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dissertation entitled

THE CENTRALITY OF A TEACHER'S PROFESSIONAL TRANSFORMATION
IN THE DEVELOPMENT OF MATHEMATICAL POWER: A CASE
STUDY OF ONE HIGH SCHOOL MATHEMATICS TEACHER

presented by

Janice Simonson Gormas

has been accepted towards fulﬁllment

of the requirements for
Curriculum, Teaching &

Ph . D . degree in Educational Policy

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Date 5/21/95?

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1/98 animus-p14

THE CENTRALITY OF A TEACHER’S PROFESSIONAL TRANSFORMATION IN
THE DEVELOPMENT OF MATHEMATICAL POWER:
A CASE STUDY OF ONE HIGH SCHOOL MATHEMATICS TEACHER
By

Janice Simonson Gonnas

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Teacher Education

1998

ABSTRACT
THE CENTRALITY OF A TEACHER’S PROFESSIONAL TRANSFORMATION IN
THE DEVELOPMENT OF MATHEMATICAL POWER:
A CASE STUDY OF ONE HIGH SCHOOL MATHEMATICS TEACHER
By

Janice Simonson Gormas

This dissertation is the result of a ﬁve year study of a high school mathematics
teacher engrossed in change; changes that added up to a professional transformation that
positioned him to provide opportunities for his students to develop mathematical power,
as deﬁned by the National Council of Teachers of Mathematics. The data analyzed,
including informal conversations, formal interviews, ﬁeld observations, the teacher’s
writings and student work, ended up revealing epistemological shifts and philosophical
shifts, informing and informed by a radically altered instructional practice.

The body of the dissertation (Chapters 2-4) describes the components of the
transformation, including a changed view of the goals of learning mathematics, a changed
philosophy of mathematics and a changed view of the nature of teaching mathematics.
These changes and the resulting mathematical power are illustrated in three vignettes
(representing three progressive stages of the teacher’s thinking and instruction) that
appear in the Introduction.

As the teacher expands his goals for student learning to include a close look at the
“why’s” of mathematics, he ends up inviting them to explore, make conjectures and to use

logical reasoning and mathematical conversations to support or reﬁne conjectures and

work toward a consensus. These signs of mathematical power were made possible by
changing his view of school mathematics to include the analysis of patterns and number
systems rather than merely developing isolated symbolic manipulation skills. Eventually,
he comes to recognize mathematics as a human construction, developed in a historical and
cultural context -- ideas that he transfers to his students mathematical activities, allowing
them the opportunity to socially construct mathematics. He encourages them to
communicate their mathematical ideas, connect their mathematical conceptions and
develop an understanding that can transfer to different situations.

The changes this teacher experienced within himself and incorporated in his
instmctional practice were heart-felt and deeply rooted. The transformation happened
largely as a response to his own frustrations around his students’ inability to make
connections. Other factors that contributed to this transformation include a school
atmosphere that promoted change as the school worked closely with a local university that
provided resources including an innovative curriculum. technological tools and
mathematics educators interested in discussing change and innovation. The real story here
is the transformation the teacher experienced, completely dismantling his views of
mathematics. teaching and learning. He then reconstructed his practice based on a new set
of beliefs that reinforced one another and were well thought-out. This transformation
positioned him to provide opportunities for all of his students to gain mathematical power.
We can learn from this experience that professional transformation of this nature can and
did happen to a mathematics teacher and is closely related to developing mathematical

power, in both the teacher experiencing the transformation and in the students.

ACKNOWLEDGMENTS

Apart from the grace of my Lord Jesus Christ, the love of God, and the fellowship
of the Holy Spirit this study would never have come to fruition. For that I am very
thankful.

I would like to extend a sincere thanks to the teacher featured in this work for
recognizing the need for change in the teaching of high school mathematics and for being
completely willing and extremely helpful in getting this version of his story told. It is his
dedication to the profession that sets the stage for such a dynamic transformation and for
that I am both grateﬁil and encouraged.

The research related to this study took place in a working Professional
Development School, an arrangement introduced by the Holmes Group (1990). The
relationship the university had with this high school served to open the door for many
conversations with the teacher and for the data collection. This connection advanced an
atmosphere at the school that promoted inquiry into practice and at the university that
allowed practice to inform inquiry. I am very grateful for the support and Opportunities
afforded me in conducting this research made possible by the PDS arrangement

I would like to thank the members of my dissertation committee, Perry Lanier,
Chris Clark, Steve Weiland, Bill Rosenthal and Dan Chazan for sharing their insight,

asking hard questions, making thoughtful contributions and engaging with me throughout

this endeavor.

iv

I owe a great deal of gratitude to Dan Chazan, who served as a mentor in many
ways throughout this study. Dan’s gentle prodding, wise advice, thoughtful questions and
on-going encouragement helped me immensely. I would also like to thank Dan, the
director of this dissertation, for an awesome experience in distance learning. As we
communicated and argued across continents over the past six months, in spite of the pain,
my learning grew by leaps and bounds with each rewrite. Thank you for your patience
and careful scrutinizing of ideas, connections and for holding me accountable to a higher
standard than I felt comfortable. I am looking forward to further collaboration around the
ideas contained in this manuscript.

My gratitude to (Mr) Bill Rosenthal extends in many directions. First, thank you
for taking the time to listen to my ideas, confront inconsistencies and for incorporating
some of these raw ideas into the courses we taught together. Second, I am very grateful
to you for teaching me about teaching and learning through, not only your words and your
endless references. but also your example as you modeled the teaching I began to envision
as ideal -- focusing on teaching students rather than course material. Third, thank you for
the hours of editing and patient help in preparing this manuscript and the ideas contained
within it.

I would like to thank my dear friend and colleague, Whitney Johnson. for listening
and sharing her wisdom at opportune times, pushing me forward and encouraging me
throughout this task.

My family deserves credit for picking up the pieces and providing a safe haven for

me as I spent seemingly endless hours back in my ofﬁce staring at the computer at all

hours of the day and night. Andy, thank you for being a loving and supportive husband.

Mariah and Thadd, thank you for believing in me, supporting me, and loving me.

vi

TABLE OF CONTENTS

INTRODUCTION ........................................................................................................ 1
NCTM Mathematical Power Deﬁnition .............................................................. 1
Implementing this Vision .................................................................................... 3
Marv’s Professional Transformation .................................................................... 6
Contrasting Vignettes of Marv’s Teaching ........................................................... 8
Method of Study and Overview .......................................................................... 19

CHAPTER 1 ................................................................................................................. 27

OVERVIEW OF MARV’S PROFESSIONAL LIFE HISTORY
Introduction ....................................................................................................... 27
Inﬂuential High School Teacher ......................................................................... 28
Teacher Education ............................................................................................. 30
First Teaching Assignment ................................................................................. 31
Teaching at Hawkins High School ..................................................................... 34
Sharing Teaching Assignments ............................................................................ 35
Relationships with the Mathematics Education Community ................................ 37
The Real Story .................................................................................................. 41

CHAPTER 2 ................................................................................................................. 43

SHIFTS IN MARV’S GOALS FOR UNDERSTANDING MATHEMATICS:
From Memorizing Rules and Algorithms to Connecting Big Ideas

Two Types of Understanding ............................................................................. 43
Marv’s Transformation to Relational Understanding Goals ................................ 45
Relational Understanding and Mathematical Power ............................................ 58
CHAPTER 3 ................................................................................................................. 6i

SHIFTS IN MARV’S VIEW OF MATHEMATICS
From Infallible Truth to Human Construction

Framework for Teacher Shifts in their Conception of School Mathematics ......... 62
Mary’s Transformation of His Conception of Mathematics ............................... 64
View of Mathematics and Mathematical Power .................................................. 85

vii

CHAPTER 4 ................................................................................................................. 91

SHIFTS IN MARV’S VIEW OF THE NATURE OF TEACHING MATHEMATICS
From Listening for Right Answers to Hermeneutic Listening

What Does it Mean to Teach Mathematics? ....................................................... 91
The Role of Listening in Teaching ...................................................................... 93
Teaching through Hermeneutic Listening and Mathematical Power .................. 102
CHAPTER 5 ................................................................................................................ 105

IIVIPLICATIONS FOR DEVELOPING MATHEMATICAL POWER THROUGH A
PROFESSIONAL TRANSFORMATION

Summary of Results ......................................................................................... 105
School/University Collaboration ...................................................................... 111
Sharing Teaching Assignments with a Teacher Asking Similar Questions ......... 113
Teacher Candidates School Experiences .......................................................... 114
Creating Curricula Considering a Philosophical and Historical Perspective ............ 116
Developing a Relational Understanding of the Mathematics .................................. 118
Focus on Student Learning ................................................................................... 119
Transformation Takes Time .................................................................................. 122
Final Thoughts ..................................................................................................... 123
APPENDICES ............................................................................................................ 126
Mary’s Professional Chronology .......................................................................... 127
Data ..................................................................................................................... 128
BIBLIOGRAPHY ....................................................................................................... 129

viii

Introduction

Mathematical power includes the ability to explore, conjecture, and reason
logically; to solve non-routine problems; to communicate about and through
mathematics; and to connect ideas within mathematics and other intellectual
activity. Mathematical power also involves the development of personal self
conﬁdence and a disposition to seek. evaluate, and use quantitative and spatial
information in solving problems and in making decisions. Students ’ ﬂexibility,
perseverance, interest, curiosity, and inventiveness also affect the realization of

mathematical power. (National Council of Teachers of Mathematics (NCTM),
1991, p. 1)

This dissertation is a case study of a teacher I will call Marv Smith, a high school
mathematics teacher, whose has made broad and signiﬁcant changes in his thinking and the
enactment of his instruction. The resulting professional transformation include changed
views of what it means to learn mathematics, a changed philosophy of mathematics, a
changed perception of what it means to teach mathematics, new experiences and
conﬁdence in his own mathematical understandings and, consequently, a transformed
teaching practice. This conﬁdence and transformed practice has positioned this teacher to
work toward creating opportunities for his students to gain mathematical power.

I am using the words, professional transformation to indicate such dramatic
changes that something similar to a metamorphosis occurred in this teacher’s perceptions
and actions within his profession. His goals for his students’ learning has changed from
memorizing procedures to making sense of patterns and relationships; his view of the
nature of mathematics has changed from truth contained in a textbook to ideas
constructed, and under continual construction, by human beings; while his classroom
changed from presentation, practice and evaluation to a forum in which students are

exploring, making conjectures, and discussing their ﬁndings, using mathematical reasoning

to argue and develop their ideas. The assessment has changed from chapter tests with
single right answers and procedures to a continual assessment that includes a great deal of
writing, in which students explain their thinking, ask questions and investigate their ideas.
The changes have left little, if any, signs of the teacher and classroom before the
transformation.

Few people experience the power of personal self-conﬁdence in the area of
mathematics. In informal conversations, unlike with reading and writing, it is perfectly
acceptable in our society to admit mathematical illiteracy and a strong aversion to the
subject. As a teacher educator these issues give me special concern. Recently, while
teaching a class of thirty-ﬁve elementary teacher candidates in a mathematics methods
course, I asked them each to share a time when they learned something mathematical or
anything from the ﬁeld of mathematics -- anytime in any area of their lives. Much to my
surprise, only three students shared such an experience; the entire rest of the class shared
horror stories related to their experiences in school mathematics. These were college
seniors soon to graduate from one of the top education schools in the country and then
become certiﬁed schoolteachers, to teach children, among other subjects, mathematics.

If college graduates are leaving school feeling a lack of mathematical power, what
are the implications for the rest of society? This is a serious question that plagues the
mathematics education community. Is mathematics an elitist subject to which only certain
people should receive access? The National Council of Teachers of Mathematics
Curriculum and Evaluation Standards (1989), developed and compiled by mathematics

educators at all levels, gives much the opposite message. In this document one of the

major tenets is that curriculum goals for K-12 mathematics are for all students. At the
secondary level, the Standards discusses core content goals to which all students,
regardless if they are college-intending, should be exposed. According to the NCTM
(1989), “The opportunity for all students to experience these components of mathematical
training is at the heart of our vision of a quality mathematics program. [T]hey will gain
mathematical power. ...[F]or each individual, mathematical power involves the

development of personal self conﬁdence” (p. 5, emphasis in the original).

Implementing this Vision

The deﬁnition the Standards uses for mathematical power, quoted above, includes
exploring and making conjectures. Exploring what? Making conjectures about what?
Before coming to graduate school, I taught high school mathematics for nine years. It
was very hard for me, as a mathematics teacher, to imagine how exploration and making
conjectures ﬁt with my vision of mathematics and teaching mathematics. My experience,
as has been the experience of most secondary mathematics teachers, had been teaching
students established mathematical methods to solve routine problems. How do you
reconcile learning and practicing methods until you become proﬁcient at them with
exploring and making conjectures. regardless of the content or teaching method you
choose? There seemed to be a mismatch between how 1 was used to teaching and the
Standard ’3 vision.

Without a change in my perceptions of doing and teaching school mathematics,
the proposals presented in the Standards did not seem applicable to my classroom. This

transformation is related to the foundational issues of what is mathematics and what it

means to learn mathematics. A renowned mathematician captures this idea in a celebrated
quote: “In fact, whether one wishes it or not, all mathematical pedagogy, even if scarcely
coherent, rests on a philosophy of mathematics” (Thom, 1973, quoted in Ernest, 1991, p.
xiii). The role of the content of school mathematics, students and the teacher in a
mathematics classroom are then based on this agreed-upon foundation. The ﬁrst two
documents of the NC 7M Standards (1989, 1991) lay out the goals for reforming school
mathematics while only subtly challenging the traditional foundation that mathematics is
absolute, an existing body of knowledge. Answers typically remain either right or wrong.
Moreover, the Standards are not explicit about the professional and philosophical
transformation in a teacher’s experience and teaching inherent to the reform being
proposed.

The NC 7714 Curriculum and Evaluation Standards for School Mathematics (1989)
discuss changing the content of school mathematics, maintaining that our changed society
and the advent of technology call for students to know different content and use different
tools. How will this give students mathematical power? It is easy to imagine that this
could be read to mean that students will become less mathematically powerful. The
amount of mathematical knowledge out there is so large and growing so fast schools
cannot keep up, so we will teach students less and make them dependent on technology.
This sounds like moving from a thinking society to a society controlled by machines.
Using the above-mentioned traditional deﬁnition of mathematics, programming the
existing right methods into a calculator or a computer may give you speed and accuracy,

but is that mathematical power? Adding and subtracting content does not, by itself,

provide opportunities for students to explore, conjecture or reason logically or to gain
mathematical self- conﬁdence. I contend that changes in curriculum content, teaching
methods and assessment alone will not empower students mathematically nor provide the
motivation for developing reasoning skills or help students build their conﬁdence in their
own ability to make sense of mathematics. According to Nelson (1995), the changes for
most teachers will involve a set of epistemological shifts.

The epistemological position inherent in the NCTM Standards is a socio-

constructivist one. Knowledge is considered to be the dynamic and

conditional product of individuals working in intellectual communities, not

a ﬁxed body of immutable facts and procedures. For most teachers,

developing a practice based on such a view of the nature of knowledge,

learning, and teaching will not be accomplished merely by adding new

techniques to their current repertoire. It will require a set of

epistemological shiﬁs -- changing their beliefs about the nature of

knowledge and learning, deepening their knowledge of mathematics, and

reinventing their classroom practice ﬁ'om within the new conceptual
framework. (p. 2)

Professional transformation, as it is used in this work. includes the epistemological shifts
mentioned above, with the addition of changing one’s philosophy of the very nature of
mathematics. and then reinventing their teaching practice in ways consistent with these
changes. Marv’s professional transformation illustrates the relationship between a
teacher’s understandings of the nature of knowledge and leaming to the classroom
experience, and the importance of having a coordinated view of learning, subject matter
and teaching to fulﬁll the ideal presented in the NC TM Standards. -- in particular.
equipping all students with mathematical power. Fostering a similar professional
transformation of this type, rather than focusing professional development and teacher

education on isolated aspects of the desired changes. seems to offer a solution to the

diﬂ’erent kind of teaching suggested in mathematics education literature (Ball, 1991;

Lampert, I989, 1992;NC7M 1989, 1991).

Marv ’s Professional Transformation

The story that follows is a story of change - changes so profound that I chose to
describe them as a professional transformation, essentially a series of epistemological shiﬁs
similar to those described by Nelson. Evidence from this study supports both that a
professional transformation occurred in this teacher’s practice and that, as a result, his
students were provided with opportunities to gain mathematical power as it is deﬁned in
the NC TM Standards (1989, 1991). Faced with the lack of successful performance
demonstrated by many of his students, this teacher initiated a dynamic change in his
teaching following a year of sharing teaching assignments with another teacher. Marv’s
teaching practice continued to be transformed in part through the partnership between his
school -- a professional development school (Holmes Group, 1990) that I call Hawkins
High School (HI-IS) -- and a nearby public university. Key components of this exposure
included differing philosophies of mathematics, a forum where he could pursue his
mathematical questions, the historical underpinnings of the mathematics he was teaching,
the use of technological tools, and a variety of approaches to the development of
curricula.

In referring to the changes in Marv’s practice as a professional transformation, 1
will be describing a phenomena that includes many speciﬁc changes to his activity as a
teacher, but is much more encompassing. The transformation I witnessed involved

coordinated changes in his beliefs and actions, changes that are widespread, even all-

encompassing. These changes have been sustained over a time period and, I believe, will
continue to be sustained, because they have not been imposed, are not experimental, and
are not dependent on any outside resources. Instead, they represent a transformed
paradigm of what it means to learn and teach mathematics. Marv is intent on pursuing the
ideas that make sense to him and bringing his practice into line with his changing
philosophies around teaching, learning and mathematics. Because this transformation is so
encompassing, although any one of the changes Marv has incorporated (e. g. innovative
algebra and calculus curricula, use of technology, conversations in the classroom, using
writing as an assessment tool, etc.) would make an interesting story, these changes in
practice are not the focus of this dissertation. Instead. after analyzing Marv’s developing
transformed philosophies that have informed and changed his practice, it became apparent
that the story demanding to be told is one of a professional transformation that addresses
his changing philosophy of the goals of student learning in mathematics (Chapter Two),
the content and activities of school mathematics. as it relates to the discipline of
mathematics (Chapter Three), and the role a teacher plays in this new paradigm (Chapter
Four). This professional transformation is illustrated and evidenced by the vignettes that
follow, as they are analyzed in light of Marv’s commentary around his questions, goals
and changing views.

One of the most striking outcomes of this teacher’s professional transformation
was his mathematical empowerment and the opportunities he now provides for his
students to gain mathematical power, according to the above deﬁnition. It is interesting

that a high school mathematics teacher would be in a position to have only limited

mathematical power. There seems to be a direct relationship between mathematical power
and a person’s vision of mathematics. One of the main features of Marv’s new found
mathematical power is related to pursuing his questions about mathematics and thus being

in a position to explore and make conjectures, rather than only make sense of someone

else’s ﬁndings.

Contrasting Vignettes of Marv ’s Teaching

In the passages that follow I will present the reader with a progression of three
diﬁ’erent lessons Marv taught in order to provide an image of the changes he has made in
his instniction. The ﬁrst lesson occurred in the fall semester of 1993, the second one was
taught in the spring of 1995 and the third in the spring of 1996. These lessons are
intended to illustrate the changes in Marv’s thinking and enactment of teaching that I have
come to see as a transformation of his teaching practice. Further analysis of these
vignettes will occur in Chapters Two, Three and Four.

On this particular day, in the fall of 1993, Marv was working with this introductory
algebra class (made up of 10th - 12th graders) solving for unknown quantities in an
equation, going over questions on homework problems (see below, from UCSMP, 1990a,

p.176)

1

8. Solve ‘3 = gay

The class was following this algebra textbook section by section. Marv was being asked

about the following question while going over homework solutions.

Marv wrote 1/5 = 1/80 y on the board

M: “What is the coefﬁcient of the unknown?”
Silence

M. “What is the unknown?”

Several Students: “y”

M. What rs the number it is being multiplied by?

S1: “One over eighty”

M: What should you do to ﬁnd “y”?

81: “Multiply by eighty?”

M: “Yes, remember you always multiply by the reciprocal of the
coefﬁcient of the unknown. Multiply what?”

A Chorus of Students: ‘Both sides”

M: What do you get?

82: “Eighty times one-ﬁfth”

M: “How do you do that?”

Silence

M: “How can we make eighty look like a fraction? Remember that if want

to somehow combine a whole number and a fraction, turn the whole
number into a fraction.”

SI: “Eighty over one”
M: “Then what?”

81: “You do eighty times one and one times ﬁve.”
M: “What do you get on the other side?”
Several students: “y”

M: “What happened to the one over eighty?”
82: “It cancels.”

M: “1 think what you mean is that when you multiply by eighty, you get
eighty over eighty. which is?”
A Chorus of students: “One”

Marv was very careful to repeat important points, rules and skills that the students should
remember to be able to successfully complete the problems on their own. In order for the
students to successfully solve for the unknown in these types of problems, they needed to
remember the process. Thus Marv repeated the rule to multiply the unknown by the
reciprocal of the coefﬁcient (see the following explanation from the textbook [UCSMP,

1990a, p. 174]) -- after he was sure everyone knew what number represented the

coeﬁicient.

 

To solve ax = b for x (when a is not zero) multiply both side
of the equation by the reciprocal of a.

In the term axta is called the coefﬁcient ofx. So to solve ax =-
rnultrply both sides by the reciprocal of the coefﬁcient of x.

It was obvious that he had done this quite well in the past because, for example, when he
asked leading questions, such as “Multiply what?,” and the class responded in unity, in this
case, “Both sides.” Students had been taught from a previous section in the book that
equivalent expressions are still equivalent if both sides are multiplied by the same number.
It was an impressive performance and it seemed that Marv’s careful attention to details
and slow deliberate presentation of ideas gave his students a calming peace during his
presentation. However, once he gave the assignment, only one or two of the students
worked on it (Field Notes, 9/18/93).

The purpose of the teaching Marv exhibited appears to me to have been to help the
students become proﬁcient at working this type of problem, a mathematical skill of sorts.
Students were to learn a process presented by the textbook and teacher, practice it and
apply it with the goal of consistently obtaining the correct answer. Marv paid special
attention to places students often made mistakes that might cause them problems in the
future, such as “canceling” rather than noticing that dividing a number by itself is
equivalent to one.

Two years later, in the fall of 1995, I was again observing Marv teach. The
conﬁguration of the students’ desks has been changed from straight rows facing the board

to a semicircle facing the center of the room. Marv handed out a worksheet developed by

HHS mathematics teachers entitled “Number Recipes #1”, see below).

10

number Recipes #1
Below are two rules:
1.. Take the input. multiply it by -3 and then add 2.

2. Take the input. multiply it by S and then take away 4.
For each rule above:

(a)
(bi

Make a table of ten inputs and outputs.

Find any inputs which will make the output be 0.

(c) Find inputs which will make the output negative.

(d1 Find two input which will make the output not a whole
number.

lei Write down any observations you have about the rule.

