Anthropological Perspectives," 35.

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imperfect. Mathematical learning requires the individual to make sense of this body of

knowledge, so she must construct her own understanding of it. This construction

happens through conversation with others.” Therefore, Ernest’s theory of objective

knowledge gives insight into how he envisions subjective knowledge, but it is not a

necessary premise. This means that his theory of learning does not depend on his

philosophy of mathematics.

Theory of Teaching

Ernest’s theory of teaching is represented by his problem posing and solving

pedagogy. Its main goals are to foster the social construction of subjective knowledge

through mathematical discovery, and to promote social justice and democratic ideals.

Ernest’s theory of teaching is merely an implementation of his theory of subjective

knowledge and his democratic aims for mathematics education. Since these do not

depend on his philosophy of mathematics, neither does his theory of teaching. There may

seem to be a natural association between a social constructivist philosophy of

mathematics and a social constructivist philosophy of mathematics education, but there is

no necessary connection between the two. This means that Ernest’s philosophy of

mathematics education can be considered on its own merits, without worrying about

commitments to a particular ontology or account of objective knowledge. Doubts about

his philosophy of mathematics need not leave his philosophy of mathematics education in

doubt as well.

Evaluation

Ernest’s philosophy of mathematics education fuses a social view of

constructivism with Lakatos’s logic of mathematical discovery. While his pedagogy is

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not necessarily Lakatosian, his underlying philosophy is built on a conversational

interpretation of Lakatos’s ideas.245 As with a Lakatosian philosophy of mathematics

education, the value of Ernest’s philosophy depends on one’s perspective. In this section

I will consider the potential strengths of his philosophy of mathematics education, as well

as possible concerns.

Strengths

One possible strength is that Ernest’s theory of learning mirrors the development

of mathematical knowledge through history. Like Lakatos’s proofs and refutations,

Ernest’s investigatory approach portrays mathematics as an evolving discipline.

However, Ernest adds the dimension of problems as a powerful way to generate

subjective knowledge. As in the mathematical community, discovery happens not only in

the context of attempting to prove, but also in the context of solving a problem. Students

learn to ask relevant questions and to mathematically problematize situations. They

pursue possible solution methods, and then compare approaches with other students.

During this process, new relationships can be discovered and new approaches can be

developed. This is an important part of mathematical practice that appears in Ernest’s

pedagogy.

Another possible strength of Ernest’s philosophy is the inclusion of democratic

values and aims for social justice. By making these explicit, he brings awareness to

issues that may otherwise go unnoticed or unexamined. A teacher might question how

her thoughts or actions perpetuate hidden bias in mathematics education, or she may

become aware that she can take steps in her classroom to socially empower her students.

245 The influence of Lakatos on Ernest’s pedagogy will be examined in more detail in the next chapter.

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However, this could be seen as out of place in a philosophy of education. As referenced

above, Matthews argues that ethics and politics have no place in an epistemological

argument. Does Ernest’s philosophy of mathematics education make a similar mistake,

and conflate ethics with epistemology? Ernest indicates that his set of values is separate

from his epistemology. While he has chosen the democratic values of western liberalism

for his own philosophy of mathematics education, he claims that other values could be

chosen to accompany social constructivism.246 He has merely adopted the values that

seem morally correct to him. This shows that he is trying to keep his ethical claims

separate from his epistemological ones. However, Ernest also claims that a fallibilist

philosophy of mathematics is necessary to his educational ideology, which includes aims

and values. He is saying that while his philosophy of mathematics does not have to lead

to a particular set of values, it is the only one that is consistent with his own ideology and

democratic ideals. While the general goal of student empowerment is not dictated by his

epistemology, he associates the need for student empowerment with the nature of

mathematics as a social institution. Inequities are linked to the social construction of

mathematics as neutral and detached, but because this system is socially constructed it

can be challenged. Ernest makes no arguments for the logical necessity of a connection

between his theory of objective knowledge and his values, but he does feel that a social

constructivist epistemology has made it possible to incorporate these values into

mathematics education.

246 Ernest, Philosophy of Mathematics Education, 265.

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Concerns

Here I will present and analyze some concerns that Ernest’s philosophy of

mathematics education may raise as it is incorporated into the classroom. These points

pertain to his pedagogical theory and its implementation.

Reliance on conversation to produce democracy. While the inclusion of

democratic values may be a strength for Ernest’s philosophy, his dependence on

conversation to foster democracy opens him up to potential criticism. Ernest proposes to

achieve democratic aims through the use of his investigatory pedagogy. Conversation

serves as the basis for these investigations, providing the means for students to carry out

mathematical inquiry, become more confident in their own skills, and uncover social

inequity in situations that often go unquestioned. Therefore, conversation plays an

emancipatory role, fostering democratic citizenship by developing both mathematical and

social empowerment in students. It seems that for Ernest, conversation serves as the

means to democracy. However, conversation can be influenced by the very social norms

that have caused students to overlook social issues in the past. Critical social theorists

such as Jürgen Habermas consider this problem, linking language to issues of social

reproduction.

For Habermas, language is a medium for communication, which provides the

means for the social coordination of activities.247 Members of a social group

communicate to establish common interests and social norms for the actions of

individuals. In his view, democracy requires “that the basic institutions of the society and

the basic political decisions would meet with the unforced agreement of all those

247 Habermas, Theory of Communicative Action, 1:101.

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involved, if they could participate, as free and equal, in discursive will-formation.”248

This unforced agreement assumes that members of society are able to converse in what

Habermas calls an “ideal speech situation.” An ideal speech situation is characterized by

unlimited discussion that is free from distorting influences, such as domination or self-

deception.249 In other words, all actors have an equal chance to make their own

statements and to critique the statements of others, free from the influences of social

power. This is consistent with Ernest’s goals for a democratic mathematics classroom—

he wants students to engage in conversations in which they each have an equal

opportunity to discuss their own mathematical thinking and to critique the ideas of others.

However, conversation in itself does not ensure the absence of domination or power

differentials. Habermas warns, “Language is also a medium of domination and social

power. It serves to legitimate relations of organized force.”250 According to Habermas,

communication occurs within a “lifeworld.” This is a community’s intersubjectively

shared worldview, or body of background knowledge, presuppositions, and convictions,

much of which is passed down from previous generations. These cultural traditions are

often seen as beyond question or criticism, heavily influencing the judgments of

individuals.251 These views can be so influential that even the groups they disadvantage

are willing to accept them. In this way, power relations become part of the lifeworld.

They go unnoticed, but affect human communication. As long as the lifeworld remains

beyond critique, it is not possible to create an ideal speech situation. In fact, as Thomas

248 Habermas, Communication and Evolution, 186.249 McCarthy, Critical Theory of Habermas, 306.250 Habermas, Zur Logik der Sozialwissenschaften, 287, quoted in McCarthy, Critical Theory of

Habermas, 183. Italics in the original.251 Habermas, Theory of Communicative Action, 1:70-71.

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McCarthy points out, real-world communication almost never constitutes an ideal speech

situation.252 Thus conversation alone will not foster democracy.

This is not to say that Ernest’s entire endeavor fails, and that attempts to develop

democratic principles in the mathematics classroom are futile. McCarthy offers that even

in the absence of an ideal situation, the concept of ideal speech remains a useful guide

and a critical standard against which to measure actual communication.253 This seems to

be what Ernest hopes to encourage in mathematics education. It must be granted that he

does address issues, such as institutionalized racism, which correspond to power relations

in the lifeworld. Ernest recommends that teachers make these issues explicit in the

classroom, encouraging conversations about cultural domination and related social

problems.254 However, he does not consider the strong influences that the lifeworld

exerts on these discussions. To defend his view against this criticism, Ernest needs to

rely on more than exposure to social issues. He must offer suggestions as to how

students can be encouraged to reflect critically on their own presuppositions and

assumptions in the course of classroom conversations. In this way, he could encourage

teachers to aspire to the characteristics of ideal conversation, even if the ideal speech

situation is not fully achieved.

Absence of formal proof. In Ernest’s social constructivism, the genesis of

mathematical knowledge is just as important as its justification. This makes it an

especially interesting basis for a philosophy of mathematics education. Mathematicians

are traditionally focused on the finished product, either a proof or a solution, and

downplay the importance of the process used to generate it. Marjorie Siegel and

252 McCarthy, Critical Theory of Habermas, 309.253 Ibid.254 Ernest, Philosophy of Mathematics Education, 272.

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Raffaella Borasi explain that while mathematicians engage in a rhetorical process of

testing conjectures when they create proofs, this creative process is hidden when the

proof is published. School mathematics reflects only this finished proof, hiding the

creation process from students and denying them an important learning tool.255 Ernest’s

goal is to change this by bringing mathematical discovery into the curriculum. However,

it seems that he may have done this at the expense of formal proof. This is interesting,

because even with an emphasis on mathematical discovery, his philosophy of

mathematics does not ignore the importance of justification. Ernest does state that the

emphasis on proof and justification in mathematics has sustained absolutist views,

obscuring the central place of problems and discovery.256 But he only suggests that the

role of discovery be emphasized, not that justification be removed. In fact, he augments

Lakatos’s philosophy to incorporate formal proof into a central role. Ernest’s theory of

teaching, however, does not seem to reflect this. While his problem posing and solving

pedagogy invokes Lakatos’s logic of mathematical discovery, the focus seems to be on

testing conjectures and possible solution methods rather than the construction of proof.

Perhaps he feels that problem posing and solving will achieve many of the same goals as

proof. What, then, are the pedagogical benefits of having students engage in proof? In

what ways might Ernest intend to achieve these benefits without formal proof?

Hersh claims that while proof in research mathematics is designed to convince, in

education its role is to explain. It may be used in a shortened, informal manner to

increase students’ understanding of a concept or application.257 A teacher could include

proof as part of a lecture, but Ernest is more interested in students constructing their own

255 Siegel and Borasi, "Demystifying Mathematics Education," 208-9.256 Ernest, Philosophy of Mathematics Education, 281.257 Hersh, What is Mathematics, Really? 59-60.

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explanations in the course of their investigations. Yackel and Hanna address the

incorporation of student proof-making into the curriculum. They suggest that in terms of

mathematics education, the uses of proof are communication and explanation. They feel

this is in line with its uses for mathematicians which, aside from formal deduction,

include interpretation, understanding, reasoning, and sense-making.258 I think Ernest

would agree with this. As I quoted earlier in this chapter, he states that proof is for

exploring the fruitfulness of new concepts and for testing conjectures and solutions.

Constructing a proof allows one to do these things by helping a person to interpret,

understand, and make sense of a concept or conjecture. Maybe Ernest omits formal proof

because he thinks investigations also achieve these things. Through problem posing and

solving, students can explore concepts and discuss them with others, construct and refine

their understanding, and organize their reasoning much in the way they would if

constructing a proof. In fact, Yackel and Hanna cite a study by Balacheff which found

that students get a better sense of what proof really does when classroom activities focus

on reasoning rather than proof. It helps students to overcome their difficulties with the

formality of proof and to understand it as a tool that establishes and communicates the

validity of a statement.259 Otherwise, students tend to get bogged down in the details of

organizing a proof, and lose sight of its actual purpose. Hanna cautions, though, that

developing reasoning skills can also be difficult and requires guidance:

It is not enough to provide mathematical experiences. It is the reflection on one’s experiences which leads to growth. As long as students see mathematics as a black box for the instantaneous production of “answers,” they will not develop the patience necessary to cope with the many and erratic paths their minds will take in trying to grasp what mathematics is about. One goal of pedagogy should be to

258 Yackel and Hanna, "Reasoning and Proof," 228.259 Ibid., 231.

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help pupils maintain the level of concentration required to negotiate a line of reasoning.260

Problem posing and solving gives students just this kind of experience. Open-ended

investigations allow students to practice narrowing their focus in the face of numerous

possible paths, to concentrate on one issue in order to develop a line of reasoning, and to

reflect on the effectiveness of their approach. Ernest may be counting on this type of

activity to develop skills that capture the essence of proof making, which seems to be a

reasonable expectation. One concern he does not address, however, is that at some point

students will need to graduate from informal argument and explanation to a more formal

form of written proof. Yackel and Hanna indicate that at some level, mathematics

education should include a transition from reasoning and problem solving to deductive

proof.261 A Lakatosian approach at least requires students to outline an initial proof to be

analyzed for counterexamples, but problem posing and solving only includes informal

explanation. Ernest proposes an investigatory approach for the entire curriculum, without

ever directly addressing the issue of proof. This could be a weakness, particularly in the

education of future mathematicians. Ernest’s views on this will be considered in the next

chapter.

Lack of relevance for non-mathematicians. One might raise the concern that

Ernest’s investigatory pedagogy emphasizes aspects of mathematical practice that are

irrelevant for most students. Those who are not pursuing careers in mathematics will

likely never need to conduct self-directed mathematical investigations, and may benefit

more from instruction in skills for work and life. Conversely, others may respond that all

students benefit from mathematical inquiry and reasoning, and they should have the

260 Hanna, "More than Formal Proof," 23.261 Yackel and Hanna, "Reasoning and Proof," 234.

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opportunity to learn to think mathematically. Ernest’s problem posing and solving, like

Lakatos’s proofs and refutations, will be evaluated differently against different sets of

goals. As in the last chapter, NCTM’s Principles and Standards document provides a

useful framework for highlighting some of these issues because it outlines a rather

comprehensive vision for the mathematics education of all students. Without endorsing

NCTM’s views, I will use its Process Standards to analyze the issues that may arise if

Ernest’s pedagogy is applied in the education of non-mathematicians. These standards

give guidelines in the areas of Problem Solving, Reasoning and Proof, Communication,

Connections, and Representation.

NCTM encourages several aspects of Problem Solving, all of which are

incorporated into Ernest’s pedagogy: building mathematical knowledge through problem

solving, solving problems in context, and developing and adjusting appropriate

strategies.262 Ernest has students consider a situation, pose a problem embedded in that

context, and choose their own strategies to solve the problem. These situations could

address abstract mathematical concepts, or be more applied. The problem solving aspect

of Ernest’s approach would appeal to groups who believe all students should be familiar

with mathematical ways of thinking, or who value the teaching of mathematical

applications in context. Some of those who wish to emphasize applications as

preparation for the workplace, however, may look for more direction from the teacher

than Ernest envisions, to ensure that particular topics are covered.

In terms of the Reasoning and Proof standard, Ernest’s approach does well in

some areas but is weaker in others. Investigations give students the opportunity to make

conjectures and develop mathematical arguments. Most evaluators would be pleased

262 National Council of Teachers of Mathematics, Principles and Standards, 52.

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with the emphasis on mathematical reasoning and thinking. However, NCTM wants

students to shift gradually into more formal types of proof: “High school students should

be able to present mathematical arguments in written forms that would be acceptable to

professional mathematicians.”263 As mentioned before, problem posing and solving does

not provide an opportunity for this transition. While it does enculturate students into

certain aspects of mathematical practice, the absence of an emphasis on proof would

probably be a concern for those who believe all students should experience mathematics

as a formal discipline.

NCTM sees Communication as explaining one’s thinking clearly to others, and

analyzing and evaluating their explanations in return.264 In Ernest’s approach, students

engage in group work which allows them to practice making a clear and convincing

argument. To make their ideas clear, students will have to practice defining concepts and

using precise terminology. While I do not think most people would see harm in these

activities, some interested groups in education put a higher priority on communication

than others. Some will see this as an opportunity for students to elaborate and refine their

ideas, while others would prefer to have students working independently to ensure they

can perform mathematical tasks on their own.

The Connections standard states that students should recognize and use

connections between mathematical ideas, and between mathematics and other contexts.265

In an investigatory approach, the teacher can choose a starting situation that allows

students to explore mathematical connections between concepts, or one that places

mathematics in the context of another discipline, such as sociology, science, or

263 Ibid., 58.264 Ibid., 60.265 Ibid., 64.

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economics. This is an important issue for Ernest, as he stresses the need to connect

mathematics to humanity and to students’ lives. Not everyone will be comfortable with

problem posing and solving as the means to making these connections, however.

Investigations could be seen as a somewhat haphazard approach to the curriculum.

Educators may worry that some students will see important connections and others will

not because their investigations may take them in different directions. Instead, some

educators may prefer that particular connections or applications be presented explicitly

by the teacher.

In the area of Representation, NCTM wants students to use mathematical

representations that make sense to them. Students in an investigatory classroom would

do this in the course of problem solving and explaining their ideas to others. However,

NCTM also stresses that students need to learn conventional representations.266 There

may be a concern that this is not central to problem posing and solving. While Ernest’s

pedagogical recommendations do not address this, his theoretical background does. He

borrows from Vygotsky the view that mathematical symbol systems are part of the

language of mathematics, and their use must be socially acquired. This part of

mathematics is an important element, because mathematical knowledge is so often shared

in written form.267 The implication is that students need to learn standard mathematical

representations in order to be able to understand the written mathematics they encounter,

and to communicate their own ideas with others. This is easily incorporated into problem

posing and solving. As students encounter new concepts and the need to communicate

about them, the teacher can introduce standard notation. Students may use their own

266 Ibid., 67.267 Ernest, Social Constructivism, 223.

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representations at first, but gradually will become more used to the standard notation as

they communicate with the teacher and with each other. Many interest groups are

skeptical about this, however. They prefer that students be taught standard

representations from the beginning, and that they be required to use them in all of their

work.

While not all stakeholders share the same goals for students who are not pursuing

advanced mathematics, this analysis reveals the strengths and weaknesses of Ernest’s

pedagogy from different perspectives. His approach is valuable in terms of reasoning,

problem solving, communicating ideas, and making mathematical connections.

However, it is weak in the areas of formal proof, standard notation, and drills of standard

algorithms. Teachers also lack direct control over the direction of their students’

activities when they use the investigatory approach. Depending on one’s priorities,

Ernest’s pedagogy may seem like an incredibly valuable learning tool, or a method

dangerously lacking in structure. This method seems very useful from a perspective

which holds that all students should learn to think mathematically, and that doing

mathematics is about exploring and discussing problems. However, it will appear

deficient to those who value the more formal aspects of mathematics, or those who prefer

a basic skills-oriented approach. In the next chapter, I will look at whether Ernest’s

pedagogical and curricular recommendations reflect his own aims and values for these

students.

