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LIBRARY
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This is to certify that the

dissertation entitled
STUDENTS' ALGEBRAIC UNDERSTANDING:
A STUDY OF MIDDLE GRADES STUDENTS' ABILITY TO
SYMBOLICALLY GENERALIZE FUNCTIONS

presented by
ANGELA S. KREBS

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Teacher Education

 

Major professor

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Date adv/ML d8 l7?

1 I ['9

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

PLACE iN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

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AA 15’9" 20in

 

 

 

 

 

 

 

 

 

 

 

 

 

1m comma-p.14

 

AS".

 

 

STUDENTS’ ALGEBRAIC UNDERSTANDING:
A STUDY OF MIDDLE GRADES STUDENTS’ ABILITY TO
SYMBOLICALLY GENERALIZE FUNCTIONS
By

Angela S. Krebs

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Teacher Education

1999

AS

The pu
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ABSTRACT
STUDENTS’ ALGEBRAIC UNDERSTANDING:
A STUDY OF MIDDLE GRADES STUDENTS’ ABILITY TO
SYMBOLICALLY GENERALIZE FUNCTIONS
By

Angela S. Krebs

The publication of the National Council of Teachers of Mathematics’
Curriculum and Evaluation Standards in 1989 was pivotal in mathematics
reform. The National Science Foundation funded several curriculum projects
to address the vision described in the Standards. After these materials were
developed and implemented in classrooms, questions arose surrounding
students’ learning and understanding. This study investigates students’
learning in a reform curriculum. Specifically, “What do eighth grade students
know about writing symbolic generalizations from patterns which can be
represented with functions, after three years in the Connected Mathematics
Project curriculum?”

The content, the curriculum, the data, and the site chosen define the
study. Initially, the study surrounded students’ algebraic understanding, but
I focused it to investigate students’ ability to symbolically generalize
functions. Although this selection is a particular slice Of algebra it represents

a significant piece Of the discipline.

I select:
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I selected the Connected Mathematics Project (CMP) as the curriculum. I
supported the authors’ philosophy that the teaching and learning Of algebra
is an ongoing activity woven through the entire curriculum, rather than
being parceled into a single grade level.

The data surrounded the solutions of four performance tasks, completed
by five pairs Of students. These tasks were posed for students to investigate
linear, quadratic, and exponential situations. I collected and analyzed
students’ written responses, video recordings of the pairs’ work, and follow-up
interviews.

The fourth choice determined the site. I invited Heartland Middle
School, a pilot site of the CMP to participate in this study. I approached a
successful teacher, Evelyn Howard, who allowed her students to participate.
Together, we selected ten students who were typical students in her
classroom to participate in this study.

In conclusion, I present two major findings Of this study surrounding
students’ understanding Of algebra. First, students who had three years in
the Connected Mathematics Project curriculum demonstrated deep
understanding Of a significant piece Of algebra. And second, teachers can
learn much more about students’ understanding in algebra by drawing on

multiple sources Of evidence, and not relying solely on students’ written work.

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over the ye-

 

ACKNOWLEDGEMENTS

I wish to first thank Dr. Sandra K. Wilcox. There is SO much for which to
be grateful to you that I can not begin to list everything here. Over the years,
you fi'eely gave your time, intellectual perspectives and emotional support.
Thank you to Dr. Glenda T. Lappan for pushing me to think differently about
mathematics education. It was through your influence that I began this
process. I also thank Dr. Dan Chazan and Dr. John P. Smith III for the
guidance you offered and Yvonne Grant for your assistance.