According to Marv, the goal of this worksheet includes giving the students the
opportunity to think about repeated calculation procedures, allowing the procedure time
to take on a life rather than imposing -3x + 2 as an abstract object right away. In Marv’s
view, taking time to think about this procedure gives purpose to the symbols, whereas if
you go to the abstract too quickly, many students are discouraged and alienated. Over the
course of the year, they will treat a function as an abstract object, but they are not read for
that yet. Part (b) on this worksheet, according to Mary, is intended to “plant a seed that
there are times that you are interested in a speciﬁc output.” Marv saw that this particular
problem, having a non-integer answer, serves to probe students’ thinking to see how they
make sense of our number system, which was particularly important since these students
were supposedly “behind” in mathematics. The students that made up this class were
upper level high school (10th -12th grade) taking introductory algebra This question
oﬁ’ers an opportunity for interaction and learning, forcing their sense-making to come to
the forefront in a context where they are not directly answering a question about the

number system. In this case, according to Marv, students’ articulating what kinds of

11

things fall between integers gives purpose to those kinds of numbers, while in the past,
many have not seen a purpose to, or even cared about numbers between integers.

Marv invited the students to work together to complete the worksheet, aﬁer which
they would discuss their work with the entire class. Some students were working in small
groups, others with a partner and a couple by themselves. The students had been

discussing number recipes -- doing operations on a variety of inputs, resulting in different

outputs.

S1: How do you want us to make the tables?

M: In whatever way makes sense to you.

S2: Why don’t you just tell us the answer?

M: Because I think you guys are smart enough to think about it.
82: Can we make up our own values for the table?
S3: But we need the values that will give us zero.
M: Will (82’s) suggestion accomplish that for you?
52: Can you just tell us what is right?

84: There is no one answer.

S2: Can we make up our own values?

M: That will work.

Some of the students started to make the tables by hand. but soon realized
it would be quicker to use a graphing calculator. Part (b), searching for
inputs that would make the output zero, seemed to be a stopping point for
all of the students. As they reached that point, the students seemed to
diverge in different directions. It turned out that the exact x-intercept
would never appear on the calculator, since it occurred when x was equal
to 2/3. the calculator only showing a part of this repeating decimal.

SS: Try fractions. (Marv has her repeat her statement, but the others
don’t pay special attention.)

S6: We tried one and it doesn’t work. We tried zero and it doesn’t work.
we tried negative zero and it doesn’t work.

M: Do you remember what outputs you got?

86: No. (They began looking at the output values.)

M: What else could you try?

S7: Negative two?

M: Yes, and I suggest that you expand your table to show the values so
you can see if there is a pattern. ‘

12

Students continue working and discussing a possible value for the x-
intercept.

M: Is zero negative or positive?

S8: It is like a divider. isn’t it? One way is positive and the way is
negative.

A different group has noticed something they want to share with Marv.

S9: Everything below .6 give you a positive number and .7 and above give
you a negative.

M.: Where do you think you should go looking for the number that give
you zero?

S9: 1 don’t know.

M: Keep thinking about it -- inputs below .6 give you a positive number
and .7 and below give you a negative. If we were thinking about money,
and everything was in cents and 1 gave you three quarters. How could you
write that amount in decimals?

S9: .75

M: Think about the signiﬁcance of that as it relates to this problem. Think
about it for a while and if you don’t get anywhere, go on and come back.
But don’t give up on it -- this is a very important idea.

Some students became very frustrated. Others began describing what was
happening (identifying the interval of the input values inside of which they
felt yielded an output of zero). Still others were busily narrowing the
interval and at least one put the table aside and began to work on solving
the problem using symbolic manipulation. The room was busy with activity,
students walking across the room to talk with other students, others
comparing notes with what they had tried. Some were very conﬁdent that
their answer of “.667” was close enough; others strongly disagreeing.
Class is convened as a whole group.

Two students put their table on the board. They used the values -1 to -10.
M: Take a look at your table and the table on the board. What do you
notice?

81: It goes up by three.

S3: None of the outputs are negative.

M: Question (b) asks for inputs which will make the output zero.

S10: We couldn’t ﬁnd any.

M: Any come up with any strategies to ﬁnd it? Anyone get close?
S8: .1

M: .1 as an input or output?
S8: Output.

13

M: Anyone get closer?

S11: I got .2, but I don’t know if that is any closer.
M. What was your input?

811:

M. Ah6yone get any closer?

S9: 1 got .002

M: What did you use as an input?

S9: .6666

M: Why did you try that?

S9: Because if I used .67 it went too far.

S5: Try another 6.

S9: you get 0.000002

M: What happens when you ﬁll up the screen?
S3: Seven zero’ s

M: Anyone get any closer?

S6: -.0001
M: What input did you use?
S6: 6.6667

M: Why did you think there was not a number that would do this?

S 10: We tried numbers.

M: What did you do?

810: Started with one and worked my way up.

M: What happened?

$10: I don’t know.

M: Everyone look at your tables and notice what happens when you go
ﬁrrther up the number system.

81: They go up by three every time.

M: If I were looking for zero, should I keep putting in larger numbers?
810: No.

M: Should I look at large negative numbers?

S6: No, look in-between zero and one.

M: (S9), can you tell us what you told me about .6 and .7?

S9: Everything .7 and above is negative and everything .6 and below is
positive.

M: (Looks at the clock) Has anyone every seen .6666666 when you were
doing calculations?

Several students: Yes

M: Does anyone know a fraction that has that approximate value?
SS: 2/3

The bell rings.

M: (Stopping their movement) Remember, the point of mathematics is not
punching a number in a calculator or ﬁnding the right answer, but the
thinking that goes on around it. You did some very important
mathematical thinking today. (Field Notes, 9/ 18/97)

14

The small group and whole class discussion ended up including many fundamental
mathematical ideas including rational numbers, number sense, estimations, tabular values
of linear functions, and algebraic manipulations of rules. (Marv later told me that S5 had
told him, “My grandmother said if you don’t know what to do in math, just undo what
you did -- so I subtracted 2 and divided by -3 and got 2/3.”)

After class, Marv talked about being pleased with students’ thinking and
engagement. He repeated what various students had shared in private and public
conversations. demonstrating a new interest in listening to his students sense-making, and
he was beginning to think about the situations he might pose for the next day to challenge,
guide or encourage their investigations. During the class the students commented more
than once that Mr. Smith never answered their questions, only asked them more questions,
but I noticed that they did not hesitate to pursue his questions.

The changes in the two classes described above take on many dimensions, from the
set up of the room to the interactions with the students. Marv is no longer describing a
certain process for students to memorize with the purpose of using it to ﬁnd one correct
answer. Although the class was in search of the x-intercept (the value of x when the
output is zero), they were not following a predetermined path to ﬁnd that value. The
paths taken were determined by the students and the students interacted with each other’s
ideas and suggestions. The students were actually making the decisions and following up
on their ideas. Marv’s goal for the students, in addition to ﬁnding the x-intercept, was for
them to be thinking about number patterns, and he seemed interested in all aspects of their

mathematical musings. As they discussed their ﬁndings in their small groups they were

15

 

forced to articulate their thinking and defend their decisions -- a form of mathematical
conversation and reasoning that is much more diﬁcult in a class that is learning a
predetermined process with only the goal of ﬁnding the one correct answer. Marv’ 3
interaction with the students was giving them space, the opportunity to think and
encouragement to pursue their ideas, as he listened closely to what they were thinking. He
was, however, directly involved in the whole-class discussion with most comments going
directly to and through him.

The next vignette comes ﬁ'om an Algebra 2 class in the spring of 1996. The
students were investigating logarithms. While studying the graphs of logarithmic
functions, they noticed that the range was all real numbers and wondered why the domain
was restricted to positive real numbers. Marv decided that one way to approach this
question was to look at the deﬁnition in the textbook, analyze what was being said in it,
then critique the deﬁnition by looking at alternative restrictions. The textbook deﬁnition
and graph appear below:

Definition:

Letb>0andb=in1.1hennisthelonaritirrnotmtntnahasab.
writtenn = 109,, m, if and only if

b"=m.

Y‘l y.z'

yﬂm, X
0‘
x-Zy

 

 

-UCSMP (1990b, p. 509)
16

The question they were considering was why the base of an logarithm had to be greater
than zero -- in particular why the base could not be negative. One conjecture was that the
output had to be positive, but that was shown to be problematic for inputs between zero
and one, where the outputs were negative. They began to think about the consequences
of allowing bases to be negative, using negative nine as a possible base. Because of the
problem of taking “even” roots of negative numbers (which does not result in a real

number), the ensuing conversation lead to a need for a deﬁnition of even and odd

numbers.

M: Did everyone hear what 811 just said? He said that you can raise -9 to any power,
you just can’t take any even root of it?

SI: Can you repeat that?

M: You can raise -9 to any power, you just can’t take any even root of it.
(Several students said they agreed.)

M: Are you agreeing only because 811 said it or do you agree?

(Many said 811, while others said they agreed. )

83: I want to know if it is true why it is true.

SS: I always get confused when you have decimals, whether they are even or odd.
M: Say a little bit more.

88: Like 9.6, is that even or odd?

82: One part is even and one part is odd.

88: Would it be even because the decimal is even?

(Lots of students talking at once.)

M: Okay, hold on. Let’s take a minute to think. This is a great question, and we need to
get responses. but we can’t talk over each other.

S3: If we could deﬁne even and odd.

S4: Can a fraction be even or odd?

S7: It takes two halves to make a whole, never mind.

M: How does this question relate to what we were talking about?

S7: If it is odd, you can’t take the square root.

S8: If you have even roots, you can’t ﬁnd them.

810: Is the deﬁnition of an even number, divisible by two?

S7: So you can divide 9 by two and get 4.5, so it is even?

M: We have to get this ironed out if we are going to know what an even root is.

17

S7: Do you know the answer?

M: I have an answer in mind, but I wouldn’t consider it to be the answer.

This is what mathematicians have to do, make deﬁnitions and see what the implications
are. 810 has given us a deﬁnition of even: An even number is a multiple of 2.

$10: That is what I said, but not what I want to say now. I think it has to be a whole
number.

M: So, tell me if this is right. An even number is a whole number multiple of 2?
$10: You also have to come out with a whole number.
M: So give me your deﬁnition of an even number.

810: An even number is a whole number, divisible by two that gives you a whole number.
That takes care of negative

M: What assumption are you making when you say divisible by two? (He didn’t write,
“that gives you a whole number.”)

S8: You have to write, “Gives you a whole number."

M: To me, divisible by two says that.

S7: An even number is a whole number when divided by 2 gives you a whole number.
(Marv writes that on the board.)

S6: So decimals or fractions can’t be even.

M: Is that an implication if we are going to accept this deﬁnition?
So, what does that say about SlO’s question about decimals?

S3: They can’t be even.

S4: So if we come up with a deﬁnition for odd numbers, then we will know.
M: That’s a good idea, let’s get a deﬁnition for odd numbers.

One of the aspects of Marv’s teaching that is most noticeably different is how he
has dispersed the authority in his classroom among the textbook, himself and the students.
This class did not use the textbook on a regular basis. but occasionally Marv handed
books out, then asked the students to read certain sections or deﬁnitions and analyze the
text by writing about it or just thinking about it. This was one of the occasions in which
he asked them to think about this deﬁnition and be ready to talk about it. This transcript
shows Marv as more of a participant in the discussion than as a focal point through which
all students’ comments went, as in the previous two vignettes. Here a discussion about a

textbook deﬁnition of exponential ﬁinctions led students to confront their understandings

18

of even and odd numbers and eventually led to an inquiry into Peano’s Axioms’. Marv
doesn’t feel compelled to control the conversation, and actually had not anticipated
students’ insistence on a deﬁnition for even and odd numbers nor their questioning the
“bottom line” of deﬁnitions, which lead to the later discussion on axioms. By allowing the
exponential functions to be the object of study, rather than the symbolic manipulation that
would produce a predetermined answer, the students are confronted with big ideas in
mathematics. Maybe even more signiﬁcant, was Marv’s response to a simple question that
most mathematicians would call gibberish ($8: “I always get conﬁised with decimals,

whether they are even or odd”) -- allowing it to become the subject of the study that gave

rise to the students’ confrontation with big ideas.

Method of Study and Overview

The study of Marv’s teaching began with observations and conversations that were
more for the purposes of studying work within the mathematics department at his school,
in general, than studying his individual teaching. We had similar interests. as we both
were thinking about how to teach high school mathematics to students not usually well
connected to mathematics or how to make high school mathematics accessible to more
students. Over a two year period, our conversations and my ﬁeld observations. reflected
dramatic changes in Marv’s thinking and the enactment of instruction. During that time I
had collected various sorts of data, including ﬁeld notes of classroom observations,

videos and interviews with Marv about his teaching. These experiences sparked my

 

' Peano’s Axioms are one formal basis for constnrcting the number system.

19

interest in thinking about doing more in-depth study and writing about these changes or,
more broadly, his professional life history.

I have been in contact with Marv over the past ﬁve years. This contact includes
teaching parallel sections of introductory algebra down the hall from him in 1993 -94,
resulting in many informal conversations; providing documentation for professional
development school (PDS) projects in his department during 1993-1996; conducting
seven formal interviews interspersed throughout the ﬁve years; making more than ﬁfty
classroom observations, including several weeks of videotaping; and studying Marv’s
writing about his teaching (see Appendix B for a table containing dates and data details).
In addition to the sources listed, I have had many informal conversations with Marv’s
colleagues, both at the school and at the university, who have served as secondary data
sources. Throughout this time, 1 did not participate directly in the culture of Marv’s
classroom or have a direct inﬂuence on Marv or his teaching. Marv and I agree that my
main inﬂuence was to serve as a sounding board of sorts as he worked at articulating his
questions, ﬁndings, changes, frustrations and revelations. My questions could have
pushed or challenged Marv, but that was not their intention nor did he received them in
that way. Although we were in the search together, he was the one on the front line and
we were both trying to make sense of his experiences and how he was thinking about
them. i

l have engaged in different levels of analysis of Marv’s teaching, including writing
various papers (Chazan, Ben-Haim, & Gormas, in press). I wrote papers for university

classes about the algebra curriculum he was using, about the collaboration at his school,

20

about the different kinds of understanding being value in his classroom, and about the
changes in his practice. I then wrote a chronology of Marv’s professional life history,
following two very long interviews related to his school, and his learning and teaching
experiences. My intention at this time was to write a teacher biography that would
somehow identify and explain the changes in his practice, while also exploring the factors
that led to them. As I studied the transcripts of his interviews and watched what was
happening in his classes, it became clear that the transformations that took place in his
thinking and instructional practice stood out as a culmination of his professional life. This
professional transformation marked such a clear transition, it became important to me to
try to come to a greater understanding of it. As I searched the literature, I found very few
similar transformations in high school mathematics teachers, which also motivated me to
dig deeper into this teacher’s experience to discern why these changes seemed like a
transformation, rather than a series of isolated changes. I eventually decided to treat the
transformation that took place in this teacher’s professional life as a case study and look
more closely at this phenomenon.

As I worked at understanding Marv’s professional transformation 1 was looking at
the following three questions: 1) What were the factors that contributed to these changes?
2) What changed? and 3) What were the results of the changes? I looked for answers to
these questions in what Marv said, what he wrote and what I was observing in his
classroom. Since the study took place over such a long period of time, some of these
answers were changing and developing continually. Then, as I searched for the evolution

of his ideas, 1 could notice patterns and ask him questions that would identify his current

21

thinking, while paying attention to how those ideas played out in the classroom. The
changes that Marv experienced in his goals for his students’ leaming, his philosophy of
mathematics and his changed view of the nature of teachings (discussed separately in
Chapters 2-4) became evident to me as I was investigating what changed. As I became
aware of how far-reaching the changes were and how they reinforced each other in ways
that made it possible for his students to gain mathematical power, these changes stood out
as the key to his professional transformation.

This study does not include a critique of the teacher’s practice, since this was not a
purpose of the study. I seriously considered this possibility during the analysis stage, but it
did not seem appropriate given the dynamic nature of this teacher’s thinking and practice.
As I considered different aspects of his practice that might be critiqued, I noticed that they
were sources of concern for him as well, and change seemed eminent. The changes that
comprised his professional transformation and their result were much more pertinent to
the intent of this study.

Part of my search to understand this teacher’s experience was to write about my
own related experiences. In an attempt to identify with his questions and recognize his
students, it was helpful to do some autobiographical analysis. This identiﬁcation and ﬁrst-
hand experience in the different roles in my life (mathematics teacher. teacher educator
and parent) indirectly inﬂuenced the analysis by helping me recognize the difference
between isolated changes and a professional transformation. Later, I decided not to
include this writing in the dissertation since it did not contribute directly to this particular

case study. It did invite me to think about changes in my thinking and practice that would

22

have been important considerations if I were to experience a similar professional
transformation as a high-school mathematics teacher.

As I studied the changes in Marv’s teaching, it became important to investigate the
validity or truth of his claims (Phillips, 1994; Donmoyer, 1990), by closely scrutinizing
classroom observations and videos, and the consistency of his claims (Atkinson, 1997), by
studying the transcripts of interviews and the various writing he has done over the past
few years. Validity was also an issue that surfaced as I talked with others, including two
professors that worked closely with this teacher, in an attempt to articulate my ﬁndings
and obtain their readings of what was happening in his classroom and what he was saying.
It was their inﬂuence that pushed me to go beyond the initial two years and investigate the
tenacity of Marv’s beginning attempts to teach differently than he was taught and ascertain
if he had continued to change is similar or different ways over the next three years. This
triangulation pushed my study beyond mere narrative. to a formal analysis of the narrative
including a look at existing structures or frameworks and the social context of this
teacher’s professional life (Atkinson, 1997).

I began to identify patterns (Donmoyer. 1990) in the changes that revealed the
fullness of what I came to call a professional transformation, as I studied the transcripts of
the interviews, Marv’s writings, my ﬁeld notes and watched the videos of his teaching. In
ways, this dissertation provides a vicarious experience, giving readers access to an
experience they might not have otherwise had (Donmoyer), but at the same time refracting
this experience through my, the researcher's, analytical eyes. By subdividing his

professional transformation into the three main areas of a teacher’s charges -- learners,

23

subject matter and teaching (McDonald’s [1992] “wild triangle” and Hawkins’s [1974] “1,
Thou, It”), I was able to separate the major transformations Marv experienced
accordingly. I studied the transcripts of Marv’s interviews for his perspective on the
changes and the factors that led to the changes, comparing his analysis to what I was
observing in his classes and what I noticed throughout this time.

Thinking about the changes according to the three aspects of learning,
mathematics and teaching worked well for a deeper analysis of each of these areas, while
at the same time posing problems because of the widespread overlap of these areas in a
teacher’s professional life. Thus the reader will notice that, for example, teaching comes
up in the chapter on mathematics and mathematics is an important part of the chapter on
learning mathematics. I decided that this overlap was both necessary and desirable since it
is artiﬁcial to make this kind of separation within a teacher’s professional life. Actually, it
is the fact that the overlap came to represent the signiﬁcance of Marv’s changes that
allows me to call it a transformation. Within each area. I looked for consistency between
what he was saying and what 1 was noticing, while across areas the changes reinforced
each other in ways that added up to a professional transformation. as discussed in Chapter
5. Marv is not the teacher he was ﬁve years ago.

As 1 related Marv’s professional transformation to the ﬁeld of mathematics
education, the NC 7714 Standards (1989, 1991) stood out as very important documents to
consider. One rarely reads mathematics education literature in which the Standards are
not mentioned. The push in the Standards for teachers to provide mathematical power for

all students drew my attention, as it had in 1989 when I was a high school mathematics

24

teacher. Realizing that I was seeing Marv’s students become mathematically empowered
caused me to analyze the relationship between his professional transformation and this
empowerment. This stood out to me as a very important result of such an encompassing
and consistent transformation.

My intention in this work is to tell my story of Marv’s story of his professional
transformation as a mathematics teacher. As a storyteller and researcher, I am at times
merely listening, hearing Marv, his students, and his colleagues talk about the teaching and
learning that help deﬁne Marv’s professional interactions in and out of the classroom, then
translating and weaving his story of transformation into current discourses in mathematics
education and the development of my own view of mathematics, learning and teaching.
Throughout the story, I allow Marv the opportunity to speak from his own perspective,
(Crapanzano, 1980), quoting him, giving him opportunity to read the transcript and
continuing our dialogue throughout the writing and rewriting (Measer & Sikes, 1992).
Yet, it is still my interpretation and analysis of Marv’s transformation.

The changes that occurred in Marv’s instructional practice and his thinking about
learning, mathematics and teaching are so signiﬁcant (so he feels, and I concur) that I
assert he has undergone a professional transformation. This transformation is similar to
the epistemological shifts that Nelson described above. The format I have chosen to tell
this story is to begin by telling some biographical background in Chapter 1, to help the
reader identify the context of Marv’s professional life, including the opportunities his
career have afforded him, some ordinary and some rather unique to his circumstances.

Next, in Chapters 2-4, 1 detail the transformation, illustrated in a snapshot by the three

25

vignettes in this chapter, by focusing on learning, mathematics and teaching, respectively,
in each chapter. The transformation in Marv’s view of and goals for his students’ learning
is discussed in Chapter 2, Marv’s dramatically changing view of school mathematics and
developments in his phiIOSOphy of mathematics provide the background for Chapter 3, and
the transformation in Marv’s perspective on the role of the teacher in a mathematics class
is the subject of Chapter 4. Within each of these chapter I include ties to the NC 7114’ 3
deﬁnition of mathematical power, demonstrating the relationship between these dramatic
changes and the development of mathematical power. The last chapter, Chapter 5,
outlines the implications of such a transformation for the ﬁeld of mathematics education

from the perspectives of a mathematics teacher and a teacher educator.

26

Chapter One

Overview of Marv’s Professional Life History

It (college math teaching) is a matter of presentation and evaluation. College
was really ridiculous - the instructional practices and everything. 80 in the
dorms I ended up helping a lot of my friends with math and I found that I had
some aptitude for that. (Marv. June 1996)

Introduction

This chapter has been included in the dissertation to identify some of this teacher’s
background in learning and teaching, as well as the events, contacts and resources that
may have contributed to the transformation that took place in his professional life. As an
overview of Marv’s professional life story, it will serve as a backdrop to help the reader
organize and place incidents discussed in later chapters. A chronology of Marv’s
professional life appears in Appendix A.

My contacts and ability to learn about Marv’s teaching and learning background
has been enhanced by the fact that he attended and student-taught at the same high school
where he now teaches, and earned his undergraduate degree and, just recently, his
graduate degree at the university that has worked closely with this school over the years.
Marv grew up in Hawkins and attended Hawkins High School, a midwestem
suburban/rural high school located just outside the state capitol and about ten miles from
the university. His thoughtfulness and tendency to question authorities are illustrated in a
story he tells about a junior high science teacher. In Marv’s view now, the teacher was

actually teaching a misconception. while at the time he determined that the teacher’s

explanation just did not make sense.

27

He hung two apples from a cord and he blew between them and it pulled

them together. He said he just got the air molecules out of the way so they

could pull together so the gravitational force could pull them together.