Difficulty of implementation. Another problem is that those who do see potential

in this approach may find it hard to implement. Implementation of Ernest’s problem

posing and solving pedagogy requires a lot of background and planning on the teacher’s

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part. She must choose an appropriate starting situation that will prompt productive

discussion. She must also have a strong enough mathematics background to recognize

mathematically interesting questions and to evaluate multiple approaches to solving the

problems that students pose. However, teachers are usually under pressure to cover

particular areas of mathematical content. A major concern with this pedagogy is its

potential lack of curricular direction. Because students are open to pose their own

problems and explore different solution methods, it is difficult for the teacher to focus the

class on a particular procedure or piece of content. It will be a challenge for the teacher

to give students enough latitude for creativity, but enough guidance to ensure they meet

mathematical objectives.

In addition, the teacher must create an appropriate classroom context. As with a

Lakatosian approach, students may at first be resistant to an investigatory classroom. The

teacher will need to create an environment in which students are expected to fully

participate and to share ideas. When students disagree, the teacher needs to help them

critique each others’ ideas in a calm, constructive way. Classroom norms must be

established in which students are comfortable acknowledging flaws in their solutions and

accepting the suggestions of other students. Ernest’s classroom approach does make this

somewhat easier than it would be with proofs and refutations. There is more cooperation

and less conflict with investigations. Rather than engaging in an argument with “every

man for himself,” students can work independently or in small groups to collect their

thoughts before presenting them to the class. There will still be some conflict when

students disagree, but it will be less aggressive. Whereas proofs and refutations has an

inherent discourse of attack and respond, investigations are set up more as a joint effort.

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Still, to implement this approach well, teachers will need to be well-trained both in the

mathematics discipline and in classroom management.

Ernest’s philosophy of mathematics education has several strengths. It is modeled

on mathematical practice, connects mathematics to applications in other disciplines, and

promotes social justice. On the other hand, it lacks an emphasis on formal proof and

includes a pedagogy that is difficult to implement. In the next chapter, many of these

issues will be reconsidered in the context of Ernest’s specific aims for different sets of

students.

Summary and Conclusion

Ernest’s philosophy of mathematics is based on the fallibilist assumption that

there is no independent, objective mathematical truth. Instead, he takes the view from

Wittgenstein that mathematics is a set of language games based in shared forms of life.

He generalizes Lakatos’s logic of mathematical discovery to describe the social

construction of the body of mathematical knowledge, with proof playing a central role in

this process. Subjective knowledge is the individual reconstruction of this body of

objective knowledge through socially situated conversation.

To form his philosophy of mathematics education, Ernest applies his theory of

subjective knowledge in the classroom while incorporating a set of democratic ideals.

Ernest’s account of subjective knowledge puts conversation at the center of his theory of

learning. He brings this and his democratic values together to develop a problem posing

and solving pedagogy. This approach is designed to enculturate students into

mathematical forms of life while promoting critical social awareness. Ernest’s

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philosophy of mathematics education is in many ways influenced by his philosophy of

mathematics, but neither logically necessitates the other.

The influence of Lakatos is less obvious in Ernest’s philosophy of mathematics

education than it is in his philosophy of mathematics. However, he sees the logic of

mathematical discovery as the mechanism for the refinement of subjective knowledge.

As discussed in the previous chapter, a pedagogy modeled on proofs and refutations

seems most suitable for training students to think like mathematicians. If one wants

students to develop mathematical abilities that can be applied in their lives and for social

action, this method may not be the most appropriate. Ernest incorporates the logic of

mathematical discovery into his educational recommendations both for future

mathematicians and for others, but in different forms. In the next chapter, I will explore

the differences in Ernest’s curricular recommendations for these two groups, and the role

of the logic of mathematical discovery in each. Then I will analyze the implications of

this distinction and its consistency with Ernest’s philosophy of mathematics education.

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CHAPTER 5

ERNEST’S DIFFERENTIATED CURRICULUM

In addition to his philosophy of mathematics education, Ernest has made practical

recommendations about the mathematics curriculum. Ernest questions the practice of

sending all students through the traditional mathematics sequence. He asserts that the

need to prepare future mathematicians should not determine the curriculum for all

students, because very few will pursue mathematical research. The needs of the majority

of students are very different from the needs of future mathematicians, and therefore

“what is needed is differentiated mathematics curricula to accommodate different

aptitudes, attainments, interests, and ambitions. Such differentiation must depend on

balanced educational and social judgements rather than exclusively on mathematicians’

views of what mathematics should be included in the school curriculum.”268 Students

should all share the same curriculum in elementary school, but once they have “acquired

basic mathematical competency,” they should be allowed to choose courses based on

their interests and future plans, or stop taking mathematics all together.269 Tracking is a

controversial topic in the mathematics education literature, so this recommendation will

likely be cause for concern for many mathematics educators.

Ernest rejects the common utilitarian justification for teaching mathematics to

students who are not training to be mathematicians. Common rhetoric is that students

need mathematical skills for future employment and for life. But Ernest argues that this

does not justify requiring them to take several upper-level, abstract mathematics courses.

Most people do not use algebra or calculus in their daily lives. Thus Ernest sees little of

268 Ibid., 273.269 Ernest, "Why Teach Mathematics?" under "Conclusion."

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the traditional curriculum as useful in daily life or in most careers.270 He claims that even

the more technical fields such as information technology, accountancy, actuarial studies,

and economics do not make use of the traditional mathematics most students have

learned in school. The necessary skills are instead taught in professional institutions or in

practice.271 Therefore, utilitarian arguments do not justify compulsory mathematics nor

should they drive the general mathematics curriculum. Instead, Ernest believes

mathematics is valuable as a tool to think about and make sense of the world around us.

Because of this, he feels that the general curriculum should provide the mathematical

knowledge necessary to appreciate the role of mathematics in human culture and to

critically analyze its uses and abuses in society.

Ernest has written explicitly about his recommendations for the mathematics

education of most students, which I will call general mathematics education. He is less

clear about the curriculum for future mathematicians, but he provides some clues in his

discussion of subjective mathematical knowledge. In this chapter, I will outline the

general curriculum Ernest envisions for most students and discuss what he intends for the

education of future mathematicians. Then I will analyze the underlying assumptions,

values, and conceptions of mathematical knowledge that are implicit in each curriculum.

Based on this analysis, I will consider whether Ernest’s recommendations are consistent

with each other and with his philosophy of mathematics education. I will also explore

other issues that may be raised by this sort of tracking.

270 Ibid., under "The Utility of Academic Mathematics is Overestimated."271 Ibid.

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General Mathematics Education

Much of what Ernest has written about the general mathematics curriculum,

including discussions in The Philosophy of Mathematics Education, is in direct response

to the British National Curriculum that was legislated in 1988 and put into practice in

1992. As part of the National Curriculum, mathematics is compulsory in British state

schools through Year 11 of formal schooling, or age 16.272 After that, students may

remain in secondary school to complete a university preparatory curriculum if they

choose. Ernest is working in this context, making recommendations for general

mathematics education during the eleven years of compulsory British schooling. The

curriculum he envisions is not specific to the British context, however. He describes how

he believes things should be rather than how they are, so his aims are general enough to

apply in any developed country.

Ernest suggests a differentiated curriculum in secondary school, which in Britain

begins with Year 7, or age 11. After this point, he recommends that the mathematics

education of students who are not training to be mathematicians should take into

consideration their particular needs and goals, which could require several different

mathematics tracks. There is also a core curriculum that he feels all of these students

should encounter, which he refers to as “mathematics appreciation” in some articles, and

“critical mathematics education” in others.273 Both names are misleading, because in

actuality his curriculum includes both components. Ernest’s vision for the mathematics

education of all students includes fostering a sense of mathematics appreciation as well as

critical mathematical awareness through the problem posing and solving pedagogy that

272 Qualifications and Curriculum Authority, "Key Stage 4 Curriculum."273 Ernest, "Empowerment in Mathematics Education," under "Social Empowerment"; Ernest,

"Why Teach Mathematics?" under "Capability versus Appreciation."

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was described in the last chapter. I will summarize Ernest’s general mathematics

curriculum, then analyze its different aspects for consistency with his philosophy of

mathematics education.

Curriculum

Ernest recommends that the general mathematics curriculum include specialized

tracks for different interests and career goals, with a common core centered around the

theme of mathematics appreciation. In the absence of solid utilitarian justifications for

learning mathematics, he suggests a focus on the discipline as an intrinsically valuable

and interesting part of human culture. The curriculum should address “big ideas” instead

of detailed procedures. Mathematics education should also empower students with the

ability to use mathematics and evaluate its use by others.

Mathematics appreciation requires that students be able to think mathematically.

This would require an understanding of the main branches of mathematics, including “big

ideas,” or interesting concepts in mathematics that affect our universe such as infinity,

symmetry, chaos, and randomness.274 Ernest feels that topics such as these can be

explored without advanced mathematical instruction. For example, a person can

comprehend the idea of infinity without knowing calculus, or randomness without being

able to calculate probabilities. He also thinks that students should understand the overall

unity of mathematics and the connections between its different branches.275

Another important part of mathematics appreciation, in Ernest’s opinion, is an

awareness of the historical development of mathematics and the different philosophical

views about its nature.276 Students should learn about the social contexts in which

274 Ernest, "Why Teach Mathematics?" under "Capability versus Appreciation."275 Ibid.276 Ibid.

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mathematics has developed and the debates over its foundations. In addition,

mathematics should be presented as an integral part of human culture “with its own

aesthetics and beauty.”277 Ernest hopes that students will appreciate its role in

philosophy, art, science, technology, and other areas. This will serve to humanize

mathematics, connecting it to society and social issues.

The final aspect of Ernest’s mathematics appreciation is an awareness of

mathematical thinking in everyday work and life, as well as the mathematics that

permeates social and political systems. Ernest wants students to recognize the part

mathematics plays in their lives, and the mathematical thinking that is implicitly required

of them in the course of their daily activities.278 In other words, students should

understand that mathematics is all around them. This part of mathematics appreciation

also includes education for social justice, as outlined in the last chapter. Students should

be able “to identify, interpret, evaluate and critique the mathematics embedded in social

and political systems and claims, from advertisements to government and interest-group

pronouncements.”279 Possible topics could include the mathematics behind voting

systems or judging the validity of statistical interpretations. This part of the curriculum is

meant to involve both critical awareness and social empowerment.

Ernest acknowledges that students will need some level of mathematical

competency to work with the concepts he proposes for the general curriculum, but he

does not think that this competency will need to be very advanced. He explains that part

of empowerment through mathematics involves “an appropriate range of mathematical

capabilities such as performing algorithms and procedures, computing solutions to

277 Ibid., under "Conclusion."278 Ibid., under "Capability versus Appreciation."279 Ibid.

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exercises, … applying general and topic specific mathematical strategies, and carrying

out plans and approaches in solving mathematical problems.”280 Thus these items should

be presented to the extent necessary for understanding broad mathematical concepts and

critiquing mathematical arguments. Ernest also states that additional topics may be

added, because teachers must address students’ needs for certain skills due to their life

goals or external assessments.281

In keeping with the philosophy of mathematics education discussed in the last

chapter, Ernest recommends the use of an investigatory pedagogy for all students. He

feels that problem posing and solving “is an activity which is accessible to all” and

should be the focus of school mathematics.282 Therefore, the different aspects of

mathematics appreciation are intended to be taught using this investigatory approach.

Students are presented with interesting situations that generate investigations in the

different branches of mathematics, including its history and philosophy. They are given

freedom to form questions and to try different approaches to answering them, working

both individually and collaboratively. Students should be encouraged to pose problems

about mathematical concepts as well as mathematical contexts in society. This will help

them to learn both the mathematical content and the critical awareness that Ernest values.

Ernest believes that these curriculum recommendations have been guided by his

philosophy of mathematics education, including his theory of mathematical knowledge

and the democratic values that have been incorporated into it. A closer look reveals that

while this is true in general, there are some important inconsistencies. First I will

examine the assumptions and values that are implicit in Ernest’s vision of the general

280 Ernest, "Empowerment in Mathematics Education," under "Mathematical Empowerment."281 Ibid., under "Social Empowerment"; Ernest, Philosophy of Mathematics Education, 209.282 Ernest, Philosophy of Mathematics Education, 283.

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mathematics curriculum, and then I will analyze the conceptions of mathematical

knowledge and the democratic ideal that underpin this vision. This information will be

used to consider whether Ernest’s curriculum is consistent with his philosophy.

Implicit Assumptions and Values

Ernest’s curriculum for non-mathematicians is based on several unstated

assumptions about mathematics and the learning of mathematics. The big ideas in his

general mathematics curriculum, drawn from pure mathematics, show that Ernest

assumes mathematics has intrinsic conceptual value for all students, not just future

mathematicians. He values education that produces academically well-rounded people,

and believes that mathematics contributes to this kind of intellectual growth. For Ernest,

a well-rounded individual should understand the unity of mathematics, the importance of

its big ideas, and its place in human culture. This will equip her to think critically about

mathematical situations and contexts. Ernest assumes that all students can grasp these

concepts without having a knowledge of advanced mathematical procedures. In terms of

his social justice aims, Ernest seems to take for granted that his democratic values are

widely accepted in the western world. He uses them to justify his ethical views, but

provides little explanation or justification as to why he has chosen them. Further analysis

of these aspects of Ernest’s curriculum will reveal his underlying conceptions of

mathematical knowledge and the democratic ideal.

Conception of Mathematical Knowledge

Ernest’s description of the general mathematics curriculum reveals an implicit

conception of mathematical knowledge in this context. For Ernest, mathematical content

knowledge is interdisciplinary, consisting of general concepts rather than detailed

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understanding of the mathematics involved. It is an understanding of the unity of

mathematics, the debates over its nature and foundations, and its presence in art, science,

and other areas of human culture. These larger ideas are much more important to

mathematical knowledge than procedural details. Ernest only sees “minimal

mathematical capability” as a necessary preparation for learning about these topics.283

For example, he believes that students should understand major concepts like infinity, but

thinks that this can be accomplished without a detailed mastery of calculus. His focus on

the big ideas in mathematics fits with his belief that the mathematics curriculum should

be a representative selection from the discipline itself. However, Ernest’s lack of

emphasis on mathematical procedures conflicts with his definition of mathematical

knowledge, which includes know-how of mathematical procedures and operations.284

Minimal mathematical capability seen only as a prerequisite to larger ideas can hardly be

viewed as a representative selection of the procedural part of mathematics. Ernest may

intend for students to gain a general sense of the underlying procedures as they learn

about big mathematical concepts, to understand how these concepts operate in the world.

Without more detail in his curriculum, it is unclear to what extent he sees this happening.

One of Ernest’s stated intentions for the general curriculum is to show students

how mathematics is intrinsically interesting and permeates the world around us, but this

uncovers a rather traditional aspect in his view of mathematical knowledge. In his

philosophy of mathematics education, he criticizes what he calls the “old humanist

ideology,” which he describes as a purist ideology that values pure mathematics and does

not see applications as the concern of mathematicians.285 He accuses its curriculum of

283 Ernest, "Why Teach Mathematics?" under "Capability versus Appreciation."284 Ernest, Social Constructivism, 136.285 Ernest, Philosophy of Mathematics Education, 174.

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elevating pure knowledge above applied knowledge and practical skills, and preparing “a

tiny minority of students to be mathematicians whilst teaching the rest to stand in awe of

the subject.”286 But Ernest’s description of the general curriculum seems to share many

aspects with the purist one. While he wants students to choose their courses based on

interests and career goals, he believes the general curriculum should include topics from

pure mathematics. The mathematical capability necessary for research mathematics may

be de-emphasized, but he presents mathematics as the overarching force that controls

every aspect of the world and our lives. Thus Ernest’s conception of mathematical

knowledge holds on to the purist view of mathematics as a privileged discipline, and it

conveys the impression that mathematics is the sole driving force in the universe. This

seems an equally effective way to teach students to “stand in awe of the subject” and to

view mathematics as imposing and inaccessible.

Ernest does want students to recognize mathematics as part of culture, to connect

to mathematics in a meaningful way and to develop an awareness of its role in our daily

lives. He believes that students should have a sense of how mathematics “permeates and

underpins” human culture, be able to identify the mathematics “embedded in social and

political systems and claims,”287 and even develop general problem solving abilities. But

as with the old humanist ideology, practical applications that might be useful in life or

work do not appear to count as knowledge. Procedures such as balancing a checkbook or

designing a stable bridge are peripheral to a theoretical understanding of the

mathematical principles involved. The emphasis is on recognizing the existence of

applications and interpreting their results, but not on actually performing them. Practical

286 Ibid., 180.287 Ernest, "Why Teach Mathematics?" under " Capability versus Appreciation."

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applications are seen as prerequisite capabilities or uses of mathematics, rather than

mathematical knowledge in their own right.

For Ernest, mathematical knowledge is a commodity that empowers the one who

possesses it. This happens on two levels. On a personal level, mathematical knowledge

gives students a sense of accomplishment, and provides access to higher education and

better employment opportunities. On a social level, mathematical knowledge provides a

conceptual background that allows students to analyze and critique the social uses and

implications of mathematics. It enables them to recognize when mathematics is being

used, either intentionally or not, to discriminate or to manipulate the public. This

recognition could be an alertness to the social implications of the mathematics embedded

in a voting system, or an awareness of statistics being purposefully manipulated in the

media to discriminate or to make a political point. Ernest envisions mathematical

knowledge as a force that empowers students to call attention to such issues and to fight

for social justice.

Ernest also reveals an underlying conception of knowledge through his

investigatory pedagogy. In the mathematical community, Ernest associates problem

posing and solving with the creation of mathematical knowledge. Recall that the

generalized logic of mathematical discovery is Ernest’s model for the creation and

justification of objective mathematical knowledge, based on a Lakatosian philosophy.

