DOCUMENT RESUME

ED 392 790 SP 036 551

AUTHOR Heaton, Ruth M.

TITLE What Is a Pattern?: An Eleme,tary Teacher's Early

Efforts To Teach Mathematics for Understanding. NCRTL

Craft Paper 95-1.

INSTITUTION National Center for Research on Teacher Learning,

East Lansing, MI.

SPONS AGENCY Office of Educational Research and Improvement (ED),

Washington, DC.

PUB DATE Oct 95

NOTE 53p.

AVAILABLE FROM National Center for Research on Teacher Learning, 116

Erickson Hall, Michigan State University, East

Lansing, MI 48824-1034 (810.93).

PUB TYPE Viewpoints (Opinion/Position Papers, Essays, etc.)

(120)

EDRS PRICE MF01/PC03 Plus Postage.

DESCRIPTORS Classroom Communication; Educational Change;

*Elementary School Mathematics; Elementary School

Students; Grade 4; Graduate Study; Higher Education;

Intermediate Grades; *Mathematical Concepts;

*Mathematical Logic; *Mathematics Instruction;

Numeracy; *Patterned Responses; Perception;

*Reflective Teaching; Teaching Guides

IDENTIFIERS *Comprehensive School Mathematics Program

ABSTRACT

After 10 years of teaching rule-driven,

procedure-based, algorithm-oriented mathematics, an elementary

teacher describes how she began to rethink her teaching style after

returning to school for a graduate degree. Her rethinking is based on

the first of four student teaching events that spanned an entire

school year in which she used the Comprehensive School Mathematics

Program (CSMP) and its pattern system. The sLart of her change was

her recognition that the CSMP teaching guide was only a guide, not a

step-by-step procedure to follow without deviance. After 3 years of

teachi,,g as a graduate student, she came to realize that mathematics

is a science of patterns, and it is the search for these patterns

that drives mathematicians. Understanding these patterns helps

children understand mathematical relationships, not just find the

right answer. Recognizing, describing, and creating a wide variety of

patterns provides the foundation for exploring these relationships in

numbers in later grades. By way of conclusion, Mark Twain's "Life on

the Mississippi" is used to construct parallels between Twain's early

learning experiences and the teacher's, mainly in the sense of his

growing understanding of the nature of what he needed to learn and

the usefulness of a text in do:ni, the work. What the teacher took

from her project was the need for a new sense of purpose, the

questions she would need to ask, and an understanding of why this

sense of purpose is important in the kind of math teaching she had

been learning to do. (Contains 46 references.) (NAV)
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Craft Paper 95-1

What Is a Pattern?:

An Elementary Teacher's Early Efforts to

Teach Mathematics for Understanding

Ruth M. Heaton

Michigan State University

National

Center for Research

on Teacher Learning

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Craft Paper 95-1

What Is a Pattern?:

An Elementary Teacher's Early Efforts to

Teach Mathematics for Understanding

Ruth M. Heaton

Michigan State University

NC -at

National

Center for Research

on Teacher Learning

Sponsored by the United States Department of Education

Office of Education Research and Improvement
Craft Paper 954

WHAT IS A PATTERN?:

AN ELEMENTARY TEACHER'S EARLY EFFORTS TO

TEACH MATHEMATICS FOR UNDERSTANDING

Ruth M. Heaton

Michigan State University

Published by

National Center for Research on Teacher Learning

116 Erickson Hall

Michigan State University

East Lansing, Michigan 48824-1034

October 1995

This work is sponsored in part by the National Center for Research on Teacher Learning,

College of Education, Michigan State University. The National Center for Research on

Teacher Learning is funded primarily by the Office of Educational Research and

Improvement, United States Department of Education. The opinions expressed in this

chapter do not necessarily represent the position, policy or endorsement of the Office or

the Department.

e 1995 by the National Center for Research on Teacher Learning

4
NATIONAL CENTER FOR RESEARCH ON TEACHER LEARNING

The National Center for Research on Teacher Learning (NCRTL)1 was founded at

Michigan State University in 1985 with a grant from the Office of Educational Research

and Improvement, United States Department of Education.

The NCRTL is committed to research that will contribute to the improvement of

teacher education and teacher learning. To further its mission, the NCRTL publishes

research reports, issue papers, technical series, conference proceedings, craft papers and

special reports on contemporary issues in teacher education. For more information about

the NCRTL or to be placed on its mailing list, please write to the Publications Clerk,

National Center for Research on Tea.her Learning, 116 Erickson Hall, Michigan State

University, East Lansing, Michigan 48824-1034.

Directors: Robert E. Floden

G. Williamson MeDiannid

Study Directors: Linda Anderson, Deborah Ball, Daniel Chazan, Helen

Featherstone, Sharon Feiman-Nemser, Mary

Kennedy, G. W. McDiarmid, Kenneth Zeichner

Many papers published by the NCRTL are based on the Teacher Education and

Learning to Teach (TELT) study, a single, multisite longitudinal study. The researchers

who have contributed to this study are listed below:

Marianne Amarel

Deborah Loewenberg Ball

Joyce Cain

Sandra Callis

Barbara Camilleri

Anne Chang

David K. Cohen

Ada Beth Cutler

Sharon Feiman-Nemser

Mary L. Gomez

Samgeun K. Kwon

Magdalene Lampert

Perry Lanier

Glenda Lappan

Sarah McCarthey

James Mead

Susan Melnick

Monica Mitchell

Harold Morgan

James Mosenthal

Gary Natriello

Barbara Neufeld

Lynn Paine

Michelle Parker

Richard Prawat

Pamela Schram

Trish Stoddart

M. Teresa Tatto

Sandra Wilcox

Suzanne Wilson

Lauren Young

Kenneth M. Zeichner

Karen K. Zumwalt

'Formerly known as the National Center for Research on Teacher Education (1985-1990), the Center

was renamed in 1991.
WHAT IS A PATTERN?:

AN ELEMENTARY TEACHER'S EARLY EFFORTS TO

TEACH MATHEMATICS FOR UNDERSTANDING

Ruth M. Heaton'

INTRODUCTION

I taught elementary students mathematics the way I learned it. My mathematics

instruction was rule driven, procedure based, and algorithm-oriented. In what would have

been my tenth year of elementary teaching, I became a full-time doctoral student in

teacher education at Michigan State University. Through work as a research assistant on a

project designed to study the relationship between educational policy and teaching

practice in the areas of mathematics and literacy.' I expanded my vision of mathematics

education. Through work on this project, I became acquainted with state- and nationallevel documents calling for a reform of mathematics education (California State

Department of Education [CSDE] 1985, 1992; National Council of Teachers of

Mathematics [NCTM] 1989, 1991). As I observed elementary teachers and watched them

try to make mathematics meaningful for students, I got my first glimpse of the difficulties

experienced teachers face as they try to teach mathematics for understanding.' I

contemplated what it might be like to teach mathematics, myself, in ways that probed and

developed meanings of mathematical ideas. I wanted to try to understand, from the

inside, what was so hard and challenging about the mathematics teaching these

2The author is grateful to the members of her dissertation committeeMagdalene Lampert, Deborah

Ball, David Cohen, Bill McDiannid, and Jay Featherstonefor their help with the development of the ideas

within this text as well as Jim Reineke for his observations and conversations about teaching. The author also

extends her gratitude to Kara Suzuka for the time and patience she put into the creation of the graphics found

within this text. None of this would have been possible without the classroom teacher and fourth-grade students

with whom she worked.

'This project is known as the Educational Policy and Practice Study (EPPS), codirected by Deborah

Ball, David Cohen, Penelope Peterson, and Suzanne Wilson.

'For numerous examples of the kinds of difficulties faced by experienced teachers I and other

researchers on the EPPS project observed, see the following case studies: Ball 1990, Cohen 1990, Heaton 1992,

Peterson 1990, Prawat 1992, Putnam 1992, Remillard 1992, Wiemers 1990, and Wilson 1990.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 1
experienced teachers were aiming to do. What would it be like to try to change my own

teaching practice?

Creating a Practice

In the following school year, the second year of my doctoral studies, 1989-1990, I

arranged to teach fourth-grade mathematics, four days a week, one hour per day, in a

Professional Development School. This was a school whose teaching staff were already

working in partnership with faculty' from the College of Education at Michigan State

University and who, for several years, had been exploring alternative ways of teaching

mathematics. One or both of the parents of students in this school were students at

Michigan State University and moved to Michigan from places all over the United States

and the world to attend the university. Of the 23 students in my class, 12 were Englishspeaking United States citizens. The other 11 were from various countries around the

world and either entered the class speaking English or learned to speak it over the course

of the school year. There were three adults who were also a part of my teaching. The

classroom teacher, an experienced teacher responsible for teaching these fourth graders all

other subjects, observed each day I taught, and we talked on a regular basis about what

she noted (Heaton, Reineke, and Frese 1991). Jim Reineke, a peer and a doctoral student

in educational psychology, observed and interviewed me from September to December

for a study he designed around understanding a teacher's and her students' changing

conceptions of mathematics. Magdalene Lampert, a university teacher educator and

mathematics teacher, taught fifth-grade mathematics in the classroom next door and

observed me teach twice each week. I observed in her mathematics class at least that

often and regularly had conversations with her about teaching.'

I began the school year making use of Comprehensive School Mathematics

Program (CSMP) (McREL 1986), a mathematics curriculum developed prior to the

current wave of mathematics reforms, yet in many ways, the pedagogy implied within it

5Deborah Ball and Magdalene Lampert were among university faculty who had already worked for

several years with teachers in this school. Both worked with teachers and students in the area of mathematics.

'See Heaton and Lampert 1993 for further discussion and analysis of our relationship around my

learning to teach.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 2
seemed consistent with "new" ways of thinking about mathematics 'teaching. Problem

solving is a central focus. Communication or mathematical discourse plays an important

role. A major aim of the curriculum is to help all children learn to reason about

mathematical ideas and understand how mathematical ideas relate to one another. CSMP

is an innovative mathematics curriculum that looked different from any teacher's guide I

had used in the past. Relying on it heavily at the start of the school year seemed like a

safe and reasonable way to begin teaching mathematics in ways that seemed dramatically

different from what I had experienced as a student or the kinds of mathematical

experiences I had offered students as a teacher.