Two things theoretically out in space should pull together. but he was

saying it was just the air molecules that was keeping them, not the fact that

he was creating rareﬁed air between them and the air pressure ﬁ'om

outside. It wasn’t that sophisticated. But I understood things well enough

to sit there and think, “What’s in the heck is this guy talking about?” It was

like that all year. As I took a physical science minor in college and looked

back on it, he created all of these misconceptions that when I was teaching

physics at a high school level and I had to deal with them daily. I knew

where they got them.(Spring 1996)

Yet, Marv’s view of course work and school in high school were related to
enjoying the recognition he received when he did well, knowing that he needed a
scholarship to afford college and getting the work done so he could participate ﬁrlly in
sports. He played varsity football and wrestled with the varsity wrestling team all three
years he was in high school. Marv said his parents were not directly involved in his
schooling experiences, or other learning experiences, recalling that they were preoccupied
a lot during his junior high and high school years in what he referred to as a long drawn
out divorce. He said that he and his brother really did not need his parents to hound them
about school -- they just did the work. If anything, related to school, he remembers them

telling him to relax more. Marv was the ﬁrst in his immediate family to attend college.

Inﬂuential High School Teacher
Marv’s experiences in high school physics gave him a rich background in thinking
about how the world works by formulating conceptual understandings using mathematics

and scientiﬁc principles. Marv talked about his high school physics teacher, Mr. Ross, and

his inﬂuence.

28

He would throw us into a situation and say, ‘All right. ﬁgure it out, make
sense of it, using everything you have until now in your schooling to try to
make sense of this.‘ Then later we would take a look at the formal stuff that
all of those dead people ﬁgured out and wrote in textbooks. Leaming did
not feel like it did in other classes. It wasn’t that hard mental crunch to
memorize everything and spit it back. You had to think for yourself and
make sense of it and write about it. (Marv, April 1996)

Marv was also inﬂuenced by his only high-school mathematics teacher, Mr. Stark,

who later served as his supervising teacher during his student teaching.

Mr. Stark had high expectations. He used the Socratic model, where he
would ask us questions and direct them right to a student to make the ﬂow
of things go. He would lead us through a series of questions to the
conclusion he wanted us to have. He helped us make connections within
the mathematics we were doing. (Marv, April. 1996)

Marv decided that his high school mathematics instruction was phenomenal compared to
the instruction he received in college as ﬁrst an engineering major and later a mathematics
major -- which he referred as nonexistent, mere presentation and evaluation. His high-
school mathematics teacher was very interested in explaining and developing his
understandings of the textbook content. According to Marv, his college instructors did
not have a goal of helping anyone understand anything -- just rehash the textbook and
send the students home to ﬁgure it out. Marv frequently found himself in the position of
helping others ﬁgure out the mathematics, which he tributes back to his high school

experience. According to Marv (April, 1996),

I saw many people out there supposedly being teachers [at the university]
that were not teaching anything. I had this experience of somebody [Mr
Stark] who really did help me understand things that were in a way much
more so than the book had to offer. Once I had that comparison, by the end
of my junior year [in college] I was thinking, ‘I seem to have an aptitude
for helping people with math and physical science stuff.’ I saw someone
who really made my life a lot easier. I was able to breeze through my
college math classes because of what I had learned in high school. As I

29

started taking calculus, between the algebraic skill that I got from Mr.
Stark and the conceptual ideas about graphs and what the areas under them
meant and what the tangents to them meant, that I got from Mr. Ross,
calculus and physics in college were a breeze.

Teacher Education

Marv later said that his teacher education program at the university planted “seeds
of conﬂict” that would cause him to question the learning happening later in his classes.
He had been in a program referred to as “Academic Learning” in the Department of
Teacher Education. He remembered that there was an emphasis on helping students make

connections with the subject matter. Marv has the following to say about his teacher

education classes:

Many of the instructors showed interest in disciplinary knowledge and how
a teacher’s expertise in that area really does affect the way he or she
teaches and makes connections within the discipline. It was rather
progressive at the time. It was not just teaching behavior models or things
like that. It was really focusing on how knowledge is communicated to
someone. This is rather strange that I am sitting here saying these things
because at the time I thought it was boring and mundane. I think they really
understood the fact you cannot teach someone what goes on in a
classroom: they have to be in the classroom to understand that. What you
can do is help someone start to form a perspective on teaching, structuring
lessons and subject matter. We talked about creating lessons that made
connections (I can’t believe I am saying all of this as much as I bashed my
education classes), but that is one of the things that ﬁrst got me thinking
differently about learning. Making these connections helped me make the
transition from as a student, when I did school work to get it done to start
thinking about really understanding it -- not just being able to do it, but
have some understanding of it. (Marv, April 1996)

He dreaded the task of making concept maps, which the teacher candidates were required
to construct, as they thought about the conceptual organization of ideas related to theirs
subject matters. However, the concept maps forced him to think about the many

connections within mathematics and reminded him of how his high school mathematics

3O

teacher had recognized connections and had worked at helping students formulate a
better, more connected, understanding. Later conﬂicts arose as he became aware that his
general mathematics students and introductory algebra were not making connections, and
his connections did not seem to work for them.

First Teaching Assignment

Marv seems to be a person who is willing to go against the odds, take a risk and
continue to hope for the best. He writes about the inﬂuence of his grandfather
(Autobiographical Sketch, Fall 1997), who taught him to work hard, but ﬁnd ways to give
to others while you are doing things that make you happy. Marv’s ﬁrst three years of
teaching were in an urban school across the country. away from his friends and family, in
an environment much different from his home town. His grandfather’s words echoed in
his ears and although it was not the easy route, he felt it was the right one. In California,
Marv would have mountains to climb and bike, the ocean swells to conquer surﬁng and
kayaking, and quiet ocean coves to snorkel -- all in the midst of mild and gorgeous
weather patterns. In contrast. given Marv‘s midwestem suburban upbringing, teaching in
a coastal urban school would present a new set of challenging, confusing and frightening
circumstances and open Marv’s eyes to a world about which he knew very little. It was an
opportunity for Marv to give to others, following his grandfather’s model and advice,
while doing what made him happy. Following is Marv’s description of his mathematics

teaching experience there:

They gave you a pamphlet that said what you taught on what days,
including the problems you assign, the worksheets you give out and the
tests you give. They were using a mastery learning cycle for testing -— you
were to give three tests on everything. The program was called “The
Achievement Goals Program.” You would get a crate of material at the

31

beginning of each quarter, including the warm-ups and everything you were
to assign. There was a person that walked around with a cup of coffee, a
16-17 year veteran teacher, who regularly looked at your little chart to see
where you were. This program was horrendous. You had no ability as a
teacher to do anything very creative or conceptual because the

administration mandated what you would do. They needed secretaries, not
teachers. (Marv, June 1996)

In the experience that is described below, Marv is pushed into isolation within his
profession. During Marv’s ﬁrst year of teaching, in California, he faced many challenges.
He was assigned to the “brightest” students, mostly coming from privileged backgrounds
in a magnet school for a physics class, to another group of students that were the least
connected to school coming from the lowest-income neighborhood for a required general
mathematics class and an honors physics class that had no prerequisites. Marv was
thinking about one of these classes in relation to his former teacher education courses and
wondering how he might make some connections to students’ prior knowledge in
introducing the topic. While still contemplating different approaches. and wanting to do it
“right,” he encountered another mathematics teacher in the copy room and asked her
about approaches she had used in the past. Later that day the principal came by Marv’s
room to ask him if he was having trouble -- she had understood from a conversation with
another teacher that Marv might not be able to handle the subject matter. Marv gave none
of his colleagues the opportunity to question his subject matter knowledge the rest of the
year. Teaching was turning into a lonely profession (Lortie, 1975).

Both summers Marv was in California, he attended NCTM Summer Mathematics
Institutes that paved the way for him to continue to think about mathematics, making

connections himself and considering how to teach in more interesting and creative ways.

32

Although Marv doesn’t remember the speciﬁc mathematics he encountered during these
workshops, he does recall them giving him a desire to know more mathematics. Marv
mentioned that his clearest memory of the immediate difference the summer institutes
made was that it gave them permission to talk about teaching mathematics differently.
They didn’t actually study the NC 7714 Curriculum Standards that had just been published,
but the purpose of the institute was to expose the teachers to ideas that were consistent
with these standards.

One of the challenges Marv faced during this time was related to his reputation for
being able to control his classes: “The better 1 got at handling my classes, the more
difficult students I was given.” He remembers one student that had refused to leave the
class when asked and when the teacher called the security guards into the classroom, “he
took both of them out.” To everyone’s horror, instead of suspending or expelling him.
they gave him a different schedule. He ended up in Marv’s mathematics class and a
violent situation ensued in Marv’s class. Later. Marv talked with the young man and
listened to his explanation. Marv’s reﬂections included. “1 could understand him and I
knew it was time to leave. When I could understand that in the context of those kids’
lives that he had to do that or his life would have been miserable or that kid and his
buddies would have come at him again if he hadn’t responded, then I knew, there was
nothing 1 could do.” His decision to return to his hometown and pursue a master’s degree
in mathematics at his university alma mater, was solidiﬁed when a part-time teaching

position back at his old high school was offered to him.

33

Teaching at Hawkins High School

Hawkins High School had been participating as a Professional Development
School (PDS), (Homes Group, 1990), entrenched in professional development work in
conjunction with the university, for the previous three years. US News and World Report
(1993) chose HHS as one of the top ten exemplary high schools in the nation the year
before. One issue that had made the headlines in 1990 was the decision to have students
come to school only 4 1/2 days per week, with the remaining half day devoted to teachers’
professional development. Marv told me that for the ﬁrst couple of years he taught at
HHS he tended to feel out of place -- so many people thinking deeply about education and
trying out neat ideas. He wondered if he ﬁt in and if he would ever be able to participate
in similar professional activities since he was “just teaching.”

The HHS mathematics teachers were very excited about a textbook series
(UCSMP, 1990a, 1990b) they were just beginning to use when Marv arrived on the scene.
One of the things that pleased these teachers was the use of applications throughout the
sections eliminating the question of, “When will I ever use this?” The fact that the
mathematics had practical applications, highlighted in each section, gave the students
assurance that the mathematics was worthwhile whether they, personally, would use it or
not. During Marv’s ﬁrst year at HHS (1991-1992), he taught part-time, including
introductory algebra and physics -- working with his old physics teacher in a parallel
section. At the same time, he was a full-time graduate student in mathematics. The next
year Marv became a full-time teacher at HHS, adding calculus to his schedule, and the

graduate work took more of a back seat. During one semester that year, he shared

34

teaching assignments with a ﬁrst-year teacher, combining two small introductory algebra
classes. One of the approaches they used in an attempt to impact their students’ learning
was to continually ask themselves, “What is it we want the students to know about this?”
In the process, they began to take the ﬁrst step in a journey toward conﬁonting the big
ideas, ﬁrst in introductory algebra, and ultimately in mathematics as a whole.

Sharing Teaching Assignments

I developed a working relationship with Marv while documenting mathematics
classes at HHS the third year he taught at HHS (1993-1994), his sixth year of teaching.
During the professional development morning each week teachers at HHS normally spent
one to two hours in department meetings, besides attending sessions that cut across
subject matters. The mathematics teachers would divide the time between full department
meetings and subject-matter small groups, representing different PDS projects and special
interests in developing projects to supplement the textbook. 1 usually participated in and
took notes for the department meeting and met with the algebra group.

The original plan for one of the PDS projects this year was to have one of the
mathematics teachers, Sherry, who was teaching both mathematics and Spanish, team
teach or share teaching assignments with Marv. Sheny had team taught with Del, the
professor from the university, the previous two years, working on innovative curricular
ideas and pedagogy for introductory algebra in an attempt to better understand the
mathematical conceptions of students who had not experienced success in mathematics in
the past. To carry out the team teaching plan with Marv, Sheny was to be released from

one of her Spanish classes, but a release teacher could not be found. So, Sherry and Marv

35

spent the ﬁrst quarter trying to do similar things with separate classes. I released Marv
ﬁ'om one of his algebra classes at the beginning of the second quarter, for the remainder of
the school year, so he could share teaching assignments with Sherry. In this position,
Marv was able to participate with Sherry in the development of curricula, discuss
individual students’ understandings and observe his students being taught. Since they
were both assigned to the class, they shared interest in and responsibility for every aspect
of the teaching and learning in the classroom.

This team-teaching experience marked an important transition in Marv’s

professional life. He wrote the following about it:

Team Teaching has provided for me opportunities that have alleviated
much of my frustration as a teacher. Traditional methods of professional
development -- one day workshops, research articles, etc. -- were a large
part of the ﬁ'ustrations I had felt as a teacher trying to improve my practice.
For me, they helped foster a notion that to become an excellent teacher, I
needed only see someone do it right and then copy their technique... I
know that for me and my colleagues at (Hawkins) High School teaching
together on a daily basis (with the same courses and students) has helped
us grow as professionals in the ﬁeld of mathematics and teaching in
innumerable ways. 1 now see good teachers as people who constantly
reevaluate their understanding of the subject. constantly struggle with
activities and lessons. and who seek their colleagues as resources. (Team
Teaching Reﬂections, March 1996)

Marv felt that the collaboration with Sherry, involving listening to students and observing
her teaching, gave him a new vision of the algebra curriculum, student learning, as well as
a pedagogy that was student-centered. The experience was an “eye-opener” for Marv in
many respects. First, Sheny did not use a textbook. nor did she have a planned
curriculum, but instead she had some ideas about the direction of the curriculum and some

worksheets that she and Del and developed over the past two years. She seemed poised

36

to write new activities based what the students were saying and doing, and her curricular
goals. The instruction seemed to center around these worksheets and the problems on
them, often only one or two, that students discussed and presented their solutions to. It
was a very diﬁ‘erent kind of pedagogy than Marv had every experienced. At times, Marv
felt he was unsure of “where things were going” or if the students would end up knowing
algebra or something else, but he was afforded the unique opportunity to observe students
interact with the mathematics, observing in ways not possible when he was teaching on his
own. He said it was like “watching your students get taught.”

During the next two school years Marv was forced to articulate the mathematics,
rationale and big ideas of a new curriculum that did not use a textbook or follow
traditional formats in introductory algebra, while he shared teaching assignments with
teachers not familiar with these ideas. It is ironic that these teachers were a former
teacher and coach to Marv while he was in high school, but in this instance he was taking
the lead, being the teacher more familiar with the new strategies. These team-teaching
experiences gave him the opportunity to question the changes in the curriculum. He could
also observe and reﬂect with colleagues regarding particular student’s ideas and his
mathematical and curricular goals.

Relationships within the Mathematics Education Community

The Mathematics Department at HHS has proven to be a productive venue for
teachers interested in mathematics education. Teachers within the department have
formed a community of mathematics educators intent on developing curricula,

incorporating their students conceptions and questions, while challenging each other as

37

they consider the goals and content, and their own questions. The atmosphere encourages
risk-taking and asking hard questions, while teachers welcome input ﬁom the university
and participate in mathematics-education seminars held there. Marv has become
acclimated to this environment and even led the HHS mathematics teachers’ presentation
at the university, attended by graduate students and professors from the mathematics and
teacher education Departments in the spring of 1977. During this presentation the
teachers discussed their goals for mathematics learning at HHS and used a video from
Marv’s calculus class to illustrate the students’ participation in their learning.

Marv has worked directly with professors within the Department of Teacher
Education. He has felt like a “partner in inquiry” as they have investigated the teaching
and learning high school mathematics. He talks about these relationships in terms of how
and the professors have contributed to each other’s thinking in important ways. He and
his students’ ideas have impacted the thinking of the university collaborators as they work
with him enacting his changing vision of high school mathematics teaching. Marv reﬂects
the many conversations with Del, the professor that introduced the innovative
introductory algebra curricula ideas. how he connected Marv to a network of mathematics
educators that had similar interests, and how he has challenged Marv’s thinking about the
nature of mathematics and its relationship to school mathematics.

Marv did some work with Michal Yerushalmy, a professor ﬁ'om Haifa University
in Israel, who is also a developer of software and intensive technology for the mathematics
curriculum, during the spring of 1996. Yerushalmy’s work includes research into the use

of technology for students to explore, make and test conjectures, while posing problems

38

for students that require interpretations of what is displayed by the technology tool
(Yerushalmy and Gilead, 1997). More than the technology or curriculum ideas, Marv says
he was impacted by her emphasis on using the students’ work and conclusions to design
activities that might bring about deeper understandings, rather than designing a series of
activities for students to step through. In the process, she has done extensive work in
thinking about mathematics and big ideas students should encounter. One of the on-going
conversations Marv has had with Yerushalmy was about the proof of the Chain-Rule
recorded in textbooks for calculus students. Marv was struck that Yerushalmy, a renown
mathematics educator, actually critiqued the textbook proof, (Marv says, “[In the past] I
had criticized the proofs for their lack of coherence to students, but not for mathematical
quality”, 1998) and she shared his conﬁrsion for improving the mathematical quality of the
proofs in the textbooks. Marv felt that his questions and ideas were met with
understanding and thus began a long-term conversation around making the proof sensible
to students, while also improving its mathematical quality. Yerushalmy has ended up
playing an inﬂuential role in Marv’s freedom and ultimate mathematical power, as he
searched his own understandings and felt at liberty to critique the “canon.”

Marv participated in a doctoral level class on the philosophy of mathematics during
the fall of 1996. This class followed a summer in which Marv became engrossed in the
idea that mathematics was actually a human construction rather than an infallible truth that
resided in textbooks. The professor for this course, Bob, a tenured mathematics professor
turned teacher educator, who was disillusioned by the facade that many mathematicians

hide behind, introduced Marv to a philosophy of mathematics that contends that

39

mathematics is socially constructed (Ernest, 1991). The following year (1997-98) Marv
worked with this professor analyzing and creating calculus curriculum. Marv marvels at
his openness and willingness to hear and value Marv’s ideas and his students ideas, as they
worked behind the scenes combining these ideas and their own knowledge of the subject
matter to develop a reasonable story line for the course. According to Marv, this
professor gave him “the conﬁdence to question mathematics.”

During the 1996-1997 school year, Marv began working with Ricardo
Nemirovsky, a researcher with Technical Education Research Center (TERC).
Nemirovsky (1994) has worked extensively to uncover students’ perceptions of
mathematical ideas. In his work Nemirovsky uses microcomputer-based lab tools that
measure and model various relationships, particularly those among time, distance, velocity
and acceleration, and transfer these relationships directly onto a graph. These tools
present students with various phenomena that force them to analyze a variety of ways to
think about these relationships. The resulting analysis uncovers many of the students
conceptions of big idea in mathematics, while giving the students the opportunity to
experiment with their ideas and conjectures and receive immediate feedback. Nemirovsky
invited Marv to take a mechanical device that connected the movement of play cars with
computer displays that graphed relationships, and use them in his classroom. According
to Marv, “the software and hardware opened up so many possibilities that I began to think
of ways of teaching and thinking about the content that could not come any other ways.”
Marv sees this technology opening up opportunities for the students to ask questions

when they would not normally do so. (or they would just ask the teacher for the “right

40

answer”). Marv contends that with the mechanical devices, the student can test their
ideas, visualize them, determine what is right without the teacher or textbook deciding
what is right and wrong, and then ask why and continue their investigation. The teacher
can then help students with their exploration and participate with them rather than act only
as an evaluator.

In 1996, Marv also met Jim Kaput, a professor at University of Massachusetts,
Dartmouth. In January of 1998 Marv, along with other high school mathematics teachers,
was invited to participate in a seminar related to SimCalc, a National Science Foundation
(NSF) funded initiative to develop software to simulate mathematics of change. Kaput’s
(1995) goal of “democratizing access to big ideas” is consistent with the work being done
at HHS, which began as a study to uncover the conceptions of students not well
connected to algebra, using a curriculum based on the operations on ﬁinctions as the
objects of algebra (Bethell, Chazan, Hodges and Schnepp, 1995). Kaput’s inﬂuence on
Marv was of a more general nature. helping him realize that there are other people
working on interesting ideas and ﬁnding it refreshing that some of those people who are
doing important things in mathematics education. appreciate the work he is doing.

The Real Story

Although Marv has had unique opportunities within the mathematics-education
community, experiences that many secondary mathematics teachers may not be afforded,
the real story of his professional life history are the epistemological shifts plus the changes
in his practice, evidenced by the three vignettes in the Introduction, that make up the

professional transformation. As a case study of one teacher, this research does not reveal

41

speciﬁc steps this teacher took, and certainly does not recommend steps all teachers
should take to transform their practice. Nor does it prescribe speciﬁc changes that should
occur. Instead, the real story of professional transformation, both to Marv and to myself,
lies in the shifts that have occurred in Marv’s thinking and perspective of 1) what it means
to understand mathematics; 2) the development and content of mathematics (both in
school and as a discipline); and 3) what it means to teach mathematics so that students are
afforded the opportunity to become mathematically empowered, and how these changed
perspectives impacted his practice. In the next three chapters, 1 will explicate these shifts
by positioning Marv’s experiences and his changed thinking within theoretical ﬁ'ameworks
discussed in current mathematics-education literature, in attempt to identify the pertinent

aspects of his professional transformation and the ways in which it led to mathematical

power for his students.

42

Chapter Two
Shifts in Marv’s Goals for Understanding Mathematics:

From Memorizing Rules and Algorithms to Connecting Big Ideas

1 made mathematics appear to be a discipline full of imposed, mysterious and
disjointed rules. (Marv. 1996)

Two Types of Understanding

What does it mean to learn mathematics? A related but very different question is,
What are some distinguishing characteristics of a person who excels at learning school
mathematics? For many, learning mathematics is the equivalent of the ability to remember
rules and procedures, then apply them correctly to produce right answers to textbook and
test problems. Exploration, creativity, inspiration or intuition are not traditionally seen as
attributes of mathematics learning in school. Logical reasoning in school mathematics has
traditionally been seen as a skill related to being able to make proper and reasonable use
of established deﬁnitions, theorems, proofs, algorithms and arguments as they appear in
textbooks -- at the exclusion of developing one’s own ideas. Again. the NC 7714
Standards (1989, 1991) advocate providing mathematical power for students. What does
mathematical power look like? According to the Standards (1989), “This term
[mathematical power] denotes an individual’s ability to explore, conjecture and reason
logically, as well as the ability to use a variety of mathematical methods effectively to
solve non—routine problems. ...[F]or each individual, mathematical power involves the
development of personal self-conﬁdence” (p. 5). How does this ability to explore,

conjecture and reason logically relate to working out traditional textbook problems? The

43

NCTM’s deﬁnition of mathematical power seems to be based on a very different kind of
learning than has been traditionally attributed to school mathematics. One way to contrast
the differences between traditional ways of learning mathematics versus those advocated
by the NCTM’s deﬁnition of mathematical power is to think about what is being
understood in each case.

Richard Skemp (1978), a renowned mathematics-education researcher and author,
discusses two different kinds of mathematical understandings: instrumental understanding
and relational understanding. According to Skemp, instrumental understanding is
“possessing and being able to use rules,” while relational “understanding is knowing both
what to do and why” (p. 9). Although an instrumental understanding is more likely to
produce consistent quick right answers. it is less likely to be remembered or transferred to
various applications. Obtaining a relational understanding is intrinsically motivating and
builds conﬁdence as the students develop a sense of schemas of mathematics that are both
dynamic and organic in quality. “by which they seem to act as an agent of their own
growth” (Skemp, 1978, p. 13). The connections necessary for building the schema are
mentioned in the NCYM Standards for Professional Teaching (1991, p. 1) deﬁnition of
mathematical power. The understandings that would allow one to make these connections
have been referred to by others as a conceptual understanding rather than a procedural
understanding (Davis, 1986; Hiebert & Lefevre, 1986). In the next section of this chapter,
1 will discuss Marv’s transformation from valuing strictly an instrumental understanding of

the mathematics for his students’ learning, to recognizing, celebrating and setting goals

for students’ relational understanding of the mathematics.