This is a process in which one investigates a question, proposes a potential solution, tests

and revises it in response to counterexamples, publishes a formal exposition of the final

approach, and uses public feedback to begin the process again. Problem posing and

solving is the mathematical inquiry leading up to formal proof and publication, providing

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the creative mechanism in this cycle. This connection between problems and

mathematical creation gives meaning to Ernest’s investigatory pedagogy in two ways.

First, the fact that he believes problems have generated the body of mathematical

knowledge reveals that he defines problems more broadly than just applications in

context. Mathematical problems can also be abstract questions or conjectures, potential

items of mathematical knowledge that need to be verified and proved. Therefore, while

some problems require solutions, other problems require conjectures and justifications.

In fact, Ernest’s de-emphasis on mathematical capability suggests that he favors the latter

type of problem, focusing on a general understanding of big ideas rather than procedural

solutions. This differs from the types of problems included in many curricula that claim

to incorporate problem solving.

Second, this connection shows that for students in the general curriculum, Ernest

conceives of mathematical knowledge as a familiarity with the inquiry that occurs in the

mathematical community. Ernest states that “[t]he mathematical activity of all learners

of mathematics, provided it is productive, involving problem posing and solving, is

qualitatively no different from the activity of professional mathematicians.”288 Students

who engage in problem posing and solving are participating in mathematical inquiry. For

Ernest, the only difference between this and professional mathematical practice is that

instead of formal publication, there is informal explanation in class. Mathematical

knowledge is not about applying routine procedures to solve problems, but is about

asking open questions and figuring out how to find the answers. It is a knowledge of how

to approach unfamiliar problems that one does not already know how to solve. In this

way, Ernest’s investigatory pedagogy is built on a conception of mathematical knowledge

288 Ernest, Philosophy of Mathematics Education, 283.

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that includes the creative part of mathematical practice. This is consistent with the active

engagement and inquiry called for in Ernest’s theory of learning.

In summary, the conception of mathematical knowledge revealed in Ernest’s

general mathematics curriculum is one of big mathematical concepts and mathematical

inquiry. It includes a mathematical background that allows one to see the importance of

mathematics in human culture, and to engage in critical thinking and analysis of

mathematical situations. Overall this is fairly consistent with Ernest’s philosophy, except

that it largely ignores the procedural aspect of his theory of mathematical knowledge and

the more formal aspects of mathematical practice. These omissions reveal that he does

not see these students as potential members of the mathematical community. He does not

believe they need to learn advanced mathematical procedures, which his theory of

mathematical knowledge holds as necessary to “make or justify mathematics.”289 In

addition, he employs an investigatory pedagogy that enculturates students into the

creative aspects of mathematical practice but not formal justification. This indicates that

he does not view these students as future producers of new mathematical knowledge. His

goal is to create informed consumers of mathematics who have some sense how the

mathematical community operates, but are kept distanced from its activities.

Mathematics is not an end in itself for these students, but rather a means to social and

intellectual empowerment.

Conception of the Democratic Ideal

Ernest also bases his curriculum on democratic values. He outlines his general

democratic principles in his philosophy of education: equity, diversity, social

empowerment, and social justice. However, some of the nuances of these principles only

289 Ernest, Social Constructivism, 136.

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emerge as he describes his curricular goals. In this context, Ernest’s conception of the

democratic ideal centers on entitlement and empowerment as markers of equity and

social justice.

Ernest believes that in a democratic society, children as well as adults are entitled

to the right of choice. He claims that children as young as eleven years old should be

given a choice about whether to continue their mathematics education, and in which

direction. “If education is to contribute to the development of autonomous and mature

citizens, able to fully participate in modern society, then it should allow elements of

choice and self-determination.”290 He also asserts that students are entitled to a

mathematics education that provides them with the material they need, and does not force

them to learn topics that are only beneficial to students with goals different from their

own. Students have the right to a mathematics education that is relevant to their lives and

goals, and will prepare them for external assessments.

Ernest also feels that students who choose to continue their mathematics

education are entitled to a curriculum that presents them with important mathematical

ideas and will help them become empowered and critical democratic citizens, which is

consistent with the aims of his philosophy of mathematics education. He sees benefit for

all students in a general understanding of important mathematical concepts and the

mathematical basis of social and political issues. Gutstein differentiates between two

kinds of mathematical literacy, functional and critical. Functional literacy prepares

people to live and work in modern society, but perpetuates social reproduction. Critical

literacy includes mathematical content, but also prepares citizens to examine

290 Ernest, "Why Teach Mathematics?" under "Conclusion."

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sociopolitical contexts and to recognize “oppressive aspects of society.”291 This helps to

describe Ernest’s conception of democratic citizenship. It does not refer to a utilitarian

view of economically productive citizens who function efficiently within the parameters

of their social status. Instead, the image of a democratic citizen that underlies Ernest’s

general curriculum is someone who understands the importance of mathematics in many

areas of life and society, and is empowered by this knowledge. A critical citizen is able

to interpret and critique the mathematics inherent in political and social systems and used

in claims in the media. When necessary, she will act toward social change. This

conception of the democratic ideal is consistent with the democratic aims in Ernest’s

philosophy of mathematics education, which include empowering students to use

mathematics to think critically about the world around them.

Ernest’s general mathematics curriculum is designed to be flexible to students’

career interests while providing a well-rounded knowledge of mathematics as a means to

social empowerment. Next I will turn to the education for future mathematicians to

determine if its underlying principles are consistent with those in the general curriculum.

Education for Future Mathematicians

Ernest’s only explicit discussion about the education of future mathematicians is

found in Social Constructivism as a Philosophy of Mathematics, where he addresses the

enculturation of new mathematicians into the discipline. In Ernest’s view, the education

of future mathematicians begins with primary and secondary schooling, which

enculturate students into the various contexts of school mathematics in order to develop

291 Gutstein, Reading and Writing the World, 5-6.

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their “mathematical powers and dispositions.”292 At the elementary school level Ernest

expects all students to share the same curriculum, which develops basic numeracy skills.

In secondary school and then at the university level, future mathematicians will continue

their socialization into mathematics and will learn more of the accepted knowledge base.

During this time, they gain “personal knowledge of mathematics, the norms of

mathematical content and rhetoric, and the social institution of mathematics.”293 At this

point, students have some mathematical knowledge and are comfortable operating within

the rules of mathematical language games. They are enculturated into mathematical

forms of life, but are not yet members of the mathematical community because they are

learners, but not yet producers, of mathematical knowledge. This is followed by an

apprenticeship in graduate school, in which students begin to participate in and be

enculturated into the mathematical community.294 They begin to undertake research and

generate new knowledge under guidance. After this stage, they are ready to begin their

careers as mathematicians.

Ernest outlines the educational path that mathematicians will take, but says little

about curricular content or pedagogy for this group after their paths diverge from other

students in secondary school. However, in his philosophy of mathematics he describes

the education of future mathematicians as the mechanism for “the onward transmission of

mathematical knowledge” and the development of “the creativity of individual

mathematicians.”295 Therefore, I can use Ernest’s philosophy of mathematics and

mathematical knowledge to construct an “Ernestian” vision of education for future

mathematicians. 292 Ernest, Social Constructivism, 239.293 Ibid., 191.294 Ibid., 190.295 Ibid., 57.

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Curriculum

Based on Ernest’s description of the preparation required to become a

mathematician, we can infer what the goals for this group of students would be. In

secondary school, mathematics students should, for the first time, study a curriculum

designed specifically for future mathematicians. This curriculum needs to provide them

with the mathematical knowledge necessary to pursue more advanced study at the college

level and for the transition to the mathematical community in graduate school. To

determine what this mathematical knowledge would be, we can look at Ernest’s model of

mathematical knowledge for the mathematical community. His model is adapted from

one outlined by Philip Kitcher, forming a view of mathematical knowledge that includes

mathematical content and forms of life.296 Ernest also sees the role of secondary school

as socializing future mathematicians into the discipline. Part of this is enculturation into

forms of life, but it also refers to a familiarity with the mathematical practice of

knowledge creation. Thus the three main areas of preparation for future mathematicians

would be mathematical content knowledge, mathematical forms of life, and mathematical

practice.

Mathematical Content

Ernest does not provide specific examples of mathematical content that should be

part of the mathematics curriculum. In secondary school, he wants future mathematicians

to take classes apart from everyone else, so they can focus on topics not needed by the

rest of the students. Because these students are being enculturated into the norms of the

discipline and are being prepared for advanced mathematical study, the content of the

curriculum would need to align with mathematical content in the larger sense. In

296 Ibid., 138-44.

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Ernest’s model of mathematical knowledge, mathematical content includes propositions

and statements, proofs, and methods and procedures.297 Propositions and statements are

the conjectures and theorems that have been accepted by the mathematical community.

These are mathematical statements that are believed to be true. Accepted proofs are also

part of content knowledge, as they explain why theorems are true and demonstrate

mathematical connections.298 The last area of content knowledge includes the standard

methods and procedures used to solve problems and perform calculations. Therefore, the

curriculum for future mathematicians would need to include representative content such

as statements, proofs, and procedures from various areas of mathematics like algebra,

geometry, and statistics. The focus is on learning more of the accepted knowledge base,

to prepare for the advanced topics that will be encountered at the college level.

In addition to topics from the accepted knowledge base, Ernest would also want

this curriculum to address the history and philosophy of mathematics. Because Ernest

believes that mathematics holds a social responsibility, he calls mathematicians to

recognize the role of humanity in its development through history and the philosophical

debates over its foundations. This brings up ethical issues, such as the use of abstract

mathematics to dehumanize social issues and the capacity of mathematics to

discriminate.299 He also feels that mathematicians need to be aware of the public

perception that mathematics is beyond question and should be left to experts. The public

should be encouraged to question and scrutinize mathematical arguments.300 Because

Ernest wants the mathematical community to acknowledge these issues, he would want

them to be included in the training of future mathematicians. 297 Ibid., 143.298 Ibid., 140.299 Ernest, "Social and Political Values," 201.300 Ernest, Social Constructivism, 274.

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Mathematical Forms of Life

For Ernest, a form of life is a social practice “with its own purposes, implicit

rules, behavioral patterns, and linguistic usages or language games.”301 Mathematical

forms of life refer to the general cultural context of mathematics, including its language

and meta-mathematical views. In Ernest’s theory of learning, the enculturation of

students into mathematical forms of life happens through sustained participation in the

social context of school mathematics. This will have begun in elementary school, but

will gain more emphasis in secondary school.

In Ernest’s model, the language of mathematics includes both linguistic and

symbolic forms of communication.302 Students begin to learn this language in elementary

school, or even before. At the secondary level, they will expand their mathematical

vocabulary and learn new sets of symbols to accompany the new areas of accepted

knowledge that they encounter. For example, this may be the first time students

encounter the language of functions. In addition, they will deepen and refine their

knowledge of mathematical language games, learning to use the language of mathematics

in a more sophisticated way. This could include a deeper knowledge of the subtleties of

logical notation, or the ability to use mathematical language and notation with more

precision. Students will learn to define objects and concepts more clearly and with less

ambiguity. This deeper understanding of mathematical language will prepare students for

the higher level of abstraction that they will encounter in university mathematics.

Another aspect of the enculturation into mathematical forms of life is an

understanding of standard meta-mathematical views. This includes, among other things,

301 Ibid., 69.302 Ibid., 139-40.

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standards of proof and definition, beliefs about the scope and structure of mathematics,

and judgments about mathematical significance.303 Students will begin to understand

both the explicit rules and implicit norms that mathematicians follow when they construct

definitions and proofs. They will also start to uncover shared beliefs about mathematical

structure, the boundaries of mathematical thought, and the value put on different

mathematical problems and lines of inquiry. Meta-mathematical views are usually not

addressed directly. Instead, students begin to get a sense for them as they engage in

socially situated conversation and encounter more advanced topics.

In Ernest’s view, mathematical forms of life cannot be learned by transfer.

Students must learn how to participate in these forms of life by actively engaging in

conversation with teachers and other students. In Ernest’s curriculum, some of this will

have begun in elementary school, but in secondary school future mathematicians should

expand their understanding of these forms of life in both breadth and depth.

Mathematical Practice

An important part of the curriculum for future mathematicians would be the

mathematical practice of knowledge creation and justification. In preparation for their

later participation in the creation and warranting of new objective knowledge, students

need to become familiar and competent with this process. In Ernest’s philosophy of

mathematics, he builds on Lakatos’s logic of mathematical discovery to model the

creation and justification of objective mathematical knowledge. In school, the analogous

process is the creation and warranting of subjective knowledge through a Lakatosian

approach. In active conversation, students engage in proofs and refutations to construct

and refine subjective knowledge. Mathematics students in secondary school will also

303 Ibid., 141.

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need to begin the transition toward more formal forms of justification and proof, and a

proofs and refutations approach will aid in this shift. Therefore, this is most likely the

pedagogy that Ernest would recommend for the training of future mathematicians.

There is evidence that Ernest also intends for future mathematicians to engage in

investigations like other students, however. He states that “a major part of mathematics

is human problem posing and solving,”304 which implies that this is an important part of

the socialization of future mathematicians. I discussed in chapter three the ease with

which proofs and refutations could be used with a problem-based curriculum. Some of

the situations chosen by the teacher for the investigations would need to suggest more

abstract questions. I have in mind the type of investigation that occurs in Lakatos’s

Proofs and Refutations. At the beginning of the dialogue the teacher alludes to an

investigation that had occurred the previous day, in which the students discovered that for

all regular polyhedra, .305 This sounds as if the students explored the

properties of the regular polyhedra, posed a question about the relationship between

vertices, edges, and faces, and generated this formula. Then a student posed the

conjecture that this is true for all polyhedra. In the dialogue, they begin with this

conjecture and proceed with proofs and refutations to explore and revise it. This is a

perfect example of how problem posing and solving can dovetail with proofs and

refutations. One day students engage in problem posing and solving about a

mathematical object or situation, forming questions and exploring possible solutions.

The next day, they use that experience to form general conjectures and to probe them

through proofs and refutations. They can use this method to debate whether a statement

304 Ernest, Philosophy of Mathematics Education, 283.305 Lakatos, Proofs and Refutations, 7.

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is true, and whether it might extend to other contexts. All the while, students are

engaging in socially situated conversations that allow them to construct subjective

knowledge and learn mathematical forms of life. When appropriate, this can be

alternated with the investigations of situations in socially relevant contexts to foster

critical awareness. On these days, instead of proofs and refutations, students would use a

less formal Lakatosian approach to debate solutions to problems they have posed.

However, these solutions would most likely be presented in a more formal manner than

that expected of students in the general curriculum.

The curriculum for future mathematicians is informed by Ernest’s account of the

mathematical community in his philosophy of mathematics, and is primarily designed to

prepare students for membership in that community. I will discuss the values and

assumptions that are implicit in this view of education, as well as the underlying

conceptions of mathematical knowledge and the democratic ideal. This will allow me to

analyze whether this aspect of Ernest’s differentiated curriculum is consistent with his

philosophy of education.

Implicit Assumptions and Values

Ernest’s underlying assumption is that a mathematician is one who has completed

a doctoral program and participates in research mathematics. He does not see others who

use mathematics in their careers, such as engineers, economists, or even secondary school

mathematics teachers, as part of the mathematical community. This shows that these

types of mathematical activities are not considered part of mathematical practice—the

production of mathematics is valued over its use. This also uncovers an inconsistency in

his vision for the education of future mathematicians. He sees the enculturation into

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secondary school mathematics as the beginning of a student’s enculturation into

mathematical forms of life. But the teachers who are providing these classroom

experiences are most likely not active participants in university level research

mathematics. Ernest does not value these teachers as part of the mathematical

community, yet he charges them with beginning a student’s enculturation into that

community’s norms and practices.

Conception of Mathematical Knowledge

Because the education of future mathematicians is based on early enculturation

into the mathematical community, the conception of knowledge draws from Ernest’s

philosophy of mathematics. His primary conception of mathematical knowledge for

these students is that it is a personal reproduction of the accepted base of mathematical

knowledge, which comprises advanced mathematical concepts and capabilities drawn

from research mathematics. This includes an understanding of the connections between

mathematics and other disciplines, but not the actual capabilities in those disciplines. As

in his general curriculum, he sees these capabilities and applications as uses of

mathematics, but not as part of mathematical knowledge. For future mathematicians,

Ernest sees mathematics as an end in itself. This conflicts with his philosophy of

education, which views mathematics as the means to social empowerment and action. In

this curriculum mathematics has ethical implications, but the goal is enculturation into

the discipline and not social change.

Ernest also sees knowledge of mathematical practice as part of mathematical

knowledge. In this context, Ernest sees this knowledge as an ability to engage in the

social process of knowledge creation and justification, through the logic of mathematical

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discovery. This includes being skilled in the use of reasoning to approach mathematical

questions, make conjectures, and create formal proofs. At this level, students will begin

to practice these activities by constructing their own personal knowledge of the accepted

knowledge base through the process of proofs and refutations, which combines reasoning

and conversation. They will also begin to transition toward the construction of more

formal proofs. In this context, enculturation provides knowledge about mathematical

practice. This is consistent with Ernest’s philosophy of mathematics education, which

sees learning as a process of enculturation into mathematical forms of life. Learning

occurs through active engagement and conversation, which facilitate the social

construction of subjective knowledge and immersion in shared forms of life.

For Ernest, the education of future mathematicians is built on a conception of

mathematical knowledge as that which mathematicians know and do. It is an

understanding of the concepts and procedures in the accepted knowledge base and the

process of creating and warranting new knowledge. Mathematics is not seen as the

means to personal or social empowerment. However, Ernest does incorporate democratic

values into this curriculum, which will be examined next.

Conception of the Democratic Ideal

As with Ernest’s general curriculum, his vision for the education of future

mathematicians contains an inherent conception of democratic citizenship. For this

group, it involves a sense of social responsibility. Mathematical knowledge is not seen as

a means to social empowerment, but its acquisition results in certain obligations. For

Ernest, enculturation into the mathematical community brings with it a democratic duty.