Studying a Practice

Three years after my efforts to teach differently, I considered options for my

dissertation research. I returned to the audio, video, and written documentation I and

others had collected around my mathematics teaching. At the time of my teaching, like

the teachers I had earlier observed, I found the teaching I aimed to do challenging and

difficult. Studying my own efforts was a way to understand and to communicate to others

interested in the enactment of these reforms what is so demanding about trying to teach

mathematics in new ways. The value of this study lies in its contribution to

understandings about teacher knowledge, learning, and development around these reforms

in mathematics education from a perspective rarely found in educational research, from a

learning teacher's point of view.

My study complements much of the existing subject-specific case research on

teacher knowledge (i.e., Ball 1993a, 1993b; Grossman 1990; Gudmundsdottir 1990;

Wilson and Wineberg 1988) grounded in the practices of experienced teachers. This

research cutting across subject areas has led to new practical and theoretical (i.e.,

Shulman 1986) understandings of teacher knowledge and its use in teaching practice.

There is a body of research on teaching concentrated in the area of mathematics. Cobb et

al. (1992) and Yackel, Cobb, and Wood (1991), for example, have attempted to clarify

what it means to teach mathematics for understanding from cognitive and sociological

perspectives through an analysis of interactions. Schoenfeld (1988) focuses on the

influence of social context as well as what it means to do mathematics (Schoenfeld, in

Michigan State University East Lan.sing, Michigan 48824-1034 CP 95-1 Page 3
press). Lampert's work (i.e., Lampert 1989, 1990) includes attention to choosing and

posing mathematical problems, developing mathematical tools for facilitating

communication between the teacher and students, and understanding mathematical

knowledge in the context of classroom discourse. Ball (1993b) has looked closely at

instructional representations, representational contexts, and pedagogical content

knowledge in the context of mathemeics teaching. Taken together, this research deepens

our understandings of what it m :ans to teach for understanding. My study enriches this

research on teaching mathematics for understanding. What my work does is highlight

some of the challenges of learning to teach mathematics for understanding or learning to

bring together, in practice, mathematical understandings of content with the development

of a new pedagogy. It sheds light on the problems, dilemmas, and difficulties associated

with a teacher's efforts to understand and make use of the interdependence between

content and pedagogy while guiding classroom interactions, choosing appropriate

mathematical tasks, making mathematical tools and representations accessible, and being

responsive to students.

The process of learning to teach remains somewhat less explored and theoretically

less developed than current understandings of what it means to teach. There is some

research in the area of learning to teach in the context of mathematics education that

spans both the practice of preservice and beginning teachers (Borko and Livingston 1989;

Eisenhart et al. 1993; Schram 1992) as well as veteran teachers attempting to make

changes in practice (i.e., Cohen et al. 1990; Featherstone, Pfeiffer, and Smith 1993; Hart

1991; Schifter and Fosnot 1993; Simon and Schifter 1991; Stein, Grover, and Silver

1991; Thompson 1985; Wood, Cobb, and Yackel 1991). A general theme across all of

these studies is the difficulty of learning to teach mathematics in meaningful ways

whether one is a novice teacher trying to develop a practice or an experienced teacher

trying to change one. What remains little understood is what makes learning to teach or

changing one's practice so hard. My study offers insight into the difficulties from the

perspective of a teacher who lived them.

In many ways, I represent a best-case scenario of teacher change or learning.

First, while my ability to understand the mathematics in children's understandings as well

as the curriculum appeared lacking at the time of my teaching, it is important to know

Michigan State University East Lansing, Michigan 48824-1034 9 CP 95-1 Page 4
that I have taken more mathematics coursework than most people Who opt to be

elementary teachers. I had successfully completed mathematics classes through calculus.

Second, I have received public recognition for my work as a teacher. In other words, I

am a good teacher. Any tendency to want to dismiss my difficulties because of my

general weaknesses as a teacher is inappropriate here. Third, I tried to make change under

extraordinary and nearly ideal conditions when compared to the ways in which the work

lives of elementary teachers are usually organized and the kinds of support which are

typically available.

BEGINNING EFFORTS TO CHANGE MY PRACTICE

What I present here is the first of four teaching events, spanning the entire school

year, which I analyzed as a part of a larger study.' The different teaching events feature

different lengths of time: a day, a moment, a series of days, a series of days followed by

a six-week break, and then the second to last day of math class for the school year. The

difficulties and confusions I confront in the teaching event described and analyzed here

are based on a lesson (9/20/89) in the third week of the 1989-1990 school year. The

challenges and frustrations that arise in this lesson are representative of what I had been

facing on a daily basis since the school year began. The first few weeks of teaching were

wrought with difficulties and frustration as I tried to enact the vision I held for the kind

of mathematics teaching I wanted to be doing. I continually felt as if I was falling short

(' my goal. I began the year heavily dependent on the teacher's guide, following it much

the same way I have followed mathematics textbooks in the past. I thought making use of

an innovative text would lead to innovative teaching. I was wrong. Making use of a new

textbook in old ways was not the solution. Over time, I had to learn to construct a new

relationship with the teacher's guie.,1. I needed to reconsider what a guide could represent,

what it is good for, what I expect of it, and how to make use of it. The description and

analysis of this teaching event illustrates the start of that learning process.

I began with my plans for teaching this specific lesson. They include my view of

the teacher's guide at the start of the school year, how I viewed an enactment of the

'See my dissertation (Heaton 1994) for an analysis of my learning around all four teaching events.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 5
reforms relative to this particular lesson, and an overview of the mathematical tasky I

wanted students to do. I provide a detailed account of what happened as students worked

the problems focused on patterns and functions which I posed. Drawing on my journal,

Maggie's observation notes, and my conversations with Jim, I offer some possible

explanations for the frustrations I felt during this particular lesson. I revisit these

frustrations three years later, bringing fresh eyes to my teaching and new insights to what

it was that I was learning. I conclude by drawing parallels between what I am learning

from this particular math lesson about learning to teach and Mark Twain's adventures

learnhig to navigate a riverboat as portrayed in Life on the Mississippi (Twain [1883]

1990).8

Following the Teacher's Guide

The lesson I did with students on this particular day came from "The World of

Numbers," one of the content strands that make up the spiral curriculum of CSMP. At the

time of this lesson, I had been following the spiral organization of the CSMP teacher's

guide from one strand to another, from one lesson to the next, with little continuity

across lessons. As I looked over this particular lesson and noticed the title, "Composition

of Functions," I wanted some sense of what this lesson was intended to be about as well

as what students were supposed to do.

There had always been a clear match between what the math textbook' said I was

supposed to teach and what students were to learn and do. In my past teaching, what my

students we:e supposed to learn was what I taught them to do. For example, if students

sIn trying to understand my own learning in the instance of practice featured here and in the other

instances found in my dissertation, I looked for metaphors in texts about others' learning from fields quite far

from educational research. These texts included Twain's 08831 1990) writing about learning to navigate a

steamboat on the Mississippi River, Sudnow's (1978) writing about learning to play improvisational jazz, and

Novack's (1990) study of improvisational dance. Their relevance to, yet distance from, mathematics teaching

made them exciting and powerful analytic tools for me. The descriptions and analyses of learning experiences

offered by these writers became useful in helping me to create new categoets, images, and language for

understanding and communicating my own experiences learning to teach.

'I use textbook and teacher's guide interchangeably through this text. The CSMP curriculum is a

collection of teacher's guides. There are no student textbooks as are found in most more conventional

mathematics cunicula.

1 Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 6
were supposed to learn to divide, I taught them the rules and procedures for computing

the algorithm, and they practiced finding the answers to many long-division problems.

Knowing when and what they understood was also more straightforward. Answers were

either right or wrong. If too many students had difficulty, I repeated my explanation cm

the rules and procedures. In my past teaching, the mere title of a math 'esson (i.e., twodigit division, three-digit multiplication, addition of fractions) gave me sufficient

information to plan, teach, and assess the successfulness of any lesson. There was never a

question about what I was supposed to teach, what students were to learn, and whether or

not we had accomplished what was planned.

In this lesson on the composition of functions from CSMP, the relationship

between what to teach, what was to be learned, my role, and what students were

supposed to do did not seem so clear to me. I saw a need to construct connections

between the activity in the teacher's guide and the mathematical ideas the activity was

intended to teach. The title of this lesson, the "Composition of Functions," gave me little

information about either. The language was foreign to my past experiences as an

elementary teacher. In my nine years of teaching, I could not recall any topic, chapter, or

lesson in an elementary math textbook that used this same language.° While I had

learns i something about functions somewhere in my own learning of mathematics, what

and why they were to be a course of study for fifth graders was a mystery to me.

I turned to the lesson summary, a description found at the start of each new lesson

in the teacher's guide.

Using arrow diagrams and the Minicomputer,

investigate the compcsition of certain numerical

functions, for example, +10 followed by +2 and 3 x

followed by 2 x. (P. 11).

'°Leinhardt, Zaslavsky, and Stein (1990) note the marginality of functions as topic in elementary math

textbooks. Open Court's Real Math (Willoughby et al. 1981) and CSMP (McREL 1986) are among the

exceptions.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 7
Reading this summary generated more questions for me than it ansWered. I knew what it

meant to fmd an answer to a math program, but what did it mean to investigate a

mathematical idea? What did it mean to investigate the composition of certain numerical

functions? What was there to investigate? What were we supposed to be looking for?

How was I supposed to assess what was learned in the investigation? Were there right

and wrong ways to investigate? Finding the right answer to a problem had signaled being

done in the past. How would I know when we were fmished? How would I know when it

was time to move on?

In an attempt to further understand what was meant by the phrase "composition of

functions," I read through the content overview of "The World of Numbers" in the section

entitled "Composition of Functions."

Several lessons in this strand deal with what

happens when you compose a sequence of functions,

that is, apply the functions in order one at a

time. These compositions lead to many general,

powerful insights into the properties of numbers

and operations. (P. xix)

After reading this, I questioned what sort of powerful insights we were searching for and

why. What constituted a "powerful in.ight"? I read a bit further and found that these

students, in previous years, should have had experiences with the composition of

functions.