I would like to thank my family and fiiends. TO my husband, Todd, I
appreciate your support and patience during this long process. TO my son
Jacob, although you may have added a year to the completion, I would not
have wished it any other way. To my parents, thank you for the inspiration

over the years.

iv

LIST OF TIA
LIST OF Fl

CHAPTER
INTRODL'A
IntroduA
Traditu

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TABLE OF CONTENTS

LIST OF TABLES ............................................................................................ viii
LIST OF FIGURES ............................................................................................. x
CHAPTER 1
INTRODUCTION—SOME PROBLEMS WITH ALGEBRA ............................. 1
Introduction .................................................................................................... 1
Traditional School Algebra ............................................................. _ ............... 2
Gatekeeping and Equity .................................................................... 5
Algebra for All ..................................................................................... 7
A Standards-Based Algebra .......................................................................... 9
Summary ...................................................................................................... 12
CHAPTER 2
WHAT IS ALGEBRA? ....................................................................................... 15
Introduction .................................................................................................. 15
Some Perspectives on Algebra ..................................................................... 16
Functions and Relations ................................................................... 18
Modeling ............................................................................................ 21
Structure ............................................................................................ 22
f Language and Representation .......................................................... 25
Patterns and the Study of Algebra .............................................................. 27
Patterns in School Mathematics ....................................................... 29
A Classification Scheme .................................................................... 36
Summary ...................................................................................................... 44
CHAPTER 3
THE STUDY ...................................................................................................... 46
Introduction .................................................................................................. 46
The Content .................................................................................................. 47
The Curriculum ............................................................................................ 47
CMP Algebra ..................................................................................... 51
The Site ........................................................................................................ 56
Heartland ........................................................................................... 57
Commitment to Reform ..................................................................... 58
Evaluating the Reform Efforts: MEAP ............................................. 59
Implementation of CMP .................................................................... 61
The Students ..................................................................................... 62
The Data .................................................................................................... 64
Data Collection .................................................................................. 64

Task Descriptions .............................................................................. 67

Task Solutions ................................................................................... 71
Data Analysis .................................................................................... 72
Summary ...................................................................................................... 74

CHAPTER 4

DIGGING DEEP—

A CAREFUL LOOK AT STUDENT INVESTIGATIONS ................................ 75
Introduction .................................................................................................. 75
Zachary and Todd ........................................................................................ 76

Borders ............................................................................................... 77
Cutting ............................................................................................... 82
Dominoes ........................................................................................... 87
Toothpicks .......................................................................................... 93
Ben and Joe ................................................................................................. 97
Borders ............................................................................................... 98
Cutting ............................................................................................. 101
Dominoes ......................................................................................... 106
Toothpicks ........................................................................................ 110
Julie and Dan ............................................................................................. 114
Borders ............................................................................................. 114
Cutting ............................................................................................. 117
Dominoes ......................................................................................... 123
Toothpicks ........................................................................................ 128
Sara and Ryan ............................................................................................ 132
Borders ............................................................................................. 133
Cutting ............................................................................................. 137
Dominoes ......................................................................................... 142
Toothpicks ........................................................................................ 147
Anna and Katrina ...................................................................................... 151
Borders ............................................................................................. 152
Cutting ............................................................................................. 155
Dominoes ......................................................................................... 159
Toothpicks ........................................................................................ 162
Summary .................................................................................................... 167

CHAPTER 5

STEPPING BACK—

AN ANALYSIS OF WHAT THESE STUDENTS KNOW .............................. 171
Introduction ................................................................................................ 171
Strategies to Formalize Generalizations .................................................. 172

Numeric Approach: Generalizing From a Table Of Data ............... 173
Geometric Approach: Generalizing about Changing Shapes ........ 183
Graphic Representations ................................................................ 185

vi

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Connections ................................................................................................ 186

Connections between Representations ........................................... 187
Connections between Tasks ............................................................ 188
Connections with Other Mathematics ............................................ 194
Disposition .................................................................................................. 195
Summary .................................................................................................... 196
CHAPTER 6
IMPLICATIONS AND LIMITATIONS .......................................................... 198
Implications ................................................................................................ 198
A Rich Understanding of an Important Piece Of Algebra .............. 198
Assessment ...................................................................................... 208
Limitations ................................................................................................. 209
Tasks ................................................................................................ 210
Implementation ............................................................................... 211
Comparison ...................................................................................... 212
Summary .................................................................................................... 212
APPENDICIES ................................................................................................ 214
Appendix A: List of CMP Units ................................................................. 215
Appendix B: Interview Questions ............................................................. 216
Appendix C: Time Spent by Students Working on Tasks
and During Interviews ......................................................... 217
Appendix D: Tasks ..................................................................................... 218
Appendix E: Task Solutions ...................................................................... 223
BIBLIOGRAPHY ............................................................................................. 239

vii

Table 1:
Table 2:
Table 3'.
Table 4:
Table 5:
Table 6'.
Table 7:
Table 8:
Table 9:
Table 10;
Table 11;
Table 12;
Table 13;
Table 14;