A democratic citizen in the world of mathematics recognizes that mathematics is value-

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laden, and believes that mathematicians have an obligation to face the implications of

those values. She promotes equity by placing value on the mathematical contributions of

other cultures, and she promotes social justice by thinking critically about how

mathematics is used in society. A critical citizen is aware that mathematics plays a role

in social and ethical situations, and that some of its values disadvantage groups such as

women and minorities. This is consistent with the aims of critical awareness in Ernest’s

philosophy of mathematics education.

What is unclear is whether this curriculum goes far enough to fully satisfy

Ernest’s democratic aims. Is the development of critical awareness in future

mathematicians enough to counteract social reproduction, when these students are

learning the same body of mathematical knowledge that Ernest accuses of causing the

reproduction? There seem to be some important elements missing in his curriculum, but

his lack of specificity about the training of mathematicians makes it hard to be sure. One

point Ernest is vague about is whether certain aspects of his general curriculum,

particularly that of social empowerment and action, should also apply to future

mathematicians. It is possible that these are general requirements for the education of all

students, which would include future mathematicians under its umbrella. He does refer

to “aims for school mathematics,”306 and to problems as central to “school mathematics

for all.”307 This seems to apply to all students, regardless of future plans. But elsewhere,

in a discussion of the “aims of the mathematics curriculum,” Ernest states, “Personal

development is vital, mathematical functionality is important and beyond this the

acquisition of mathematics per se is of little general significance.”308 The lack of value

306 Ernest, "Why Teach Mathematics?" under "Conclusion."307 Ernest, Philosophy of Mathematics Education, 283.308 Ernest, "Social and Political Values," 198.

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placed on mathematics itself in this comment implies that general aims are intended only

for students who are not training to be mathematicians. This is supported by the fact that

Ernest never mentions future mathematicians in his writings about the general

curriculum. When he does discuss the education of mathematicians, there is no reference

to his general curriculum recommendations. I offer that the source of this tension is that

while Ernest’s outline of the general curriculum is meant to apply to those who are not

studying to become mathematicians, it is shaped by an overarching set of aims that he

feels are universally important. It may be that some of his goals, such as mathematical

power and critical thinking, are not seen as relevant for future mathematicians because he

assumes they will already be addressed in the course of their studies. Based on the values

and democratic principles that recur in Ernest’s philosophy and curriculum, I believe that

he would want future mathematicians to develop the same desire for social change that he

describes for other students.

This analysis of Ernest’s differentiated vision for mathematics education has

raised several issues about the mathematical knowledge and democratic ideal depicted in

each curriculum. I will now look at these curricula as part of a whole system, and

consider the implications of having such a differentiation in the mathematics education of

secondary students.

Critical Comparison

Taken as a whole, Ernest’s vision for secondary mathematics education is of a

differentiated curriculum based on students’ interests and career goals. The main

distinction that is relevant for this discussion is between the curriculum for future

mathematicians and the part of the general mathematics curriculum that is common to

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everyone else. The intrinsic values, conceptions of knowledge, and conceptions of

democratic citizenship in the two curricula are similar in several ways but also uncover

important differences. Both curricula reveal a view that mathematics is intrinsically

valuable, and that it is intertwined with its history and philosophy. In addition, they both

incorporate elements of mathematics appreciation, such as the big ideas of mathematics,

the connections between its branches, and its role in human culture. An enculturation

into mathematical practice is also present in both curricula, through a process of active

knowledge construction. However, this enculturation takes different forms. For non-

mathematicians, it is a socialization into problem posing and solving. In the curriculum

for future mathematicians, there is an enculturation into the more formal side of

mathematical language and reasoning through proofs and refutations. While the

conceptions of mathematical knowledge in the two curricula are similar, the students are

being prepared for different roles. Ernest views mathematicians as creators of new

mathematical knowledge, while he sees non-mathematicians as users of existing

knowledge.

Another important difference is found in the critical awareness aspects of these

curricula. In the general curriculum this centers on empowerment, while in the

curriculum for future mathematicians the focus is on social responsibility. Ernest wants

non-mathematicians to analyze and critique the way mathematics is used in society.

These students are being prepared to join the group of empowered citizens who are

outside of the mathematical community. There is a sense of social resistance against the

systems and the people who, whether consciously or not, use mathematics to

disadvantage or manipulate others. Future mathematicians, on the other hand, are being

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prepared to join the mathematical community. Ernest calls for mathematicians to be

critically aware insiders, examining their own behavior and assumptions. They need to

acknowledge that mathematics is value-laden, and that mathematicians are responsible

for the way in which it alienates women and minorities. This is a more self-reflective

version of critical citizenship than that in the general curriculum. At first glance this

difference seems rather small, addressing two aspects of the same basic democratic ideal.

However, it separates those inside the mathematical community from those outside,

creating a power differential. There is the implication that mathematicians need to

protect everyone else by ensuring that their discipline does not discriminate or

dehumanize. Everyone else needs to learn enough mathematics to recognize its misuse

and advocate for themselves, but they are learning the language of a group that does not

include them. It is like traveling to a foreign country and knowing enough of the

language to be aware of someone taking advantage of you, but not being fluent enough to

be accepted as a local resident. Like this foreign traveler, Ernest views the non-

mathematician as disadvantaged.

The implicit values, assumptions, and conceptions of mathematical knowledge

and the democratic ideal in Ernest’s two curricula have several features in common.

Members of the mathematical community are seen as the only creators of mathematical

knowledge, but this knowledge has intrinsic value for all citizens. Mathematics is

embedded in many aspects of our lives and culture, sometimes with negative

implications. The role of the democratic citizen is to be aware of these implications and

to work for social change. However, Ernest’s differentiated curriculum is inconsistent

with the aims of his philosophy of mathematics education. His visions of equity and

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social justice are undermined by the power differential created by the insider-outsider

roles inherent in his curricula. The general curriculum encourages non-mathematicians to

become personally and socially empowered, while the curriculum for future

mathematicians portrays non-mathematicians as people in need of protection. Further

equity issues are raised when Ernest’s differentiated curriculum is viewed in the larger

context of tracking. In the next section, I will look at these issues and their implications

for Ernest’s recommendations.

Tracking Issues

Tracking is a highly controversial topic in the education literature, and many of

these debates have focused specifically on mathematics. Many concerns about equity

have been raised, including questions about access to quality education and fair course

placement. Here I will consider some of these issues and their implications in the context

of Ernest’s differentiated mathematics curriculum.

Tracking systems in U.S. schools are often categorized as curricular or ability.

Curricular tracking is common in secondary schools, and it places students into a certain

sequence of courses based on some assessment of their ability and career goals.

Common tracks are college preparation, vocational education, or general education.309 At

some schools these tracks dictate a block of subjects, while at others the mathematics

track is separate. Ability tracking is grouping by common ability levels, and can take

several different forms. In secondary schools, it can appear as a parallel set of courses

that have the same name but different content. For example, there could be a higher track

with a geometry course that includes formal proof, and a lower track with an informal

309 Tate and Rousseau, "Access and Opportunity," 276.

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geometry course that does not. Alternatively, students might take the same courses but in

different years.310 Some might take geometry during their first year of high school, while

others take it their third or fourth year. Both types of tracking have been heavily

criticized in the literature as inequitable. Many studies conclude that poor students and

racial minorities are disproportionately placed in lower tracks.311 This is supported by

findings that one of the factors in these placement decisions is educator perceptions about

ability and motivation, and that these perceptions are affected by a student’s racial or

social group.312 Once placed in a lower track, students often receive an inferior

mathematics education. Jeannie Oakes cites her own research as well as many other

studies which indicate that lower-track mathematics classes cover less demanding topics

with less emphasis on problem-solving tasks, focusing instead on fragmented chunks of

material and rote memorization.313 Findings such as these have led many researchers to

believe that the disadvantages to students in the lower tracks are not worth any

advantages that might be reaped by students in the higher tracks.

Ernest is also wary of tracking, stating that the norms and assessments used often

cause bias that disadvantages minority students.314 He is against ability tracking, as he

finds the concept of “mathematical ability” to be problematic in two ways. First, it

assumes that ability is inborn and fixed, that a low achiever in mathematics must always

be a low achiever. Ernest counters this assumption with research findings that low

achieving students can raise their performance levels when engaged in socially and

culturally relevant mathematical activities.315 Second, Ernest is concerned that labeling

310 Ibid.311 Ibid., 292.312 Oakes, Multiplying Inequalities; Oakes and Guiton, "Matchmaking," 28.313 Oakes, Multiplying Inequalities, 81, 89.314 Ernest, Philosophy of Mathematics Education, 267.315 Ibid., 245.

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students as low achievers in mathematics can be self-fulfilling.316 Tracking students by

ability may encourage students in the high tracks and leave students in the low tracks

with little motivation. Therefore, Ernest sees mathematical ability as largely a social

construction, created by students’ experiences and the perceptions of others.317 He finds

ability tracking to be unfounded and dangerous, often harming students from minority

groups or lower social classes.

Ernest instead proposes a form of curricular tracking, in which the mathematics

track is chosen separately and is not tied to tracking in any other subjects. The choice

would be based solely on student interests and goals, and not on any measurement of

student mathematical ability. There would be a track for future mathematicians, as well

as tracks for other career goals. This might include different college preparatory tracks

for mathematics-related fields such as engineering or information technology, as well as

non-mathematics fields such as the humanities, and vocational tracks for students not

planning to go to college. On the surface, at least, Ernest’s vision alleviates some of the

concerns about tracking. Because students would choose their track based on their

interests, they would not be held back by perceptions about their ability. Content would

be tailored to student goals, but there would not be a parallel track offering less

mathematical content than its counterpart. All courses would provide meaningful,

thought-provoking content relevant to the students’ lives or future careers. All students,

regardless of curricular track, would engage in independent thought and problem solving.

Ernest feels that his curriculum would provide a more democratic method of course

selection, eliminate the overrepresentation of poor and minority students in vocational

316 Ibid., 244.317 Ibid., 208.

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tracks, and remedy the inferior pedagogy based on rote memorization and meaningless

procedure that students in vocational tracks often encounter. However, he is idealistic in

some respects and ignores some important factors.

Ernest’s vision of a curricular tracking system in which student interests prevail

seems unrealistic. Student choices can be unduly influenced by the recommendations of

teachers and counselors. Studies have found that poor and minority students receive

limited academic advice about course options, and that women and minorities receive

less encouragement to pursue careers in scientific fields.318 Therefore, students who are

unsure about their career interests may be nudged in one way or another based on

stereotypes about race and social class. For example, in a case study of three high

schools in California, Jeannie Oakes and Gretchen Guiton found that many teachers and

counselors thought of Asian students as the most suited to academic coursework.

Latinos, and to a lesser extent African-Americans, were often thought to be better served

by occupational training because these students were not likely to go to college.319 These

types of perceptions affect how students are advised, and can be quite influential in how

young people see themselves. In these cases, student choice is basically meaningless.

There is also danger in allowing students to choose their curricular track so early.

Remember that for Ernest, secondary school begins in year 7. Not only might these

children not be ready to make the best decision for their futures, but their choices may

already be affected by social inequity. In a review of research, Oakes found that many

non-Asian minority students leave the “scientific pipeline” in elementary school. Even

this early, they are more likely to be placed into special education and less likely to be

318 Oakes, "Opportunities, Achievement, and Choice," 184.319 Oakes and Guiton, "Matchmaking," 17-18.

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placed into enriched programs.320 Minority students are also more likely to attend poor

schools that suffer from a lack of school funding, which results in inferior opportunities

and educational quality.321 Whatever the reasons, the elementary school experience of

many low-income and minority students restricts the mathematics courses they are

prepared to take in middle and secondary school. Therefore, these students are much less

likely to choose a mathematics-intensive course of study. Worse, Ernest would allow

them to opt out of mathematics all together. Not only would this rob them of any

mathematics content they might benefit from, but it would also deprive them of the

critical awareness and social empowerment aspects of the curriculum that Ernest feels are

so important. It seems like social justice would be better served by encouraging students

to continue in a mathematics track that would not only keep their future options open, but

would also develop their critical skills and possibly allow them to discover interests that

have been suppressed by earlier experiences. In this case, Ernest’s principle of

democratic choice is inconsistent with his desire to overcome social inequities in

mathematics education. Rather than perpetuate the cycle, it seems that Ernest should

want the secondary curriculum to encourage women and minorities to pursue avenues

that have previously seemed closed.

An issue Ernest does not mention at all is that of the resources necessary to

provide so many high quality curricular options. Schools would need extra funding for

the development of curricular materials, recruitment and salaries for quality teachers, and

professional development. Teachers would need an advanced knowledge of mathematics

in order to teach Ernest’s mathematics appreciation topics. Schools in poorer areas are

320 Oakes, "Opportunities, Achievement, and Choice," 160.321 Ibid., 180.

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already much less likely to offer a range of mathematics options that includes calculus,

and have the hardest time recruiting qualified mathematics teachers.322 Many cannot

afford to offer more courses, or to provide the level of professional development needed

to foster effective implementation of an investigatory pedagogy in all classes. With

resources already constraining the variety and quality of mathematics offerings, it seems

unlikely that less affluent schools would be able to implement Ernest’s curriculum,

further exacerbating social inequity.

Even with ample resources and completely free choice for students, tracking

could still be problematic. With ability removed from the placement criteria, students

could still assign a hierarchy to the curricular tracks. Society places higher value on and

attributes more difficulty to some career tracks than others, and this could carry over to

perceptions of the mathematics curriculum. The track for future mathematicians would

be at the top. Even with Ernest’s robust vision for the non-mathematics curriculum, other

tracks may be seen as having “less mathematics,” and therefore as being easier. The

tracks for students not going to college would most likely rank at the bottom. This would

create an unintended power differential between students in the various curricular tracks.

There is no easy solution, however. Requiring all students to take a college preparatory

sequence is unattractive as well. Students who are not training to be mathematicians are

then required to take courses well beyond the level required for their goals, possibly at

the expense of more relevant courses. Ernest is trying to avoid this, because he sees it as

undemocratic. Advanced mathematics courses would not provide much benefit, in terms

of utility or even personal enrichment, to students going into careers that are not

mathematics-intensive. Another option would be to have all students take the same

322 Oakes, Multiplying Inequalities, 40, 46.

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courses, but allow students not training to be mathematicians to stop at a certain point in

their schooling to focus on other areas. Ernest would not like this idea because courses

for future mathematicians need to include more abstract topics, language, and procedures.

Some of the students who do not want to be mathematicians will be bored by this, and

some will struggle with these more formal features of the curriculum. In both cases,

these students could become frustrated and therefore miss the larger concepts. They

would benefit more from a course that takes their interests into consideration, focusing on

the larger concepts without addressing unnecessary procedural details. This brings us

back to the option of curricular tracking.

As far as tracking options go, it is possible that Ernest’s design would avoid some

of the pitfalls of traditional curricular tracking. He has attempted to take the needs of all

students into consideration, and has designed a curriculum intended to empower those

who are usually disadvantaged. However, there are many issues that he has not

considered. An empirical study would need to be done to discover how the pros and cons

of his tracking design play out in practice.

Summary and Conclusion

Ernest recommends a differentiated mathematics curriculum for students after

elementary school, in order to tailor instruction to students’ interests and goals. For

students who are not training to be mathematicians, he finds utilitarian arguments for

mathematics education to be unsound. Therefore, his general mathematics curriculum

focuses on the intrinsic value of mathematics and its connections to the world around us.

He also incorporates a critical awareness component designed to foster social

empowerment and action. The curriculum for future mathematicians prepares students to

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join the mathematical community, providing advanced mathematical content and early

enculturation into mathematical practice. It also addresses the social responsibility of

mathematics and mathematicians.

The intrinsic value of mathematics and the need to promote democratic values are

woven into both sets of curricular recommendations. Both make use of an active

pedagogy based on the logic of mathematical discovery. However, there are equity

concerns with any proposal for tracking in education. Ernest’s curricular tracking sets up

a power differential between mathematics insiders and outsiders which is not consistent

with his philosophy of mathematics education. It also allows students who are already

socially disadvantaged to end their mathematics education after elementary school, rather

than attempting to counteract the negative effects of previous social and educational

experiences. Finally, Ernest fails to address the financial resources necessary for

effective implementation of his curriculum at all levels.

In chapter four I analyzed Ernest’s philosophy of mathematics education, and in

this chapter I have addressed his curricular recommendations. In the next chapter, I will

illustrate the implications of this analysis by exploring the issues that would arise if

Ernest’s view were applied to developmental mathematics at the community college

level. I will consider how his approach would frame the particular issues that arise in this

context, as well as potential drawbacks and resistance that could be faced.

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CHAPTER 6

IMPLICATIONS IN A DEVELOPMENTAL MATHEMATICS CONTEXT

In this chapter, I will examine the application of Ernest’s framework to

community college developmental mathematics. Community colleges have a history of

providing educational opportunity. This is a somewhat contentious claim, as some

researchers worry that the opportunities available at a two-year college actually

encourage students to pursue less ambitious goals.323 Nonetheless, open-admissions

policies at community colleges do provide educational opportunities to many students.

However, many arrive unready for college-level mathematics courses. Developmental

mathematics is seen as an obstacle they must overcome before achieving their

educational goals. Because Ernest’s philosophy of mathematics education focuses on

both the teaching of mathematics and the democratic obligations of mathematics

education, it provides a useful perspective in this context.

For at least the last twenty years, there have been researchers concerned that the

field of developmental education lacks a theoretical framework. In 1990, Donald Blais

wrote that in the developmental mathematics literature, “it is difficult to find discussions

related to knowledge, learning or teaching that are primarily philosophical or theoretical

in nature. In other words, there has been little debate over fundamental beliefs

underlying educational practices.”324 In 1999, participants at a meeting on future

directions in developmental education expressed the concern that developmental

education practices are driven by “historical accident and local pragmatics” rather than

323 Clark, "'Cooling-Out' Function."324 Blais, "Constructivism Applied to Algebra," 27.

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the thoughtful application of theory.325 The need is not just for theories of cognitive

development and pedagogy, though these are vital. A philosophical framework is also

important, to guide decision making and the use of other theories. The application of

philosophy allows faculty and program directors to articulate their underlying beliefs and

goals, and to uncover their hidden assumptions about developmental education. It

provides a framework in which to analyze the intended and unexpected implications of

these beliefs and assumptions, and to consider alternatives. Philosophical analysis also

provides a tool for evaluating current practice for alignment with articulated beliefs and

goals.