Your students' extensive experiences with the

composition of functions in the CSMP Upper Primary

Grades curriculum led them to many insights that

involved the development of algorithms, the

discovery of number patterns, and efficient mental

arithmetic techniques. A goal in this strand is to

review these discoveries and to apply composition

to new situations and problems. (P. xx)

Michigan State University East Lansing, Michigan 48824.4034 CP 95-1 Page 8
The fact that students might be somewhat familiar with the content of the lesson I was

about to teach was of some comfort. But, if this was true, would they remember what

they had learned? And, what about the handful of students in my class who were new to

the school? What could I count on in terms of any students' past experiences?

Most of the CSMP lessons were designed to be completed in a single class period.

In the first few weeks of my teaching that year, I found that lessons took much longer

than CSMP or I planned. CSMP typically allowed one day for each lesson while I spent

at least two days, sometimes longer, on each one. I had begun to feel like I was falling

behind. I looked for logical ways to divide the single day CSMP lessons over a couple of

math classes. One way to lessen the feeling of falling behind was to become more

realistic about moving ahead.

This particular lesson on functions involved a sequence of problem around

addition, subtraction, and multiplication. I decided to do the three problems from the

teacher's guide that dealt with addition and subtraction (defmed as Exercise 1) one day

and multiplication (clefmed as Exercise 2) the next. Even with splitting this lesson over

two days, I remained skeptical as to whether or not I would actually get through what I

had planned. These days, my judgements about pace felt off. This was unsettling. As I

found myself falling further behind in terms

of numbers of problems finished and pages

covered in the teacher's guide, I began to +10

question if these were reasonable ways to

measure progress ia my current practice.

Why were things taking so much longer?

What were realistic expectations? Was I

moving anywhere? In those first few

weeks, I experienced a strange sense of

feeling behind in a context where I felt

unsure of where I was supposed to be

going.

The problems in the teacher's guide

under the heading of "composition of

+12

starting number ending number

Figure 1

Michigan State Univetsity East Lansing, Michigan 48824-1034 1 CP 95-1 Page 9
functions" looked like the example in Figure 1. The labels of the arrows varied from one

problem to the next. At the time I planned the lesson, the table seemed easy enough for

students to complete. I was certain they could follow the arrows, one red and one blue in

the teacher's guide, and come up with numbers to fill in the table. Once the table was

filled with numbers, the students were to notice patterns in the relationship between the

input (starting) numbers and the output (ending) numbers. I decided to do this first

problem together with the whole group and assign the remaining two problems in the

teacher's guide for the students to work on their own.

starting number ending number

Figure 2

+cl

starting number ending number

I planned to do just what appeared in the teacl-/.1.'s guide. Students would fmci starting

and ending numbers to complete the tables, identify the composition arrow, an arrow

which would encompass the other two, for each table of numbers, and notice patterns.

Plans to Enact a Vision of the Math Reforms

In my early weeks of teaching. CSMP was a guide for my practice as were ideas

that I got elsewhere for the kind of math teaching that I wanted to learn to do. When I

came across the question of patterns in the script of the CSMP lesson, my familiarity

with the math reforms at the time, based on my observations in Maggie Lampert's fifthgrade math class and what I had read in the Mathematics Framework (CSDE 1985),

helped me to see this as an important question even though a study of patterns was

Michigan State University East Lansing, Michigan 48824-1034 t t) CP 95-1 Page 10
foreign to my past experience as a teacher or student. The importance of the study ofthe

relationship between patterns and functions was described this way:

The study of mathematical patterns and functions enables students to

organize and understand most observations of the world around them. It

involves discovery of patterns and relations, identification and use of

functions, and representation of relations and functions in graphs,

mathematical sentences or formulas, diagrams, and tables. (P. 10)

I had observed Maggie engage fifth graders on a regular basis in discussions of patterns.

I was eager to have the opportunity to attempt a discussion of patterns with these fourth

graders. I had become intrigued with the idea of mathematical discussions and wanted to

get students talking. I thought patterns would give us something to talk about.

Before I became acquainted with the reforms, I never laiew there was anything

that would lend itself to a discussion in math class. I had learned through observing

Maggie's class and reading the Mathematics Framework that it was possible to discuss

mathematical ideas. I could ask students to explain how they solved a problem and why

they solved a problem in a particular way. I had led discussions in other subjects" and

could not imagine that mathematical discussions would be that much different to conduct,

now that I knew there were ideas to discuss.

At the onset of this school year, I expected that moving away from a primary

fociiv on rules and procedures would deepen students' understandings of mathematics and

push at the limits of my own mathematical experiences and understandings. I was

worried about the mathematics I did not understand. I was reminded of the limits of my

understanding as I searched for the meaning of "composition of functions" within the text

of CSMP. These concerns about my lack of mathematical understanding did not,

however, keep me from going ahead with this lesson. With the teacher's guide in hand, I

felt more or less prepared. Even though I was unclear exactly what I was supposed to

teach, I thought having something to do would get us through the hour. The teacher's

guide provided me with problems I thought students could do, questions for me to ask,

"1 engaged students in discussions of real litervare in language arts and held discussions in social

studies making use of the Man: A Course of Study (MACOS) curriculum materials.

Michigan State University East Lansing, Michigan 48824-1034 t; CP 95-1 Page 11
and even student responses I might hear. I

hoped that working on three different tables

would fill the time and asking the question

of patterns would encourage a different way

of thinking about numbers and spark a

discussiontwo aspects of the reforms I

was aiming to make prominent in my

practice that day.

A Table of Numbers

At the start of class, I wrote the problem in Figure 3 on the chalkboard. I said,

"I'd like you to take a look at the chart. Think of a number. Everyone just think of a

number." Bob'' inquired about the parameters for choosing a starting number. "Between

what?" he asked. We barely began and already there was a question. I had not considered

putting conditions on the problem nor was there any indication in the teacher's guide that

I should. Why was Bob asking the question? Given my own uncertainty about why we

were doing what we were doing, the last thing I wanted was for a student to question me

about the task. I said to hi. n, "Just think of a number and add 10 to it and then add 2

more to it." After a few moments of silence while students thought of numbers, I asked

for someone to give the beginning number they chose. I started with Richard, whose hand

was in the air.

+10

starting number ending number

Figure 3

Richard: 99

Ms. Heaton:' And what did you end with?

Richard: 111

Ms. Heaton: And can you tell us how you got that?

Richard: Well, I did 99 plus 10 equals 109 and then plus 2 is

like 9 plus 2 equals 11.

'2A11 students are identified with pseudonyms.

"Please note that I refe. to myself as "Ms. Heaton" in the excerpts of transcripts. I do this because this

is how I was referred to by students. In excerpts of transcripts of conversations with Jim Reineke, I refer to

myself as "Ruth." Jim and I are peers and refer to one another by first names. This switch in how I refer to

myself is a complicated issue and related to issues of voice and identity. For further discussion of these issues,

see Chapter 3, "Learning About Learning to Teach" (Heaton 1994).

Michigan State University East Lansing, Michigan 48824-1034 I 7 CP 95-1 Page 12
We followed a similar routine for adding

the first four pairs of numbers to the table

(Figure 4). Asking students the question,

"How did you get that?" was my idea. It

did not appear in CSMP, but it was a

question I had heard Maggie ask in order to

prompt her fifth graders to talk. Rather than

just have students give me beginning and

ending numbers, I added this question

thinking that asking it would get students

talking about how they solved the problem. What happened was that individuals, like

Richard, responded with procedural explanations that did not seem to hold anyone's

interest. Why was this happening? When Maggie asked the question of her students, they

always offered interesting explanations.

As the teacher's guide suggested, I varied the process of filling the table with

numbers by asking some students to give ending numbers first, followed by an

explanation of how they generated them. The teacher's guide alst, suggested that I call on

someone different to offer these explanations. These variations in the task and ways of

calling on students appealed to me. I thought that starting with the ending number might

make the explanations about how students got from one number to the other more

interesting and calling on a variety of students would increase the number of people

talking. Maybe this would help the talk seem more like a discussion in which many were

engaged. Here is a sample of the interactions that occurred.

starting number ending number

99 111

8000 8012

250 262

1 13

Figure 4

Ms. Heaton:

David:

Ms. Heaton:

David:

Ms. Heaton:

David:

Ms. Heaton:

Jennifer:

Can somebody give us their ending number? The number

they ended with? David?

19

And what was your starting number?

7

Can you tell us how you got that?

Because 7 plus 10 is 17 and 17 plus 2 is 19.

Okay, can someone else give an ending number?

215

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 13
Ms. Heaton: Can someone else tell us what she might have started with?

Bob?

Bob: 203

Ms. Heaton: Okay, how did you get that?

Bob: Because 203 plus 10 is 213, 213 plus 2 is 215 and I have

another number.

I continued calling on students and filled in s .ng and ending numbers until I ran out

of space on the chalkboard. We ended up with a table of numbers (Figure 5), but my

efforts to generate a discussion led us nowhere.

Do You See Any Patterns? +10 Disappointed with the lack of

interesting talk that had occurred thus

far, I held high hopes for the next part

of the activity in which I planned to ask

students to notice patterns. I followed

the script in the teacher's guide and said

to the students, "I want you to look at

these numbers. Do you see any

patterns?" For the next 15 minutes,

students proceeded to give their ideas

about patterns. I called on Valerie first. Figure 5

She said, "Each of them have a

beginning number and then they have an ending number that is

starting number ending number

99 111

8000 8012

250 262

1 13

7 19

203 215

-4%9,- 5000

4988

12 more." I asked her

how she knew the ending number was 12 more. She said, "Because you have to add 10;

you fmd a number, you add 10 to it and then 2." I added the green arrow, the

composition arrow, and said, "If I were to put in another arrow here (Figure 6), what

would I put, plus what?" Valerie responded, "Plus 12." I added the label to the arrow and,

given my goal to initiate a discussion, proceeded to find out what others were thinking.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 14
Pili said, "I see, I think I see a

pattern. It is 80, 80." She pointed to the 80

in 8000 and the 80 in 8012 as she said,

"Eighty right here and 80 right here."