44

Marv ’s Transformation to Relational Understanding Goals

When in high school, Marv was actually exposed to a teacher who valued a
relational understanding. The subject was physics and the teacher, Mr. Ross. When Marv
reﬂects back on that experience, he marvels at the fact that it didn’t inﬂuence his teaching
until much later. When Marv was a student teacher, he saw his high school physics
teacher as an enigma. Early in his career, he did not know how Mr. Ross’ unique way of
teaching ﬁt with the rest of what he was learning and experiencing. Only later did he
realize it “was exactly the way he wanted to do things." Mr. Ross did not view his role as
physics teachers as someone who should hand out knowledge to his students or even tell
them what they needed to know. His practice was to set up situations for his students to
explore and investigate, considering what they knew, using sources he made available,
including himself. By being led by their questions and ﬁndings, and sharing them with
each other, they were being given opportunities to learn physics. Marv has come to view
Mr. Ross’ model as an effective way to approach learning mathematics.

Marv’s description of the mathematics teachers he had experienced who inﬂuenced
the direction of his early teaching was much different.

My previous experience had shown me teachers that could persuade

children to engage in typical textbook curriculum by their wonderﬁrl

personalities (my college methods instructor), their ability to instill fear

(many of my former teachers), or out of respect for their long-standing

reputation as an excellent teacher (my high school mathematics teacher).

As a relative novice, I had no long-standing reputation, so I tried on the

other two hats, but performed neither well. My attempts were embarrassing

failures. 1 could not impose accountability standards that forced children

into completing assignments: nor could I throw ‘cool’ problems and
mathematical tricks at them to entice their curiosity. My disposition and

45

approach to mathematics ran counter to both of these models.
Unfortunately, I could not appreciate any other vision of what a successful
teacher might look like in the classroom. It takes time to develop a
signiﬁcant appreciation of any teaching style or technique; sit-and-get
workshops, snapshot in-services and professional teaching articles did not
change my practice. They gave me ideas for a lesson or two only. The only
time tested role models I had were my practice teaching mentor, university
methods course instructor and teachers with whom I shared classrooms.
(Marv, January 1996)

Marv’s instruction and assessment before 1994 reﬂected a focus on helping his
students develop an instrumental understanding. Referring to the ﬁrst vignette in the
introduction, Marv stressed rules, without giving students opportunities to know why the
rules were sensible. This is typical of school mathematics, where students are asked to
memorize the process for “carrying,” “borrowing,” “solving” linear and quadratic
equations, “plugging” values into area and volume formulas, “writing” two colurrm
geometry proofs, etc., as spelled out in the textbook, without the supporting framework
for analyzing or knowing why what they are doing works. Traditionally, a student like
Marv, who is able to excel in an instrumental understanding of mathematics, is not only
very successful in school mathematics, but encouraged to become a mathematics teacher.

What happens when students question the rules, or for whatever reason do not
commit them blindly to memory and continue to wonder about the mathematics? Often
they are told they are not ready for algebra and teachers continue to work on hammering
the arithmetic rules into their minds. This was my experience when teaching at the high-
school level. The test to decide if a student was ready for and able to learn algebra was an
arithmetic test with a signiﬁcant emphasis on operations of fractions. Students who did

not pass the test were put into a pre-algebra course that stressed arithmetic operations,

46

especially fractions. Students not well-connected to school mathematics were the
population on whom Del and Sherry were focusing when developing the introductory
algebra curriculum at HHS. Marv was ﬁ'ustrated by his work with these students and their
seeming inability to develop an instrumental understanding of the mathematics.

As a student, Marv saw learning as equivalent to “doing school.” His ability to
“do school” successfully could explain his ﬁustration and confusion over students that
were not able to perform similarly. Consequently, education courses on learning theories
seemed irrelevant to him. He reported that he had not even thought about or developed a
conceptual understanding about what it meant to learn apart from doing school, making it
difficult to consider different learning theories as an undergraduate in the teacher
education program. During his ﬁrst ﬁve years of teaching, his goal for students was to get
them to “do school” successfully, which could be translated into developing an
instrumental understanding of the mathematics. The success that really matters and that a
classroom teacher can inﬂuence, in Marv’s eyes at that time, was successful completion of
assignments and doing well on tests and exams. Marv pointed this out to me when the
topic of developing the students’ self-esteem, a popular topic at HHS, would come up.
Marv felt that his contribution to his Students’ self-esteem would be to help them become
successﬁrl in his classroom.

This focus could explain some of Marv’s inclination toward improving his
mathematics instruction. He envisioned his job description in a more narrow sense than
many teachers. He thought his relationship with students should deal exclusively with

their interaction with the work of the class. This was his area of expertise, and to become

47

a counselor or social worker to his students, he decided, would do them more of a
disservice.

Focusing on their learning served Marv well as a motivation for improving his
instructional practice. Essentially, it ﬁustrated him that his students were not able to “do
school” in the ways that he did. One of the major motivations to change was his
frustrations with the students’ performance or ability to apply and retain the material
covered in class (i.e. developing an instrumental understanding). He talked about bringing
up something covered a few weeks earlier, only to discover that there was nothing there --
students looked at him blankly. Marv’s continued response to this ﬁustration was to
examine his teaching practice rather than blame the students. This tendency could well be
related to his teacher education experience in placing responsibility for students’ making
connections on the teacher’s planning and the experiences afforded the students, an
emphasis Marv discussed.

Marv’s claim that his teacher education course work did eventually impact his
teaching is an interesting exception to research and the image of university education
courses on teachers’ learning to teach (Feiman—Nemser. 1983). Teachers are quick to
report that only actual school experiences taught them anything about teaching. They
learned to teach by teaching, and most maintain that university classes provided theory not
connected to the real work of teaching. Marv might also have said something similar early
in his career, because the ideas related to making connections and leaming did not effect a
change in his thinking at the time, but only later when he became cognizant of what kinds

of learning his students were experiencing. His recognition that many students were not

48

retaining an understanding that could be used later and applied in different settings was
not something he easily dismissed as the students’ problem. Recognizing the diﬂ’erences
between what he had discussed in his teacher education course work and his experiences
during his ﬁrst few years of teaching, he says that “seeds of conﬂict” were sown during
those courses. He recognized that if students were not making these connections, it had
something to do with the way they were being taught. It is possible that the seeds planted
during teacher education were able to grow because of Marv’s experience sharing
teaching assignments -- in a way modeling for Marv or providing him with a school or
ﬁeld experience that illustrated many of the ideas and methods Marv eventually embraced.

Marv’s preoccupation with the students’ instrumental understanding is
demonstrated in the following clip from his classroom. The homework question was to
solve a simple proportion problem in which 3/5 = x/ 100. My notes say,

Amazingly enough, although they had gone through this process two days

before, no one in the class knew how to solve it. So Marv told them to turn

to page 242 in the book and read it. Several in the class expressed

frustration over the reading. Marv addressed that problem and talked to

them about getting more out of the reading -- asking questions, drawing

diagrams, do some calculations. ...He then went on to show them how to

solve proportions with unknowns. (Field Notes, December 1993)
Marv’s main concern was that students didn’t remember the process and that they

couldn’t follow the explanation and example in the textbook to reproduce the process.

The textbook (UCSMP, 1990a, p. 242) introduces the “Means-Extremes Property” (see

below).

49

 

Why does this property “work”? Why is it called the “Mean-Extremes Property”? These
questions are not addressed in the text. The goal of this procedure appears to be an
instrumental understanding, knowing both what to do and when. Even the term “Means-
Extremes” was not discussed in the textbook, other than stating that “The numbers b and
c are the means of the proportions” and “The numbers a and d are the extremes of the
proportion” (p. 242). The fact that students could not remember material covered two
weeks before and that many of them failed semester exams were sources of great
ﬁustration for Marv. According to Marv, part of the frustration could be traced back to
his teacher education experience in which it was emphasized that students should be
making connections -- within the mathematics and to their own understandings. This was
not happening and it was not clear to Marv how to teach the material in the textbook and
allow students to make connections.

Besides his own frustrations, another spark that seemed to ignite 'Marv’s
transformation in his goal for his students’ learning was the engagement and connections
he saw them making in the class he was sharing with Sheny. These were students who
were in introductory algebra while in the tenth, eleventh and twelfth grade -- and for many
this was not their ﬁrst attempt to pass the class. Yet, he was hearing them and seeing
them make vital mathematical connections and understandings that he wished more of his

calculus students had made. One of those connections was the distinction between

different types of rates of change.

50

Marv talked about feeling rather uncomfortable at ﬁrst when Sherry would assign
a single problem, spend a day or two on it (sometimes even longer) and not necessarily
assign any homework, such as the one presented in the second vignette in the
Introduction. In this example, the students worked on their ideas and the initial discussion
during one class period. Although they had discussed less than half of the worksheet,
many very interesting ideas had surfaced during the small group discussions and the whole
class discussion that would give Marv some ideas about their conceptions of numbers,
non-integers in particular. Prior to this, Marv thought that homework and doing a larger
quantity of problems was a way of holding students responsible for their learning. Later

he gave the following reﬂections:

I started to see that these problems had so much to them that the kids
could really dig into them. They could ﬁnd alternative methods of solving
them and thinking about them. If they had come out of the textbook, they
could have ﬂipped to the back and assumed there was only one correct
answer. Even then if you talk about another way to approach it, they say it
is confusing -- they just want you to show them one way. With these
problems, a student could talk about it graphically, another one could be
looking at the table and even their results could be different or it could be

expressed differently. They could talk about what they were thinking and
write about it. (Marv, Fall 1994)

Marv noticed that Students at different levels of understanding could participate
equally well and stay engaged. The goal was not to work toward a single right answer,
but instead to give a forum for individual students or small groups of students to express
their understandings to the class and participate in mathematical conversations. The goal

or focus was a relational understanding of the mathematics -- knowing both what to do

and why.

51

The conversations between the students, both in small groups and whole-class
modes, took on a different dimension than Marv had experienced in his other algebra
classes. Marv felt that the difference was related to the situations they faced. These
“situations” represented the problems Sherry used to confront the students with -
questions or problems that might push their thinking related to the topics being discussed.
The situations were either written by Sheny or ones that she and Del had used the two
previous years while teaching the same course. As they analyzed these situations, the
students had something to work on and ﬁgure out; they needed to bounce ideas oﬁ‘ one
another. Time spent presenting ideas did not have to be teacher centered. Students would
go to the board to help the others understand how they were thinking about it.

Although the situations held the students’ interest, Marv noticed that Sheny’s
focus was not the material but the students, as she listened to what they were saying and
doing. They were both continually amazed by the sophisticated and thoughtful responses
and mathematical ideas the students considered. Sherry recalled a time when Marv was
talking with the class about the effect the rate of change had on the graph of a linear
relationship. He held his pencil diagonally as a line and asked what would happen in the
rate of change increased. The class indicated that he should tilt the pencil more. As he
tilted it more and more, they indicated that the rate of change would be continually getting
bigger, until he held it completely vertical. Someone said that would be impossible. He
asked, “Why?” A student spoke up and suggested that the class think about the x-axis as
“time” and y-axis as “distance.” The student went on to say that a line that goes straight

up and down would be like going somewhere in no time. Here was a student thinking

52

about lines for what they are - - a representation for a relationship between an
independent and dependent variable changing at a constant rate, developing a relational
understanding of the mathematics rather than memorizing the rule the vertical lines have
an undeﬁned slope. This student noticed that a vertical line cannot represent a linear
function and was able to articulate why using distance as it relates to time.

Marv contrasted the conversations happening in his other algebra class that was
following the book. He said that comments from students in that class most often were,
“How do you do this?,” “I don’t get this.” or “What’s the answer?” Other comments in
class would be short single-word answers to his questions. Beyond that, the conversations
were not related to mathematics. In his class with Sherry, the students would argue with
each other or present alternative ideas. Marv was thrilled with the connections he saw
students making -- for example, connections between tables of functions and the related
algebraic expressions. which they called rules. that enabled them to make conjectures
about the behavior of the graph. As they investigated the mathematics. they were using
graphs. rules and tables as tools for a relational understanding, rather than merely as the
goal of an instrumental understanding. One of the differences between these classes was
the type of understanding being valued. In a class that values an instrumental
understanding, knowing only the process and when to use it puts the students in the
position of recipients of schemas already developed, left with a simple isolated schema that
is merely a process or set of rules. In a class that values a relational understanding,
students are building more complex and connected schemas through their present

understandings, while being exposed to new experiences in the class.

53

Marv’s changing view of the understanding he wanted for his students is illustrated
in the second and third vignettes in the Introduction. In the second vignette, Marv allows
the students the opportunity to use any method that makes sense to them to ﬁnd the x-
intercept. Here the students are invited to make connections to what they know, look for
patterns and develop their own schemas that will help them ﬁnd an answer. Marv never
gives them a procedure or rule to follow. The students are thinking about the behavior of
logarithmic functions, odd and even numbers and deﬁnitions in the third vignette. Again,
Marv’s goal is the development of their collective understandings around the graph and
textbook deﬁnition of logarithmic functions, inviting them to build a sensible schema, not
relying on himself or the textbook to be the sole authorities or disseminators of ﬁxed
procedures in the classroom. Consistent with Skemp’s claim for relational understandings,
the students in this class remember this conversation two years later, and refer to it in a
calculus class, remembering even the details of the conversation.

This issue of authority in the classroom is discussed more in Chapter Four.
Teaching with only the goal of an instrumental understanding is based on the authority of
the textbook and the teacher. while the expectation of a relational understanding invites
the students to assume some authority over their own understandings. This issue calls for
a transformation of the teacher’s role in the classroom, the topic of Chapter Four of this
dissertation.

What follows is a students’ explanation of a typical day in Marv’s introductory
algebra class near the end of the ﬁrst year he taught without a textbook

Well, usually we will do our work, then the next day we come and we will

go over what we have done and we will, like, talk about it. He will ask us
questions that we will need to answer and he will put things on the board

54

and have us ﬁgure it out. He will never tell us the answer, and that gets

ﬁ'ustrating. But we will say, “Tell us.” And he’ll say, “Figure it out.” Then

he will give us a new thing that is a little bit harder than what we had

before to work on. He will go around the classroom and help us, then the

next day we will go over it and we will move on to something a little bit

harder. We usually work by ourselves, but talk a lot about it. We don’t

cheat which is really weird, because in math classes I have had before,

people copy each other. We talk with our neighbors and stuff. helping each

other out. Everybody participates. I guess it is because, like, Mr. Smith

makes us participate, because he throws questions out at us all of the time.

And it makes us think about it and everybody can give their ideas and, like,

not really be aﬁ'aid because like I said, we’re all friends and nobody is

aﬁ'aid to say what they think. (Student #8, Student Interviews, 1995)
From this student’s perspective, the students are very active in thinking, talking and
participating in the work of the class. It is interesting that the student differentiates
between copying each other’s work and helping each other out. I overheard an interesting
conversation between two students in Marv’s class that supports this difference. One
student questioned another student regarding a situation being presented on a worksheet.
Although I did not hear how the question was phrased, the student being questioned said,
“Yes. I answered that one. Tell me how you are thinking about it.” The ﬁrst student went
on to talk about what she was thinking and what she found puzzling. They discussed
different sides to the issue without once considering what might be the “right” answer or
even the answer the teachers had in mind.

My notes reﬂect my amazement. My previous experience was similar to this
student’s: Students who “helped” other students told them what to do or how they did it,
with little or no questioning or interaction. When the goal is an instrumental

understanding, questions are about a process, either the most efficient process or the one

being covered at that time. while in this case the goal was making sense of the issue,

55

examining one’s own ideas in relation to others’ ideas, including tradition and convention,
with the ultimate objective of developing a relational understanding.

Marv discusses the transformation in the similar terms. In reference to the ﬁrst six
years of his instnictional practice, Marv (Synthesis Paper, unpublished) said,

It involved lecturing about a concept that students had yet to think about,

or demonstrating algorithms for which no need has been acknowledged by
the intended learners. I made mathematics appear to be a discipline full of
imposed, mysterious and disjointed rules.

An instrumental understanding, apart from a relational understanding, involves
memorizing just such niles. Although Chapter Four will focus more directly on Marv’s
teaching, the learning goals affect the way one teaches. By “beginning in the leamer’s
world,” Marv sees his instruction as allowing the students’ questions, interests and ideas
to motivate an investigation of the material he designs to build the ideas of the course.
When focusing on an instnimental understanding of the mathematics, investigations were

not needed as much as practice. In the following quote Marv describes a different process

when the goal is a relational understanding:

With an inquiry based approach to new content, followed by a socially
constructed analysis of the students’ ﬁndings, students are drawn into a
more interesting, connected study of mathematics that is far from
superﬁcial. Their study builds from ideas that they can identify and, the
questions their mathematics seeks to answer are explicit -- because the
students ask them. This process also adds credibility to a standard of
mathematical rigor that is unappreciated by mathematics students whose
teachers artiﬁcially impose this tradition or constantly provide the rigorous
arguments themselves. As students work toward consensus, validating or
rejecting results, thorough arguments become necessary and valued. In this
process, generalizations are made, mathematical technique and structure
are built, and established mathematics can be studied more critically than in
a traditional setting. As a result, more students engage in a conscientious,
coherent study of mathematics, which I believe enables them to continue to

study the subject as well as apply it in other areas. (Marv. Synthesis Paper,
unpublished)

56

In this process, the students’ relational understandings of the content are the focus rather

than speciﬁc content goals or benchmarks, as understood by the teacher or documented in

the textbook or discipline.

Marv is now operating under that assumption that if mathematics is reasonable,
students will eventually come to fairly similar conclusions or questions. This will either
force them to push further or consult with what others have done with the same ideas. He
is thinking about the difference between investigations initiated by the students and those
orchestrated by the teacher -- seeing the need for both. Thus it is clear that Marv does not
feel that a relational understanding should stand in opposition to an instrumental one, but
instead they should complement one another, a view shared by Skemp (1978). Marv
voices this perspective and concerns in the following quote.

As students mathematize problem situations, they engage in the
construction of mathematics. It may not always be the exact mathematics I
know and intend. but I believe that the process itself is an effective vehicle
for helping students better appreciate the mathematics developed by others.
What I saw as problematic at one time -- students agreeing on and adapting
alternative approaches, deﬁnitions, etc. -- actually provides a context for
rigorous, mathematically sophisticated discussions and authentic reasoning
from students as they study pro-existing mathematics. I see my students
more willing to delve into. critically analyze and attempt to understand the
workings of other mathematical systems after they have created their own
approach or exhausted their own attempts at abstracting mathematics from
related contexts. Managing this process is difficult. ...The dynamics of
human interactions are complex as mathematical ideas move from one
individual’s suggestion, to face group criticism, and on to acceptance or
rejection by the community at large. Relinquishing the traditional authority
held by a teacher is not easy either. Finding a balance between giving input
into student conversations and guarding against unintentional manipulation
of the dialogue that might subvert the process is not easy either. (Marv,
Fall 1996, “Social Constructivism Happens,” TE 950)

57

 

Transforming one’s teaching to value a relational understanding presupposes a view of
mathematics that is also relational (Skemp, 1978). Relational mathematics involves
developing a schema that is dynamic and connected rather than a series of skills that tend
to be limited and isolated, as is the case in instrumental mathematics. Marv’s
transformations related to his views and understandings of mathematics will be discussed
in Chapter Three of this dissertation.

Relational Understanding and Mathematical Power

Memorizing procedures or algorithms implies that the subject matter is static --
given the right steps and skills, problems are solved and right answers are found. Often
these steps involve mathematical “tricks” that must be memorized or algorithms invented
by someone else that are guaranteed always to “work.” When this is the major focus, it
often results in a “learning situation that leads to the quickest most comfortable route to
mastered habit and attitude. using prescribed and applied knowledge (Schwab, 1978, p.
173).” When this represents the learning goals of the teacher, this habit and attitude
inhibits the application of ideas beyond the immediate situation, while stiﬂing sense-
making and creativity that allows one to engage in reasoned logic or gain mathematical
power.

Often, textbooks and curriculum packages are designed with this type of learning
in mind. Students are led, either by examples shown by the teacher or those in the
textbook, to ﬁnd answers. then they are given problems to which they must ﬁnd the
answers using similar strategies. The teacher’s role is to understand the examples and help

the students solve the problems. In most cases, teachers do not need to understand the

58

mathematics beyond what is presented in the material, thus they are basing their
instruction on an instrumental understanding of the material. The expectation is that the
students would become programmed and molded to produce someone else’s answers
using someone else’s ideas to answer someone else’s questions.

“Perhaps the most important single reason why students give illogical answers to
problems with irrelevant questions or irrelevant data is that those students believe that
mathematics doesn’t make sense” (Frankenstein & Powell, 1994). Students would come
to believe that mathematics does not make sense if they are never expected to use their
sense-making abilities (or ignore them) to think about or engage in the problems. This is
the type of knowledge, an instnimental understanding, that was valued in Marv’s school
mathematics classes and university mathematics experiences and that he valued in his
beginning years of teaching.

Relational (Skemp, 1978) or conceptual understanding (Hiebert & Lefevre, 1986)
is based on the idea that students need to know both the how and why behind
mathematical concepts and procedures. Here students build their understandings by
making connections to their current understandings and related topics and building a
schema that is transferable to other situations (Skemp, 1978, p. 13). There is not a
predetermined procedure to follow; instead, given a situation students begin with their
ideas, and then are challenged by other’s ideas. convention, tradition and their continued
thinking. Marv observed this kind of understanding being sought and happening in
Sherry’s class. More emphasis is placed on the process of developing mathematical

arguments and articulating mathematical ideas when one is in search of a relational or

59

conceptual understanding. Marv came to value relational or conceptual understanding as
he transformed the focus of his instruction to the students and their ideas, with curriculum
goals providing the background and guiding his decisions in the use of student ideas and
concerns and in the design of mathematically rich situations.

One of the results of Marv’s transformed learning goals for his students was an
atmosphere that encourages and provides for students to gain mathematical power.
Students are not likely to gain mathematical power, according to the deﬁnition in the
NC 7M Curriculum and Evaluation Standards, when the goal of their learning is an
instrumental understanding. Apart from a relational understanding, understanding
mathematics is memorizing a complex network of rules, many of which have been
developed merely for convention, creating an internal consistency within mathematical
operations (e. g., a negative times a negative is positive; a° = 1). Relational understanding,
born out of the students’ opportunity to explore, question and develop a schema of the
mathematics. in community with other students and in light of traditional mathematics and
the historical development of mathematics. and includes an instrumental understanding, is

much more likely to contribute to the NCTM’s deﬁnition of mathematical power.