To contribute to the theoretical literature in this area I will explore the

implications of Ernest’s framework for developmental mathematics at the community

college level, considering both his philosophy of mathematics education and his

curricular recommendations. For the purposes of this discussion, community colleges

will refer to public two-year colleges. While some researchers use the terms

developmental and remedial interchangeably in the context of pre-college level

coursework, others feel that they carry different connotations. These educators feel that

remedial implies a deficiency on the student’s part, and prefer the term developmental to

communicate a philosophy of personal growth.326 This reflects my own thinking, so I

will use the latter term except in direct quotes or when differentiating a remedial view

from a developmental one. Developmental mathematics will include those courses

generally considered to be preparatory, but beyond the level of adult basic education:

prealgebra, beginning and intermediate algebra, and geometry. While there is a

325 Collins and Bruch, "Theoretical Frameworks," 19.326 Grubb and Associates, Honored but Invisible, 179.

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considerable amount of literature on the different types of students who take these

courses, my focus is on students who have a high school diploma or its equivalent, and

are enrolled in a program that is not mathematics-intensive but requires them to take

mathematics courses. They are unlikely to use higher level mathematics in their careers,

but need a certain level of competency as defined by their different programs: to

understand basic personal finances, to perform calculations for construction or nursing, or

as preparation for a transfer-level general education course, for example. Students take

developmental mathematics courses for a variety of reasons. Some did not take these

courses in high school, some have taken these courses but did poorly, while still others

have passed these courses in high school but did not show mastery of the content on a

placement test. A large number do not think these courses should be required for their

majors.327 This study will contribute to discussions about the justification of mathematics

education for developmental students, the meaning of educational opportunity, and

implications of different approaches to mathematics education in this context.

In this chapter, I will explore how Ernest’s view can be applied to developmental

mathematics at the community college level. First I will draw from the literature to

provide a brief background of the community college and developmental mathematics in

the context of open access and educational opportunity. There is a large body of

literature surrounding equal opportunity and ways it can be defined, but I will limit my

discussion to a brief example that provides context for the application of Ernest to the

community college. Then I will relate my analysis of Ernest in the previous chapters to

this setting and consider how his perspective might be used to reframe this issue. Next, I

will demonstrate how his framework can be used to evaluate current policy and practice.

327 Grubb and Cox, "Pedagogical Alignment," 95.

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Drawing examples from different aspects of the discipline, I will examine developmental

mathematics standards, required textbooks, and instructional practice through Ernest’s

lens. Finally, I will consider possible obstacles to implementing Ernest’s approach in this

setting.

Community Colleges

The role of developmental mathematics in the community college must be

examined within an historical context of educational opportunity. Prior to the appearance

of community colleges, legislation laid the groundwork for higher education

opportunities for more students. In 1862, the passage of the Morrill Act introduced a new

philosophical purpose for higher education: practical subjects such as agriculture and the

mechanical arts.328 These land-grant institutions opened the doors to new groups of

students who previously had not attended college. Even more students gained access

with the passage of the second Morrill Act in 1890, which cut federal funding to any state

that discriminated in higher education.329 While this did not require integration, it did

result in separate land-grant institutions for African-Americans. The Morrill Acts laid the

groundwork for the idea that higher education was not just for the children of elite

landowners, and that it could be used to learn practical skills.

The first community college is generally considered to be Joliet Junior College in

Illinois, which began in 1901 as a postgraduate extension of Joliet High School.330

Community colleges began to appear across the country to serve several different

purposes. Some were set up as vocational schools, while others were junior colleges, set

328 Markus and Zeitlin, "Remediation," 167-68.329 Casazza, "Who are We?" 3.330 Vaughan, Community College Story, 31.

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up to provide the first two years of a college education. In some communities, junior

colleges allowed students to start college without traveling great distances to state

universities. There was also a push from some university faculty to funnel the first two

years of general education courses to junior colleges, so that the universities could focus

on upper-level coursework and research.331 Thus in some cases, community colleges

were started to expand educational opportunities, while in other cases they were actually

started to restrict access to the universities.

After World War II, the Servicemen’s Readjustment Act of 1944, better known as

the G.I. Bill, provided federal funding for the higher education of millions of returning

servicemen and women. Colleges and universities were flooded with students

representing a range of ethnic backgrounds, ages, and academic abilities. Many of these

students needed some sort of extra help to prepare them for college-level coursework.332

Partially in response to the needs of these students and the overcrowding in universities,

President Truman appointed the President’s Commission on Higher Education to

determine how to expand higher education opportunities for more people.333 The

Commission’s report, published in 1947, calls for free education for all students through

the fourteenth grade. In order to accommodate all of these students, it recommends the

creation of more publicly-funded junior colleges to form a nationwide system. The

Truman Commission report emphasizes that this educational opportunity was to be for

everyone, stating, “Equal educational opportunity for all persons, to the maximum of

their individual abilities and without regard to economic status, race, creed, color, sex,

national origin, or ancestry is a major goal of American democracy. Only an informed,

331 Witt et al., America's Community Colleges, 7.332 Markus and Zeitlin, "Remediation," 169.333 Witt et al., America's Community Colleges, 130.

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thoughtful, tolerant people can maintain and develop a free society.”334 The discussion

continues, outlining the need to remove economic barriers to education, and to end

discrimination in college admissions.335 The Commission paid special attention to racial

and religious discrimination. It also called for an expansion of services at junior colleges,

to accommodate the different needs and goals of students. These colleges were to be

flexible to local needs, and to offer a comprehensive range of services such as adult

education, vocational and transfer programs, and internship programs.336 The

Commission suggested that these schools be called community colleges, to reflect their

community-based nature.337

The Truman Commission report did much to shape the current idea of a

comprehensive community college, and gave these schools a clearer identity as providers

of educational opportunity. However, stakeholders continued to disagree about whether

the primary function of a community college should be preparation for university transfer

or vocational education. Federal legislation such as the Manpower Development and

Training Act of 1962 and the Vocational Education Act of 1963 provided funding for

community colleges to greatly expand their career and adult education offerings.338 This

essentially secured the role of community colleges as providers of these two programs in

addition to the well-established transfer programs. The curricular functions of

community colleges today look much as the Truman Commission envisioned. While

specific offerings depend on local needs, they usually include preparation to transfer to a

334 President's Commission on Higher Education, Organizing Higher Education, 3. Italics in the original.

335 Ibid., 26.336 President's Commission on Higher Education, Equalizing and Expanding, 67.337 President's Commission on Higher Education, Financing Higher Education, 5. Some schools

were already using the term, so it was not created by the Truman Commission.338 Hardin, "History of Community College," 221, 294.

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baccalaureate program, vocational-technical education, continuing education,

developmental education, and community service.339

As a result of their historical development, community colleges have come to

serve many roles, which include providing businesses with trained workers, providing

alternatives to university attendance, and providing higher education opportunity to a

wider population. Most have open-access admissions policies, and they have come to be

seen as democratizing institutions because they have

reached out to attract those who were not being served by traditional higher education: those who could not afford the tuition; who could not take the time to attend a college full time; whose ethnic background had constrained them from participating; who had inadequate preparation in the lower schools; whose educational progress had been interrupted by some temporary condition; who had become obsolete in their jobs or had never been trained to work at any job….340

This is supported by recent statistics. During the 2003-2004 school year, about 15% of

community college students identified themselves as black and 14% Hispanic, 53% were

age 24 and older, 61% were independent of their parents, 35% were independent with

children, 69% attended college less than full-time, and 41% worked full-time.341

Community colleges tend to serve a more diverse population in terms of ethnicity, age,

and commitments outside of school than four-year institutions. The role of community

colleges as democratizing agents has not gone unchallenged, however. In 1960, Burton

Clark reported on the “cooling-out” function of these institutions. His claim was that

rather than providing marginalized students with higher education opportunity,

community colleges grant admission to students who cannot meet college-level standards

of performance. These students’ hopes to complete a college education are gradually

cooled-out through a process in which they are encouraged to change from a transfer 339 Cohen and Brawer, American Community College, 20.340 Ibid., 28.341 U.S. Department of Education, Profile of Undergraduates, 25-28, 52, 123.

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course of study to a vocational program, are gradually faced with increasing evidence of

poor performance, and are eventually counseled to lower their aspirations to an alternate

but lower-status career.342 While Clark refers to students in transfer programs, this

argument could apply to any community college student who arrives with one set of goals

and is gradually pushed toward a less challenging program. In addition to the cooling-out

criticisms, various studies between 1970 and the present have cited evidence that

community colleges accentuate social stratification, have high dropout rates, and fail to

provide marginalized students with access to baccalaureate programs.343 The critics

believe that real opportunity in higher education has only been achieved if students of

low socioeconomic status enter four-year programs at the same rate as other students.

They argue that community colleges have failed as providers of this opportunity. Many

researchers feel differently, however. Arthur Cohen and Florence Brawer argue that the

negative statistics are the result of policies allowing students to choose their own

curricular path rather than being placed based on a talent assessment. While this may

allow some students to pursue programs they cannot complete, it also provides

individuals with the opportunity to elevate their social class.344 Clark, revisiting his

cooling-out theory twenty years after its original publication, explains that equity is

actually a larger social problem, and that community colleges have no attractive

alternatives to the cooling-out process. In general, the options would require either the

revision of open-access policies or the lowering of standards for transfer coursework.345

Neither alternative seems desirable to Clark. Like Cohen and Brawer, he views cooling-

out as a necessary part of equal opportunity. Students are given the chance to overcome 342 Clark, "'Cooling-Out' Function," 571, 574-75.343 Cohen and Brawer, American Community College, 376-82.344 Ibid., 394.345 Clark, "'Cooling-Out' Revisited," 18-21.

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their social status or educational history through open-access policies. Many students

have benefitted from this opportunity and have actually raised their aspirations as a result

of attending a community college.346 For those whose plans are beyond their abilities, the

cooling-out process provides gentle guidance toward a more attainable goal. It presents

alternatives rather than declaring students as failures.347 Those who defend community

colleges believe that educational opportunity has been provided if all students are given

access to the same programs, and if there is a chance that some will achieve higher goals

than they would have otherwise. The debate about educational opportunity at community

colleges is a philosophical one. It raises issues about how we define equity, equal

opportunity, and fairness. My intention is not to take sides in this debate, but rather to

place my discussion of Ernest and developmental mathematics in the context of this

issue.

Developmental Mathematics

Developmental mathematics is a key factor in issues of educational opportunity.

At community colleges, approximately 43% of students take at least one developmental

course, and almost 80% of those enrollments are in mathematics.348 Developmental

students make up over half of the mathematics enrollments at these schools.349 However,

many are unsuccessful in their developmental coursework. In a study of developmental

education, Robert McCabe found that “[m]athematics is the greatest hurdle for deficient

346 Ibid., 25.347 Ibid., 20.348 U.S. Department of Education, Profile of Undergraduates, 133, 137.349 Lutzer et al., Statistical Abstract, 138.

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students.”350 In this section, I will discuss how Ernest’s framework can contribute to

conversations about developmental mathematics and its role in educational opportunity.

Ernest’s philosophy of mathematics education provides a way to reframe issues

pertaining to educational opportunity in terms of developmental mathematics instruction.

The cooling-out debate and related discussions focus on access and transfer, looking at

who is admitted to transfer programs and the number of students who successfully

transfer to a baccalaureate program. Clark presents the problem as a disconnect between

admissions requirements and standards for performance, the only options being the

adjustment of one or the other. There is no discussion of possible strategies to help

students satisfy those requirements. The underlying assumption in this argument is that

student ability is fixed, and that nothing can be done to help low-achieving students reach

college-level standards. Ernest allows community college educators to instead examine

what is happening in the developmental mathematics classroom to close the gap between

aspirations and achievement. Ernest believes that mathematical ability is not fixed, but

rather is greatly impacted by social context and the way students are perceived and

labeled by others.351 Achievement can be affected by environment, experiences, and

other social factors. Therefore, when students are placed into developmental

mathematics, it should not be assumed that they are unable to learn the material and

achieve their educational goals. From Ernest’s perspective, community colleges should

instead look at how classroom experiences impact the learning and achievement of these

students. For him, real educational opportunity has not been realized unless students

have access to a mathematics education that provides authentic mathematical

350 McCabe, No One to Waste, 40.351 Ernest, Philosophy of Mathematics Education, 208.

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experiences, allows for the social construction of knowledge, and encourages awareness

of social issues. I will elaborate on this in the context of current practice.

In general, there is evidence that the predominant approach in community college

developmental mathematics classrooms is a very traditional one. Standard lecture is

consistently the most common instructional method.352 Behaviorist learning theory has

been very influential in developmental mathematics, dominating instruction despite the

fact that it has come under great scrutiny in educational circles. Many instructors focus

on drilling mathematical operations and procedures.353 Both in and out of the classroom,

most developmental mathematics programs use some combination of programmed

instruction, computer-assisted instruction, written study guides, and mastery learning.354

When group activities are introduced into the classroom, they tend to involve worksheets

that provide skill-based practice. From Ernest’s point of view, the traditional approach is

flawed in several ways. First, it is not based on authentic mathematical experiences.

According to Ernest, course content should expose students to important concepts in

mathematics, but the traditional curriculum focuses on operations and procedures.

Students do not get to explore topics that are intrinsically interesting or embedded in

human culture. In Ernest’s opinion, the traditional curriculum underestimates students’

ability to grasp important concepts and robs them of rewarding mathematical

experiences.355 Second, because the traditional approach does not allow for the social

construction of meaning, Ernest believes that it does not promote learning. It does not

account for the way mathematical knowledge is formed, and therefore does not provide

students a real opportunity to learn. Finally, Ernest sees the traditional approach as 352 Lutzer et al., Statistical Abstract, 148; McDonald, "Developmental Mathematics Instruction," 9.353 Grubb and Associates, Honored but Invisible, 189-94.354 Boylan, "Foundations of Developmental Education," 2.355 Ernest, "Why Teach Mathematics?" under "Conclusion."

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undemocratic because it creates citizens who are unaware of social issues that affect their

lives. He believes students must be given a real opportunity to question social situations,

which to him means they must be exposed to social and political claims and taught to

scrutinize mathematical evidence in the media. For Ernest, educational opportunity

requires access to meaningful mathematical content, instruction that incorporates social

construction, and activities that encourage critical analysis of mathematical claims.

Ernest’s framework can be used to question what it means to provide effective

mathematics instruction and to achieve democratic outcomes. As outlined in chapters

four and five, Ernest provides an approach to mathematics education that is both

philosophical and pedagogical. Philosophically, he describes mathematical learning as a

process of social construction, requiring conversation and active engagement. He sees

the democratic role of mathematics education as providing every student the opportunity

to develop mathematical literacy, a sense of the relevance of mathematics in their lives,

and a critical awareness of its uses and abuses in society and politics.356 Pedagogically,

Ernest makes recommendations about how to provide this opportunity. For him,

effective mathematics instruction must incorporate interaction, self-directed inquiry, and

active problem solving to foster the social construction of knowledge. To achieve

democratic outcomes, he believes students must be exposed to socially relevant issues

and engage in critical mathematical analysis. Through activities that foster the social

construction of knowledge and critical engagement in the classroom, he hopes that

students will come to see mathematics as less intimidating, more accessible, and more

relevant as a tool for overcoming social inequity. Ernest provides a framework for

356 Ibid.

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reconsidering how we view the role of developmental mathematics instruction in helping

students to achieve their goals and in educating democratic citizens.

Ernest’s framework provides a comprehensive approach to mathematics

education, but it also provides a tool for evaluating policy and practice. Even if one does

not agree with all of Ernest’s points, this framework can highlight particular aspects of an

issue that may otherwise not be considered, such as assumptions about student ability,

student learning, and the democratic role of mathematics education. To illustrate what

this type of analysis might reveal, I will provide examples from three different areas of

practice and explore them through Ernest’s lens. First I will consider standards by

examining the developmental mathematics standards of the American Mathematical

Association of Two-Year Colleges (AMATYC). Then I will use a study of

developmental mathematics textbooks in Illinois, conducted by Vernon Kays, to

demonstrate the evaluation of course materials within Ernest’s framework. Finally, I will

move within the classroom by means of a case study by Donald Blais, and I will use it to

analyze classroom pedagogy.

AMATYC’s Developmental Mathematics Standards

The United States has been in the midst of a standards and accountability

movement in education for many years. It began at the K-12 level, but its influence has

extended to higher education. AMATYC responded in 1995 with the development of

Crossroads in Mathematics: Standards for Introductory College Mathematics Before

Calculus. These standards were reaffirmed and complemented with a set of

implementation recommendations, Beyond Crossroads: Implementing Mathematics

Standards in the First Two Years of College, in 2006. AMATYC’s standards are not

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legally binding, but they are meant to provide guidance to two-year colleges in the

development of mathematics curricula. The organization attempts to address the needs of

all community college students with their diversity of backgrounds, educational paths,

and career goals. It is AMATYC’s feeling that it can serve all of these students with one

set of standards, because it believes that there is a common set of topics and skills that all

students should encounter.357

The Crossroads standards define introductory college mathematics as

developmental mathematics, college algebra, trigonometry, and precalculus, as well as

terminal courses such as introductory statistics and finite mathematics.358 AMATYC

provides general standards for all of these courses, split into three categories. There are

Standards for Content, which describe the main content areas of mathematics that should

be touched on in all educational programs. There are also Standards for Intellectual

Development which outline skills to be learned by students, with a focus on problem

solving. Finally, there are Standards for Pedagogy which suggest instructional methods.

AMATYC recommends that these sets of standards be integrated with “the viable

components of traditional instruction” to ensure students acquire necessary skills and

learn mathematical abstraction.359 These three sets of standards are roughly analogous to

Ernest’s content recommendations, theory of learning, and theory of teaching,

respectively.