Table 15:

Table 16;
Table 17;
Table 18:

Table 19

 

 

Table 1:
Table 2:
Table 3:
Table 4:
Table 5:
Table 6:
Table 7:
Table 8:

Table 9:

Table 10:
Table 11:
Table 12:
Table 13:
Table 14:

Table 15:

Table 16:
Table 17:
Table 18:

Table 19:

LIST OF TABLES

Spring Data ..................................................................................... 21
Table of Square Numbers ............................................................... 31
Cost Of Phone Call .......................................................................... 34
Perimeter Dots on Squares ............................................................ 37
Exponential Table ........................................................................... 38
Allowance Table .............................................................................. 39
y = x2 Constant Differences ............................................................ 40
y = x3 Constant Differences ............................................................ 40
y = x2 Constant Differences in Total Squares Table ..................... 41
Walking Times Table ...................................................................... 42
CMP Algebra Units ........................................................................ 54
Heartland Middle School Reform Table ........................................ 58
List Of Students ............................................................................... 63
Two Cases Of Dominoes ...................................................................... 123

Range of Values for the Independent Variable in

Tables Made by Students ............................................................. 168
Symbolic Rules Found by Students ............................................. 170
Cutting Table with Factors Of 2 Represented .............................. 181
Rules for White Tiles in Borders .................................................. 184
Symbolic Rules Generated for Simple Cases ............................... 206

viii

Table 20: .
Table ‘Zl: l
Table 22: .
Table ‘23: A
Table 24'.
Table 25:
Table ‘26:

Table '27:

Table 28;

Table 29

Table 30
Table 31;
Table 32;

Table 33:

Table 34:

Table 35:

 

 

Table 20:
Table 21:
Table 22:
Table 23:
Table 24:
Table 25:
Table 26:

Table 27:

Table 28:

Table 29:

Table 30:

Table 31:

Table 32:

Table 33:

Table 34:

Table 35:

Symbolic Rules Generated for More Complex Cases .................. 207
Order of CMP Units ...................................................................... 215
Approximate Number of Minutes Spent on Each Task .............. 217
Approximate Number of Minutes Spent on Each Interview ...... 217

Figure Number and Blue Tiles in Borders solution .................... 224
Figure Number and White Tiles in Borders solution .................. 225
Cut Number and Sheets of Paper in Solution Of Cutting ............ 226
Cut Number and Sheets of Paper in Cutting solution

up to 10 cuts .................................................................................. 227
Cutting solution with exponential notation ................................. 228
Cut Number and Sheets of Paper in Cutting solution

for 4, 5, 10, 20, and It cuts ............................................................ 228
Cut Number and Inches Thick in Cutting solution ..................... 229
Cut Number and Area in Cutting solution .................................. 229
Maximum Spots and Possible Dominoes ..................................... 232
Maximum Spots and Possible Dominoes in Dominoes

Case 2 solution ................ . .............................................................. 235
Figure and Perimeter in Toothpicks solution .............................. 237
Figure and Total Toothpicks in Toothpicks solution ................... 237

Figure l: l
Hme:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
Figure 7:
Figure 8:
Figure 9:
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
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Figure 1:
Figure 2:
Figure 3:
Figure 4:
Figure 5:
Figure 6:
Figure 7:
Figure 8:

Figure 9:

Figure 10:
Figure 11:
Figure 12:
Figure 13:
Figure 14:
Figure 15:
Figure 16:
Figure 17:
Figure 18:
Figure 19:

Figure 20:

LIST OF FIGURES

Graphical Representation to Find Solutions ................................. 20
Spring Illustration .......................................................................... 22
Spring Graph .................................................................................. 22
Distributive Property Of 3 x (4 + 5) ................................................ 23
Distributive Property of (3 x 4) + (3 x 5) ....................................... 23
Square Number Pattern ................................................................. 29
Perimeter Dots on Squares ............................................................ 37
Linear Graph .................................................................................. 38
Allowance Graph ............................................................................ 39
Polynomial Pattern Of Figures ....................................................... 41
Walking Times Graph .................................................................... 43
Heartland MEAP Data ................................................................... 60
Borders Graphic .............................................................................. 68
Toothpicks Figures .......................................................................... 71
Zachary's work on Borders ............................................................. 78
Todd's work on Borders .................................................................. 79
Zachary’s work on Cutting ............................................................. 83
Todd’s work on Cutting ............. 84
Zachary's work on Dominoes .......................................................... 88
Todd's work on Dominoes ............................................................... 89

Figure 21:
Figure 22:
Figure 23:
Figure 24:
Figure 25:
Figure 26:
Figure 27 :
Figure 28:
Figure 29:
Figure 30:
Figure 31:
Figure 32:
Figure 33:
Figure 34:
Figure 35:
Figure 36:
Figure 37:
Figure 38:
Figure 39:
Figure 40:
Figure 41:

Figure 42:

Zachary's work on Toothpicks ........................................................ 94

Todd's work on Toothpicks ............................................................. 95
Ben and Joe’s work on Borders ...................................................... 99
Ben’s Borders diagram ................................................................. 101
Ben's work on Cutting .................................................................. 102
Joe's work on Cutting ................................................................... 103
Ben’s initial response to question 2 a in Cutting ........................ 104
Ben's work on Dominoes ............................................................... 107
Joe's work on Dominoes ................................................................ 108
Ben and Joe’s work on Toothpicks ............................................... 111
Julie and Dan’s work on Borders ................................................. 115
Julie's work on Cutting ................................................................. 119
Dan's work on Cutting .................................................................. 120
Julie's work on Dominoes ............................................................. 124
Dan's work on Dominoes .............................................................. 124
Julie's work on Toothpicks ........................................................... 129
Dan's work on Toothpicks ............................................................. 130
Sara and Ryan’s work on Borders ................................................ 134
Ryan’s tracing in Borders ............................................................. 136
Sara and Ryan’s work on Cutting ................................................ 138
Sara's work on Dominoes .............................................................. 143
Ryan’s work on Dominoes ............................................................. 144

xi

Figure 43:

Sara and Ryan’s work on Toothpicks ........................................... 148

Figure 44: Anna’s figure 3 in Borders ............................................................ 153
Figure 45: Anna and Katrina’s work on Borders .......................................... 154
Figure 46: Anna and Katrina’s work on Cutting ........................................... 156
Figure 47: Anna and Katrina’s work on Dominoes ....................................... 160
Figure 48: Anna and Katrina’s work on Toothpicks ..................................... 163
Figure 49: Katrina’s Answer to Question 3 in Dominoes .............................. 185
Figure 50: Dot Pattern Graphic ...................................................................... 189
Figure 51: Dot Pattern Written Response ..................................................... 189
Figure 52: Ryan’s sketch during Dot Pattern interview. .............................. 190
Figure 53: Connections between Tasks ......................................................... 191
Figure 54: Borders Task ................................................................................. 219
Figure 55: Cutting Task ................................................................................. 220
Figure 56: Dominoes Task .............................................................................. 221
Figure 57: Toothpicks Task ............................................................................ 222
Figure 58: Borders Figures ............................................................................ 223
Figure 59: Figure 4 in Borders solution ......................................................... 224
Figure 60: General Rectangle in Borders ...................................................... 225
Figure 61: Cutting Graphic ............................................................................ 226
Figure 62: Dominoes Graphic ......................................................................... 230
Figure 63: Sketch of Dominoes with 0-2 Spots—Case 1 ............................... 231
Figure 64: Additional Dominoes with 3 Spots ............................................... 231

xii

Figure 63
Figure 65
Figure 67
Figure 65
Figure 6E:
Figure TC
Figure 71
Figure 72
Figure 73