60

Chapter Three

Shifts in Marv’s Philosophy of Mathematics:
From Infallible Truth to Human Construction

As I struggled with these classroom experiences, continued to collaborate with
my peers and read educational literature, 1 came to the conclusion that it was my
mathematical philosophy that needed questioning, not my intended approach to
teaching and learning. (Marv. December 1996)

Mathematics can be understood to be an existing body of knowledge, on which
mathematicians continue to build and it can be thought of as the process of developing and
analyzing these and other quantitative and spatial relationships. School mathematics (the
content that students encounter in grades K-l6) has traditionally been presented as an
existing body of knowledge. It is comprised of isolated skills and algorithms that students
must memorize and become proﬁcient at, while applying them in speciﬁc situations,
developing an instrumental understanding. The relational understanding of mathematics
that Marv began to value is dependent on a view of mathematics that is relational.“z It
might be helpful in this section to draw a distinction between school mathematics and
mathematics as the classical discipline and existing body of knowledge. As a student and
in his beginning years of teaching Marv saw these two as one and the same, school
mathematics merely providing the skills that could prepare one for the study of

mathematics as a discipline. At a certain point, Marv recognized that school mathematics

 

2 Skemp (1978) introduces the idea of relational mathematics and says that “learning relational
mathematics consists of building up a conceptual structure (schema) from which its possessor can (in

principle) produce an unlimited number of plans for getting from any starting point within his schema to
any ﬁnishing point (p. 14).”

61

was more ﬂexible, ideas could be investigated in a variety of ways, but the goal still
needed to be an established set of rules and skills that make up the discipline of
mathematics. As Marv’s philosophy of mathematics changed, viewing it as a human
construction rather than an infallible truth that was discovered, he adjusted his enactment
of school mathematics accordingly. He came to realize that just as mathematics was
socially constructed over time, his students could construct their understandings of the
content over time, and that because it was reasonable, his students would come to many of
the same conclusions. And, just as mathematicians construct mathematics by pursuing
their questions, students can develop a more connected understanding of the content as
they pursue their questions. Thus, Marv’s transformed philosophy of mathematics directly
affected the content to be considered and mathematical activity that went on in his
classroom.
Framework for Teacher Shifts in their Conception of School Mathematics

As teachers are confronted with reform ideas of teaching mathematics differently
than they were taught, emphasizing a different kind of understanding, they must rethink
their own understanding of the mathematics and the content of school mathematics. A
teacher’s philosophy of mathematics affects the activity of a mathematics classroom and
ultimately the opportunity the students have to gain mathematical power. “In fact,
whether one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rests
on a philosophy of mathematics” (Thom, 1973, quoted in Ernest, 1991, p. xiii). Deborah
Schifter (1995), who currently works at the Center for the Development of Teaching at

Education Development Corporation, where she directs an National Science Foundation

62

ﬁinded project entitled “Teaching to the Big Ideas,” has set up a framework for
interpreting teachers’ changes in the conception of school mathematics as they move
through reforming their practice. In some respects, it represents a move from instrumental
mathematics to relational mathematics. A summary of each stage is as follows:
First Stage: School mathematics is facts and procedures; discrete and
ungrounded; learned mechanically and mechanically applied. Truth is

simply incarnate in textbooks and instructor

Second Stage: School mathematics includes students ﬁnding patterns,
solving problems and making conjectures

Third Stage: Students follow up their ideas by investigating the
reasonability, generalizability, and validity.

Fourth Stage: Teaching is organized so students confront the “big ideas”

of the mathematics curriculum -- the organizing principles of mathematics.

It is clear from my data that Marv went through similar stages in his transformation,
perhaps with the addition of perhaps a ﬁfth stage during which Marv gives himself and his
students permission to critique existing bodies of mathematics. deﬁnitions and school
mathematics curricula. In describing Marv’s changing views of school mathematics I will
reference Schiﬁer’s stages.

While changing his view of school mathematics, Marv faced a tension in his
thinking and enacting his practice between staying true to the discipline of mathematics,
and at the same time, valuing and pursuing his students’ ideas and questions (Dewey,
1902; Elbow, 1986). This was partially a question of coverage (preparing students for
standardized tests and future course work), but also it was a personal issue of integrity for

Marv. It was important to him that the students not be compromised in their mathematics

63

knowledge. This tension was not resolved for Marv until he changed his philosophy of
mathematics. He changed from an absolutist phiIOSOphy of mathematics to what Paul
Ernest (1991), a renowned mathematics education theorist and author, coins a social
constructivism philosophy of mathematics. According to Ernest, in the absolutist
philosophy, mathematical knowledge “consists of certain and unchangeable tru ” (p. 7),
while the social constructivism philosophy “draws on conventionalism, in accepting that
human language, rules and agreement play a role in establishing and justifying the truths of
mathematics. mathematics knowledge and concepts develop and change” (p. 42). This
shift in Marv’s thinking seemed important in order for him to enact changes in school

mathematics content and activities consistently and in good conscience.

Marv ’s Transformation in His Conception of Mathematics

As illustrated by the ﬁrst vignette in the Introduction. for the ﬁrst ﬁve years of
Marv’s teaching career. he followed closely the order and content of the textbook. He
later claimed that as he went through school. college and the beginning years of his career,
he was convinced that the mathematics was what was in the textbook.

I remember having the anxiety related to people being out there still
creating mathematics. so the body is going to just keep getting bigger and
bigger. So mathematics was just the stuff that was stored away in
textbooks. That is the stuff you had to learn if you were going to learn
math. It was completely a fonnalist. absolutist kind of view that said that
the stuff in the textbooks weren’t chosen conventions and techniques and
things, but that is the way the world is and that is the way the world works.

That was my conception coming out of college and for quite a while
teaching. (February 1998)

64

This closely parallels Schiﬁer’s ﬁrst stage. Marv’s vision of an effective mathematics
teacher was one who developed and executed clear explanations of the content in
textbooks. Marv’s teacher education courses and his high school mathematics teacher left
him with images of students making connections, but he saw this being a result of the
teacher’s clear explanations of procedures (essentially the teacher’s connections), not
something initiated by or directly related to students’ ideas.

Marv’s ﬁrst experience that would challenge his view of mathematics was the
NCTM Summer Institutes in California. He later reported that the major impact these
institutes had on his thinking was related to recognizing the patterns in mathematics,
giving him a desire to know more mathematics. He envisioned graduate mathematics
classes giving him the opportunity to pursue his questions and make deeper connections
himself. It is possible that these NCTM institutes caused him to consider both the second
and third stages of Schifter’s framework, recognizing that if he was going to have his
students investigate patterns and their ideas. he must have that experience himself ﬁrst.
Later, Marv was greatly disappointed by his experiences in graduate school mathematics,
where once again his questioned were not considered. it was more presentation, practice
and evaluation of an existing body of knowledge, according to him. He was not given the
opportunity to become mathematically empowered, to “develop personal self-conﬁdence
and a disposition to seek, evaluate. and use quantitative and spatial information in solving

problems and in making decisions” (see deﬁnition at the beginning of the Introduction) in

this setting.

65

Marv began to seriously consider the content of school mathematics, in particular,
introductory algebra while sharing teaching assignments with Sherry, in the midst of being
introduced to new curriculum ideas for introductory algebra. It might be helpful to
consider again the two vignettes appearing in the Introduction, only this time focusing on
the mathematics. In the ﬁrst scene from 1993, Marv is having the students ﬁnd the value
of an unknown quantity in the linear equation, 1/8 = 1/50 y. The students have been
presented with a set procedure to ﬁnd unknown quantities, often referred to as “solving
linear equations.” Here Marv’s conception of school mathematics is clearly in line with
Schifter’s ﬁrst stage where the mathematics is “facts and procedures learned
mechanically and mechanically applied.” In traditional introductory algebra textbooks, the
ﬁrst half of the textbook is devoted to working with unknown quantities in linear
situations, while in the second half of the text, the unknown becomes a variable in a
function. The ﬁrst half is. in many ways, a continued review of arithmetic operations with
an emphasis on developing symbolic manipulation skills, presumably to go from the
speciﬁc to the more general. However, when “x” or “y”, in this case. is an unknown
quantity it is actually a very speciﬁc question. much like arithmetic problems in earlier
grades.

The approach to introductory algebra being developed at HHS (Chazan, 1993;
1996), is different in fundamental ways. In this approach, the unknown quantity “x” is a
treated primarily as an independent variable (rather than a speciﬁc unknown quantity),
whose changing value affects the dependent variable, y,” or the output. The objective is

to be able to identify attributes about the relationships between quantities that are

66

changing. The resulting functions become the objects of study and can be studied for
patterns, tendencies and classiﬁed accordingly using their rule, table and graph as tools in
the investigations. Thus in the second vignette, the students are working with the
situation, “Take the input, multiply by -3 and then add 2,” with the majority of the
discussion centering around the question of the inputs that will make the output equal to
zero. This could have been written similar to above, -3x + 2 = 0, with the directions being
to solve for x. However, in this activity the goal is much different than ﬁnding this
unknown value. The students end up studying this relationship and paying attention to
what happens to the outputs as the inputs vary, while noticing patterns and general
tendencies that might transfer to other functions, by using tables and graphs, as well as the
rule. In this vignette, the students are using these tools to study the relationships, while
exploring the mathematics and utilizing logical reasoning -- actually becoming
mathematically empowered. The mathematics in the second vignette is closer to the
mathematics found in Schifter’s second or third stage, where some students are ﬁnding
patterns, solving problems and making conjectures. while some are investigating the
validity and reasonability of their ideas. It is also designed so students will confront a big
idea in mathematics -- reasoning by continuity, consistent with Schifter’s fourth stage.

The design of the algebra curriculum at HHS, which does not reside in a textbook
or curriculum package, is related to Schifter’s fourth stage -- considering functions and
related relationships as a big idea in mathematics. The teachers at HHS are building a data
base of problem situations and activities they have used in their quest to help students
confront the big ideas of algebra. The third vignette in the Introduction demonstrates

Marv’s openness to Shifter’s fourth stage, where he allows the students to inquire,

67

conjecture and investigate ﬁrst, the deﬁnition of even numbers and eventually “deﬁnitions”
-- an organizing principle of mathematics.

Rather than talking about right and wrong answers, the norm in Sherry’s and
Marv’s class was to notice different answers and discuss how and why they were different.
This approach is consistent with Schifter’s third stage in which students are investigating
the validity of their ideas. For example in a class I observed, the students were discussing
the classiﬁcation of functions. Each student had been given a different rules for various
functions, including linear, quadratic, exponential, cubic and quartic functions, and asked
to make a table and graph it. The students then presented their graphs and tables to the
class mentioning things they noticed, including the shape of the graph, whether it was
increasing or decreasing and anything about the table they found interesting. Later the
students, in groups of three or four, classiﬁed the graphs in groups that made sense to
them. The ﬁnal step was to agree as a class how to classify the functions. This provided
the opportunity for students to argue about what they noticed and attempt to convince
others. Throughout the work on classifying functions, there was no emphasis on right or
wrong, yet the students were thinking about ideas that were more sensible or reasonable.
One student noticed that the graphs were not always accurate since they were sketched by
hand. She was able to convince the class, by looking at the table, that a cubic graph, see
the graph and table below, was both increasing and decreasing, although the graph that
had been sketched looked like it was only increasing with the middle part having no

changes. Some students were convinced. others questioned the table. They continued to

investigate.

68

f(x)=-2x3+11x2-x-8

 

 

X Y
-10 3102
-8 1728
-6 826
-4 300
-2 S4
0 -8
2 18
4 36
6 -50
8 -336
10 .918

 

(UCSMP, 1990b, p. 636, 906)

When many of the students changed their minds, they received affirmation from Sherry --
emphasizing her view that changing one’s mind is evidence of learning.

Marv’s reﬂections show that he had wrestled with this idea of discussing incorrect

answers in the past.

In math conferences and education courses you always hear about how
fruitful it can be to discuss incorrect answers or incorrect processes. Yet,
when you have something like 2x + 5 = 9, a wrong value for x does not
give you a whole lot to discuss. Students are not that willing to talk about
the fact that they made a mistake somewhere. Yet they are very interested
in trying to ﬁgure out why the function 2x + 5 yields diﬁ‘erent outputs than
2(x+5) and start looking for reasons for the diﬁ‘erences. (Spring 1995)

When the goal of mathematics is ﬁnding the correct answer and attempts are either right
or wrong, as it is in Schifter’s ﬁrst stage, students become preoccupied with strategies

that will prevent them from failing (Holt, 1982). These strategies prohibit them from

thinking about the mathematics and what makes sense.

69

Marv talks about a speciﬁc series of events that spurred a transition in his own
thinking related to school mathematics curricula. Sherry was out for a couple of weeks,
for a medical situation, and Marv was left on his own with the algebra class and no
textbook. He had been doing some planning with Sherry so he had ideas about how she
and Del made curricular decisions. He knew he needed to consider old problems, think
about the goals of the course, and then write problems or situations that made sense to
him - considering where the students were and the direction in which they needed to go.
This represents a major shift in Marv’s thinking related to school mathematics content and
related activities for students. Based on the format of the class, Marv felt free to create
curriculum based on his understanding of the concepts being studied and his students’
sense making. As he tells it. a story of liberation emerged.

It was amazing. I was actually in control of what was going on, rather than

having this textbook guideline that I was chasing behind trying to sort

through it all. I could think about what I wanted the kids to be able to do

or get out of a problem, then design a problem accordingly. (Spring 1995)

Here Marv’s thinking is beginning to resemble Schifter’s fourth stage in which the
teacher is considering how to organize the teaching so that students will be confronted
with the mathematics he sees as important. Now when Marv thinks back about the
comfort he and his students found in having a textbook the year before, he equates it to an
abused spouse becoming used to the abuse. He said, “Kids hate textbooks, yet they will
cry for them if you take them away from them.” According to Marv,

The thinking is taken away from the kid. The authors can say what they

want, but as soon as you lay out the mathematics and say, ‘Here, practice

it,‘ or ask a question about how their model ﬁts or how they set it up, they

are doing the thinking. The questions that are about thinking are always

based on your accepting the theorems and deﬁnitions in that section, with
the only extension of the author’s ideas being an application. It’s nothing

7O

like, ‘Here is a situation, write about the mathematics here and the patterns
you notice.’ (Marv, June 1996)

In addition to taking thinking away from the kids, at least one study has shown that pre-
packaged curriculum programs can also take thinking away from the teacher. The project
designers of IMPACT, a project designed to inﬂuence and support teacher change in
teaching mathematics (Campbell & White, 1997) happened onto a very interesting
phenomenon that addresses this subject. The project began with two sessions of a
summer institute to help teachers take a deeper look at teaching mathematics for
understanding, as opposed to helping students develop mathematical skills. When the ﬁrst
session began, the project designers did not have any “ready-to-use” materials. They
focused on concepts and then helped teachers ﬁnd or design material that would help
students make sense of the concepts. The teachers were involved in curriculum design.
For the second session of the same content, the new group of teachers were presented
with “ready-to-use” materials. In the authors’ words, “This was a mistake.” They
discovered that the teachers did not engage in the mathematical concepts. were not able to
recognize the depth of the mathematical ideas and did not understand the mathematics
when they weren’t involved in the design of the materials. This consideration of the big
ideas of related concepts is, again. Stage Four of Schifter’s framework.

The freedom that Marv experienced in designing curriculum alleviated many of the
ﬁustrations he had experienced in trying to help students think beyond ﬁnding a single
correct answer or memorize a procedure that did not necessarily make sense to them. As
the designer, he could present the situation in ways that would encourage different

responses. while inviting them to use familiar mathematical tools and representations to

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investigate and push their thinking. In the process of designing the situations, Marv

needed to think through the mathematics himself, investigate the connections and his

understandings.

Marv had moved into Schifter’s fourth stage as he began to reconsider his own
ideas about the curriculum of introductory algebra and how to design appropriate
situations for the students. He actually changed his mind about the essence of the
mathematics in school mathematics as evidenced by the following quote:

The course called algebra to me was teaching a bunch of techniques for
solving equations and simplifying equations. an assembly line of stuff
that kids had to become proﬁcient at before they could move through the
curriculum. Now I see these techniques as tools in a process of studying
different kinds of patterns and relationships. It’s the patterns and
relationships that I see are the fundamental ideas right now. I love the
symbolic theoretical mathematics that comes out of stuff. I think they are
very cool and elegant. I think they make a great model and structure. But,
that is what it is now. Before I thought it was infallible -- this is the way it
is. With that kind of attitude I can see why people get mystical about
mathematics. The Pythagoreans worshipped it. The fundamental laws, for
example, circumference equals pi times diameter, are a lot more interesting
when you think about the distance around a circle is always in a constant
ratio to the distance across the circle. I’d rather a kid know that and if they
know that they are going to experience pi, even though they might not
know c=rrd, or c equals two times that little Greek thing times r.
...Mathematics is just a model and a structure with which you should be
free to play. (Marv, Spring 1995)

Here Marv is beginning to consider mathematical concepts that cut across different
mathematics courses and the ideas that form the connections. He is questioning his past
practice of marching through a textbook, section by section, implying that this approach
will leave students without connections, while providing only isolated skills put in short

term memory. Marv feels that one of the most important lessons he has learned over the

72

past few years has been “if the teacher does not have an understanding of the way their
content ﬁts together -- the content is useless to students” (February, 1998).

Yet, he still wrestled with ideas related to how students would discover or uncover
what is found in textbooks. Moving away from the idea that the mathematics was in the
textbooks was a very hard transition for Marv. One of the major dilemmas for Marv was
helping students make the connections he expected them to make. He knew now how to
get students thinking about the mathematics and noticing interesting phenomena, but he
felt that he often had to force his ideas and direction on them in order for them to cover
the material or reach the conclusions that would prepare them for the next step. This
question became more intense for Marv as he took on the role of lead teacher in the fall of
1994, during which he was expected to model teaching the emerging introductory algebra
curriculum, examining linear functions while treating x primarily as a variable, without a
textbook.

It was the year after Marv shared teaching assignments with Sherry, and he was
assigned to teach introductory algebra with a teacher 1 will call Sherman, a veteran teacher
at HHS. Sherman had taught mechanical drawing many years at HHS and had even been
Marv’s teacher for a shop class when Marv was a freshman in high school. Throughout his
years at HHS. Sherman had also taught one or two mathematics classes -- introductory
algebra or general mathematics each year. He was intent on pushing his students to be
responsible and had developed an accountability system in which he had established
weekly contact with many of the parents of students in his algebra classes. Sherman saw a

connection to success in mathematics, and ultimately success in life, and a high work ethic.

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Del had suggested that Sherman share his work with parents with Marv, while Marv share
the new curriculum ideas with him. Meanwhile, Sherry was sharing teaching assignments
in both introductory and advanced algebra with Matte, the department chair at HHS. I
was involved in documenting these team teaching situations for PDS work, a study of the
beneﬁts or tensions in team teaching situations (Chazan, Ben-Haim & Gormas, 1998).

Many of the conversations Marv and Sherman shared for the ﬁrst halfof the year
were centered around content goals and coverage. They were both concerned that
students would leave the class not prepared for traditional geometry or advanced algebra -
- as these subjects were presented traditionally, from the canon of the textbooks. Sherman
was feeling disoriented because he could not go to a book or a ﬁle of materials that would
outline the content of the course. Consequently, he began to gather the material for each
unit of study into a notebook, organizing the activities in the order they were presented to
the students and even working ahead. lining up activities from the previous year.

Later in the year. Marv tells about his discomfort with the direction of the course,
as he began realize that he had fallen into a pattern of trying to force students into his
connections and rich understandings of the material, so they would be prepared for the
next activity. “1 would have a vision of where I wanted things to go and I would hold to
it steadfastly for a while, not realizing how stiﬂing that is to students.” Marv’s
recognition that he was pushing the students through the curriculum, compiled from the
previous year, came ﬁrst by looking at his students’ faces, seeing blank looks, and then

listening to what they were saying.

When I was doing that, I could see it in the faces of the kids that it was not
natural. 1 was being hypocritical about what I was saying. I don’t know if
they were thinking that deeply, but they would become ﬁustrated and start

74

using the same vocabulary that they did when I was using the book, like, “I
don’t get it -- period.”

The revelation of what was happening came one day when Marv was handing out an
activity to the students. He glanced down while one of the students read the activity out
loud and realized, to his embarrassment, that the names of the students used in the
examples were ﬁ'om the algebra class the previous year, and he realized that the activity
was based on the students’ ideas from the that class and not the current class. This was a
subtle, but clear, message to Marv that he was not using his students’ ideas or insight, but
was instead expecting them to chase after his ideas and come to his understandings, or to
be at the exact same place as the class the previous year.

Even though Marv and Sherman reached the place in their conversations that they
discussed what the students were doing and saying rather than the curriculum content
(Marv, Spring 1995), the dilemma persisted throughout the year for Marv. How do you
help students discover what they need to know for school mathematics -- essentially
textbook content? He found himself allowing the students time and space to discuss the
mathematics coming out of situations. and he knew leaming was happening. But,
ultimately, if they didn’t “discover” what he wanted them to get out of it, he would tell
them and have them practice it. The whole idea of students making discoveries bothered
him. He said, “I don’t think kids are going to sit there and have their own little epiphanies
about mathematics - like ‘I don’t want to keep writing 4 x 3 x 2 x 1 so I’ll put an
exclamation point at the end.’ I think some people have some vision that it (exploration

or discovery) is like that.” At the same time, he wasn’t sure how to get kids to “discover”

75

what the curriculum was expecting. Eventually, Marv decided that this was a futile and
unnecessary expectation.

By the end of the 1995-1996 school year, Marv, at his own admission had become
fairly good at designing interesting situations that would engage students in pertinent
mathematics, in-depth enough to result in lively class discussions. This was an indication
that Marv’s conception of school mathematics had moved along in many respects to
Shifter’s third stage, and often the fourth stage, in which students were ﬁnding patterns
and making conjectures and then debating their ideas using reasoning skills. These class
discussions, at every level from introductory algebra through calculus, would involve the
students in varying aspects and perspectives of the mathematics that would be rich in
content, but often left Marv wondering what to do with the students’ ideas. What good
were the conversations if when all was said and done, the students needed to be herded
into the traditional structure? Marv’s reﬂections about this dilemma are summed up in the

following quote:

I knew that in order for them to have any hope of acquiring anything useful
or carrying anything with them, they had to spend time making sense of the
stuff before I showed them anything or taught them anything directly. But I
was still kind of unsure about the conversations. I wasn’t really sure what
curriculum was all about. I thought it was about writing textbooks or
deciding just what the kids were doing, then the teaching was something
else. Looking back now, there was some conﬂict between what ﬁguring
out I was suppose to be doing with the kids in the classroom and the
content of what 1 was suppose to be doing - pedagogy versus curriculum.

1 don’t think I would make that distinction any more. (February 1998)

Del, the university professor working with HHS math teachers, responded to

Marv’s dilemma by giving him some books to read, one of which was Proofs and

76

Refutations: The Logic of Mathematical Discoveg, by Mire Lakatos (1976). That
summer, while hiking in Ireland with his wife, Marv read the book and found it amazing.

When I read Proofs and Refutations in Ireland that one summer, I was

walking around I in a daze, thinking, ‘Wait a minute this is all just a house

of cards. People just put stuff together and you can change that.’ It is not

just changing how you make sense of it or change how you prove this or

prove that. There is not just one proof for anything -- which is where I was

at before that. You can construct something completely different ,

depending on your deﬁnitions and axioms. It is completely absurd for

mathematicians to say that there is an absolute foundation that they go
back to or build off from. (February, 1998)

In this famous work, Lakatos writes a ﬁctional dialogue between a teacher and
students around the Ellen’s polyhedral formula; V - E + F = 2. In the dialogue, the
teacher presents a “thought-experiment” which he poses could serve as a proof of the
formula. The students caste doubts and ﬁnd exceptions to the teacher’s proof, while also
developing cases for or against each other’s ideas. In Lakatos’ introduction to the story,
he points out the pattern of proof, then refutation of that proof, resulting in a
“decomposition of the original conjecture into sub-conjecture,” which represents a more
accurate view of mathematical proof. according to Lakatos. than a deductive guarantee of
certain truth (Lakatos. p. 13). The heart of the story is tied to the footnotes in which
Lakatos outlines the historical signiﬁcance and the historical mathematicians represented
by the arguments in the discussion.