These general standards are followed by guidelines for interpretation in each area

of introductory college mathematics. My focus here will be on the recommendations as

they apply to developmental mathematics. AMATYC sees developmental mathematics

357 American Mathematical Association of Two-Year Colleges, Crossroads in Mathematics, xii.358 Ibid., ix.359 Ibid., 5.

159

as the common mathematical foundation for all students, regardless of their different

curricular paths. It is intended to provide a broad, solid basis for further mathematical

studies and for life. Some students master these topics and skills in high school, but

community colleges provide developmental courses for students who arrive without a

solid mathematical foundation.360 I will provide a summary of the Crossroads standards,

with an emphasis on the recommendations for developmental mathematics. In order to

demonstrate how Ernest’s philosophical framework may inform practitioners, I will use it

to analyze these standards and discuss their implications.

Standards for Content

AMATYC believes that there is a set of core content areas that all students should

be familiar with, to provide context for mathematical problem solving. These

suggestions are analogous to the recommendations Ernest makes for content in general

mathematics education. Specific content items are not listed, as they will differ

depending on the student’s major or the goals of a particular course. However,

AMATYC provides seven general categories that should be addressed to different extents

in each program, including developmental courses. Of the seven content standards, three

are emphasized primarily at the developmental level. The first is Number Sense, which

involves arithmetic as well as an intuition about number properties, estimation,

proportionality, comparison, and the reasonableness of answers.361 The second standard

is Symbolism and Algebra. This offers a transition from the concrete operations of

Number Sense to a more abstract understanding of number properties. It is characterized

by abstract symbols and equations, but not rote manipulations and algorithms.362 Next is

360 Ibid., 24.361 Ibid., 13, 26.362 Ibid., 27.

160

the Geometry standard, which calls for spatial and measurement sense. This refers not

only to the recognition of shapes and formulas, but also to the ability to visualize and

transform objects.363 These three standards are intended to foster an intuitive sense of

mathematical structure and geometric relationships. In Ernest’s framework, this would

align with the basic mathematical capability that he sees as a prerequisite for further

general mathematics study.

The remaining four content standards present areas that should be introduced at

the developmental level, but will be emphasized more heavily in college-level

coursework: Functions, Discrete Mathematics, Probability and Statistics, and Deductive

Proof. The first three are seen as problem-solving tools. Functions can be used for

algebraic modeling or statistical analysis. In developmental courses, students are

expected to learn to recognize families of functions, their general properties, and multiple

representations. They should encounter discrete mathematics topics such as Venn

diagrams, permutations and combinations, recursion, and matrices. In addition, students

should learn about basic probability laws, data collection and organization, and

descriptive statistics.364 These topics are meant to present students with different contexts

and perspectives for mathematical problem solving. The final content standard is

Deductive Proof, which is to provide an introduction to mathematical justification. While

formal proof will not be a focus before advanced mathematics courses, at the

developmental level students are expected to be familiar with “valid and invalid forms of

mathematical arguments.”365

363 Ibid., 13.364 Ibid., 14, 27-28.365 Ibid., 28.

161

From Ernest’s perspective, mathematical content in the classroom should convey

the “big ideas of mathematics,” the connections between its branches, its historical

development, and multiple views about its philosophical foundations. It should also

allow students to see the central role of mathematics in culture and its presence in daily

life, while helping them to develop a critical understanding of the uses of mathematics in

society.366 Like Ernest’s recommendations, AMATYC’s standards do not outline specific

content, but rather provide general topics around which the curriculum should be

designed. For AMATYC, these topics are intended to give students an overall sense of

the nature and utility of mathematics, building the mathematical sense necessary for

reasoning, drawing conclusions, and understanding relationships. This is meant to

provide a framework for problem solving, emphasizing the “meaning and use of

mathematical ideas” rather than “rote manipulation.”367 AMATYC’s approach brings

many branches of mathematics together into one general framework. This can help

students to see how different aspects of mathematics are connected, and how they can be

integrated to solve a problem. By applying this framework to many different types of

problems, students are exposed to mathematics in other disciplines and in everyday life.

However, while AMATYC’s content recommendations align with Ernest’s in these ways,

they do not embrace the richness and wonder of mathematics that Ernest would like to

convey to all students. AMATYC’s focus on developing a framework for problem

solving portrays mathematics as a utilitarian discipline, useful in applications but not

valuable in its own right. While Ernest does see critical thinking and problem solving

skills as important, he also wants students to explore important topics that reveal the

366 Ernest, "Why Teach Mathematics?" under "Capability versus Appreciation."367 American Mathematical Association of Two-Year Colleges, Crossroads in Mathematics, 12.

162

intrinsic value of mathematics. For him, topics such as chaos, infinity, and randomness

are not only key concepts in advanced mathematics, but also capture the mystery and

beauty of the discipline.368 He believes these topics are accessible to students, even

without advanced mathematical capability. Even though AMATYC recommends a

curriculum that includes important topics from areas such as discrete mathematics and

statistics, it does not encourage teachers to expose students to the sorts of rich

mathematical concepts that Ernest has in mind. Therefore, AMATYC’s Standards for

Content do not foster an appreciation for mathematics and its intrinsic value. In Ernest’s

view, these recommendations restrict students’ exposure to meaningful mathematical

content.

Standards for Intellectual Development

The Standards for Intellectual Development describe seven intellectual skills that

AMATYC believes students should learn in the course of their mathematics education.

Each standard focuses on a different aspect of mathematical thinking. AMATYC

envisions that practicing these skills will enrich the learning process for all students.

While the standards refer to “skills,” they actually reflect an underlying learning theory

that can be analyzed through Ernest’s lens.

The first three Standards for Intellectual Development deal with different aspects

of mathematical inquiry: Problem Solving, Modeling, and Reasoning. AMATYC’s

description of problem solving states that it should be “substantial,” and that students

should “use problem-solving strategies that require persistence, the ability to recognize

inappropriate assumptions, and intellectual risk taking rather than simple procedural

368 Ernest, "Why Teach Mathematics?" under "Conclusion."

163

approaches.”369 Students should pose questions, analyze situations, and draw

conclusions. Problems should be relevant and interesting, providing context and purpose

for the learning of mathematical content and skills. Modeling is a particular type of

problem solving tool which fits mathematical formulas to real-world phenomena and

data. Students should be able to create a mathematical model, then determine how

accurate it is and whether it would apply in other situations.370 Reasoning allows students

to develop and test conjectures. According to AMATYC, they should be able to justify

their conclusions and to evaluate the mathematical arguments of others.371

The next three standards describe skills or ideas that students should learn in the

course of their mathematical studies: Connecting with Other Disciplines,

Communicating, and Using Technology. First, students should “develop the view that

mathematics is a growing discipline, interrelated with human culture, and understand its

connections to other disciplines.”372 AMATYC encourages teachers to show how

mathematical ideas have developed, and how mathematics is connected to other

disciplines such as the sciences, economics, art, music, literature, and history. The

standards also state that students should acquire mathematical communication skills,

including vocabulary and notation, and learn to use technology as a problem-solving

tool.373 The final standard in this section is Developing Mathematical Power, which

integrates the previous standards. It calls for students to participate in meaningful

problem solving that will develop mathematical ability as well as confidence and

369 American Mathematical Association of Two-Year Colleges, Crossroads in Mathematics, 10.370 Ibid.371 Ibid.372 Ibid., 11.373 Ibid.

164

persistence.374 This expresses a learning theory that requires the incorporation of all the

previous Standards for Intellectual Development.

Ernest’s theory of learning, as described in chapter four, is based on the social

construction of meaning. He believes that language is the vehicle of thought, or the

medium for learning and concept formation. Students learn mathematics by participating

in mathematical language games which are embedded in the forms of life that exist in the

mathematics classroom. These language games include vocabulary, notation, and rules

for use. Through conversation students learn this mathematical language, which allows

them to compare ideas with other students and to test their theories. Some personal

understandings are corroborated, and others are adjusted in light of new information or

the suggestions of others.375 In this way, active engagement is necessary to learning.

Students must explore mathematical topics and discuss their ideas with others. In many

respects, AMATYC’s Standards for Intellectual Development are consistent with

Ernest’s theory of learning. AMATYC encourages active engagement through problem

solving that requires students to pose questions and analyze situations, and the use of

technology to facilitate inquiry. Students are also expected to interact with others,

discussing their solutions and evaluating the reasoning of other students. However, the

importance of communication is not emphasized in the way Ernest would like.

AMATYC describes communication as the ability to use appropriate mathematical

vocabulary and notation, and to understand mathematical arguments.376 This does

promote the learning and use of mathematical language, but it does not capture the

central role of conversation in Ernest’s theory of learning. His is a much more

374 Ibid., 12.375 Ernest, Social Constructivism, 220-21.376 American Mathematical Association of Two-Year Colleges, Crossroads in Mathematics, 11.

165

sophisticated view of communication than AMATYC’s, which reduces communication to

a skill that must be mastered. For Ernest, one cannot merely learn to communicate.

Instead, one must communicate in order to learn. Through conversation students learn

mathematical language games, which allow them to form new conceptual ideas and

negotiate meaning. It is through this process that subjective mathematical knowledge is

constructed. Thus AMATYC’s Standards for Intellectual Development are consistent

with the active engagement and student interaction required by Ernest’s theory of

learning. However, they are not based on his underlying theory that sees conversation as

the vehicle for learning.

Standards for Pedagogy

The Standards for Pedagogy outline principles that should be incorporated by

teachers into their classroom teaching. AMATYC states that these standards “are

compatible with the constructivist point of view” and that they “recommend the use of

instructional strategies that provide for student activity and student-constructed

knowledge.”377 After a brief summary, I will analyze them using Ernest’s theory of

teaching.

There are five Standards for Pedagogy. The first is Teaching with Technology,

which asks teachers to make use of technology that will enhance learning and prepare

students for technology in the workplace.378 These resources can be used as the main

method of instruction, or for supplemental assistance. Second, AMATYC encourages

Interactive and Collaborative Learning, with a focus on learning to communicate about

mathematics. This includes group discussions and projects, which can help students

377 Ibid., 15.378 Ibid.

166

develop an understanding of the language of mathematics and allow them to practice

using its signs, symbols, and vocabulary.379 For developmental students, AMATYC sees

collaborative learning opportunities as “critical to providing positive learning

experiences” because they reduce the power differential between teacher and student.380

The hope is that this will allow students to develop confidence in their own mathematical

skills. The third standard is Connecting with Other Experiences, which corresponds to

Connecting with Other Disciplines in the Standards for Intellectual Development.

Teachers should design classroom activities that build on students’ experiences, making

connections between branches of mathematics as well as connecting it to other disciplines

“so that students will view mathematics as a connected whole relevant to their lives.”381

Students should be shown how mathematics is present in different cultures, and the

contributions different cultures have made to the discipline. The fourth pedagogy

standard is Multiple Approaches. This calls for teachers to model the use of multiple

representations for mathematical concepts, including numerical, graphical, symbolic, and

verbal representations. Students should be encouraged to use alternate representations

and to consider multiple approaches to complex, open-ended problems.382 The final

standard is Experiencing Mathematics, which ties together the other Standards for

Pedagogy. It states that teachers should assign open-ended projects and other

experiences that will allow students to gain confidence, to practice using mathematics,

and to learn to be independent thinkers.383

379 Ibid., 16.380 Ibid., 28.381 Ibid., 16.382 Ibid.383 Ibid., 17.

167

Ernest’s theory of teaching addresses two issues: the teaching of mathematics and

the education of critical democratic citizens. Ernest believes that the learning of

mathematics requires conversation and active engagement, so the teaching of

mathematics must incorporate both. Therefore, teachers should engage in mathematical

discussions with their students and encourage students to interact with each other.384 To

facilitate active engagement as well as the education of critical citizens, Ernest advocates

an investigatory pedagogy based on mathematical inquiry. This teaching approach

presents students with mathematical situations and allows them to pose problems and

investigate possible solutions. These self-directed projects require students to be actively

engaged in mathematical investigation and problem solving. They are also designed to

foster critical awareness and social empowerment. Teachers should choose materials that

reflect the contributions of non-western cultures to mathematics, so that students are

aware of these contributions and how they are often ignored in mainstream

mathematics.385 In addition, Ernest wants teachers to choose socially relevant topics for

mathematical investigation to foster critical awareness. These topics may use

mathematics to uncover social inequity through the analysis of issues such as global

poverty, or they may reveal the misuse of mathematics to make claims in the media,

“from advertisements to government and interest-group pronouncements.”386 Through

group work, students build confidence and develop a sense of personal mathematical

empowerment.387 Ernest hopes that this personal empowerment, coupled with critical

awareness, will lead to social empowerment and student involvement in political or social

movements.384 Ernest, Philosophy of Mathematics Education, 208.385 Ibid., 265.386 Ernest, "Why Teach Mathematics?" under "Capability versus Appreciation."387 Ernest, Philosophy of Mathematics Education, 209.

168

AMATYC’s Standards for Pedagogy recommend a pedagogical approach which

is consistent with many of Ernest’s principles. Teachers are encouraged to assign

activities that allow students to see mathematical connections and the role of mathematics

in their cultures and lives. There is a focus on active problem solving in context, both

individually and collaboratively. AMATYC wants students to experience mathematics,

or in Ernest’s terms, to be immersed and enculturated into mathematical practice.

However, from Ernest’s perspective these standards lack two important features. First, as

with the Standards for Intellectual Development, AMATYC does not incorporate

conversation and social interaction in the prominent way that Ernest does. Collaboration

is offered as a way to teach mathematical language and to foster mathematical confidence

in students. However, AMATYC does not see it as critical to the learning of

mathematics. Where Ernest sees interaction as crucial to the social negotiation of

meaning and the construction of subjective knowledge, AMATYC sees it merely as an

enhancement to problem solving activities. Second, AMATYC’s Standards for Pedagogy

do not include the critical mathematics education that Ernest describes. Ernest wants

students to learn “to identify, interpret, evaluate and critique the mathematics embedded

in social and political systems and claims” through an investigatory pedagogy.388

AMATYC is more focused on preparing students to “use mathematics effectively in their

multiple roles as students, workers, citizens, and consumers.”389 While Ernest believes it

is important for students to develop mathematical skills for employment and daily life, he

also wants students to feel empowered to analyze the mathematics embedded in social

contexts and to act for change when they see inequity. AMATYC’s pedagogical

388 Ernest, "Why Teach Mathematics?" under "Capability versus Appreciation."389 American Mathematical Association of Two-Year Colleges, Crossroads in Mathematics, 24.

169

recommendations are consistent with Ernest’s in many ways, but they do not share his

goals of social construction through conversation and critical democratic engagement.

Overall Analysis

An analysis of the Crossroads standards through an Ernestian lens reveals that

they are consistent with Ernest’s philosophy of mathematics education in many respects.

However, they do not fully realize Ernest’s vision of educational opportunity in

mathematics through access to meaningful content, instruction that incorporates social

construction, and activities that encourage critical analysis of mathematical claims. The

stated purpose of these standards is to improve mathematics learning at two-year colleges

and to encourage more students to study mathematics.390 More specifically, the aim is to

make mathematics meaningful and relevant, so that all students leave college with a

certain level of mathematical knowledge. Ernest would agree with these aims, but he and

AMATYC have different conceptions of mathematics learning and democratic values. I

will examine each in turn.

Ernest and AMATYC agree that mathematics is a growing discipline that

connects to many other disciplines and areas of life. They both believe that learning

mathematics involves both mathematical content and mathematical practice. Students

should be exposed to key mathematical concepts from several different branches of

mathematics, but they must also become familiar with mathematical practice in the form

of problem solving and mathematical reasoning. However, Ernest believes that

mathematical content should include big mathematical ideas while AMATYC

recommends procedures and problem solving tools. Ernest and AMATYC also differ on

how they see the construction of mathematical knowledge in the classroom. AMATYC

390 Ibid., xii.

170

focuses on active engagement to develop content knowledge, mathematical skills, and the

ability to communicate in mathematical terms. But these activities are designed to help

students learn more efficiently by structuring lessons in a way that makes mathematics

meaningful to them personally. This creates a picture of individual students becoming

personally engaged and constructing individual meaning. Ernest, on the other hand,

attributes a much more important role to language and interaction. Conversation does not

just develop communication skills, but is the medium for knowledge construction. Ernest

agrees that active engagement is important, but he sees language as the critical factor in

the learning of mathematics.

Ernest and AMATYC also have different conceptions of what it means to educate

democratic citizens. AMATYC hopes to create productive citizens, with a

developmental curriculum that increases students’ educational and career options.391 To

Ernest, the mark of a democratic citizen is critical awareness and action rather than

productivity. He is not interested in creating efficient workers. Instead, he feels that a

democratic education will provide students with the ability to analyze mathematical

situations in society for bias or discrimination. Both Ernest and AMATYC believe that

students should become empowered through mathematics, but for Ernest this entails

critical awareness, while for AMATYC it involves social mobility in terms of career

opportunities.

From Ernest’s point of view, the weaknesses in Crossroads are mostly issues of

omission. The standards do not contradict Ernest, rather they fall short of his vision for

mathematics education. Through this analysis it can be seen that while AMATYC claims

that its standards are consistent with constructivist principles, there are deeper issues to

391 Ibid., 9, 25.

171

be considered. AMATYC incorporates constructivist activities into its standards, but not

the rich conversation and interaction seen as necessary by social constructivism. These

standards also do not encourage the analysis and critique of the mathematics

underpinning social institutions and political claims. From Ernest’s perspective, these are

major weaknesses.

An analysis such as this can help an instructor or a department to evaluate the

strengths and weaknesses of AMATYC’s standards by identifying potential areas of

concern. This, in turn, can guide implementation. A department might decide to keep

these issues in mind as they design a developmental mathematics program, implementing

AMATYC’s standards in a way that also addresses Ernest’s concerns. For example,

critical mathematics activities and an emphasis on conversation could be integrated into

the recommendations of the Problem Solving standard. On the other hand, another

department could decide that its main concern is the preparation of students for

employment, and thus critical awareness is not a priority for its developmental courses.