Hearing Pili's idea, I wondered what sort of

patterns we were looking for. Defmitely not

+10

Figure 6

44

1144:1144:41:41:4IiiiiI444:4141.131::141"1.

ones like this. She was right. There was an 8 and a zero in each of those numbers but

taken together, they did not mean "80." The 8 was actually 8000 and the zero was in the

hundreds place. In another attempt to promote

discussion, I asked students to comment on Pili's idea. This call for comments from other

students was a pedagogical move I had observed Maggie make with her students. In

addition to being able to explain their own ideas, I wanted to teach students to listen and

build on the ideas of others. I thought that what Pili had said seemed like a meaningless

pattern and I hoped that one of her classmates might say something to challenge her idea.

I repeated words not found in CSMP but ones I had often heard Maggie ask.

"Okay, what do other people think about that?" I called on Jennifer, who said, "I agree

with her." This, unfortunately, was not the response I wanted. I hoped Jennifer would

disagree with Pili. I could have asked

Jennifer why she agreed with Pili, but I

was not in a mood to hear explanations for

what I thought was a meaningless pattern.

Wasn't it clear that Pili was wrong? Should

I just come out and tell her so? But wasn't

doing that counter to my goal for students

to construct their own meanings? How did

the categories of "right" and "wrong" fit

into my desire to give students space to

make their own sense of ideas? Should I

accept all of students' ideas, even the ones I

thought were wrong?

starting number

99

ending number

111

Figure 7

250

1

7

203

-4088-

4988

262

13

19

215

5000

Michigan State University East Lansing, Michigan 48824-1034 :2 o CP 95-1 Page 15
I moved on without pursuing Jennifer's comment or dealing with this bigger issue

of right and wrong answers. I addressed the class, "Okay, are there any other patterns that

you see?" I called on Lucy, who said, "There is 8 and then 8 going across and then 2 and

then 2 going across." She went to the chalkboard and pointed out what she was talking

about. She pointed to the 8s at the beginning of 8000 and 8012 and the 2s at the

beginning of the pairs of numbers 250, 262 and 203, 215 (Figure 8). She was right.

These numerals matched one another, but what did this observation have to do with the

question of patterns in the context of functions? Reluctantly, I returned to Pi li who was

waving her hand frantically, ready to give another "pattern." She said, "I see three zeroes

here and here," and she underlined the three zeroes in 8000 and 5000 (Figure 9). I was

worried. This "pattern" seemed even more irrelevant than her last one. Maybe I should

have come out and said what I thought of her ideas. She attempted to draw connections

between a pair of numbers not even in the same row. Since the ordered pairs were

generated independent of one another, I Imew there was no mathematical reason why

there would be meaningful connections between numbers not found in the same row.

Maybe Pi li would see for herself the unreasonableness of what she was doing if I could

her to talk about her idea. I asked her to explain. She said, "In 8000 there are three

zeroes and in 5000 there are three zeroes." But what did this have to do with patterns?

iLettedir:2 + +10 +2.

starting number

99

ending number

111

''.r.VsYtk.).., ,=c.(7.'s ec -

1

7

13

19

.40388.

4988

Figure 8

5000

starting number

99

ending number

111

8012

262

13

19

215

Figure 9

250

1

7

203

4888

4988

Michigan State University East Lansing, Michigan 48824-1034 21 CP 95-1 Page 16
What was a pattern? I knew that what I was hearing were not patterns. Jennifer followed

Pi li's comment and said, "I have something similar to hers, but it is not exactly the

same." She went to the board and underlined the zeroes in the hundreds' place in 8000,

8012, 4088, and 5000 (Figure 10).

I felt a mix of emotions. On the one

hand, I was excited that a student had

listened to another student and was making

an attempt to build on a classmate's idea;

on the other hand, I was growing frustrated

as I watched one irrelevant pattern lead to

another. In fact, Jennifer just included a

number that did not even belong on the

chart. It was there with a line through it

because it had been revised. The idea of

taking all ideas seriously, revised or not,

was something else I had picked up from

Maggie. Seeing Jennifer making use of the error, however, made me question my

decision not to just erase it like I would have done in the past. Was that revised number

just adding to the confusion and the fact that it was revised insignificant to what we were

doing? Were there some times when visible evidence of revised thinking was more

important than others?

These questions about revisions seemed like the least of my worries. How could I

stop this generation of meaningless patterns without just reverting back to my past ways

of telling students they were wrong? I was trying to search for mathematical sense in

what students were saying, but was this an instance where there was none to be found? I

was curious how Jennifer would describe the pattern she noticed, so I asked, "What

would you say that pattern is? How would you describe it?"

+10 +2,

star dng number

99

ending number

111

250

1

7

203

262

13

19

215

Figure 10

4988

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 17
Jennifer:

Ms. Heaton:

Jennifer:

One here and one here, and one here and one here.

I don't understand what you mean by one here. What are

you showing me?

There is a zero in the hundreds' place and one zero in the

hundreds' place, and there is one zero in hundreds' place and

one zero in the hundreds' place.

Hearing Jennifer talk of place value gave me a momentary sense of relief from the

meaninglessness of all of this. I continued to ask myself, however, whether any of what

she or the other students were saying had to do with patterns or functions. What was a

meaningful pattern? What was a function? I called on Richard. He explained, "I have

another pattern. Can I go up there? Right

here is three zeroes in a row and three ones

in a row, and then three zeroes in a row." I

asked what the pattern was. He said, "000,

111, 000" (Figure 11). I felt troubled as I

listened to Richard. I knew these were not

patterns in a mathematical sense but what

were they? And, more importantly, what

were mathematically meaningful patterns in

this table of numbers? These students

noticed "patterns" in pairs of numbers in

the same row, pairs of numbers that

spanned rows, and now senseless patterns

in numbers in each of the columns. I knew that "meaningful" patterns could not be found

in columns of numbers in which pairs of numbers had been generated by beginning or

ending numbers chosen at random.

After more than 15 minutes of these patterns, when I could bear this "discussion"

no longer, I announced, "I want to move on to another example." I wrote the following

problems (Figure 12) on the chalkboard for students to work on.

+10 +2,

starting number

99

250

1

7

203

-4888-

4988

Figure 11

ending number

::47:751

8012

262

13

19

215

7-73.7777,_

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 18
starting number ending number

Figure 12

starting number ending number

As the students worked on these problems, I had a few moments to think and asked

myself once more, What is a pattern? I knew that I had already made plans to teach the

second half of this lesson the next day. Could I bring myself to come back the next day

and face this teaching and, quite possibly, the question of patterns again? I left school

feeling perturbed with myself for having trusted the teacher's guide and annoyed with

myself for having thought I entered this lesson prepared to teach it.

REFLECTING ON TEACHING

That evening, I was distraught and troubled by the day's events. My mind was

filled with questions as I tried to sort out the difficulties I experienced that day. Was I

teaching? What were students learning? What were they supposed to be learning? What

would teaching and learning in this situation look like? Why was I unable to get an

interesting discussion started? Why was I asking students the question of patterns

anyway? What was a pattern? Did the patterns students noticed have any relevance?

What is a function? Why didn't students come up with responses that matched the

teacher's guide? How was I supposed to be making use of this teacher's guide?

In my interview with Jim Reineke that afternoon, I said:

I felt like I was floundering today.. . . I was asking myself the question,

What do I mean by a pattern? What does the word "pattern" mean? I am

not sure I have an answer to these questions, but that's what I was

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 19

A
thinking as I stood up there, listening to these kids. And the sort of student

responses the book says will come upthat CSMP puts in, what ,ids

would respond when you say, "What's a pattern?"none of that came up. I

felt like I didn't know what to do to move a discussion about patterns.

(Reineke's interview of author, 9/20/89)

I had trusted that the teacher's guide was going to help me with this work. I felt like it

was letting me down. It was, at once, too much of a guide and not enough of a guide. It

had given me enough guidance to lead me to believe we could do this activity without

providing some broader sense of its purpose. The only reason I had for doing the lesson

was because it was the next thing in the teacher's guide. In the midst of teaching, I found

myself feeling lost and helpless with no way to respond to students.

As soon as I heard students' responses, I realized I did not know what I meant by

"pattern" or really why I was asking the question. I initially asked the question because it

was in the teacher's guide and, given what I knew about the reforms, it had seemed like a

good idea to me. But students' responses did not match what the teacher's guide had

predicted. It frustrated me to have an intuitive sense about what was not a pattern but no

real sense of what would be sensible patterns or how to g,et students to notice and discuss

them.

The predicted responses in the teacher's guide were not useful to me in the

situation. Here is what appeared in the script of the teacher's guide.

T: Look closely at this chart. What patterns do

you notice?

S: An ending number is always larger than the

starting number.

S: If you start with an even number, you end

with an even number. If you start with an odd

number you end with an odd number.

S: An ending number is always 12 larger than the

starting number (p. 12).

Michigan State University East Lansing, Michigan 48824-1034 t)r CP 95-1 Page 20
I had these responses in hand when I went into class. In the past, I would have been

intent on listening for these responses. If I had not heard them, I would have encouraged

students to produce these "right" answers. Like many good "traditional" teachers, I have

had years of experience asking convergent questions. I know how to lead students to the

"right" answer. But, in this situation, I wanted to be open to the idea that students'

answers might vary from the teacher's guide. How was I to know which students'

responses were reasonable variations of the responses found in the sci-ipt and which were

not? How would I figure out how to be responsive to these variations? Was there a way

that I could be respectful of all students' ideas and at the same time let them know that

many of their observations were irrele7ant to the question of patterns? How could I

respond to students' ideas in ways that would push their thinking? The script in the

CSMP teacher's guide was designed with questions for me to initiate, but I felt frustrated

by what little help it offered me in figuring out what to do next in the situation,

especially when it seemed that what students were saying did not match w 'Qat the

developers of CSMP had anticipated they might.

My frustrations went beyond students' responses not matching the ones in the

teacher's guide. I was bothered that the talk that was happening felt out of my control.

Moreover, I did not have any sense of what it would mean for it to be in my control. I

wanted to do somethhig to guide a discussion but, not knowing what to do, I remained

silent. Inside, I was struggling. I wanted us to have a meaningful discussion. I knew that

we were not having one, yet I did not know what we could have talked about or how I

could have shaped what my students were saying into a meaningful discussion. While

troubled by my students' responses to the question of patterns, I was also wrestling with

my own response. I asked myself over and over again, What is a pattern? I had not seen

this question asked or answered in the teacher's guide, but it was a question that

concerned me.