Lakatos’ purpose was to discuss the philosophy of mathematics or the
methodologi of mathematics (p.3). Yet, he does this using people rather than symbols or
formal proofs. He says that the core of this case-study is to “challenge mathematical

formalism” (p. 5). One quickly recognizes the fallibility of mathematics inherent in its

77

development. Deﬁnitions and axioms were adjusted to purposely restrict the situation and
leave out things so mathematicians could agree on a coherent structure.

Even after realizing the importance of allowing students the opportunity to make
sense of concepts, Marv had still felt uncomfortable when students veered too far away
ﬁ'om established mathematics. Marv describes the change in his thinking as follows:

After “Proofs and Reﬁrtations” and some other stuff, I kind of faded away

from the learning thing and starting thinking about if I am having kids

discuss then it is perfectly okay and perfectly reasonable and perfectly legal

for them to come up with their own deﬁnitions, conceptions and structures

of axioms and everything else. (February 1998)
The “learning thing” that Marv is referring to above was his tendency to hold onto speciﬁc
learning goals for his students. The third vignette in the Introduction of this dissertation is
a wonderful example of students coming up with their own deﬁnition, as they wrestled
with the deﬁnition of an even number. During this scenario, Marv is allowing is students
the opportunity to develop mathematical power and exhibit to him the value of organizing
his teaching in ways that students are allowed to question and confront what Shifter refers
to as “the organizing principles of mathematics” in her Fourth Stage. The students’ quest
for understanding and insight remained a major factor in Marv’s professional
transformation. It is interesting that this teaching event happened prior to Marv’s reading
Lakatos. Marv claims that Lakatos would not have made sense, in the context of the

classroom, if he had not experienced his students having very similar arguments and

developing proofs and refutations, or agreeing on deﬁnitions.

78

An interesting conﬁrmation that his view of mathematics had changed ﬁ'om

something ‘out there to be discovered’ was solidiﬁed in the Fall of 1996 when an intern

was observing Marv’s calculus class.

When I really appreciated that was last year when an intern was sitting in
on a calculus class and he said, “Oh my goodness, they almost discovered
the derivative.” No, they were doing something that is perfectly logical.
They were looking at rates of change over smaller and smaller intervals and
they realized it would be a pain to keep having to do that at every single
point, it would be a whole lot easier if we could generalize this in some
way so that we could get a function that would do it. That is not
discovering the derivative. The ‘derivative’ isn’t matter or some kind of
energy. I don’t think there is anything like that. It’s not something
hanging out there in space that whoever runs the universe, God or
whatever, has as part of his operating principles. It is a human
construction. Once he said that, that was after I read “Proofs and
Refutations,” I started looking at a lot of other things. The deﬁnition of
continuity -- you are not saying that there is a special class of functions out
there that we can do all of this wonderful mathematics on. It is ‘we want
to get rid of all of this stuff that is hard and confusing and causes us
problems, so that we can do mathematics on it’. That is what the deﬁnition
of continuity if all about -- to kick all of the other stuff out of your

conversation. That is a really different perspective on what that is.
(February 1998)

This major transition in Marv’s thinking provided insight into the question that had
bothered him for several years. “How do you get kids to discover what is in the
textbooks?” After having spent the summer considering that what was in the textbooks
was just a particular view of a piece of humanly constructed mathematics, Marv was
thinking more about the concepts and why people had agreed to construct it in certain
ways. Realizing that the process of ﬁnding a derivative is not a truth to be discovered, but
instead is one way of talking about the phenomena of changing rates, represented a new

perspective on mathematics. This experience helped Marv realize what he came to see as

79

the similarities between school mathematics and school mathematics activity and the

discipline of mathematics and its development.

During the fall of 1996 Marv took a doctorate level teacher education course on
the philosophy and epistemology of mathematics at the university This course served to
push Marv’s developing philosophy of mathematics and challenged him to transfer his
ideas of mathematics being fallible into his classroom and affected both his curricula and
pedagogy. By the end of the course, Marv had developed some new theories about his
students’ mathematical discussions and their construction of mathematical structures.
Although still concerned about managing the process, Marv was beginning to value the

students’ alternatives structures and how that process will open up traditional structures to

the student’s world of understanding.

As students mathematize problem situations, they engage in the
construction of mathematics. It may not always be the exact mathematics I
know and intend, but I believe that the process itself is an effective vehicle
for helping students better appreciate the mathematics developed by others.
What I saw as problematic at one time -- students agreeing on and adapting
alternative approaches. deﬁnitions, etc. -- actually provides a context for
rigorous. mathematically sophisticated discussions and authentic reasoning
from students as they study pre-existing mathematics. I see my students
more willing to delve into, critically analyze and attempt to understand the
workings of other mathematical systems after they have created their own
approach or exhausted their own attempts at abstracting mathematics from
related contexts. Managing this process is difficult. The dynamics of
human interactions are complex as mathematical ideas move from one
individual’s suggestion, to face group criticism, and on to acceptance or
rejection by the community at large. Relinquishing the traditional authority
held by a teacher is not easy either. Finding a balance between giving input
into student conversations and guarding against unintentional manipulation
of the dialogue that might subvert the process is not easy either. (Marv,
Fall 1996, “Social Constructivism Happens,” TE 950)

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Marv became intrigued by the idea that mathematics is socially constructed
(Ernest, 1991) and began to imagine how that might affect his work with students and
school mathematics. In his own words, “I read Ernest and social constructivist stuﬁ‘ and
got a better handle on it being perfectly okay if the kids are talking about some stuﬂ’ and
my role was to help guide them to come to some kind of a consensus and conclusions and
build their own mathematical structure - construct their own mathematics.” Marv is now
quick to say that mathematics is a human construction, which gives him and his students
permission to take part in the process, while taking into consideration established
mathematics. According to Ernest, “The social constructivist view is that the objects of
mathematics are social constructs or cultural artifacts” (p. 57). Ernest confronts the
notion that mathematics is merely scientiﬁc proven objective knowledge and explains his
position regarding the social construction of mathematics in the following passage:

The social constructivist view is that objective knowledge of mathematics

is social, and is not contained in texts or other recorded materials, nor in

some ideal realm. Objective knowledge of mathematics resides in the

shared rules, conventions, understandings, and meanings of the individual

members of society, and in their interactions (and consequently, their social

institutions). Thus objective knowledge of mathematics is continually

recreated and renewed by the growth of subjective knowledge of

mathematics, in the minds of countless individuals. (p. 82)

Realizing that his students did not have to reproduce exactly what was in the
textbooks was emancipator for Marv. Marv tells of an interesting example of his decision
to not push them to textbook conclusions, but depend ﬁrst of all on his own growing
understanding and experience of the content and curriculum, then value students’ ideas

that differ from convention. Marv had decided to postpone study of the Product Rule in

his calculus class until late in the year, deciding to focus ﬁrst on the Chain Rule in an

81

attempt to let this concept “sit in their minds for a period of time.” While working with
Michal Yerushalmy, he had been considering the ramiﬁcations of allowing the students to
gain a more grounded understanding of the Chain Rule. He consciously held oﬁ’
introducing the Product Rule until it could be connected to ‘Integration by Parts.’ Later,
when he could hold oﬁ’ no longer, he was introducing the traditional proof of the Product
Rule, which included some “magical steps” (steps that are often referred to as “tric ” that
someone sometime put together to make the proof work). One of his students then spoke
up, noticing that by taking the natural log and using the Chain Rule, the Product Rule
made perfect sense. Here was a proof that was reasonable to the entire class, with no
magical steps. This experience conﬁrmed Marv’s belief that “students learn more and
analyze other people’s mathematics much more thoroughly if they are ﬁrst given a chance
to tackle situations with mathematics they already know or create new mathematics” (Feb.
1998)

Marv does feel a responsibility to be constantly aware of the differences between
his students’ constructions and traditional constnictions and conventions. Then only if the
differences seemed signiﬁcant does he expose the students to historical and traditional
thought on the subject, avoiding setting up tradition as the ultimate authority. Marv
summed up his emerging view of students’ mathematical constructions as follows: “I want
kids to spend time forming conceptions about stuff before they try to tackle any kind of
traditional stuff. A lot of times we don’t have to do that (tackle the traditional stuff)

because whatever they come up with is close enough. I use to think you had to mold it

into that, I don’t think you have to do that anymore.”

82

In order for Marv to move from Schifter’s stage three to stage four in his
conceptions of school mathematics, he needed to consider his philosophy of mathematics
and what counts as mathematical knowledge. He began to separate the existing body of
knowledge and the processes that were included in its development, realizing that rather
than an infallible truth it was a human construction. This step gave him permission to step
outside of tradition and study the big ideas or organizing principles of mathematics
without being totally bound by existing textbooks. He could then take his students’ ideas
more seriously, as he developed his understandings of the content around these principles.
In the following quote, quoted earlier when considering Marv’s goal for students’

understanding, Marv describes his understanding of the social construction of mathematics

in his classroom:

With an inquiry based approach to new content, followed by a socially
constructed analysis of the students’ ﬁndings, students are drawn into a
more interesting, connected study of mathematics that is far from
superﬁcial. Their study builds from ideas that they can identify and. the
questions their mathematics seeks to answer are explicit -- because the
students ask them. This process also adds credibility to a standard of
mathematical rigor that is unappreciated by mathematics students whose
teachers artiﬁcially impose this tradition or constantly provide the rigorous
arguments themselves. As students work toward consensus. validating or
rejecting results. thorough arguments become necessary and valued. In this
process, generalizations are made, mathematical technique and structure
are built. and established mathematics can be studied more critically than in
a traditional setting. As a result, more students engage in a conscientious,
coherent study of mathematics, which I believe enables them to continue to
study the subject as well as apply it in other areas. (Marv’s writing, 1997)

In this process. the students’ understandings of the content are the focus rather than

speciﬁc content goals or benchmarks, as understood by the teacher or documented in the

83

textbook or discipline. An example of this process for Marv, described below, was his
recent development of curriculum for his calculus class.

Over the summer of 1997 Marv spent signiﬁcant time thinking about the big ideas
of calculus and how they were connected. Marv has determined that his role as creator of
curriculum is more of “getting the ball rolling” than setting specific goals that his students
must reach. The content then becomes the investigation that ensues, initiated by the
problem being studied. This would compare to an assertion made by Ernest (1991), that
while problem solving is a convergent activity, mathematical investigations are divergent
(p. 285) . Ernest goes on to propose that the problem and the investigation make up the
content of school mathematics.

In consultation with a mathematics professor, Bill Rosenthal, who published a
textbook on calculus (Rosenthal, 1989), Marv decided to have his students encounter
rates and accumulated quantities in a context before doing any of the formal established
algorithms associated with calculus. In the design of the ﬁrst few activities of calculus the
next year, Marv was inﬂuenced by a question that Ricardo Nemirovsky asked him related
to the mechanical device mentioned in the Introduction. in which two cars are controlled
mechanically while their distances, rates, etc. are recorded electronically, and displayed
graphically. Nemirovsky gave one car a variable velocity and asked, “How fast would the
other have to travel in order to reach the same place in the same time, if it moved at a
constant pace?” The implications of such a task are very interesting and for Marv, “the
thinking it initiated can be credited for a good deal of my new personal understanding of

calculus.” He began the year in calculus by riding a stationary bike, giving the students his

84

pedaling RPM every 15 seconds and then asking them to ﬁgure out how many times his
legs rotated.

Later, Marv was excited to share about a group that found a method for taking
rate information and getting an accumulation. Marv related that the group felt really
comfortable and brought up arguments along the line, “you need to average the beginning
and the end rate, and multiply by the interval and then add those all together.” Marv said
he was tempted to push that into trapezoids, but instead marveled at their insight and at

how quickly they applied and made sense of the method they developed.

Philosophy of Mathematics and Mathematical Power

Marv’s transformed view of the nature of mathematics and school mathematics
puts him in the position to help his students develop mathematical power, as he plans his
teaching around confronting students with the organizing principles of mathematics. As in
the NCTM’s deﬁnition of mathematical power, noted on the ﬁrst page of this document,
his calculus students in the above example were being given the opportunity to “explore,
conjecture, and reason logically; to solve non-routine problems; to communicate about
and through mathematics; and to connect ideas within mathematics and other intellectual
activity.” Rather than rote memorization of symbolic manipulations, these students are
being asked to think about the content and put together reasonable conjectures. Then they
must work through their ideas. using reasoning skills to address the non-routine situations
Marv presents to them. This is possible because Marv has changed his philosophy of

mathematics, recognizing it as a human construction. rather than a compilation of proven

indisputable facts and procedures.

85

Armed with a new perception of the nature of mathematics, Marv has become
more mathematically empowered himself. He now feels free to follow his ideas and
pursue his questions, building his understanding or doing mathematics. He had a renewed
desire to know mathematics and understand the connections and big ideas. Marv became
more ﬂexible in his teaching as he created and enacted curriculum and as he turned his
attention to students’ ideas and used students ideas; and the activity in the classroom
changed from individual seat work to dynamic conversations. In the following
paragraphs, 1 will expand these ideas that connect Marv’s transformed view of
mathematics and mathematical power, to his ability to provide opportunities for his
students to become mathematically empowered.

“The issue, then, is not, What is the best way to teach? but, What is mathematics
really all about?...Controversies about...teaching cannot be resolved without confronting
problems about the nature of mathematics” (Hersh, 1979, p. 34). As Marv adjusted his
philosophy of mathematics, recognizing it as a human construction, he also changed his
philosophy of mathematics education. deciding that mathematics is socially constructed,
both historically and in his classroom. These changes challenged him to work on his own
sense making, a necessity for teachers according to Skemp (1978, p. 13), “So nothing else
but relational understanding can ever be adequate for a teacher.” Marv has a changed view
of himself as a learner. He now sees learning as a disposition, a curiosity of sorts that puts
you in position to make sense of the world, rather simply a matter of searching for one
correct answer. He says he is more apt to think about connections and he is not content

just being able to do things, because he wants to know why they work or if there is a more

86

appropriate representation. Marv continues to show these signs of mathematical
empowerment. According to Marv (1997, unpublished Synthesis Paper), “The teacher
must know mathematics very well - content, history, various branches, applications and
so on.” Marv believes that mathematical integrity must be maintained and could be
sacriﬁced if mathematics teachers were to allow their own understandings to become static
rather than continually growing.

Marv’s students show many signs of being mathematically empowered. Marv
noticed that when students keep their ideas for as long as possible, meaning that their
ideas are seen as conceptions rather than misconceptions, they are in the best position to
appreciate established mathematics. Their empowerment seems to be a direct outcome of
Marv’s freedom to think about and investigate the mathematics himself before designing
curricula, and allowing the story to unfold for his students, accordingly. This freedom
came about as a result of his challenging and changing his philosophy of mathematics,
coming to see it as a social construct and then engaging in the construction of his ideas
with colleagues. This idea that teachers must be involved in learning if they are to pass
this onto their students is supported by Cochran-Smith (1991), in the following quote:

One key assumption that underlies our process of alternative image

building is that, for schools to be positive learning environments for

students, they must be positive learning environments for teachers as well.

Teachers who are not free to construct their own activities. and assess their

own competence will be unable to create those possibilities for students.

Teachers who do not have self-esteem and a sense that they can control

their own destinies will ﬁnd it difficult to foster those beliefs in others (p.
223).

Marv felt much more control of his students’ opportunities to learn and much more

positive about his own involvement in that learning process when he began to design

87

activities for the students based on his perceptions of the students’ connections and needs
with respect to his developing curricular goals.

Moving away from school mathematics content being isolated skills (symbolic
manipulation) to ideas described by patterns and relationships, with the symbols merely
providing structure, marked a major shift in Marv’s thinking. According to Polya (1950),
"Mathematics has two faces. Presented in a ﬁnished form, mathematics appears as a purely
demonstrative science, but mathematics in the making is a sort of experimental science.
A mathematical theorem must be guessed before it is proved (p. 512)." On the other
hand, ”The school curriculum treats mathematics as a collection of discrete bits of
procedural knowledge” (Ball, 1991). These differing views of the epistemology of
mathematics or teacher’s conceptions of mathematics is a powerﬁrl inﬂuence that informs
a teacher’s practice (Thompson. 1992). According to Hersh (1979), “Mathematics deals
with ideas. Not pencil marks or chalk marks, not physical triangles or physical sets, but
ideas which may be represented or suggested by physical objects” (p. 22). In Marv’s case,
a critical piece of his transformation was related to his thinking about the nature of school
mathematics. allowing him the ﬂexibility to create learning situations that often resulted in
mathematically empowering his students.

The mathematical activity in Marv’s classes has changed in many ways. One of the
attributes of mathematical power is the ability “to communicate about and through
mathematics.” Marv came to accept the view that mathematics is socially constructed, a
human construct, which has caused Marv to realize that his students’ mathematical

constructions are a valid and an imperative part of their learning mathematics. Given this

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freedom, as Marv says, “they proceed ‘naturally’ to do mathematics.” According to
Marv, this happens in his classes during the conversations. This is evidenced by the
conversation in the third vignette in the Introduction. As the students work toward a
consensus of meaning, they ﬁnd it necessary to define even numbers. Then the
conversation produces a fonrm for what Marv calls “epistemological shiﬁs and balances”
amongst the students and himself. This manifestation of mathematical power is possible
because of Marv’s transformed philosophy of mathematics in which he now views
mathematically as fallible and socially constructed.

As Marv’s focus moved from the textbook contents to the big ideas or organizing
principles of mathematics he became much more attentive to student ideas. Marv’s
emerging view of mathematics as fallible and as a human construction has given him the
permission he needs to investigate mathematics historically based on his own
understandings, analyze textbooks critically, and design curricula for his students that are
consistent with his own understandings. At the same time. Marv is working toward
addressing questions his students are posing and inviting them to pose new questions.
When a teacher like Marv becomes more directly involved in the teaching and learning
process, such as designing the activities based on students’ questions and sense making, as
well as engaging students in their own relational learning, it represents a move to
“abandon the ideal goal of deposit-making and replace it with the posing of the problems
of men in their relations with the world” (Freire, 1970, p. 66). Marv’s analogy of feeling
like he was hitting his students with a stick when making them do textbook assignments

gives life to the following statement about a violent situation:

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Any situation in which some men prevent others from engaging in the
process of inquiry is one of violence. The means used are not important; to
alienate men from their own decision-making is to change them into
objects. (Freire, 1970, p. 73)

When teachers feels like they are chasing after a textbook that contains the canon
of mathematics, followed by little evidence of learning, they ends up victirrrizing students.
The experience of being out of control, as a pawn, under the control of faceless textbook
writers that know neither of the teacher nor of the students, yet who were making all of
his curricular decisions, is a picture of oppression. The liberation and mathematical power
that Marv experienced as he began to think deeply about the mathematics and about the
nature of mathematics was mirrored in his students’ deep thinking and interest in pursuing

mathematics questions, signs of developing mathematical power.

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Chapter Four

Shifts in Marv’s View of the Nature of Teaching Mathematics:
From Listening for Right Answers to Hermeneutic Listening

Listening is a part of what 1 do and I did not think that it needed to be, other
than listening to see ifthey are getting it or not. I have to listen more critically
now to know what they are saying, to [mow about what they are becoming

confused or what might be an interesting thing to pick up and run with. (Marv.
June I 996)

What Does it Mean to Teach Mathematics?

Another area in which Marv has gone through a major professional transformation
is his vision of what it means to teach mathematics. Paul Ernest (1989) says that a “model
or view of the nature of mathematics teaching [is one of the] key belief components of the
mathematics teacher” (p. 250).3 Early in his career, Marv’s view of teaching was
explaining procedures in a thorough and understandable manner. The procedure or idea
being explained was his sense of it or his interpretation of the textbook’s presentation.
After spending time listening to the students while sharing teaching assignments with
Sherry, Marv began to see the importance of ﬁnding out what the students understand and
how they are making sense of the material as an important aspect of teaching. In Marv’s

words,

I used to think that I was up here (indicates a height above his head) and
the kids were down here (lowers his hand to lower than his waist). My job
was to bring them up to where I was. Now I do not see it that way at all. I
think I need to locate them, then come along side of them and travel with
them to a place of greater understanding. I often end up taking a route I
hadn’t previously considered, stopping off at different points and reaching
a variety of destinations. (Marv, Fall, 1996)

 

3 The other two Ernest mentions are the teacher’s “view or conception of the nature of mathematics” and
the teacher’s“ model or view of the process of learning mathematics” (p. 250).

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The nature of teaching has changed for Marv. It seems that the difference pointed
out in his quote above is an important distinction if students are to be given the
opportunity to gain mathematical power. If the teacher’s job is to “bring students up” to
his or her place of superior understandings, what is the place of exploration, conjecturing
or logical reasoning, as spelled out in the NCTM’s deﬁnition of mathematical power? If
instead the teacher needs to “locate students,” “come alongside students” and “travel with
them,” even taking routes or reaching places not previously considered, teaching is a much
different undertaking. There is a much greater emphasis in this picture of teaching on
what the students are saying, thinking and concluding as an invaluable and critical piece of
teaching. Whereas, in Marv’s previous understanding of what it meant to teach, the
emphasis was to merely evaluate the students’ responses, according to what the he, as
their teacher, had explained and their ability to get a single correct answer.

Eventually, Marv saw teaching in terms of interacting with students in ways that
help students appreciate established mathematics in the process of socially constructing
mathematics the class will use and upon which they will build. As stated earlier. Marv
marvels at what he calls the "epistemological checks and balances that go on in the
conversations” in his classes, as the students build on, argue against and investigate each
other’s ideas. One way to think about the transformation in Marv’s view of teaching is to
analyze the changes in his enactment of the role of teacher in the classroom, while also
noting what he says about his vision of teaching. One of the most dynamic ways in which
Marv has demonstrated this transformation in his thinking about the nature of teaching has

been the changing role that listening has played in his classroom. .As he envisioned his

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role diﬂ’erently, it became imperative that he hear and understand his students’ sense
making. In the next section I will analyze the transformation that took place in Marv’s

enactment of his role as mathematics teacher in terms of the changes in his listening

patterns in his classroom.

The Role of Listening in Teaching

In a recent work. Brent Davis (1997), an assistant professor of curriculum studies
at the University of British Columbia, uses listening as a lens to view mathematics
teaching. Davis divides the ways teachers listen into the following three categories: I)
Evaluative listening; 2) Interpretive listening; and 3) Hermeneutic listening. Evaluative
listening is listening for the purpose of judging the rightness of an answer or listening for a
certain response. Interpretive listening happens when a teacher asks questions to gain
information about the sense students are making. The students’ responses, right or
wrong, are then used by the teacher to help students come to the same understanding as
the teacher. In the third category, hermeneutic listening, the teacher becomes a
participant, interacting with the students in building an understanding. Authority is
dispersed amongst the class while the teacher and students jointly explore mathematical
issues.