Either way, Ernest’s framework has allowed a more thorough consideration of issues

such as the construction of knowledge, the requirements of democratic citizenship, and

what it means to provide educational opportunity.

Standards such as AMATYC’s provide broad guiding principles for curriculum

design. I will next consider developmental course materials, which embody curricula and

suggested educational approaches in a more concrete way for instructors and students.

Specifically, I will look at developmental mathematics textbook selections and their

implications.

172

Developmental Mathematics Textbooks

Textbooks are an important part of mathematics education. There is considerable

evidence that mathematics textbooks establish curricular content as well as classroom

pedagogy.392 Research by Calandra Davis corroborates this at the community college

level. In her study, she interviewed both full-time and part-time mathematics instructors

with various levels of qualifications and teaching experience. All reported that the

textbook factored into their lesson plans.393 In fact, most of the instructors reported that

reading the textbook was their first step in lesson planning, to ensure they were using a

consistent approach. Davis concludes that the textbook determines whether these

instructors will teach a particular topic in a traditional or reform-oriented way.394 Kays

states that the role of the textbook may be even more pronounced in developmental

mathematics programs, because the majority of these courses are taught by part-time

faculty who often have less time to devote to instructional planning.395 With this in mind,

Kays conducted an analysis of all textbooks being used in community college beginning

or intermediate algebra courses in the state of Illinois for the Fall 2003 semester. His

findings show that the textbooks driving the developmental algebra curriculum at these

schools are primarily focused on the replication of skills demonstrated in the text. I will

summarize the framework Kays used for his analysis and consider the implications of his

findings through Ernest’s lens.

Kays’s study analyzes textbooks within a framework that categorizes pedagogical

intent as habituation, enculturation, or construction. These categories are drawn from the

work of David Kirshner, connecting theories from educational psychology to pedagogical 392 Kays, "National Standards, Foundation Mathematics," 37.393 Davis, "Instructional Attitudes," 72.394 Ibid., 104.395 Kays, "National Standards, Foundation Mathematics," 3-4.

173

research in mathematics education.396 The extension of this framework to textbooks is

made possible through the use of a content analysis procedure adapted from the Third

International Mathematics and Science Study (TIMSS). This procedure analyzes the

pedagogical intent of textbooks through its macro structures, which shape interpretations

of the author’s pedagogical perspective and the learning opportunities presented to the

students. Kays analyzes three macro structures: the role of the narrative text, the manner

in which students are expected to engage with the text, and the type of student

performance required by the problem sets.397 Each can be categorized within Kirshner’s

framework, allowing an analysis of the overall pedagogical intent of the textbook.

Habituation draws on behavioral psychology and is characterized by a focus on

the improvement of mathematical skills through repetitive practice.398 Topics are

presented in small, linear pieces. In Kays’s analysis, the narrative text is said to represent

habituation if its primary role is to provide explanation through detailed, worked

examples, replicating a traditional lecture. Content is grouped by topic, creating a

disconnect between one chapter and the next.399 Students are expected to engage with the

text by working through detailed, skill-based examples, and the exercises at the end of

each section primarily ask students to replicate these skills by solving problems similar to

the examples.400 Habituation represents a fairly traditional textbook with clear exposition

and routine problem sets.

Enculturation is based on sociocultural theory and problem solving perspectives.

The classroom is seen as a microculture, and the priority is engaging students in

396 Ibid., 25.397 Ibid., 95.398 Ibid., 26.399 Ibid., 96, 187.400 Ibid., 188-89.

174

mathematical activities that will help socialize them into the mathematical community.

Problem solving is viewed as the hallmark of mathematical practice, so it is central to

enculturation pedagogy.401 Therefore, textbooks focusing on enculturation emphasize

problem solving and use collaboration to build a mathematics community in the

classroom. Students “will develop problems, make conjectures, formulate possible

solution processes, and present and justify solutions to their mathematics learning

community…. Together, students develop the language and habits of a mathematical

community.”402 For Kays’s categorization, the narrative text in an enculturation-based

textbook is less directive, with themes that connect across chapters. There are more

complex problems than in an habituation text, with fewer completely worked out

examples.403 The text encourages students to pose questions and models analysis and

problem-solving strategies, with results presented using mathematically correct language

and symbols. The expected student engagement is active, as students are guided through

investigations of problems in context. The problem sets are often extensions of these

investigations, requiring students to reason collaboratively about open-ended problems

with multiple solutions.404

In Kirshner’s framework, construction refers to the radical constructivism of von

Glasersfeld.405 Thus the focus is on the construction of mathematical concepts through

cognitive perturbations and accommodations, as described in chapter two of this paper.

While some constructivists encourage problem solving, Kays interprets Kirshner’s

construction to involve more writing activities. As with enculturation textbooks, Kays

401 Ibid., 30.402 Ibid., 97.403 Ibid., 96-97.404 Ibid., 187-89.405 Ibid., 35.

175

explains that construction textbooks will use the narrative text to guide students through

an investigation.406 Instead of open-ended problem solving, however, these investigations

are highly structured to help students develop a conceptual understanding of a particular

concept or skill. Students engage with the text by working through the activities and

forming oral or written reflective responses. Rather than traditional problem sets, student

performance takes the form of extensive writing activities. Reflective writing and

explanation are used to help students refine their understanding of the concepts and skills

they have learned.407

Of these three categories, only enculturation textbooks are consistent with

Ernest’s philosophy of mathematics education. They make use of an investigatory

pedagogy very similar to what Ernest has in mind. Like the problem posing and solving

he describes, the approach in an enculturation textbook is to encourage students to pose

their own questions and to experiment with different solution methods through open-

ended problem solving activities. In addition, students are expected to collaborate and to

practice using mathematical vocabulary and symbols. As in Ernest’s philosophy, the

focus is on the enculturation of students into mathematical forms of life, specifically

mathematical language and mathematical practice in the form of problem solving.

Using Kirshner’s framework, Kays’s study coded the macro structures of each

textbook being used in beginning or intermediate algebra classes at Illinois community

colleges in the Fall 2003 semester, and categorized them by overall pedagogical intent.

There were 47 textbooks total. Of these, Kays’s results show that 46 focused primarily

on habituation. The remaining textbook was categorized as construction.408 Through

406 Ibid., 97.407 Ibid., 187-89.408 Ibid., 141.

176

Ernest’s lens, this is particularly troubling. The construction-based textbook has some

promising features, in that these books focus on conceptual understanding and encourage

students to write and speak mathematically. Students also participate in both individual

and group activities, but they tend to be structured rather than open-ended. In addition, a

construction textbook is one that uses personal reflection as the central component in the

learning process. It does not encourage the level of collaboration and open-ended

problem solving that Ernest’s social constructivism requires. Of more concern, however,

is the dominance of habituation-based textbooks. These books provide highly structured

exposition and worked examples, leaving little room for exploration of concepts.

Students are expected to work through these examples and take detailed notes. The

problem sets mirror the examples in the text, requiring students to replicate algebraic

skills to arrive at one correct answer.409 From a behaviorist point of view, this type of

textbook is seen as highly effective. Students are provided with clear examples, and each

skill or topic builds incrementally on the previous one. This approach provides

maximum guidance to help students master algebraic skills. From Ernest’s perspective,

this promotes rote memorization and a lack of conceptual understanding. Disjoint topics

hide the connections between different mathematical concepts. Habituation encourages

independent practice of basic skills rather than the collaboration and complex problem

solving that Ernest describes. From a learning perspective, Ernest sees this approach as

an ineffective way to foster the social construction of subjective mathematical

knowledge. Students do not engage in mathematical reasoning or investigations, nor are

they socialized into mathematical practice. Through the lens of Ernest’s democratic

principles, habituation produces citizens who do not see the mathematics permeating their

409 Ibid., 187-89.

177

lives and social systems. They instead learn to see mathematics as rigid and procedural,

and therefore they do not question its use in politics or the media. For Ernest,

habituation-based textbooks are objectionable because they fail to promote the process of

social construction, and they do not encourage the development of critical democratic

citizens. Thus he would be deeply concerned that in Illinois, nearly all community

college beginning and intermediate algebra courses are using these types of textbooks.

This type of analysis provides an alternate framework for evaluating potential

textbooks. Traditionally, habituation-based textbooks have been viewed as effective in

developmental mathematics courses because they break the material up into small pieces,

and provide students with clear examples and expectations. Evaluation through Ernest’s

lens provides another perspective that mathematics faculty may not have considered

before, giving a theoretical basis for the importance of collaboration and problem solving.

Some might be prompted to explore enculturation-based options in their textbook

searches. Others may continue to find useful elements in the habituation approach, but

also see a need to incorporate aspects of enculturation into their classrooms. For

example, an instructor may feel that it is important to provide students with a textbook

that includes clear, fully worked examples of skill-based problems. But the instructor

may also believe that open-ended problem solving activities are critical to the

development of conceptual understanding. She could use habituation-based textbooks

and provide opportunities to practice algebraic skills, while also setting aside class time

for mathematical investigations. While not all mathematics faculty members will find

Ernest’s perspective valuable, in conjunction with traditional measures it provides a

178

richer evaluation tool that could influence textbook decisions or classroom

implementation.

While textbooks have a strong influence on curriculum and pedagogy, it is

classroom instruction that has the most direct impact on students. This is also the area in

which instructors have the most control. Therefore, the final aspect of developmental

mathematics that I will explore is the implementation of pedagogy.

Developmental Mathematics Classroom Pedagogy

To illustrate the philosophical analysis of pedagogy through Ernest’s lens, I will

consider a study conducted in developmental mathematics classrooms. Blais’s research

examines developmental algebra teaching at the college level, including case studies of

three instructors. One of these instructors, Ms. Anderson, is a part-time community

college instructor teaching beginning algebra. Blais provides the following excerpt from

a class he observed:

MS. ANDERSON: We are going to be talking about the slope of a straight line. In other words, the way it slants or the angle that it goes up and we can define slope of a line [writes]

We are going to have a formula for it in a few minutes. But let’s say here’s a new line, we haven’t done this one yet. [writes]

[Plots the line]

179

(0, 2)

(4, 0)0

(0, 2)

(4, 0)0

1 1( , )x y

2 2( , )x y

x

y

Now, if we take any set of two points, … in other words let’s take (0, 2) and (4, 0). We can find out what the change in y over the change in x would be and the actual formula we are going to be using is

Notice this is down, this is not an exponent, this is a subscript, it is

labeling a kind of y. Let’s make this point one, so this is and this

is . [The points (0, 2) and (4, 0) are labeled and as shown in the graph below.]

x

y

All right? Now, I just follow my formula [writes]

So, for this line m = –1/2. Now, what that tells us, remember that a negative fraction means that either the numerator is negative and the denominator is positive or … in other words, they are different and so we call the fraction negative. So this means, remember it’s the y on top, now we take the negative as being the top. Then in this particular line that we just graphed, from any point, for everyone of those down, it goes over two and you could make a little staircase out of it, I mean you could have seen that right? And we could keep going here—approximating—that one’s a little long… This is what slope means.

STUDENT: What does m mean?

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MS. ANDERSON: Slope. It’s the letter we use. You do need to know this formula:

What would happen if had been called instead?[Ms. Anderson demonstrates that it would have been the same.]Now this [points to the line in the graph above] is called a negative slope, when a line is slanting in this direction, anywhere around here it’s a negative slope. A positive [slope] is slanting in the other direction.410

According to Blais, this example is representative of a typical lesson in Ms. Anderson’s

classroom. Her goal is to help students develop the ability to think about mathematics in

a logical, step-by-step fashion.411 Blais feels that her teaching encourages memorization

of rules, and focuses on modeling the correct use of rules, formulas, and procedures. His

analysis of her lesson concludes that while this method is effective for teaching algebraic

skills, it encourages imitation of demonstrated procedures rather than developing

conceptual understanding.412

Ernest, too, would criticize this teaching excerpt, which represents a fairly

traditional approach to mathematics teaching. Ernest believes students must encounter

mathematical concepts in context through active engagement. The example the teacher

uses is removed from any context. This makes it difficult for students to see how the

concept of slope is important to mathematical thought, how it connects to other concepts

in mathematics, and how it appears in other disciplines. Slope is the basic skill

underlying the concept of rate of change, which is important in both the social and

physical sciences. However, students will not get a sense of this from Ms. Anderson’s

lesson. In addition, the students are not actively engaged in any sort of activity. There is

not even much interaction between students and the teacher as she presents her lecture. 410 Blais, "Constructivism Applied to Algebra," 128-30. Comments in brackets are in the original.411 Ibid., 120.412 Ibid., 131, 134.

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For Ernest, conversation is critical to the construction of subjective knowledge and the

negotiation of meaning. The students in this teaching excerpt are not given the

opportunity to engage with the material or to interact with each other.

An important part of Ernest’s philosophy is the enculturation of students into

mathematical practice. For him, mathematical practice involves the creation and

justification of mathematical knowledge. Students should engage in mathematical

investigations, posing questions and explaining their solutions to others. In Ms.

Anderson’s lesson, students are not enculturated into mathematical practice in Ernest’s

sense. Rather than allowing students to explore the concept of slope through problem

posing and solving, she immediately explains what slope is and how it can be calculated.

Ms. Anderson sees mathematical knowledge as a body of memorized formulas and

procedures, and her lesson portrays mathematical practice as their recall and use. For

Ernest, formulas and procedures are just the skills that are a necessary prerequisite to

engaging in mathematical practice. Mathematical practice itself is the creation of new

ideas through mathematical investigation. In Ernest’s view, Ms. Anderson is teaching

basic skills but is not teaching her students what it means to engage in mathematical

activity.

Ernest believes that educating critical democratic citizens is also a crucial part of

mathematics education. Students should be given the freedom to pose their own

questions about socially relevant situations to encourage them to critique the use of

mathematics in politics and other arenas. Ms. Anderson does not provide an opportunity

for students to form questions, nor does she embed the concept of slope in a context that

encourages critical analysis. Instead, her students are encouraged to accept mathematical

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facts as given, without considering their origins or their uses. They do not connect

mathematics to their lives, nor do they recognize its role in social contexts. From

Ernest’s perspective, Ms. Anderson’s pedagogy does not empower her students, either

personally or socially.

Ernest does not provide specific examples of how to implement his

recommendations in the classroom. However, an alternate lesson involving slope could

be designed within his framework. For example, the lesson would need to be centered on

some sort of critical awareness activity, connecting mathematics to students’ lives.

Students could be asked to work in groups, and to find data on the internet for different

geographic areas in their state. This data could include demographic statistics such as

ethnicity, income, and highest level of education. Ernest’s investigatory pedagogy

requires that students be allowed to pose their own questions, so the teacher could not tell

students exactly what to do with this data. However, some guidance could be given if

students were unsure where to start. If you compare two sets of data, are there any

patterns? How could you make it easier to see these patterns? Each group of students

should be encouraged to try different methods of organizing their data, to discuss their

strategies with each other, and to choose a strategy that seems to provide the most

information. This will foster social construction and negotiation of meaning as students

discuss what can be learned from different strategies. Some students will create scatter

plots relating two sets of data. The concept of slope and rates of change, and possibly

linear regression, will emerge as they examine patterns of increase and decrease. For

example, students may notice that poverty rates increase when the percentage of residents

who are considered ethnic minorities increases, or that poverty rates decrease with higher

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rates of college degrees. Students might be drawn, then, to compare the correlation

between poverty and college degrees with the correlation between minority status and

college degrees. Are the rates of change similar? Or does one category seem to have a

stronger effect than the other on the percentage of college degrees in an area? Other

types of data could be used, depending on the students’ interests, to explore other social

issues in their state. Not only will students connect the concept of slope to its meaning in

context, but they will also encounter difficult questions about equity as well as social

correlations that might inform strategies for change. This will develop critical awareness

and, ideally, social empowerment for action. While this greatly oversimplifies issues of

implementation, it gives an idea of how a lesson might look if it were designed using

Ernest’s recommendations.

While Ernest does not provide any sample lessons as exemplars, there are other

sources that instructors can look to for guidance in planning lessons on various topics.

Several researchers have designed classroom projects based on principles similar to

Ernest’s. Ole Skovsmose, whose work in critical mathematics education is cited by

Ernest, provides some examples in his book Towards a Philosophy of Critical

Mathematics Education. Skovsmose’s projects begin with a mathematical situation, or

“scene” in his terms, that will be the context for investigations over the course of many

class periods.413 Most of these units include questions introduced by the teacher, rather

than problems posed by the students as Ernest envisions. However, they are designed to

encourage sustained group inquiry and critique of “authentic, real-life applications of

mathematics,” which closely resembles Ernest’s investigatory pedagogy.414 The projects

413 Skovsmose, Critical Mathematics Education, 91.414 Ibid., 141.

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address issues such as energy consumption, welfare, and political issues in the students’

community. Students are given time to explore some aspect of the given situation, and

then they are asked to use mathematics to analyze a particular issue. This incorporates

both the problem solving and critical aspects of Ernest’s investigatory approach.

Gutstein has designed an entire curriculum based on teaching for social justice, which he

describes as a pedagogical approach with two goals. It teaches students the mathematics

skills needed for further study, while also engaging them in activities that teach them to

interpret social issues mathematically and to use mathematics to work for social

change.415 Gutstein uses topics such as the negative effects of real estate development,

distribution of world wealth, and racial profiling to engage students in group problem

solving with the goal of empowering them to challenge situations they see as unjust. As

with Skovsmose’s projects, questions are posed by the teacher but otherwise make use of

the inquiry, problem solving, and social critique that Ernest envisions. While projects

such as these do not align perfectly with Ernest’s philosophy, they do follow most of his

recommendations. Community college developmental mathematics instructors wanting

to implement Ernest’s approach could use them as a starting point and adapt them as

needed.