The script in CSMP offered me no assistance with responding to the students or to

the questions I asked myself. If I had been more certain of the point of tl e lesson, I

might have been more willing and able to move away from the script and ask the

students a different questionmy questionor I might have had a way of responding to

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 21
their responses. As it was, the script in my hands, intended to help.me in r- role as the

teacher, felt disconnected from my students and myself and the sense any of us were

making of the question of patterns.

Questioning the Question

When I asked students if they noticed any patterns, my response was to ask

myself, What is a pattern?' I asked students one questionWhat patterns do you

notice?and myself anotherWhat is a pattern?repeatedly throughout this lesson, yet I

never considered asidng students the question I asked myself. I was surprised when Jim

brought up the idea in our interview.

Ruth: The question for me became, What is a pattern? And, Are

these things that the kids are giving me patterns?

Jim: So, why didn't you ask that?

Ruth: Why didn't I ask . . . ?

Jim: The class.

Ruth: What?

Jim: What is a pattern?

Ruth: That would have been a good question. That is the question

that I had on my mind.

As I said the las' sentence, I heard myself laugh on the audio tape. I think I was realizing

how slow I was to see that what Jim was suggesting was that I ask students the question

that had been on my mind. It did not occur to me, prior to this moment in my

conversation with Jim, that asking it would have been a reasonable thing to do. I did not

even consider it an option. Why? One reason might have had to do with my concern for

time and my own need to get through the lesson I had planned. How could I have

justified spending time on what patterns were when, according to CSMP, students were

supposed to be searching for them? The teacher's guide defined the task as noticing

patterns, not defining them. Would the students have gotten through the task of noticing

14Recognizing the question of 'What is a pattern?" in the context of asking the question of noticing

patterns as well as struggling with the fundamental nature of that question seems similar to the question of "What

is a whole number?" which comes up in a lesson one week later. Maggie and I wrote about that lesson (9/27/89)

in -Learning in Hear Voices: Inventing a New Pedagogy of Teacher Education" (Cohen, McLaughlin, and

Talbert 1993).

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 22
patterns if we had stopped to defme them? I began, however, to question how worthwhile

the task of noticing patterns had been without some shared understanding of what we

were looking for and why.

I was relieved to find out that I was not the only one who questioned the meaning

of patterns. Jim said that he had asked himself the same question. Maggie had also

observed this lesson and, as I read her observation notes for the day, I saw that she had

also been playing around with the defmition of a pattern. What follows is an excerpt

from her notes:

Asking "Do you see any patterns?" What do you want to get out of that?

Was the idea to get someone to say that the ending number is 12 more

than the beginning number? There are relevant and irrelevant patterns (i.e.,

like 8-2-2 and 8-2-2). Hard to exactly explain what I mean by "irrelevant

patterns." There are things you could get out of almost every pattern the

kids come up with . . . What the kids are saying are more like

observations than patterns. I think you were trying to get at this a bit when

you asked, "Does that pattern apply to any other set of numbers?"

(Lampert's observation notes, 9/20/89)

During the lesson, I assumed that I was the only one puzzling about patterns. It was

comforting to know that I was not alone and that, from Maggie's perspective, there was

not an easy answer. The definition of a pattern is related to questions of relevance.

Questions of relevance are related to questions of purpose and questions ofpurpose are

related to the mathematics (i.e., both mathematical ideas and ways of knowing) to be

learned. Where might this discussion have gone if I had taken my own questions of

patterns seriously? How might things have turned out if I had asked the question that

made sense to me rather than the one with little meaning for me found in the teacher's

guide?

I went into this teaching thinking that I did not know enough mathematics and

that I was going to be learning a lot. That was in the abstract, before I started this

teaching. Reality hit when I was faced with a question for which I really did not know

the answer. Could I ask students a question that was genuine for me?' Admitting to

'51 had had the experience of asking genuine questions as a teacher and learner with students in the

context of Man: A Course of Study (MACOS). For example, 'What makes man human?"the question arovad

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 23
myself and others that I had something to learn clashed with my view of my role and

responsibilities as a mathematics teacher. I was the teacher. I was supposed to ask

questions for which I had the answers. Or was I? Could I legitimately ask a question as a

learner while also being the teacher? Could I take the risk? I needed to learn to set aside

the teacher's guide, value what I did not know, and trust myself to ask a different

questioneven if it Ineant raising questions about questions in the teacher's guide.

The Limitations of Following a Teacher's Guide

My frustrations with the teacher's guide in this lesson prompted me to reconsider

why it was I thought following this teacher's guide was a good idea in the first place.

The content and pedagogy represented by the CSMP teacher's guide looked different

from any curriculum materials I had used in the past. Since I wanted to move away from

"traditional" teaching, "nontraditional" curriculum materials seemed like a useful tool for

making fundamental changes in my view of content as well as how to teach it. I was

coming to see that making the changes in my teaching to achieve what I was aiming for

was not going to be as easy as merely following the CSMP teacher's guide as I had

followed teacher's guides in the past.

How was I making use of the CSMP teacher's guide in my teaching? As in the

past, I used the CSMP teacher's guide to make my decisions about what and how to

teach. In this lesson, what to teach was defmed as the "composition of functions" and

how to teach was to "investigate." I lacked a good understanding of both of these at the

time. In my past teaching, what I did was the same as what I wanted students to learn

and there was less room for interpretation of either. Now everything seemed wide open to

interpretation and much more complicated.

With little success, I had tried to figure out the point of the lesson by using the

teacher's guide when I planned this lesson. At the time, I ended up putting aside my

which the entire curriculum is organizedis a question for which I did not have an answer when I began

teaching MACOS and one I continue to ponder. Schoolhouse Politics (Dow 1991), especially Chapter 4 on

teacher workshops, supports the idea that other teachers found themselves, often accompanied by much anxiety, in a similar position.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 24
questions about the mathematical idea of composition of functions and what students

were supposed to learn and proceeded with giving students the problems from the text.

The problems looked doable without an understanding of some larger purpose, and I

trusted that doing them would lead to something meaningful. Once I asked the questions,

I questioned my decision to continue discusshig patterns without understanding why they

were important: How could the teacher's guide have helped me to be better prepared for

this lesson? Were there ways that these problems could have been situated in terms of

broader mathematical goals that I understood? What would I have needed to know

outside of the teacher's guide to have felt prepared and gotten the most out of these

problems and the question of patterns? Were there aspects of this teaching that CSMP or

any textbook could not prepare me for?

Another set of frustrations accompanied my use of the teacher's guide when I

tried to generate a discussion and interact with students. The script of the teacher's guide

offered me one possible route through the material. It offered enough detail of the

activity to get students and I started even though I did not understand exacJy where I

was going or why. This had worked fine in my past math teaching, and it would have

been fme this time if what had happened in my class had matched exactly what was

anticipated by CSMP. But this was not the case. At the time, I thought there was a

complete mismatch between the anticipated student responses in the teacher's guide and

what students said. In my journal, I wrote:

What my kids came up with did not resemble these responses . . . I am

wondering what the role of the teacher's manual is in this type of teaching.

A lot of the space in CSMP is taken up with examples of student/teacher

interaction. I'm not finding this particularly useful and at times

troublesome. When I read it, I think I have an idea of what is going to

happen in class and today was an example, last Thursday was too, of

things not going as I anticipated. I base my plans on the book but that

doesn't seem to be working (Author's joIrnal entry, 9/20/89).

This frustrated me to no end. Were there alternative routes to the one in the teacher's

guide? If so, what were they, and how could I connect them to the varied sense students

were making of the question of patterns?

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 25
Before this year, I would have steered us right back on the Course defmed by the

textbook regardless of how students responded to my questions. Now, however, I wanted

to be doing things differently. I wanted to be responsive to students' ideas. Was there a

way I could do that and still get to where we were headed by choosing alternate routes

prompted by students' sense-making? What sort of guide could give me enough of a feel

for where I was headed to guide my decisions about how to get there, yet do so in ways

that were responsive to these students' ideas?

What Is There to Talk About?

While I was teaching, I was frustrated with my attempts to generate a discussion.

Even though students were talking, both as they filled the table with numbers and as they

searched for patterns, it felt to me like there was nothing to talk about or what they

proposed was meaningless. In the first part of the lesson, the discussion fell flat, and in

the second part of the lesson, the substance being discussed seemed irrelevant. Teaching

as telling dominated my past practice as a teacher. At the time of this lesson, only a few

weeks into the school year, I was detemiined to change my ways of interacting with

students and to avoid doing anything that resembled telling. I thought asking questions

was a way to make this happen. I even inserted questions of my own into the script of

the teacher's guide. There was still no discussion.

Given my lack of success at generating a discussion through questions, I began to

think that there had to be more to my role than just asking questions. There was

something to be learned about how to comment on students' ideas in the situation. How

do I decide what to say, about what, to whom, and when? I discovered that I could get

students talking with the questions I read about and heard Maggie use in her class. These

questionsDo you see any patterns? How did you get it? What do others think?along

with the questions in the CSMP guide were pedagogical tools for getting people to talk.

Asking them was a way for me to try on the role of a different sort of teacher. The

questions got students talkhig but then what? Asking the questions was one thing.

Knowing what to do with the responses was quite another.

I began to think that the decisions I could see I needed to make were related to

where I wanted to go with a lesson which was dependent on an understanding of the

Michigan State University East Lansing, Michigan 48824-1034 31 CP 95-1 Page 26
purpose for attempting a discussion in the first place. If I had known why we were

looking for patterns, I might have been able to consider the relevance of the students'

patterns. When I asked a question about patterns in my journal, Maggie commented in

the margin, "Isn't a larger question, Why are we looking for patterns in math class in the

first place???" (Lampert's annotation of an entry in author's journal, 9/20/89).