Marv now sees his practice as dependent on all three kinds of listening, whereas in
the past he had only recognized evaluative listening as an important part of his job. At
times he is merely trying to assess or evaluate how his students are understanding certain
concepts; at other times he is listening closely to what they say and interacting in ways that

might affect their understandings; and at still other times Marv is joining the students in

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their quest for understanding, allowing their ideas to develop, learning with and from
them.

In Marv’s description above, he indicates the changes in his perception of what it
means to teach mathematics by thinking about the teacher’s role relative to his or her
students. His comment illustrate the role of listening in his changed view of teaching --
now he feels he needs to “locate students” (evaluative listening), “travel with them”
(interpretive listening) and “take a route he hadn’t previously considered” (hermeneutic
listening). Several months earlier, he made the following comments about his job as

teacher:

I see my job as something completely different now. Going in, I thought it

was telling kids and doing it in an understanding or clear or creative or fun

way -- that was what my job was to be. My expectation of what I would

consider a good classroom is completely different. I now view myself as

someone who has to set up situations for kids to grow with and pull stuff

out of. with an overall backdrop of topics and concepts the mathematics

department has decided that each class would discuss. I see my role as

ﬁnding ways to put kids in srtuations where they are going to be able to

really talk about the material and learn. (Marv. June 1996)

A current goal of Marv’s instruction is to create situations. as discussed in the last
chapter. that help students confront the big ideas of the course. while at the same time
enabling the students to “really talk about the material and learn.” In order to create these
situations, Marv must hear and understand the students’ conversations. Marv’s
transformation in his vision of teaching mathematics changed from strictly an evaluative
listener to include interpretive and hermeneutic listening. When his focus was on

explaining the procedures. he was only listening to hear if the students’ answers were right

or wrong. Listening to interpret the students’ understandings is similar to when Marv said

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he could get the students to talk, but didn’t know what to do with what they were saying.
As Marv began to hear and interact with his students and work with his students in the
process of socially constructing mathematics, his view of teaching was transformed, as is
evidenced by the hermeneutic listening going on in his room (see the third vignette in the
Introduction).

Sometime into the second semester of sharing a teaching assignment with Sherry,
Marv would repeatedly have an exciting story to tell about what was happening with
students in their class. He related these stories to me when I came to teach the algebra
class down the hall. He found the thinking the students were doing amazing -- tenth and
eleventh graders, tracked into an introductory algebra course -- making connections to
calculus concepts! On one occasion, they were discussing different possibilities for the
rate of change of polynomials of higher degree, eventually agreeing that this concept only
made sense at speciﬁc points, essentially referring to an instantaneous rate of change or
the ﬁrst derivative in calculus. Being in the position of sharing a teaching assignment, and
with Sherry leading at this point, Marv was able to observe and listen to the students’
ideas.

It is interesting to note that Marv’s excitement and interest in changing his
approach to teaching introductory algebra was partly motivated by hearing students’ sense
making. One aspect of Marv’s role that he had not seriously considered in the ﬁrst few

years of his practice was listening.

Listening is a part of what I do and I did not think that it needed to be,
other than listening to see if they are getting it or not. I have to listen more
critically now to know what they are saying, to know about what they are
becoming confused or what might be an interesting thing to pick up and
run with. (Marv, June I996)

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These comments by Marv show his transition in his conception of the role of the teacher
from strictly evaluative listening, as deﬁned by Davis (1997), to hermeneutic listening in
which teachers become participants in the investigation, with that participation informing
their instructional decisions. Hermeneutic listening has become a critical aspect of Marv’s
teaching role. He must hear his students’ ideas, connections and current understandings in
order to develop strategies to give the students appropriate mathematical experiences to
build their conceptions of the mathematics. John Dewey writes in similar ways about the

place of experiences in teaching.

[W]hat concerns him [sic] as a teacher is the ways in which that subject

may become a part of experience; what there is in the child’s present that is

usable with reference to it; how such elements are to be used; how his own

knowledge of the subject matter may assist in interpreting the child’s needs

and doings and determine the medium in which the child should be placed

in order that his growth may be properly directed. (1902, p. 201)
Teachers could come to know this “medium” only by actively listening to the child while
the child is involved in an interactive environment around situations and problems that can
be discussed and analyzed. Mathematically “rich” problems by themselves, apart from
hermeneutic listening, will not put the teacher in a position to help the child grow
mathematically.

Marv tells a couple of stories that illustrate his impression of the negative impact
teaching mathematics by telling or informing, with only evaluative listening, can have on
students. Del, the professor who taught at HHS, told one story that fascinated Marv and

which he repeats. He recalls that Del was teaching the same group of students for algebra

and Bible at a Jewish school. What was amazing was that when the students were in Bible

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class they were much more willing to debate and discuss issues. while in algebra class, the

book was the complete authority. In reaction to this story, Marv said, “I thought that was

so interesting that math teachers are so successful at beating kids into submission that

there is this higher power -- higher than everything else and it is this answer in the back of

the mathematics textbook.”

telling:

In the following tale. Marv gives an interesting analogy to teaching mathematics by

It’s kind of like when we were in Greektown going to the Detroit Institute
of Art with my wife’s family. My father-in-law said, “Okay, follow me.” He
jumped in the car and off he went. He knew how to get there; he knew the
short cuts, how to drive in Detroit traffic so you do not get stopped at
lights. I could see him doing it and I was following, but I did not
understand what or why he was doing it. I think that is what goes on in the
classroom sometimes. How many times have you sat in a lecture hall and
someone is just blabbing away? They are experts in the ﬁeld and they know

about what they are talking. You hear their words, but you cannot see their
images. (June 1996)

Marv’s desire to allow students the authority to build their own images is

evidenced by the following exchange in which his listening could be interpreted as

hermeneutic listening. The students were working on a situation in which grandparents

set up a savings account for their grandchild with $1000 and added $50 for each birthday.

S]: Mine was different, I had (0, 1050) because they got $50 the ﬁrst year.
82: Zero is when they started with $1000 and one is their ﬁrst birthday when

they got $50.

M: Do you agree? Did you want to change your mind?
Sl: No, I still think I am right.

M: Can you convince him?
S]: No, 1 can’t say it but I know what I am thinking.

M: Okay, keep thinking about it and try to think of a way to verbalize it. We

will come back to it.

(Video, Marv’s Algebra I, May 1995)

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It seems that Marv’s intent here to allow this student to think for himself. Marv respects
the student’s idea although he is unable to verbalize his argument. I have noticed while
observing Marv’s classes that students and their ideas are not discounted because they are
not what he was thinking or what he has seen in the past. Instead, he seems to believe that
the students are operating with reason and that since mathematics is assumed to be
reasonable, their ideas will be reconciled with the discipline without their being forced to
memorize someone else’s answers. This vignette illustrates Marv’s willingness to share
the authority in the classroom as he engages in hermeneutic listening, expecting to learn
himself as he listens to the students’ reasoning.

It is difficult, if not impossible. to clearly differentiate between curriculum and
instruction. as they are so very dependent on each other. How a teacher teaches is very
dependent on the content and what counts as knowledge. Marv’s method of instruction
before 1994 was to clearly explain the processes. concepts and steps needed to reproduce
what he had said or what was in the textbook. with the only listening being evaluative. As
Marv began to investigate and develop algebra and later calculus curricula, he also began
to develop his instruction in ways that put the students’ conceptions in the center ring,
forcing him to engage in interpretive listening and then hermeneutic listening. The
textbooks, mathematics traditions and conventions. and his own understandings became
sideshows, important factors to be considered and compared with his students’
understandings, but these were not under the main spot light during classroom instruction.

Instead, his students’ ideas and understandings have taken center stage.

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In a video of Marv’s calculus class, the students are considering a question that
ends up leading them to ideas related to the Fundamental Theory of Calculus. Marv is
seated at his desk in the comer interjecting his questions and insight, while the students are
engaged in debates, making presentations and pursuing each other’s ideas, building a
mathematically sound case for their ideas. Traditionally, this theorem is presented,
explained, examples are demonstrated by the teacher and students practice, applying the
resulting algorithm to more examples. The risk is kept at a minimum, especially at this
stage, when all but a small handful are happy to accept “truth” without question. As Marv
said in relation to his ﬁrst teaching assignment -- a trained secretary could perform the job.

When a teacher controls the curriculum and not the students, a higher level of
intellectual engagement is required. The teacher must continually dig deeper into her or
his own understandings, the discipline and the students’ conceptions -- looking for
connections and building more complex and involved structures. This process involves
hermeneutic listening, listening that is searching for meaning with the students. Teaching
that moves in this direction trusts that students and mathematics are both reasonable and
will interact accordingly. Teaching that trusts students and pursues an intellectually sound
basis for the content is motivated by love for all students, believing that they deserve an
education, an opportunity to build their mathematical understandings and choices in
society according to their desires and interests. It is inconceivable that Marv would say
that he did not listen before his transformation. But he listened for what he wanted to
hear: ﬁrst the correct answer (evaluative listening), and then later for what the students

might say that could help him move the class along to his goals for the content

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(interpretive listening). Marv is very explicit in the following quote that he has
“transformed his role to a listener.” Here Marv is discussing his use of hermeneutic

listening as the students’ ideas change the direction of the course and impact his thinking

and planning.

To equip students with a mathematical knowledge base that can give them
cultural capital in our technological society, I must see to it that student
ideas move each class closer and loser to an understanding of the world of
the practicing mathematicians and people who require experience with
formal mathematics. These considerations have transformed my role to a
listener. As students deliberate and work toward resolution of problems, I
must continually assess their understanding, note questions or difﬁculties,
and use my observations artﬁilly to develop subsequent lessons that will
keep the class’ progress centered on curricular goals. These must include
traditional mathematical structures so that students are not disadvantaged

when they move to settings that expect conventional structure. (Marv,
Synthesis Paper, 1998)

Marv is discussing a balance between allowing the class to be completely student-centered
and remaining true to the discipline. He feels that the key for him, in his role as teacher, is
to listen in a participatory manner. so he can truly give the students the opportunities they

need to both build their understandings and to understand the existing structures in the

ﬁeld.

“He never answers our questions.”

“He never tells us anything. He just wants us to ﬁgure it out.”

“Instead of answering our questions, he just asks us more questions.”

“His questions make us think.” (Student Interviews, 1995)
These comments have become normal occurrences in Marv’s math classes. It is unsettling
for some students when a mathematics teacher is resolute in not explaining procedures for

getting right answers. Eventually, Marv’s students realize that this technique is not a

fluke; he really is not going to answer their questions and tell them exactly what to do to

100

get the answers he expects. He never even tells them exactly what he expects, and he
seems satisﬁed with whatever they decide to try or investigate. He does ﬁnd some things
they do more interesting than others and he does share insight ﬁom classical mathematics,
but he does not present himself as the person with all the right answers or the sole
authority. Harking back again to his teaching before 1994, this was quite a change for
Marv, as is illustrated with the vignette in the Introduction from the fall of 1993. At that
time he was intent on ﬁnding ways to make his explanations and understandings clear to
his students so that they could produce similar results. For the ﬁrst ﬁve years of his
career, Marv was only an evaluative listener.

Listening is also a critical part of Marv’s role as it relates to assessment. The
mode of assessment that Marv likes to talk about rests on his listening carefully to how the
students understand the situations and related mathematics. He recognizes assessment for
the purposes of evaluation or assigning a grade as part of his role, but still struggles with
creating authentic assessment tasks that would indicate what the students have learned and
can do. He thinks about assessment daily rather than only on review or test days as he did
in the past. He says, “The assessment I put my faith in is what kids tell me when we talk.”
Again, hermeneutic listening is the key for Marv. Following are some of the questions
Marv frequently asks his students:

“What do you think you might want to do?”

“How did you or are you thinking about that?”

‘(Why737

“Why does that make sense to you?”

“Why do you think that is true?” (Field Notes, 1995-96)

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These questions are asked in a personal sense. Each of the “you’s” is addressed to a
speciﬁc person, as Marv listens herrneneutically and works with the students to understand
their thinking. Marv’s questions invite students to become involved in the thinking and
developing their mathematical conceptions by making conjectures and developing
reasonable arguments. Students’ responses give him vital information about how they are
making sense of the ideas. This is teaching as listening, listening as teaching. It has
resulted in students becoming increasingly mathematically empowered, both as the year
progresses and as they advance in mathematics classes. Marv is amazed at how conﬁdent
and thoughtful his calculus classes are compared to a few years earlier. He also marvels at

how his calculus students recall conversations from his mathematics classes two years

previously.

Teaching through Hermeneutic Listening and Mathematical Power

John Holt (1982) says, ”I doubt very much if it is possible to teach anyone to
understand anything, that is to say, to see how various parts of it relate to all other parts,
to have a model of the structure in one’s mind. We can give other people names and lists,
but we cannot give them our mental structures; they must build their own” (p.145). Can
we give others our cognitive maps? In the book Making Connections: Teaching_and the

Humgn Brain (Caine & Caine, 1991) (a book that Marv claims has greatly inﬂuenced his

 

thinking), the authors claim that cognitive maps are always very complex and personal,
being dependent on one’s past and present experiences. So, although we can give people
opportunities for their own experiences, we cannot give them our experiences or the

understandings we have developed. Teachers cannot give mathematical power to their

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students. However, by participating with students in conversations that include
hermeneutic listening, students are afforded opportunities to gain mathematical power.
Marv has learned the value of including students in his instruction by listening carefully to
their sense making and adjusting his instruction accordingly. Giving students the authority
to build their own mathematical schema, while using the teacher’s expertise to create
situations that will confront them with the big ideas is made possible by hermeneutic
listening.

The changing view of the role of a teacher that seems to pervade in Marv’s story
moves in the direction of an intellectual/moral obligation rather than a clerical/procedural
enactment of a job (Hansen, 1995). Like Giroux’s (1997) cry for educators to move away
from the language of critique related to social injustices and redeﬁne their practices in a
language of transformation and hope, this story illustrates a move toward seeing the need
for school mathematics reform discourse to be clothed in transformation and hope.
Marv’s story is a beautiful and explicit example of what happens when a teacher shifts
control, from trying to control student behavior to controlling curriculum and treats his or
her students as intelligent and capable human beings. As the teacher has more control of
the cun'iculum, they are in a better position to both hear the students and incorporate the
students’ ideas and ultimately inﬂuence the students.

One very interesting result is that students are given opportunities to gain
mathematical power and conﬁdence as they pursue their own questions and those of their
classmates and teacher. In Marv’s case, these investigations were directly linked to the

way he now he listens to his students. He no longer feels obligated to lead them or

103

facilitate their conversations to reach speciﬁc goals; instead, he participates in ways that
turn his classroom into an intellectual forum in which he is a listener and a learner, while
creating the situations that will develop a story line of the subject and helping students
gain a relational understanding of mathematics.

Marv’s vision of what it means to teach students mathematics has shifted from his
performance in explaining the procedures to designing educative experiences, by being a
student of the subject matter and an aggressive listener to his students. Although listening
is one of the things he does differently, it is an indication of the epistemological shift that
now says that teaching must be directly related to learning, not merely “presentation and
evaluation” as Marv described his experience in his college mathematics courses. In a
mathematics classroom that values giving students opportunities to gain mathematical
power, the students are engaged in “exploring, making conjectures and participating in
logical reasoning,” yet they will not create something out of nothing (Dewey, 1902). So,
the responsibility of the teacher, according to Marv. is to use their ideas and his expertise
to create mathematics, while analyzing, critiquing and coming to better understand

classical mathematics. Listening closely to students ideas is an important role of the

teacher in this scheme.

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Chapter Five

Implications for Developing Mathematical Power
through a Professional Transformation

Mathematical power includes the ability to explore. conjecture. and reason
logically; to solve non-routine problems; to communicate about and through
mathematics; and to connect ideas within mathematics and other intellectual
activity. Mathematical power also involves the development of personal self-
conﬁdence and a disposition to seek. evaluate. and use quantitative and spatial
information in solving problems and in making decisions. Students ’ﬂexibility,
perseverance, interest. curiosrty. and inventiveness also affect the realization of

mathematical power. (National Council of Teachers of Mathematics (NCTM),
l99l,p.l)

Summary of Results

The shifts in Marv’s view of learning, mathematics and his role as a teacher, taken
together with the changes he has made in his practice (as evidenced by the three vignettes
in the Introduction), describe Marv’s professional transformation. This sort of
professional transformation is very rare and it cannot be reduced to a concrete set of
actions or changes to practice. Instead, the fundamental components of Marv’s
transformation were shifts in his beliefs and the way he thought about his work as a
teacher, together with his changed practice. These shiﬁs put Marv in a position to provide
opportunities for his students to gain mathematical power.

In particular, holding a goal for student learning that includes developing a
relational understanding of the mathematics sets an agenda that fosters the ability to
connect ideas within mathematics, a component of mathematical power. Exploring and
making conjectures around mathematical ideas are skills that reﬂect mathematical power.
They make sense only in the context of a perspective that views mathematics as a human

construction rather than absolute truth. The Standards emphasizes communication as an

105

important part of mathematical power, which implies a kind of interaction made possible
by participating in hermeneutic listening rather than seeing mathematics teaching as a mere
demonstration of the “right” way or the most “eﬁcient” way. Thus the key components
in Mary’s professional transformation are central in developing mathematical power.

Fostering the abilities spelled out in the above deﬁnition of mathematical power
(exploration, making conjectures. logical reasoning, mathematical communication and
connections) in a group of high school mathematics students was made possible in Marv’s
classes as a result of his professional transformation. As many mathematics teachers have
discovered, allowing students the freedom to explore mathematics does not mean they will
happen onto conventions and ideas as they appear in textbooks. The intention and goal
for having students explore mathematics needs to be consistent with the teacher’s goal for
the students’ learning and the teacher’s philosophy of mathematics. In Marv’s case, the
fact that his calculus students were allowed to explore their ideas related to accumulated
rates was provided because of Marv’s belief that students need to build a relational
understanding of the mathematics, together with his belief that their understanding does
not have to look exactly like what is recorded in textbooks. Prior to Marv’s transformed
goal for his students’ learning and view of mathematics. exploration did not have an
apparent purpose and was frustrating to students and the teacher, as Marv could only
allow them to go so far and then he needed to tell the students what he wanted them to
know.

As students are exploring ideas related to problem situations or following up on

problems they have posed. they are positioned to make conjectures. An example of a

106

conjecture from one of Marv’s students (as it appears in the third vignette), is the student
who suggested that “You can raise -9 to any power, you just can’t take any even root of
it.” This conjecture led to a discussion during which students were communicating their
mathematical understandings, using logical reasoning. This was possible for several
reasons. First, Marv was having the students analyze and critique a textbook deﬁnition
rather than just accepting it at face value. Second, he conceived his role as being
responsible for listening carefully to their ideas and thinking through the mathematics with
them, while addressing their ideas and questions. And third, his goal for their learning was
to conﬁ'ont them with organizing principles of mathematics. (ﬁinctions as objects of
studies, domain, as a attribute of a function, which led to an investigation of the students’
number sense) so they could develop their own relational understanding of the
mathematics. When the learning goal is strictly an instrumental understanding, the teacher
would accept the textbook deﬁnition at face value, have the students memorize it and
encourage them to look at the graph to identify the domain represented. This combination
and coordination of Marv’s transformed beliefs and the resulting instructional enactment
created a forum for students to become mathematically empowered.

Earlier I recounted the time when one of Marv’s calculus students, using his
understanding of mathematics, suggested a proof for the Product Rule for Differentiation
that was both original and reasonable. This student’s inventiveness and conﬁdence
(aspects of developing mathematical power) in proposing a proof that was different than
the textbook proof, was an illustration of the mathematical power being realized in Marv’s

classes. Again, this was possible because Marv had come to realize that the proof was

107

dependent on logical reasoning and was a human construction, rather than seeing the
textbook proof representing indisputable truth. He followed the student’s suggestion and
incorporated his idea because he had become a hermeneutic listener.

Marv’s students often spend several days investigating one problem or one
mathematical question. They have learned that thinking and examining their ideas is much
more desirable and valuable than quick instant answers. As his students construct
mathematics. they exhibit perseverance in their search for understanding, a characteristic
of mathematical power. These exhibitions of mathematical power seem to be directly
related to Marv’s professional transformation. As he continues to analyze and develop
curricula for his classes based on his changed view of the nature of school mathematics, he
positions himself and his students to be confronted with the story line he has developed
based on classical mathematics. At the same time, Marv is allowing the class the
opportunity to construct their own relational understanding of the mathematics, as he
enacts his transformed role by engaging in hermeneutic listening.

As I searched to understand Marv’s professional transformation, the encompassing
nature of the changes in his thinking and the ways these dramatic changes affected his
instruction played important roles. He was considering the big picture, perspectives of his
profession, including his students and their understandings, the discipline and nature of
school mathematics and how one teaches consistently with one’s philosophy of
mathematics. He was not inﬂuenced by a program of reform or a speciﬁc result, but
instead by his own changing vision and a desire to stay true to what he believed. The

relationships between the changes spelled out in the middle three chapters are so

108

intertwined, it was challenging to tease them apart and analyze each separately. Yet, they
represent very different aspects of his instructional practice. I would maintain that it was
their relatedness or the consistency represented in his transformed goals for students’
learning, his philosophy of mathematics and his view of teaching and the related changes in
his practice that added up to a professional transformation. The changes actually
reinforced each other. Isolated changes or changes that are inconsistent with one’s
practice would not qualify as a transformation in the same way. For example, one sees
many instances in which teachers incorporate reform ideas in their classrooms (e. g. group
work or allowing the students to talk about their interests or use of innovative curricula),
but because the learning goal is still an instrumental understanding or students are
presented with a predetermined process to reach the answer in the back of the book,
mathematical power is not being fostered.

Marv seemed to gain conﬁdence in his own relational understandings of the
mathematics. and became more willing to explore and search for a connected
understanding of mathematics -- all of which are signs of his increased mathematical
empowerment as he transformed his philosophy of mathematics and goal of learning
mathematics. After experiencing the freedom of pursuing his questions and critiquing
conventional mathematics, Marv was anxious to give his students similar experiences.

Marv’s desire for this students to have a rich connected understanding of the
material challenged his philosophy of mathematics and his understanding of the nature of
school mathematics. His feeling that the students should own their understandings and not

merely replicate what he thought or what was written in the text pushed him to consider

109

his role as their teacher and again his philosophy of mathematics. If mathematics was
what was in the textbook, Marv was conﬁised about what to do with students’ various
understandings and ideas. By challenging his beliefs and practices in all three areas, and
doing it in a heart-felt way that impacted the enactment of his instruction and his basic
belief system about the subject matter and profession, Marv was in a position to
experience a professional transformation and position his students to gain mathematical
power.

What does this imply for “non-Marv’s”, professional development or teacher
education, as we try to support teachers and teacher candidates in ways that they might
experience a similar transformation and subsequently teach differently than they were
taught. Fundamentally, it seems that all of the following three questions are very
important for teachers to consider, either explicitly or implicitly, realizing that their beliefs
in each area will impact the other areas.

I) What is the goal for student learning?

2) What is mathematics?

3) What does it mean to teach mathematics?