Figure 74

 

Figure 65:
Figure 66:
Figure 67:
Figure 68:
Figure 69:
Figure 70:
Figure 71:
Figure 72:
Figure 73:

Figure 74:

Triangular Array Of Dominoes with up to 2 Spots ...................... 232
Triangular Array in 3 by 4 rectangle ........................................... 233
Triangular Array in (n+1) by (n+2) rectangle ............................. 233
Sketch of Dominoes with 0-2 Spots—Case 2 ............................... 234
Sketch Of Dominoes with 0-3 Spots—Case 2 ............................... 234
Toothpicks Graphic ....................................................................... 236
Toothpicks Figure 5 ...................................................................... 236
Toothpicks Figure 6 ...................................................................... 236
Horizontal Toothpicks .................................................................. 238
Vertical Toothpicks ....................................................................... 238

xiii

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CHAPTER 1

Introduction—Some Problems with Algebra

Introduction

The discipline of mathematics is in a state Of rapid change and growth,
but the commonly held View Of school mathematics has not evolved in
parallel or in conjunction with these advances. One aspect of mathematics is
the discipline of algebra. Many people still consider algebra as the study Of

letters. I recall the reaction from one student after we found a solution of
3 . . . .
y = 74‘ for an equation. She threw up her hands In disgust, and said that She

thought y was always two. To her, learning algebra meant Simply learning a
correspondence between the letters of the alphabet and the numbers. TO
others learning algebra involves manipulating letters.

The N CTM Curriculum and Evaluation Standards (1989) and other
reforms recommend a move away from the traditional algebra curriculum
and teaching practice towards a discipline and instruction which is more
inviting and meaningful to students. Traditional algebra is Often seen as a
gatekeeper. Without successfully completing an algebra course, many
students are denied access to certain careers. Standard school practices are

rooted in traditions several centuries Old and cannot prepare students for

their math
reform belie
access.

One r»
projects. \V
what can a
understan

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students.
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1989). T‘:

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their mathematical needs Of the 21st century (Steen, 1990). Supporters of the
reform believe that algebra can be a discipline where all students have
access.

One response by the government was to fund several curriculum
projects. With money being given to these curricula, the question becomes
what can a curriculum based on these reforms contribute to a students
understanding of algebra.

In this chapter I begin by describing a traditional algebra classroom. It
is standard in both content and teaching. This raises concerns regarding
access to algebra that has ignited a movement to encompass algebra for all
students. Next, I consider aspects of a curriculum that develop algebra
supporting the vision presented in the NCTM Curriculum and Evaluation
Standards (1989). In conclusion, I pose the question surrounding the

algebraic understanding of students in a curriculum such as CMP.

Traditional School Algebra

School algebra has not changed much in the past fifty years (Thorpe,
1989). The modal algebra classroom looks the same today—in both
curriculum and teaching—as it did many years ago. The content in an
Algebra I course is fairly standard and can be inferred by examining a typical

text. Most algebra books consist of chapters divided into sections and sections

into lesson:
expected tc
included at
equations.
mitture w:
trinomials
with ratiOA
two equat'
roots: and
square (L'
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into lessons. Each lesson typically spreads over two facing pages and is
expected to be covered in one class meeting. The typical topics or chapters
included are Operations with positive and negative numbers; solving linear
equations, linear inequalities, and proportions; age, digit, (1 = rt, work, and
mixture word problems; Operations on polynomials and powers; factoring of
trinomials, monomial facts, and special factors; simplification and Operations
with rational expressions; graphs and properties of graphs; linear systems of
two equations with two variables; simplification and operation with square
roots; and solving quadratics equations by factoring and completing the
square (U siskin, 1987). A lesson typically Ofi‘ers a handful of examples,
followed by a number Of exercises for students to practice the new skill or
procedure demonstrated in the examples. There is much repetition in this
approach. Students do many exercises which follow a similar format,
although the exercises may get progressively more complex as students
proceed down the page.