A sense of purpose might have helped me to figure out what there was to talk

about or possible ways to help students think about their ideas. I opened the discourse

with questions I had heard Maggie ask, but I did so without a sense of where I wanted to

go. I found myself with lots of students' ideas before me, plenty of my own thoughts in

my head, but no way to respond, and the CSMP teacher's guide that carried me this far

was of little use. I did not know what to do to make a discussion out of what students

were saying. I did not know how to respond in ways that would push their thinking. I

also lacked a sense of where I wanted to push their thinking and why. What was it that I

was trying to teach about the composition of functions? What was important for students

to learn? I did not have answers to these questions. I was trying to get a discussion of

patterns going without a sense of what there was to discuss.

THREE YEARS LATER: REVISITING MY FRUSTRATIONS

At least part of my frustration at the time was directed at the teacher's guide and

what I thought was its inability to help me respond to students. In going back to this

lesson three years later, I wanted to understand more about my relationship with the

textbookwhat I expected of it or what I thought it would provide me with or do for me

while I was teaching, the trust I had in it and the people who wrote it, and the way I

made use of it in class. To do so, I have reexamined my frustrations and tried to

understand what it is that I could have known or understood that would have eased or

avoided my frustrations. What mathematics was I supposed to be teaching? What was

mathematically significant about the patterns students gave me? What is a pattern? What

is a function? What is the connection between patterns and functions? What is

mathematically significant about the predicted student responses in the teacher's guide?

What was the intent of this lesson from CSMP's perspective? How do the mathematical

Michigan State University East Lansing, Michigan 488244034 CP 95-1 Page 27

3 :2
ideas in this lesson connect to broader mathematical ideas? How does doing the activity

address what it is that students are supposed to learn? What is important for students to

learn?

In the section that follows, I draw on resources outside of the CSMP teacher's

guide to answer these questions. It is unclear to me whether or not the authors of CSMP

expected me to know prior to teaching this lesson what I now understand. From my

conversation.s with people who know about CSMP and the written commentary on the

curriculum, I have no doubt that the people who designed the individual lessons, the

content strands, and the curriculum had a sense for the mathematical connections and

relationships at many different levels that I am only now coming to understand. This

mathematical structure, however, was not presented in ways that were accessible to me,

nor am I certain that it was intended to be. It seems that the developers may have

overestimated the sheer difficulty of the mathematical ideas represented by the

curriculum.

In the course of pursuing an understanding of the mathematics for myself, I have

discovered a set of connections and relationships among mathematical ideas in CSMP

that were invisible to me at the time I taught the lesson. If the authors of CSMP thought

I understood what follows, I am uncertain how they thought I would have learned it. If

they assumed the information was in CSMP, it was not in a form that I found accessible.

The areas of mathematics I pursue are organized under three broad questions. Why are

patterns and functions an important topic to teach? Why teach about patterns and

functions in a particular way? What sense might students make out of a study of patterns

and functions?

Learning the Importance of Patterns and Functions

The presence of a lesson on the "composition of functions" within CSMP implies

that it was something worth teaching. But why is this topic worth teaching? What is the

importance of learning about functions? What is a pattern? Why study patterns? What is

meant by the composition of functions? How do patterns and functions connect to

broader mathematical ideas? Why does CSMP link patterns and functions? What is the

relationship between patterns and functions? I had begun to ask myself various forms of

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 28
these questions at the time of the lesson. Now, more than three ye.4s later, I have begun

to construct some answers.

Why Spend Time on Patterns? I have recently begun to appreciate that

mathematics is a science of patterns and a search for patterns drives the. work of

mathematicians. Steen (1990, p. 1) writes, "Seeing and revealing hidden patterns are what

mathematicians do best." In an effort to broaden and deepen students' understandings of

mathematics, reformers advise a similar search for patterns in mathematics by all students

at every grade level (National Research Council 1989; NCTM 1989; CSDE 1985, 1992).

For example, the Professional Standards for Teaching Mathematics (p. 4) notes that as

"teachers shift toward the vision of teaching presented by these standards, one would

expect to see teachers asking, and stimulating students to ask, questions like 'Do you see

a pattern?" Patterns, according to the Mathematics Framework (CSDE 1992, p. 108),

"help children to see order and make sense of underlying structures of things, situations,

and experiences. Patterns help children predict what will happen." A hunt for patterns

expands the concept of doing mathematics beyond the search for a single right answer

toward an understanding of mathematical relationships, an aim of the current reforms in

mathematics education. Recognizing, describing, and creating a wide variety of patterns

provides a foundation for exploring mathematical relationships in numbers in later grades.

The Curriculum and Evaluation Standards for 3chool Mathematics put it this way:

From the earliest grades, the curriculum should give students opportunities

to focus on regularities of events, shapes, designs, and sets of numbers.

Children should begin to see that regularity is the essence of mathematics.

The idea of a functional relationship can be intuitively developed through

observations of regularity and work with generalizable patterns. (P. 60)

In the context of a study of patterns, the doing of school math bears strong resemblance

to the work of real mathematicians. The study of patterns now permeates the latest

version of the Mathematics Framework (CSDE 1992) as a "unifying idea" across content

strands. There are good reasons why patterns have come to take a prominent role in

ciementary mathematics curricula.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 29

3 '41
What matters in the study of mathematics is not so much which particular

strands one explores, but the presence in these strands of significant

examples of sufficient variety and depth to reveal patterns. By encouraging

students to explore patterns that have proven their power and significance,

we offer them broad shoulders from which they will see farther than we

can. ( .een 1990, p. 8)

What Is a Pattern? There was no indication in the teacher's guide that my

question about what a pattern is was reasonable. To have asked it would have required

me to value my own question above what I found in the teacher's guide. This would have

required a confidence that I did not have at the time. In the throes of changing my

practice, my self-confidence was at a low point. I thought looking to the CSMP teacher's

guide as an authority for what and how to teach was a safe way to proceed with making

changes in my math teaching. Doing the teaching was hard enough without taking on all

of the responsibility for decisions about what to do next. I assumed that the teacher's

guide knew better than I what questions were reasonable to ask. At the time, I also

assumed that my questioning the defmition of a pattern was related to my being new to

investigatiens of patterns. I realize now that noticing patterns and figuring out what a

pattern is are related tasks. Contemplating one informs the other. Constructing the

definition of a pattern, as you notice patterns, is part of the work of figuring out whether

a pattern is relevant or irrelevant.

Regularity and predictability are the two fundamental characteristics of a pattern.

Something is a pattern if you can observe regularity in it. The regularity allows you to be

predictive about the pattern's behavior. A numerical pattern with regularity and

predictability enables you to describe a relationship between two variables. Identifying a

pattern allows you to manipulate one variable and predict what will happen with the

other. A relationship between two variables with this kind of regularity and predictability

is a function.

I have learned several ways to think about this relationship called a function. One

way is to describe this relationship of variables as a relationship between sets as shown in

Figure 13. If for any member of one set (Set A) you can describe the predictability that

enables you to tell with certainty the corresponding member of a second set (Set B), the

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 30
relationship between the two sets is a function. In other words, given f (x) = y. For any

x (X1, X2, X3, . . . ), you can predict any y (yA, yB, yB, . . . ).

'Set A Set B

A FUNCTIONAL RELATIONSHIP BETWEEN Two SETS

Figure 13

Leinhardt, Zaslavsky, and Stein (1990, p. 27) offer a similar explanation of this

relationship using slightly different language. A function is a "special type of relationship

or correspondence, a relation with a rule that assigns to each member of Set A exactly

one member of Set B." A well-defmed function will allow one to say what y value will

go with a particular x. For each x there is only one value of y. Each element in the

domain (Set A) corresponds to only one element in the range (Set B). When starting with

a particular value, you can predict what the output will be with certainty. Functions carry

with them the notion of predictability and regularity, the essence of pattern.

Why Spend Time on Functions? Understanding the composition of functions

encourages students to build flexibility into the way they look at numbers. Studying

functions in elementary school is a way to prepare for algebra or "the study of operations

and relations among numbers through the use of variables" (Karush 1989, p. 4). It is also

a way to prepare for operating "with concepts at an abstract level and then applying them,

a process that often fosters generalizations and insights" (NCI'M 1989, p. 150). CSMP is

a curriculum oriented around functions. It attempts to build some conceptual

understandings of algebra without using terminology through work with arrow roads.

From this foundation, the expectation is that students in later grades will be able to do

Michigan State University East Lansing, Michigan 48824-1034 C, CP 95-1 Page 31
the kind of abstract generalizing required in the explicit study of algebra. A major goal is

to build the kind of thinking and flexibility that will make a transition to the more

advanced world of mathematics easier, especially aspects of mathematics having to do

with functions.

The overall goal of CSMP and this lesson was to learn to look at calculation in

different ways. Composing functions offers an alternative way of thinking about

computation through the practice of mental strategies for the purpose of combining

different numbers. For example, adding 12 can be thought of as adding 10 plus 2. A

developer of CSMP admitted that the problems given in the CSMP teacher's guide, + 10

followed by + 2, + 7 followed by - 3, and + 9 followed by 4 are not particularly strong

examples of good practice in mental arithmetic. The numbers are easy to calculate and

you do not have to have very complex mental strategies to figure them out. A more

difficult example might be to add 38 and think of it, for example, as the composition of

two functions, + 40 followed by - 2.

Looking back on these problems now, I can see how mental arithmetic could be

emphasized, but this was not apparent to me as a goal when I read through the teacher's

guide. When I asked students to explain how they figured out starting and ending

numbers, it could have been understood as an attempt to make public students' mental

strategies for figuring out these problems. The calculations they did, however, were not

too difficult. Therefore, the mental strategies that students revealed were not too

interesting for a student to report or the rest of us to consider. Perhaps larger and more

difficult numbers to compose would have made completion of the table more interesting

and the goal of building flexibility in mental strategies more obvious to me.

Learning the Purpose of the Task

What is the meaning of the task of searching for patterns in a table? Why are

tables useful in learning about functions? What were students supposed to notice when

they looked for patterns in the table? These were not questions I asked myself at the time

I was teaching. At that point in time, I assumed that by doing the lesson, students would

learn something. Rather, I hoped they would. Someone else had decided it was a

worthwhile task. Three years later, I decided, once again, to fmd out for myself what was

() Michigan State Univetsity East Lansing, Michigan 488241034 CP 95-1 Page 32
important about it. I approached my

investigation of the meaning of this task

with two questions. First, what sort of

representation of a function is a table?