This case study analysis illustrates the implications of Ernest’s philosophy of

mathematics education for developmental mathematics pedagogy. Ernest’s framework is

not intended here as a model of perfection, but rather as another way of thinking about

developmental teaching. It provides an alternative perspective that asks teachers and

program coordinators to examine their own instructional aims and to explore the

implications of current teaching methods. Instructors may decide to model their entire

415 Gutstein, Reading and Writing the World, 23-31.

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curriculum on Ernest’s recommendations, or to integrate them with more traditional

teaching methods. For example, some teachers may prefer to begin with a lesson similar

to Ms. Anderson’s one day, and then follow it with an investigatory activity the next.

Even if one does not agree with Ernest’s point of view, his framework highlights issues

such as teaching for conceptual understanding and democratic citizenship. Educators can

learn from this type of reflection and use it to guide curriculum and pedagogy.

Obstacles to the Application of Ernest’s Framework

There are several obstacles to the application of Ernest’s framework in the

community college developmental mathematics context. His approach offers a different

way of thinking about teaching and learning, which would shift the focus from

quantitatively measured outcomes to knowledge construction and the development of

critical democratic citizens. This shift will not be immediately embraced by all those

involved. Difficulties will arise surrounding implementation, faculty support, and

assessment of outcomes.

Many of the difficulties for implementation of Ernest’s pedagogy are not unique

to this context. As mentioned in chapter four, these types of lessons are time consuming

to plan, and difficult to implement due to the freedom students are afforded in their

mathematical investigations and the classroom norms that must be established. This

implementation is made even more difficult, however, by the diversity of students in a

community college developmental mathematics classroom. As outlined earlier,

community colleges draw a diverse student body in terms of ethnicity, age, academic

background, and educational goals. To a greater extent than other classes, developmental

classrooms reflect this full range of diversity. Students are more likely to be enrolled in

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developmental mathematics if they identify as an ethnic minority, are above traditional

college age, come from a low-income family, or have a disability.416 The social dynamic

that occurs in such a heterogeneous classroom presents obstacles to the full

implementation of the conversation and collaborative inquiry in Ernest’s approach. In

chapter four, I described Habermas’s theory of the lifeworld and its effect on ideal speech

situations. This is even more of an issue in light of the diversity found in the community

college developmental mathematics context. Social norms and implicit power relations

will affect the freedom and openness with which students engage in conversation and

collaborate on projects. These issues can arise in the interaction between students of

different social groups, but other differences can also impose constraints on conversation.

Students who are far apart in age may find difficulty conversing as peers. Those who

have weaker academic backgrounds may not feel comfortable sharing their ideas freely

with students who they perceive as mathematically talented. These power relations are

difficult to overcome completely. However, as McCarthy argues, the ideal speech

situation is still an ideal worth working toward.417 The basis of Ernest’s philosophy in

conversation allows instructors to address these issues directly by opening a dialogue

between students.

Faculty who want to implement Ernest’s approach in developmental mathematics

classes will need support as they learn how to manage this new classroom dynamic.

W. Norton Grubb and Associates found that while most innovation in developmental

education is the result of a collective effort, community colleges often have a culture of

instructor isolation.418 Faculty do not have widespread opportunities to collaborate and

416 U.S. Department of Education, Profile of Undergraduates, 138.417 McCarthy, Critical Theory of Habermas, 309.418 Grubb and Associates, Honored but Invisible, 199, 283-85.

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support each other in instructional planning or implementation of new ideas. This is

compounded by the fact that at community colleges, developmental courses are most

often taught by part-time faculty.419 Part-time instructors are even more isolated, making

them less likely to collaborate with others or to volunteer for new initiatives. They

usually have multiple outside commitments and their salary is based solely on teaching.

Therefore, part-time faculty tend to have less time and less incentive to participate in

professional development or to coordinate with other faculty.420 This is a major obstacle

to the implementation of Ernest’s framework in developmental mathematics, which can

only be overcome through efforts to increase faculty collegiality and to better integrate

part-time faculty as members of the department.

Another obstacle to the implementation of Ernest’s framework is the set of long-

held beliefs many instructors have about developmental mathematics education. There

may be resistance to the investigatory aspect of Ernest’s pedagogy. As Grubb observes,

changes in teaching require instructors to overcome deeply rooted ideas about teaching

and learning. This does not happen easily or quickly.421 Currently, behaviorist thinking

is very influential in developmental mathematics. Many instructors believe that its

measurable objectives, systematically organized content, and frequent assessment provide

students with the structure and organization necessary for effective learning. In fact,

these instructional practices are often cited as characteristics of effective developmental

programs.422 In mathematics, the result is that the goals of these courses are often seen as

remedial rather than developmental—the content is seen as a set of skills and procedures

419 Roueche and Roueche, High Stakes, High Performance, 26.420 Grubb and Associates, Honored but Invisible, 332-34.421 Ibid., 247.422 Boylan, What Works, 78, 87.

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that must be drilled and mastered.423 Ernest’s investigatory pedagogy may be seen by

faculty as a distraction from important basic skills. In addition, faculty may believe that

developmental students lack the mathematical ability to benefit from a problem-based

curriculum. The meaningful content and mathematical inquiry in Ernest’s approach may

be perceived as too difficult for use in developmental mathematics classrooms. However,

there is growing evidence that project or problem-based curricula with a focus on student

interaction are both accessible and beneficial to students at this level. In a study of

college students in a developmental mathematics course, Laura Coffin Koch found that

students taught using a collaborative problem-solving approach scored higher on a skills

post-test and passed the course at a higher rate than students taught using traditional

methods.424 While the pedagogy in Koch’s study is based on the views of von

Glasersfeld, her findings indicate that a move away from a behaviorist model is most

likely a positive one. Ernest’s focus on conversation would only enhance these benefits.

David Pugalee found that, in an at-risk high school algebra class, student discourse

supported the development of meaningful mathematical knowledge.425 This is supported

by AMATYC in its standards for developmental mathematics. Though it will take time

and effort to develop widespread support among faculty for an investigatory pedagogy,

the value of these instructional methods is becoming more widely acknowledged in

professional journals and among members of professional organizations.

For some community college educators, the critical aspect of Ernest’s approach

may be more difficult to support than its basis in problem-solving and conversation.

Many faculty believe that the democratic responsibility placed on community college

423 Grubb and Associates, Honored but Invisible, 190.424 Koch, "Revisiting Mathematics."425 Pugalee, "Algebra for All."

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developmental mathematics is to provide students with mathematical skills that will open

educational and career opportunities. As mentioned in the discussion of AMATYC’s

standards, this is the view that AMATYC holds. Unlike Ernest, these educators do not

feel that this responsibility includes an obligation to foster critical awareness of social

issues. Many do not see these issues as relevant to mathematics education, or fear their

controversial nature. Ernest acknowledges these difficulties. He explains that the

dominant view of mathematics as neutral and value-free causes instructors to reject social

and political values as part of the mathematics curriculum.426 Some educators fear that

introducing these topics into the classroom will be cause problems. Students who

disagree with each other may be unable to engage in rational discussion, causing conflict

and argument. In addition, the exploration of social issues may be seen as promoting a

personal or political agenda. Ernest cites governmental intervention in some countries

when social responsibility was incorporated into the high school physics curriculum.427

However, the community college developmental mathematics context is different from

that of elementary and high school. While instructors may share some of the same

reservations that Ernest discusses, at the college level they are afforded a little more

freedom to address difficult or controversial ideas. Recently, there has been a trend in the

literature encouraging more attention at the community college to civic education and the

engagement of students in critical thinking and reflection on social and moral issues.428

Proponents believe that educational experiences in college should encourage higher-level

critical thinking and the development of informed opinions. In developmental education,

426 Ernest, Philosophy of Mathematics Education, 211-12.427 Ibid., 211.428 For example, see American College Personnel Association and National Association of Student

Personnel Administrators, Learning Reconsidered, 16; Higginbottom and Romano, "Efficacy of Civic Education," 24; O'Banion, Learning-Centered Community Colleges, 17.

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in particular, some researchers have begun to embrace transformation theory, a critical

education theory put forth by adult educator Jack Mezirow. Transformative education

encourages students to critically reflect on the context and assumptions associated with

information they receive, and to critically self-reflect on their own assumptions and

beliefs.429 So far, this has been implemented less in mathematics than in the humanities

and social sciences. Ernest’s framework allows faculty to incorporate these principles

into developmental mathematics education, giving students the opportunity to use

mathematics to engage in critical inquiry and to draw conclusions based on the analysis

of evidence.

In addition to the support of faculty, full implementation of Ernest’s approach in

developmental mathematics would require the support of administrators and other

stakeholders. Community colleges receive the majority of their revenues from state and

local governments, as well as student tuition and fees.430 Therefore, they need the support

of government officials and members of the community in order to fund projects and

programs. Especially in developmental education, where many policy makers believe

programs are duplicating the high school curriculum, there has been an increasing

demand for accountability in terms of student outcomes, efficiency, and productivity.431

However, the student outcomes that are emphasized in Ernest’s approach are not easy to

assess in the standard ways. The success of developmental programs tends to be

evaluated based on student pass rates in current courses, pass rates in subsequent college-

429 American College Personnel Association and National Association of Student Personnel Administrators, Learning Reconsidered, 10-11; Higbee, Arendale, and Lundell, "Access and Retention," 7; Mezirow, "On Critical Reflection."

430 U. S. Department of Education, Education Statistics 2007, 488.431 McMillan, Parke, and Lanning, "Remedial/Developmental Education Approaches," 23.

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level courses, and graduation and transfer rates.432 While it is desirable to have these

types of outcomes in developmental mathematics, this is a very limited view that does not

reflect qualitative differences in student development or thinking. Two different

approaches may produce roughly equivalent outcomes in terms of pass rates and

graduation rates, but have very different outcomes in terms of depth of understanding and

critical thinking skills. In particular, these traditional measurements do not readily

differentiate between mastery of skills and Ernest’s goals of authentic mathematical

experiences, social construction of knowledge, and critical awareness of social issues. To

garner support for Ernest’s approach, the value of what is going on in the classroom will

need to be made more transparent. A recent push for higher education to move toward a

learning-centered focus433 has resulted in recommendations that are applicable to this

context. It is suggested that assessment, both at the program level and in the classroom,

focus on student learning and include tools such as questionnaires, observations of

student behavior, portfolios, data from group work, and case studies. Rubrics should be

developed that describe levels of growth and learning.434 This type of evaluation requires

time and resources, but is valuable for helping administrators and stakeholders to better

understand the impact of particular teaching approaches. These qualitative assessments

would allow faculty implementing Ernest’s framework to document the depth of

understanding students gain from conversation and mathematical inquiry, as well as

student growth in terms of critical awareness and empowerment.

432 Blais, "Constructivism Applied to Algebra," 7; McMillan, Parke, and Lanning, "Remedial/Developmental Education Approaches," 23-24.

433 See American College Personnel Association and National Association of Student Personnel Administrators, Learning Reconsidered; Barr and Tagg, "From Teaching to Learning"; O'Banion, Learning-Centered Community Colleges.

434 American College Personnel Association and National Association of Student Personnel Administrators, Learning Reconsidered, 23.

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While there are many obstacles to implementing Ernest’s approach in the

community college developmental mathematics context, it is not an impossible goal. The

process will require hard work, dedication, and patience, but to me the benefits for

students seem worth the effort. Ernest offers students a richer view of mathematics than

traditional methods by introducing them to interesting mathematical concepts in the

world around them. This view also emphasizes that mathematics is not a set of disjointed

content items, but rather a process of inquiry and problem-solving. In addition, Ernest’s

approach provides students with a more engaging educational experience than traditional

approaches. Developmental mathematics students often have histories of social

disadvantage or failure in previous mathematics classes. Many feel stigmatized for being

placed into developmental courses. Ernest’s investigatory pedagogy may help them to

see mathematics in a different way, and to become more engaged in class. Ideally, this

will lead students to feel mathematically empowered, perhaps for the first time. The

critical aspect of Ernest’s pedagogy can help these students to analyze the social

conditions that may have played a role in their lives, providing them with the social

empowerment to work toward changing their own circumstances. Students who have not

experienced any kind of marginalization will benefit as well, as they learn to become

critically aware of social situations and to advocate for change when it seems necessary.

Through this experience, all students have the opportunity to become more confident in

their mathematical ability and more engaged as democratic citizens. While full

implementation of Ernest’s approach may be overwhelming, integrating it with some

traditional methods would accomplish many of the same goals. For example, some

lectures and skills-practice can be integrated into a classroom otherwise guided by

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Ernest’s philosophy. With thoughtful planning, these activities can be used in

conjunction with investigations without compromising the mathematical and critical

value of the latter. Ernest’s approach requires one to ask questions about the purposes of

developmental mathematics education at the community college, the nature of

mathematics learning, the characteristics of democratic citizenship, and the meaning of

educational opportunity. By focusing on the underlying philosophy rather than a

checklist of instructional practices, faculty can find a balance in their teaching methods

that allows them to present structured information while still preserving Ernest’s

worthwhile vision of mathematical engagement and social empowerment.

Summary and Conclusion

The community college developmental mathematics context is unique because of

its historical context and its student characteristics. With a history of offering a variety of

educational programs and a policy of open-admissions, community colleges draw a

student body that is diverse in many ways, with a variety of educational goals. Many

faculty members are unsure how to address this diversity, and they fall back on familiar,

traditional instructional approaches. The application of Ernest’s framework allows us to

analyze the implications of classroom practice for issues of educational opportunity in

this context. It also provides a theoretical basis for analyzing policy and practice in

developmental mathematics at the community college level.

The application of Ernest’s framework in developmental mathematics would

require major changes from the traditional view on course content and instructional

methods. More importantly, it would highlight areas for discussion and reflection. This

analysis would require faculty and administrators to examine what it means to provide

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effective mathematics instruction and to educate democratic citizens. Whether Ernest’s

philosophy and recommendations were adopted or not, they would allow for the

thoughtful contemplation of aims, justifications, hidden assumptions, and possible

directions for the future. It would encourage careful planning rather than haphazard

decision making.

The experiences students have in community college developmental mathematics

classrooms are shaped by many factors, including the recommendations of professional

organizations, the selection of course materials, and classroom instructional methods.

Ernest’s framework provides a lens for their evaluation and an analysis of their

implications. I have illustrated what this might look like by analyzing the AMATYC

standards for developmental mathematics, the types of developmental mathematics

textbooks used in Illinois community colleges, and an excerpt from a developmental

algebra lesson in a community college classroom. Not only does this analysis reveal the

level of alignment with Ernest’s philosophy, but it also demonstrates how Ernest’s

framework can highlight issues about mathematical learning and democratic aims. I

believe my analysis contributes to discussions about the justification of mathematics

education for community college developmental students, the implications of current

practice, and educational opportunity in the community college developmental

mathematics context.

The purpose of this dissertation was to analyze the theoretical underpinnings and

implications of Ernest’s social constructivist philosophy of mathematics education. I

outlined the different varieties of constructivism to provide theoretical context for

Ernest’s philosophy, but also to demonstrate how his version of constructivism differs

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from the others. Ernest’s social constructivism departs from what is generally referred to

as constructivism in the mathematics education literature, which draws on von

Glasersfeld’s radical constructivism. I discussed important theoretical influences on

Ernest’s philosophy, namely Vygotsky’s social theory of mind, Wittgenstein’s

philosophy of language, and Lakatos’s quasi-empiricist philosophy of mathematics.

Ernest draws from these ideas to form a version of constructivism that incorporates

language and social interaction as the means of constructing knowledge, and the logic of

mathematical discovery as a model for mathematical practice and learning. I analyzed

Ernest’s differentiated curricular recommendations for general mathematics education

and the education of future mathematicians. This revealed that while Ernest’s general

conceptions of mathematical knowledge and the democratic ideal are similar for these

two groups, he views mathematicians as producers of knowledge and the general public

as users of mathematical knowledge. This characterization seems to place research

mathematicians in a position of power over the lives of others. Finally, I explored the

implications of Ernest’s framework for community college developmental mathematics.

I discussed what his view would contribute to conversations about educational

opportunity and how this framework might be used to evaluate policy and practice at this

level.

This project contributes to the literature in philosophy of education, mathematics

education, and developmental education. Ernest’s work is fairly unique in providing a

comprehensive philosophy of education that is specific to mathematics. So far, there

appear to be no in-depth analyses of his view. My analysis adds to the literature in this

field and provides insight into possibilities for application. It also provides a much-

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needed philosophical framework for the analysis of developmental mathematics in the

community college. Future research could apply Ernest’s philosophy of mathematics

education to policy issues in different areas. While learning theories in the literature

consider how learning happens, Ernest’s philosophy also places learning in a larger

context of aims and goals. This framework could contribute to discussions about

graduation requirements, standards, tracking, and a number of other issues. Another

direction for future research is implementation in the classroom. Ernest makes general

curricular and pedagogical recommendations, but he does not provide examples of course

materials or classroom excerpts. It would be helpful to have a more detailed mathematics

curriculum that aligns with Ernest’s philosophy. A set of social constructivist classroom

activities would help instructors implement Ernest’s ideas, and empirical research could

investigate the success of this implementation in terms of Ernest’s goals.

Philosophy is a valuable tool for the analysis of issues in mathematics education.

It provides a framework in which to discuss appropriate aims and justifications for the

teaching of mathematics, theories of teaching and learning, and the implications of

policies and classroom practices. Ernest’s view, in particular, highlights important issues

about the nature of mathematical knowledge and the democratic purposes of mathematics

education in public institutions. This theory provides a useful perspective to frame

discussions about the philosophy of mathematics education and its applications.

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AUTHOR’S BIOGRAPHY

Erin Cecilia Wilding-Martin was born in Jacksonville, Illinois, on August 16,

1978. In 1999, she earned a Bachelor of Science degree in Mathematics and Computer

Science from Illinois College, graduating summa cum laude. She was awarded the

Charlotte Hart Prize in Mathematics and was inducted into the Phi Beta Kappa Society.

After earning a Master of Science in Mathematics from the University of Illinois at

Urbana-Champaign in 2000, she began doctoral work in the Department of Educational

Policy Studies. She is an Associate Professor of Mathematics at Parkland College in

Champaign, Illinois, where she has taught since 2001.

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