One intention of this kind of reﬂection and investigation would be to move teachers and
prospective teachers out of the realm of inﬂuence that motivates them to teach like they
were taught or maybe more accurately, how they perceive they were taught. By
articulating their beliefs and understandings they will come face to face with
inconsistencies, both between their answers to the different questions and between their

beliefs and practice. Throughout the rest of this chapter, I will consider different contexts,

110

activities and circumstances in which practicing teachers and teacher candidates might be,
either directly or indirectly, confronted with these questions. I will consider ways in which
teachers might be supported in considering their beliefs and the related shifts in their
thinking necessary to experience a professional transformation and position themselves to

provide more students with more opportunities to gain mathematical power.

School/University Collaboration

The teacher’s learning community, a key factor in Marv’s professional
transformation, presents us with a complex and important set of circumstances for the
professional life of a teacher. Teachers typically spend many hours a day interacting with
groups of students. with little time structured around professional collaboration. The half-
day built into the weekly schedule at HHS provided a window of opportunity for
professional development work, including confronting big ideas in mathematics, curricula
development, and discussing students sense making with colleagues. In Marv’s case, the
community expanded beyond the walls of the school and included the university and even
international mathematics education scholars. The resources provided by this greater
community impacted his practice and ensuing professional transformation. Encouraging
teacher candidates and practicing teachers to pursue relationships with university teacher
educators and education scholars is one way to work at removing these walls and bringing
the two worlds together.

The collaborative culture within HHS has been developed, at least in part, through
the alliance formed with the university in relation to its standing as a PDS (Professional

Development School). As a well-functioning PDS, the intentions of this design have been

111

carried out at HHS. These intentions include “a school for the development of novice
professionals, for continuing development of experienced professionals and for the
research and development of the teaching profession” (Holmes, 1990, p. I). These
ongoing relationships carried out as a partnership, rather than top-down directives or
projects, have served both communities in pursuing serious inquiry related to policy,
subject matters, teaching and leaming. Marv feels that he is a “partner in inquiry with the
university”, and as a teacher, he represents a perspective that university researchers need.
At the same time, he feels that his interactions with Del, who brought the curriculum ideas
to HHS and has stayed in communication with Marv, and the network that has followed,
has afforded him unique opportunities to learn how others are thinking about mathematics
education. More than anything else, these contacts gave Marv permission and conﬁdence
to pursue his questions and ideas - the opportunity to gain mathematical power.

One possible lesson learned from Marv’s experience is that researchers and pe0ple
in academia need to take teachers seriously. Is it possible that many teachers have
frustrations and questions similar to Marv’s, but no one is listening to them or they do not
feel they have the right or opportunity to question their practice? It seems important to
ﬁnd out the questions that plague practicing teachers and give them opportunities to
pursue them, to dig deeply and confront their beliefs in collaboration with other educators
with similar concerns. Hargreaves and Dawe (1990) came to a similar conclusion in their
study, implicating that strategies for professional development should "shift away from
university or college-based courses targeted at the individual teacher and intended to raise

his or her level of intellectual awareness and ability to reﬂect, to more school-centered

112

 

forms of professional development which recognize, bring together, and build upon the

skills, experience, and insights that teachers already have” (p. 229). Marv was the active
agent in the professional transformation he experienced, not the curriculum, technology,
outside resources or opportunities, etc.. Although they all made important contributions

to the changes, the transformation did not happen in isolation in any one of those areas.

Sharing Teaching Assignments with a Teacher Asking Similar Questions

Having the same teachers teach the same students at the same time offered many
advantages. Marv summarizes his reaction to the experience as follows:

Teaching with someone gave me a common experience in mathematics so
that we had a common language to develop our ideas. We could talk about
what the students had said and the ideas that had surfaced and because we
were both there we didn’t have to ﬁll in every detail or try to picture the
context. Otherwise, you are hearing about someone else’s experience and
in your mind placing it in the context of your own classroom and practice.
Team teaching pushes you to think about your teaching and why you are
doing things. It gives you time to sit back and watch your students being
taught. while you listen and think. You have opportunities for someone
else to tell you what they saw and were thinking while you were teaching.
The greatest professional beneﬁt that I see is that in an attempt to
understand a different way of teaching or the beneﬁts of a different
curriculum, you have the opportunity to see what goes on in the classroom.
Looking at test scores or professional education articles gives you rather
limited information. It is the change in attitude that the kids have about
mathematics. what they are doing in the classroom, and how they
understand the material. I was talking with one of our special ed.
teachers who was at an elementary school and we were talking about
teaching mathematics differently. She said she talked about it and talked
about it with teachers, but did not get it until she team taught. That is the
same conclusion we have come to. (Marv, spring 1995)

Sharing teaching assignments has encouraged collaborative in-depth study of the
mathematics within the department at HHS. According to Marv, “I think that one of the

best things about team teaching is that you're getting together with other math teachers

113

 

and just being able to talk about math and talk about new ideas (F all 94) . ” This
collegiality has made curricula creation and development a reality at HHS, as the teachers
have worked together to think about the big ideas they want their students to encounter at
each level and then share activities or situations they have developed to provide
opportunities for that to happen. Teacher learning and professional collaborative
relationships are tied together in much of the literature on collegiality (Cohn & Kottkarnp,

1993; Engestrom, 1994; Rosenholtz, 1989; Little, 1990, 1993; Hargreaves & Dawe, 1990;

Grifﬁn, 1991; Richert, 1991).

Teacher Candidates ’ School Experiences

Marv does not hesitate to talk about the important transition in his thinking and
instruction that took place while sharing teaching assignments with a teacher who was
modeling a type of instruction Marv found compelling, while developing curricula based
on a philosophy of mathematics that allowed for ambiguity and diversity. While in this
situation, Marv was able to develop a vocabulary and a sense of teaching mathematics
differently than he had been taught or than he had seen. The value of this experience for
Marv implies that sharing classrooms with teachers modeling the kinds of teaching
educators are endorsing could prove invaluable to teacher candidates’ development

What message are we sending to teacher education students when we talk to them
about teaching for a relational understanding and then expose them to teaching practices
that focus only on an instrumental understanding? Is it possible for these teacher
candidates to transfer what they are learning to an unknown or imagined setting and make

sense of it? What picture comes to their minds when we talk about mathematics students

114

 

engaging in conversations around the mathematics concepts if, as in most cases, it is a
phenomena they have never experienced? Even when we give them similar experiences in
a university classroom, they assume that holding conversations about mathematics is a
reasonable activity for a group of mathematics majors, but not so for a group of freshmen
studying introductory algebra. To most teacher candidates this is a completely foreign
idea and will become a perceived reality only if they experience it ﬁrsthand. If we place
teacher candidates with practicing teachers who are considering these questions, we can
expect that they also might reconsider their beliefs. Or, in some cases, this collaboration
can work the other way around, where interns asking these questions can impact
practicing teachers.

A colleague and 1 were able to study the effect of contrasting ﬁeld experiences
recently when a secondary mathematics teacher candidate divided his ﬁeld time during his
senior year methods class between two very different settings (Gormas & Rosenthal,
1997). The candidate said that the conversations and experiences we had in the method’s
class about teaching mathematics “did not make sense” until he experienced it in the
second placement, during which he listened to students share their ideas (making
conjectures), disagree with each other (take part in mathematical reasoning while
communicating about and through mathematics) and reach a consensus (problem solve
using logic) about mathematical concepts. What really amazed the education student was
that these high school students were 10th-12th graders in an introductory algebra course,

most qualifying for “special education” case loads, necessitating a special education

teacher’s presence during the instruction.

115

 

Creating Curricula Considering a Philosophical and Historical Perspective

How does a mathematics teacher’s philosophy of mathematics affect his or her
view of curricula? If, as Marv and I both had thought, mathematics is restricted to the
information, skills and processes presented in the textbook, there is not much work or
thinking to do apart from describing the contents of the textbook. On the other hand, if
mathematics includes the development, connections and ideas related to the various
concepts, as Marv came to perceive mathematics, it is a dynamic ﬁeld that includes the
cultural, historical and on-going construction of these concepts.

Should teachers serve merely as dissenrinators of prepared curricula? If teachers
are to play a more active role in the classroom, facilitating discourse and on-going
assessment, should they be engaged in creating curricula for their students? This approach
assumes that teachers have been confronted with the big ideas of mathematics and have
engaged in the development of their own relational understandings. Marv’s professional
transformation included serious study of the mathematics content. as he considered the
philosophy and history of mathematics and the big ideas within the subjects he was
responsible for teaching. What resources do teachers need to partake in this type of
activity? What are the implications of this for non-Mary’s?

As teacher educators recognize the importance of helping teacher candidates move
into Schiﬁer’s fourth stage of changing their conception of school mathematics, they
might consider organizing their teaching so that the teacher candidates encounter the big
ideas of mathematics. If we challenge teachers and teacher candidates to consider story

lines of algebra. geometry and calculus, they will be in a position to consider the

116

relationships, connections and build schema of the mathematics intended for students to
learn. This kind of thinking becomes foundational if we ask teachers to create curricula
that will help their students understand relational mathematics. Mary’s experience in
struggling with handling students’ ideas in light of the intended cuniculum suggests that
teacher candidates could beneﬁt greatly from investigating their philosophy of
mathematics as well as the historical context of the mathematics. As long as people
consider mathematics strictly as an infallible truth existing in a canon somewhere, rather
than a human construction developed over time, memorizing and mimicking makes more
sense than constructing an understanding, that is relational, as well as instrumental.
Paying attention to the historical development of mathematics could help teacher
candidates realize the ﬂexibility, diversity and multiple approaches to mathematics that are
viable and reasonable.

An important piece of Marv’s professional transformation that is interesting to
consider is the historical and cultural development of mathematics. Marv’s reading of
Lakatos (1976) introduced him to a philosophy of mathematics that views the construction
of the discipline of mathematics as a human development, and therefore fallible. While
investigating the history of mathematics, Marv began to recognize the many approaches to
mathematics that are viable and put him in a position to both appreciate the conventions
and traditions, as well as the freedom to critique them. This brand of mathematical power
was then passed on to his students as he gave them similar opportunities based on their

questions and discoveries. Considering the historical development of subject matter could

117

 

be an important element in confronting the three big questions mentioned at the beginning

of this chapter.

Developing a Relational Understanding of the Mathematics

The mathematics teachers at HHS have worked hard at developing their relational
understandings of the mathematics they teach. The mathematical depth of their
conversations seems related to their perceived responsibility for creating and developing
mathematics curricula for their students (Hargreaves, 1992). As they consider the story
line for each subject matter, they confront the big ideas themselves and then work on
creating situations to help their students confront these big ideas (Schiﬁer, 1995). An
intern I worked with recently said, “My CT (collaborating teacher) never talks about the
mathematics, only schedules and management issues. Even in department meetings, the
discussions are centered around purchasing materials and attending conferences.” This
seems reasonable if the mathematics instruction is restricted to presentation and evaluation
of the information presented in the textbook.

How can we help pre-service and in-service teachers build their own relational
understandings? The idea that mathematics is more than and different than what appears
in textbooks causes concern for mathematics teacher educators. According to Ball
(1991), learner-focused mathematics teaching requires a broad knowledge base in
mathematics to recognize and capitalize on opportunities that arise or apply mathematical
ideas and procedures. Most mathematics teachers see mathematics only in the realm of
school mathematics -- arithmetic, algebra and geometry textbooks (Thompson, 1992). A

high school mathematics teacher recently said to me. “Mathematicians and textbook

118

 

writers see high school teachers as too dumb and incapable of thinking deeply about the
mathematics, so they encourage us to demonstrate watered-down, simplistic skills that will
somehow help high school students succeed at higher level mathematics without
understanding any of it.” It isn’t clear what has caused this situation, but it is clear that
mathematics teachers, for the most part, seem comfortable in this position, while at the
same time frustrated when students progress is slowed down or halted by their lack of
understanding. One inference from Marv’s early view of algebra being a series of skills
that students need to get proﬁcient at before moving on, is that teachers tend to see the
mathematics only in the context of textbook examples and skills. Marv’s experience
implies that getting teachers involved in creating curriculum may push their thinking
beyond amassing skills, to a relational understanding of the mathematics. Designing
curricula, that takes into considerations big ideas and organizing principles of

mathematics, seems like an important foundation for teacher candidates.

Focus on Student Learning

One way of thinking about this issue is in terms of authority. How should
authority be dispersed in a classroom between tradition and convention (as represented in
the textbook), the teacher and the students? The question of authority or criticism of the
idea that the teacher is the sole holder of knowledge (Freire, 1970) in the classroom is at
the heart of many subject matter reform issues and professional development efforts. Part
of Marv’s transformation in his practice has been to alter his teaching strategies from
telling as the expert, to listening to students’ thinking and ideas -- putting a high premium

on their knowledge and sense making abilities. According to Marv’s calculus students’

119

 

comments on an end-of-the-year evaluation (Spring 1995), students ﬁnd learning
mathematics much more satisfying and enjoyable and feel prepared to continue in higher
level mathematics after this type of mathematics experience. In their evaluations, they
discuss the difference between trying to ﬁgure out the textbook and teacher’s meanings to
discussing their understandings, sharing the authority, and ﬁnd the latter much more
helpﬁrl in making sense of the concepts. The classical view of mathematics teaching,
clearly explaining procedures and facts ﬁom mathematics textbooks, will not empower
students mathematically if mathematical power is developed through conjectures,
exploration, analysis and proof (NCTM, 1991). These processes invite teachers and
students to actively participate in the discipline as well as study what others have
developed and the existing body of mathematics knowledge.

One type of communication available to teachers and teacher candidates in all
school settings is listening to students. as a type of teaching method, rather than focusing
on their own performance or even the collaborating teacher’s performance. As the teacher
candidates hear the students’ sense making, they are enacting on-going assessment and
having the opportunity to observe that the learners are capable of reasonable, logical,
mathematical thinking. This practice of listening might begin with more evaluative
listening during the teacher candidate’s time of mere observation and can be expanded to
include hermeneutic listening that is more conducive to learning as the teacher candidate
takes on more teaching responsibilities. Marv’s transformation from strictly evaluative

listening, to including interpretive and hermeneutic listening implies that teachers need to

develop in this area.

120

 

Teacher educators have the unique job of addressing practical and theoretical
aspects of the profession simultaneously (Labaree, 1996). Teacher education, as it is now
enacted, even in the best programs, places an inordinate amount of focus and energy on
teacher performance. Final evaluations for teacher candidates are based on performance --
their ability to lead and manage a class in “active” learning experiences over a period of
time. Perhaps part of the implication of Marv’s story is imagining a change in focus in
teacher education programs toward discovering what it means to learn something by
studying learners in context and making connections to their conceptions and the discipline
by listening. It is signiﬁcant that Marv found himself taking on the disposition of an on-
going learner to become a better teacher rather ﬁnding better ways to perform. Dewey
(1904) states the necessity of this type of disposition if a teacher is to expect intellectual

engagement from his or her students.

Only a teacher thoroughly trained in the higher levels of intellectual method
and who thus has constantly in his own mind a sense of what adequate and
genuine intellectual activity means, will be likely, in deed, not in mere
word, to respect the mental integrity and force of children (p. 329).
Marv discovered that it takes time to develop a Deweyan sense of ‘adequate and genuine
intellectual activity’ on the job. Even as a veteran teacher, he was well into the second

semester of a class in which student learning was the focus, before he was actually hearing

the students and feeling comfortable with his own understanding of the curriculum.

Transformation Takes Time

I ﬁnd it very encouraging and compelling that seeds planted during Marv’s teacher

education program were able to take root and contribute to his professional

121

 

transformation. What are the delayed effects of teacher education courses? Teacher
candidates desire answers similar to what they have gained in the rest of their education --
concrete procedures that will produce speciﬁc results. Marv argues that this is perfectly
reasonable, based both on their previous schooling and their experiences related to
learning. Changing one’s perceptions constitutes challenging assumptions and offering
alternative experiences. In Marv’s case, when the idea was considered over time, as a
teacher, it became something with which to reckon. This brings attention to the
importance of in-service teacher education continuing the work of helping teachers make
epistemological shifts (Nelson, 1995), building on their experiences in the classroom,
while challenging their assumptions related to teaching and learning. These experiences
give the teacher candidates opportunities to “change their beliefs about the nature of
knowledge and learning, deepen their knowledge of mathematics, and reinvent their
classroom practice from within the new conceptual framework” (Nelson, p.2).

People who enter the teaching profession do so after having gone through over a
thousand hours of an “apprenticeship of observation” (Lortie, 1975). Their decision to
teach is based on their school experience and. in making that decision. they are imagining
teaching as they were taught. In the case of mathematics teaching, there is an enduring
tradition of following a textbook section by section. Instruction has also followed a cycle
which involves demonstrating how to do complete problems in the next section, assigning
homework problems from the textbook and answering homework question. The NC 7714
Curriculum, and Teaching Standards (1989, 1991) were written to change this pattern

and reform school mathematics, in an attempt to empower all students mathematically,

122

(see deﬁnition in the Introduction). This case study has addressed how a teacher
changed, or transformed his practice, including his basic beliefs about it, in ways that
empowered him to ﬁrst think about and then teach mathematics differently than his own

school experiences. The changes that resulted in a transformation happened over a period

of time in the midst of relevant experiences.

Final Thoughts

Marv’s story is a story of transformation -- a kind of transformation that has been
rare, especially in mathematics and especially at the high school level (Cuban, 1993).
Teachers have a view of teaching, learning and mathematics that has been handed down to
them by the society in which we live. Nearly everyone has a deep loyalty related to how
school should be conducted and what it means to teach -- both in and out of school, which
pushes one toward traditional ways of teaching (Cohen, 1988). Society is also interested
in controlled results, which can only come from controlled practices. Sharing the
authority with students, framing problems and being forced to provide intellectual
rationales rather than being a voice for the text makes reform teaching a high risk and
taxing adventure (Cohen). Yet, if we continue the same patterns without somehow
confronting teachers with their beliefs and providing opportunities for transformations, we
are perpetuating a problem that affects the depth to which the mathematics students have
an opportunity to learn and the number of students who participate in mathematics
instruction. The majority of our teachers and students will remain mathematically
disempowered. The truth is “as a society, we seem to be educating only a tiny fraction of

the population to be mathematically literate” (Ball, 1997). By recognizing that the

123

 

structure and format of traditional mathematics instruction should not serve to sort
students, this story of Marv’s professional transformation is a testimony -- “bearing
witness to motivate listeners to participate in the struggle against injustice” (Giroux,
1997). Here the injustice is against the many students who do not have the opportunity to
become mathematically empowered.

This case of professional transformation leading to mathematical power was not
merely a series of changed teaching techniques, although many changes occurred in
Marv’s practice, but was instead a transformed view of the profession as a whole,
including the important aspects of his role within it. Marv transformed his basic belief
system, related to the teaching and learning of mathematics. Through a variety of
particular opportunities and circumstances that led to this professional transformation,
Marv gained greater mathematical power, and was then positioned, as a teacher, to give
his students opportunities to gain mathematical power.

So. is a transformation like this necessary for teachers to teach for mathematical
power? That is a good question to consider. but not within the scope of this work, since it
is a single case study. Yet, there is much to be learned from this transformation and the
resulting mathematical power. First and foremost is that a teacher can and did experience
a professional transformation of this nature. Second, the goal of each student having the
opportunity to develop mathematical power. which doesn’t seem feasible under a
traditional instructional paradigm or techniques, became a reality as a result of this
teacher’s professional transformation. Third, the changes that have gone on in this teacher

and his practice are heart-felt. not experimental or outwardly imposed, and therefore, there

124

does not seem to be much chance he will revert back to the teacher he was, which says

something about the long term affect of this kind of a transformation.

125

APPENDICES

126

Appendix A

Marv ’s Professional Chronology

Spring of 1984 - Graduated from HHS

Fall of 1984 - Entered university - School of Engineering

1987-88 Accepted to university’s College of Education - Academic Learning Program; Student taught at
HHS

Summer of 1988 Graduated from university - BS. in mathematics, minor in physical science; Secondary
Teaching Certiﬁcate

1988-89 lst year teaching - California: two physics. an extremely remedial 7th grade arithmetic and 8th
grade general life science

1989-90 2nd year teaching - California: physics and math
1990-91 3rd year teaching - Califonria: all rrrath

[991-92 4th year teaching - part time at HHS: algebra one and physics and full time mathematics
graduate student at university

[992-93 5th year teaching - HHS: physics. calculus and algebra one. ﬁrst semester released to tutor

algebra in after school program. second semester team taught algebra with ﬁrst year teacher,
Katie

1993-94 6th year teaching - HHS: algebra l’s. statistics. calculus: released from an algebra one class to
team teach with Sherry

-The year 1 met Marv

l994-95 7th year teaching - HHS: team teaching algebra one with Sherman (former junior high school
shop teacher). algebra two. calculus

1995-96 8th year teaching - HHS: team teaching algebra one with Brad (former high school coach),
algebra two. calculus: university intern in algebra one class: took Master’s level mathematics

education course through university’s Mathematics Department. visited Boston to learn about
algebra software

1996-97 9th year teaching - HHS: algebra one. algebra two and calculus: university senior 4 hours/week.
took mathematics education doctoral level course through university’s College of Education
(Mathematical Ways of Knowing with Mr. Bill Rosenthal): started work with mechanieal devices
in project with TERC. Ricardo Nenrirovsky

[997-98 10th year teaching - HHS: algebra two. calculus. released one hour to work on TERC project;

two university seniors: taking two master’s level courses through university’s College of
Education

127

Appendix B
Data

Over the past four years I have gathered much information in an attempt to make

sense of Marv’s story. The main source for this information has been Marv himself,

including interviews, his own writings, ﬁeld notes and videos of his classes (see the table

that follows). I have also interviewed his students. collected student work and their

written evaluations of his classes, talked with his colleagues and collected writings from

perspective teachers observing and participating in his classes.

 

 

19934994 [994-1995 1995-1996 19964997 1997-1998
Field Notes 9/8: 9/13 (lst 9/28: 11/16: 9/18: 9/29'. 10/4'.
(Each date period): “/18; 11/30; 10/5'. 11/1'. 11/16;
represents one 9/13 (6th 12/21: 1/25: 11/17: 11/30:
class period) period); 9/17: 3/6: 3/29 1/17'. 2/20: 2/26:
9/23: 9/28: 2/27; 3/18: 3/27:
10/12; 11/3: 419:4/1715/6:
11/17; 11/18; 5/7; 5/31
11/19'. 12/2;
3/18 (2nd
period)“. 3/14 -
3/ 18 (6th
period): 3/21
Student Work Calculus class Alg 2 Tests
cvaluanns
Videos 3/14-3/ 18 (6th 5/20 - 5/24 (First 5/ 12 - 5/16
period) and third class (Fourth period)
PeriodS)
Interviews 5/6 10/10: 4/6 4/10: 6/15 2/27‘.
Social
Mary’s Team Teaching Constructivism Synthesis
Writings (Fall) (Dee) Paper
Philosophy of
Teaching
(Spring)

 

128

 

 

BIBLIOGRAPHY

129

 

BIBLIOGRAPHY

Atkinson, P. (1997). Narrative turn in a blind alley? Qualitative Hgalth Research, 7(3),
325-344.

Ball, D. L. (1991). Teaching mathematics for understanding: What do teachers need to

know about subject matter? In M. M. Kennedy (Ed), Teaching academic subiects
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