Second, what does it mean to look for starting number ending number

patterns within such a table? The problems

under the heading of "composition of

functions" in CSMP looked like the

problems in Figure 14. The labels of the

arrows varied. Once the table was filled THE FIRST PROBLEM I GAVE STUDENTS

with numbers, the students were to notice Figure 14

patterns. My current understandings of the

purpose of the search for patterns within this problem come from my own investigation

into its meaning by using resources outside of the CSMP teacher's guide.

What were students learning about functions when they completed the table with

starting number ending number

99 111

8000 8012

250 262

1 13

7 19

203 215

4688- 5000

4988

THE BEGINNING AND ENDING NUMBERS GENERATED BY STUDENTS

Figure 15

Michigan State University East Lansing, Michigan 488241034

t CP 95-1 Page 33
the numbers as shown in Figure 15? To complete the table, studentS chose inputs and

predicted outputs or chose outputs and predicted inputs. Picking any starting number, they

could be certain of the ending number by adding 12 to it. They could also predict the

starting number by subtracting 12 from any ending number. The only uncertainty came if

a student had difficulAy performing the addition or subtraction. When students started

with the ending number, they performed the inverse function, arrived at by subtracting

f(x)=x+10

+2,

ieroscai

g(x)=x+2

+10 +2

g (f(x)) = (x + 10) + 2

THE COMPOSITION OF FUNCTIONS

REPRESENTED ALGEBRAICALLY

Figure 16

first 2 and then 10. Doing enough of these, students could have learned about inverse

relationships among functions. Commutativity is also a possible focus. The students could

have asked the question: If the two functions are applied to a starting number in the

opposite order, do you get the same ending number? In other words, if I do + 2 followed

by + 10, do I end up in the same place as + 10 followed by + 2? The table is normally

thought of as the simplest representation of a function. In the example of + 10 followed

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 34
by + 2, the table represents a composition of functions. The first oiieration, add 10, gives

an output that becomes the input for the operation, add 2. The first function is f (x) = x +

10. The output of this function is symbolized by f (x). The input is x. The second

function is g (x) = f (x) + 2 where f (x) is the input for g (x). It can be written

algebraically as show in Figure 16. In the case of the composition of functions, the two

functions are composed such that the outputs for the first function become the inputs for

the second function.

Looking for the kinds of patterns that lead to an understanding of functional

relationships involves multiple steps as well. The two columns of a table come in rows

which have a beginning number and an ending number. The beginning number

determines the ending number because the labels of the arrows allow me to predict an

ending number from a beginning number or vice versa. To notice patterns as they relate

to functions, I begin by making observations about a pair of numbers in one row. If these

observations hold true fbr more than one row, they are patterns.

There is another layer of

sophistication to the analysis of +10

patterns in tables. Looking, for

example, at a set of starting and

ending numbers not chosen

randomlyin the example,

Figure 17, the starting numbers

are all multiples of 3I can look

for patterns down the columns as

starting number ending number

3 15

6 18

9 21

12 24

well as across the rows. By STARTING AND ENDING NUMBERS

AS MULTIPLES OF 3

putting the beginning numbers in

order and choosing them with Figure 17

some regularity, I can make use

of patterns in the column of starting numbers to predict the column of ending numbers.

Since the outputs depend on the inputs, I can look to see if there are patterns in the ways

that the outputs change when there is a certain orderly change in the inputs. I could ask,

for example, Is there a constant difference between the outputs when there is a constant

Michigan State University East Lansing, Michigan 48824-1034

1 CP 95-1 Page 35
difference between the inputs? Patterns in a table where the numbers are ordered and not

randomly chosen have the potential to be more complex because they represent a

coordination between changes in inputs and outputs. Patterns and functions are also a

way to consider classification. Functions are objects and, as objects, they can be

classified and soiled. Different functions will have different kinds of patterns. For

example, the function 4- 12 has certain patterns. The functions + 10 or + 2 will have

other patterns.

Learning to Make Sense of the Search for Pattern

What sense can be made out of the question of looking for patterns in the context

of a table of numbers? To answer this question, I look at both the responses that are a

part of this lesson in the teacher's guide and students' responses made during the lesson.

How did I understand these responses in the teacher's guide then, and how do I

understand them now? What sense did I make out of the students' responses then, and

how do I interpret them now? What do the inconsistencies between my perceptions and

understandings then and now say about what I am learning?

Responses in the Teacher's Guide. If the CSMP dialogue is intended to be a

guide and a way to prepare for what might happen along the way, an understanding of

the text seems important. What I found is that just reading the text did not guarantee that

I understood it the way CSMP may have intended. In my explorations of the intentions of

CSMP, I discovered a wealth of mathematical ideas and connections within the teacher's

guide that were not apparent to me at the time and only recently became visible to me in

a conversation with a math educator involved in curriculum development and familiar

with the development and design of CSMP.

I focused my conversation with a developer of CSMP on the three student

responses listed in CSMP in response to the question of noticing patterns (Figare 18).

When I taught the lesson, I was under the impression that these responses were ones I

should expect from students because these represented how other students have responded

to the question. What I have since discovered is that the student responses that appear

throughout CSMP come either from the developers' experiences in classrooms with

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 36
T: Look closely at this chart. What patterns do

you notice?

S: An ending number is always larger than the

starting number.

S: If you start with an even number, you end

with an even number. If you start with an odd

number you end with an odd number.

S. An ending number is always 12 larger than the

starting number. (P. 14)

STUDENT RESPONSES TO THE QUESTION OF PATTERNS FROM CSMP

Figure 18

students, or they represent ideas that are mathematically interesting from the perspective

of the person or people responsible for writhig the text of the lesson. I learned that the

responses in this particular lesson were mathematically interesting ones and not ones that

I could necessarily expect from students. I wish that the teacher's guide had included

information on where these responses had come from and why they were considered

mathematically interesting.

Here is how I now understand the responses in CSMP. The third response in the

text, that the ending number is always 12 larger than the starting number, is a response at

the level of a generalization and the most related to the function notion. This response

represents an important mathematical idea through its description of the relationship

between the starting and ending numbers. The first response, that the ending number is

always larger than the starting number, is an observation that is a more general variation

on the third response. The second response, if I start with an even number, I end up with

an even number, or the parity of the odd/even relationship of the starting number and

ending number, is an interesting observation. If I add 11 rather than 12, the parity would

be different. The significance of each response and how they are related to one another

are things I have learned but not from the teacher's guide. If I had known any of this

going into the lesson, would it have helped me along the way? At this point I can only

hypothesize. I think the answer to this question would be yes.

The questions of how much, what kind, and in what ways to provide information

for the teachers create ongoing dilemmas in curriculum development, and they are

Michigan State University East Lansing, Michigan 48824-1034 zei CP 95-1 Page 37
problems that the developers of CSMP have considered. When I voiced my concern of

not having had access to certain information and ideas through the teacher's guide, the

CSMP curriculum developer explained concerns about giving too much background on

the responses which might make the use of the guide too tedious for teachers. What is

the balanct between enough and too much information in a teacher's guide? In this

particular example, how useful are the suggested students' responses without the

additional information I learned on my own? If I had had some understanding of these

responses, I think I would have had a better sense of what made for sensible answers. I

expected the students' responses to vary, but without an understanding of the significance

of the mathematics, I had no idea what reasonable variations might be.

What were students seeing? What am I able to see and hear in students' responses

now that I was unable to see or hear at the time I was teaching? I was curious to see if it

really was the case that what students had said in response to the question, "Do you see

any patterns?" was unlike any of the students' predicted responses in the teacher's guide.

In examining this lesson three years later, I have gone back to the video tape of this

lesson and replayed it with the scripted lesson from CSMP in one hand and a transcript

in the other.

Valerie's Response. Figure 19 is an excerpt from the teacher's guide placed side

by side to the excerpt from the lesson's transcript where I asked the question of patterns.

As I look at the teacher's guide and the transcript, I see that Valerie, the first

person to respond to my question of patterns, gave one of the answers that is in the

teacher's guide. I even responded to her, added the composition arrow, and labeled it with

plus 12. Yet, I continued with what resulted in a pointless discussion of patterns, and I

ended that class feeling frustrated that students had not responded in any of the ways

predicted by the teacher's guide. What does this discovery of the similarity of Valerie's

comment to what appears in the teacher's guide say about what I understood to be my

frustrations at the time and what I have learned since then that might help me to

understand my feelings? I wanted to try to understand rather than discount the sense I

Michigan State University East Lansing, Michigan 48824-1034 ,1 3 CP 95-1 Page 38
The CSNIP Script

T: Look closely at this

chart. What patterns

do you notice?

S: An ending number is

always larger than

the starting number.

If you start with an

even number, you end

with an even number.

If you start with an

odd number, you end

with an odd number.

S: An ending number is

always 12 larger than

the starting number.

(P. 14)

Figure 19

Transcript 9/20/89

Ms. H.: Okay, I want you to look at these

numbers. Do you see any patterns?

Valerie?

Valerie: Each of them have a beginning

number. and then they have an

ending number that is 12 more.

Ms. H.: Okay. And how do you know that it

is 12 more?

Valerie: Because you have to add 10. You

fmd a number, you add 10 to it, and

then 2.

Ms. H.: If I were to put in another arrow

here, what would I put? Plus what?

Valerie: Plus 12.

+10

D:000 4'

+12

CSMP STUDENT RESPONSES COMPARED TO VALERIE'S RESPONSES

made out of the situation at the time. How could it be that I thought all students'

responses to the question of patterns were meaningless and bore no resemblance to what

was in the teacher's guide when Valerie's response was really such a close match?

One explanation that comes immediately to my mind was my focus on having a

discussion. If this was the case, then I was not really focused on the right answer, and it

might make sense that I did not "hear" Valerie or want to "hear" her. If I had, she was

the fust person to respond, and her "right answer" might have ended the discussion. What

more was there to talk about? It also seems quite possible that I might have "heard"

Valerie but did not really understand what she was telling me. At the time, I was

Michigan State University East Lansing, Michigan 48824-1034 4,1 CP 95-1 Page 39
uncertain about the point of the lesson. If the question of noticing Patterns relates to the

reasons why one would look for then, it might follow that I was listening for the

responses in the teacher's guide, but I did not really understand their significance to the

question of patterns in this context. I did not really know what I was listening for. My

ability to hear Valerie now may have something to do with what I have learned about

patterns and functions since that year of teaching.

Valerie's response, that each of the beginning numbers was an ending number that

is 12 more, seems on the surface to be rather mundane. But I appreciate now ways in

which it is mathematically significant. Her response is a generalization that defmes the

composition of the two functions in this problem. If I had asked her to write down what

she was saying in the shortest form possible, she would probably have written something

like, "number + 12." This could be thought of as the equation "x + 12." I now see that

looking for patterns in a table is like looking for the equation that defines the function.

This is a move in the direction of algebra. Valerie's generalization meets the criteria of

regularity and predictability, the two key characteristics of patterns. It is a statement that

holds true for the pairs of numbers in all

rows. You cf3uld choose any starting

number and predict with certainty, using

Valerie's pattern, the ending number.

Other Students' Responses. What

about the responses of the other students?

Were those patterns? If not, what were

they? What the other students were

observing were not mathematically relevant

patterns. They were observing regularities

in the numbers. Take the "pattern" a student

noticed as shown in Figure 20 for

examplethere are three zeroes in 8000 as

well as 5000. There is nothing predictable

starting number ending number

99 111

8012

262

13

19

215

250

1

7

243

4686)-

4988

A STUDENT'S EXAMPLE OF A PATTERN

Figure 20

Michigan State University East Lansing, Michigan 48824-1034 45 CP 95-1 Page 40
about this observation. It does not tell me anything about what else is happening with

numbers in the table. If I made use of the criterion that to fmd patterns I must start out

with observations about a pair of numbers in one row, the responses where students made

observations about pairs of numbers not in the same row are eliminated immediately as

patterns. The "pattern" identified in the table of Figure 20 is an example of this. Even

when students were looking at pairs of numbers in the same rows, the sorts of regularities

they were noticing were not predictive.

When students explained how they found beginning and ending numbers to fill in

the table, they gave procedural explanations of how they got from one to the other. Their

explanations, which I first interpreted as dull and pointless, reveal the meaning of the

composition of functions and the relationship between outputs and inputs. The following

examples illustrate this point.

Richard: Well, I did 99 plus 10 equals 109 and then plus 2 is like 9

plus equals 11.

Lucy: One plus 10 is 11 plus 2 is 13.

David: Because 7 plus 10 is 17 and 17 plus 2 is 19.

Bob: Because 203 plus 10 is 213, 213 plus 2 is 215.

If their procedures were represented in a chart of inputs and outputs, they would look as

they appear in Figure 21.

Input Output Input Output

99 109 109 111

1 11 11 13

7 17 17 19

203 213 213 215

THE RELATIONSHIP BETWEEN INPUTS AND OUTPUTS

Figure 21

Perhaps there was something interesting to notice in their responses to the question of

how they got from their beginning number to their ending number. Mathematically, the

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 41
composition of functions is a layer of complexity that is quite significant even though the

mental arithmetic necessary to fill in the chart was rather simple for fourth graders.

Bob's question about what idnds of number they should choose as beginning

numbers pointed out to me that there were no constraints on the numbers to use in the

chart as the task was presented in the teacher's guide. When either the starting or ending

number is chosen at random, as was the case in the problem I gave students, the chance

of noticing interesting patterns is reduced. Constraining the choice of numbers and

ordering the numbers chosen could help to make obvious some interesting patterns

between rows and columns which might otherwise go unnoticed.

LEARNING THE SHAPE OF THE RIVER

Mark Twain's writing about what he had to learn to navigate the river in Life on

the Mississippi ([1883] 1990) offers me images of what it is that I needed to learn as I

attempted to make changes in my math teaching.' In this section, I construct parallels

between his early learning experiences and mine mainly in the sense of his growing

understanding of the nature of what he needed to learn and the usefulness of a text in

doing the work.

In this particular lesson, I found myself dependent on the CSMP teacher's guide. I

trusted that it was going to help me do the kind of mathematics teaching I envisioned. In

a similar way, Twain began his adventures on the river with a notebook in his hands on

which he depended. Twain, like me, thought this text was going to be the key to

navigating the river. He wrote about what was in his notebook and his sense of its

contents: "I had a notebook that fairly bristled with the names of towns, 'points,' bars,

islands, bends, reaches, etc.; but the information was to be found only in the

notebooknone of it was in my head" (p. 43). For both Twain and I, our work while

trying to follow these guides did not go as planned. We both became frustrated. Within a

short time on the river, Twain found that the information in his notebook was insufficient

for the navigation he needed to do. I found the same to be true in my efforts to make use

16As was noted earlier, Twain's experiences offered me one set of images and language for helping me

to understand and describe the process and particulars of learning to teach mathematics for understanding. In my

dissertation, I play, additionally, with images and language of improvisational jazz and dance.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 42 4 7
of the teacher's guide. Following its directions was not helping me teach in the way I

imagined it would when I found myself trying to interact with students around

mathematical ideas. Twain ([1883] 1990) wrote of the frustration his reliance on his

notebook caused him:

The boat came to shore and was tied up for the night . . . I took my supper

and went immediately to bed, discouraged by my day's observations and

experiences. My late voyage's notebook was but a confusion of

meaningless names. It had tangled me all up in a knot every time I had

looked at it in the daytime. I now hoped for respite in sleep; but no, it

reveled all through my head till sunrise again, a frantic and tireless

nightmare.' (P. 48)

I can understand how he felt. The teacher's guide and the sense I made of it seemed to

hinder my teaching more than help it. I was naive to think that the teacher's guide could

carry aspects of this teaching that I was learning were my responsibility. For example, I

needed to have an understanding of the mathematical purposes for what I was doing, a

mathematical sense of why I was asking the questions I was asking. Simply following the

script in the teacher's guide was not working. Without a sense of purpose or an

understanding of what mathematics was important for my students to learn, I was lost as

far as being able to figure out what to do next.

Twain learned much about navigating the river from Mr. Bixby, an experienced

riverboat pilot on the Mississippi River and Twain's teacher. Mr. Bixby encouraged

Twain to keep a notebook and intended for it to be useful. But how did he expect him to

make use of it? What role did he think it should play in Twain's navigation of the river?

What did the writers of CSMP know that made them think that what was in the teacher's

guide would be helpful to me? What was it that either Twain or I needed to know to

make use of our texts? Was it the text that needed to change or the way we made use of

it? Or both?

I think that Mr. Bixby knew much more about the river than Twain realized, and I

doubt that Mr. Bixby, an experienced riverboat pilot, ever imagined that Twain would

17Author Heaton's emphasis, not Mark Twain's.

Michigan State Univetsity East Lansing, Michigan 48824-1034 CP 95-1 Page 43

11A. V
rely so heavily on his notebook. In my case, I did not want to be sb dependent on the

textbook, but in the heat of the moment, in the face of uncertainty, I fell back on old and

familiar ways of using a math textbook. I trusted it. I followed it without an

understanding that was meaningful to me of what was important for students to learn and

why. Like Mr. Bixby, I do not think the developers of CSMP ever expected me to cling

so tightly to the text. That was part of my problem. But to stray from the text in

purposeful ways, without wandering too far from important mathematics, I needed a

much stronger and clearer sense of purpose for what I was doing. I started out searching

for that when I planned the lesson. When I failed to turn up much that seemed useful, my

past experiences pushed me to go ahead with the lesson anyway. That is when I found

myself in trouble, unprepared and ill-equipped.

Twain ([18831 1990) wrote about what he learned about the way he needed to

know what was in his notebook:

I have not only to get the names of all of the towns and islands and bends,

and so on, by heart, but I must even get up a warm personal

acquaintanceship with every old snag and one-limbed cotton-wood and

obscure wood pile that ornaments the banks of this river for twelve

hundred miles; and more than that, I must actually know where these

things are in the dark. (P. 47)

The idea that he learned he needed a "personal acquaintanceship" with the river seems

related to what I learned about the sense of purpose I needed to teach mathematics for

understanding. The authors of CSMP had a much greater sense of what was important for

students to learn and why particular concepts were important than was accessible to me

in the way the teacher's guide was constructed and I interpreted it. The information in

Twain's notebook was a representation of the river. My teacher's guide was a

representation of the mathematical terrain. Mr. Bixby knew the river. The authors of

CSMP knew the terrain. What would Twain's notebook look like if it represented what

Mr. Bixby knew about the river in a way that was accessible to Twain? Could it be

represented in a text? What would it look like for the teacher's guide to represent what

the developers of CSMP knew about the mathematics in a way that was accessible to

me? What would it mean to design a textbook educative for teachers?

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 44
After Twain had been learning particulars on the river for i while, Mr. Bixby

shared his view, in more general terms, about what it was that Twain was learning and

why it was important: "You learn the shape of the river, and you learn it with such

absolute certainty that you can always steer by the shape that's in your head, and never

mind the one that's before your eyes" (Twain [18831 1990, p. 54). From this particular

lesson, I learned that I needed to develop a sense of purpose for what I was teaching that

was my owngrounded in the mathematics others believe is important for students to

learn but connected in meaningful and dynamic ways to my own mathematical

understandings and what it is that particular students understand. I learned, specifically,

about what it would mean to have this sense of purpose for teaching patterns and

functions by looking closely at why patterns and functions are important to teach, why

this particular lesson was important, and what students could learn from it. But what I

have learned goes beyond the specifics of patterns and functions. What I take from this

lesson is the need for a new sense of purpose, the questions to ask myself to go about

constructing a sense of purpose for whatever mathematical idea I am trying to teach, and

an understanding of why this sense of purpose is important in the kind of math teaching I

am learning to do. What I have learned is that the river or the mathematical terrain has

a shape that I must learn to enable myself to navigate my way around in it. What is

important is my understanding of the mathematics I am trying to teach for that will help

me "hear" students and build connections from their ideas to the mathematics I want

them to learn.

Michigan State University East Lansing, Michigan 48824-1034 CP 95-1 Page 45
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