The curriculum also has commonalties in differing sites. This traditional
school algebra follows a “layer-cake” approach (Kaput, 1993; and Davis,
1993). The curriculum typically consists of some form of Algebra I in the
ninth grade or earlier,l followed by Algebra II. Unfortunately, with this
traditional approach most of a student’s algebra learning is confined to at

most two years of school. By keeping students’ learning in separate distinct

 

1 Algebra I is being taken by more and more eighth grade students to allow them the time to
take the advanced mathematics courses as juniors and seniors.

layers. or 0
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employr

layers, or courses, the development Of ideas across multiple areas of
mathematics is hampered (Kaput, 1993; and Steen, 1990).

Teachers in these traditional classrooms Often follow a format of
instruction that parallels the typical text. The teacher might begin the class
with a review Of questions fi'om the previous day’s homework. After these
concerns have been addressed, the class moves on to the next lesson. The
teacher demonstrates a procedure by completing several examples for the
class to see. This might be done interactively with input fiom the whole class,
but the teacher typically has a predetermined agenda to show students how
to progress through the problem using her method. After the teacher
completes her select examples she might assign some seatwork, a kind of
supervised practice, to monitor whether her students understand the
procedure. Once the students appear to be on the right track, the teacher
allows the rest Of the class time for students to complete their assignment Of
exercises from the text. These assignments typically consist of more exercises
similar to the examples demonstrated. A word Of caution might be offered to
the students that the later problems might contain some tricky or
challenging aspects.

The N CTM Board of Directors (1994) summarize three major flaws in
the traditional algebra courses. First they note that the focus on pencil and
paper manipulations is Often divorced fi'om any meaningful context. In fact,
these skills that are developed are not necessarily what students who are

employment-bound or college-bound need in a technology world. The directors

add that
understa
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add that the traditional curriculum does not encourage an informal
understanding of algebraic ideas in grades K-8 that could prepare students
for future investigations. Finally, they acknowledge that the concepts and
methods Of algebra are isolated from other strands Of school mathematics:
statistics, geometry, and discrete mathematics. Students do not have the
Opportunity to integrate their learning of mathematics.

The traditional curriculum and teaching of algebra do not foster success
for most students; neither is it clear how some of the content is meaningful or
worthwhile to students. Instead of drawing in students, traditional algebra is
viewed as a gatekeeper that effectively excludes certain students from future
studies and/or careers. Traditionally minority groups and women are the
ones who are most Often filtered out. There is a commitment by many policy
makers, professional organizations, and individual educators to help all

students develop algebraic competence.

Gatekeeping and Equity

American culture places a high value on algebra. The course “algebra”
listed on a student's transcript sends a message about this student's
mathematical experiences and perceived competence to future employers,
admission counselors, and others. In 1990 the United States Department of
Labor reported that the number of mathematics courses taken during high
school is the strongest predictor Of earning nine years after graduation. In

schools, algebra is typically a prerequisite for geometry. Nearly all students

who plan
their rao
algebra.
Students
these doc
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who plan on attending college take geometry in high school regardless Of
their race or ethnicity (Pelavin and Kane, 1988). When students do not take
algebra, they greatly limit the number of future choices and close some doors.
Students who are successful with algebra leave high school with many of
these doors Open.

Moses (1993) called algebra the new gatekeeper of the twenty-first
century. Students who have passed algebra are afforded many Opportunities
that would not otherwise be available. Davis (1993) noted that the United
States is becoming more Of a bi-modal society relative to income. Although
education is not solely responsible for this distinction, it plays an important
role. According to Davis, one aspect of this separation is the segregation Of
“those who know algebra versus those who do not.” Students who find success
in algebra have more opportunities available than those who do not. Algebra
is Often seen as a filter that only allows a select number of students to pass
through. Those who are blocked are Often prevented fi'om achieving specific
goals.

In the NCTM’s Board of Directors (1994) statement, the authors
summarized: “First year algebra in its present form is not algebra for
everyone. In fact, it is not the algebra for most high school graduates today.”
If algebra is used as a gatekeeper, then we need to consider who is being
excluded fi'om the advantages algebra afiords. Unfortunately it is most often
minority students and women who are likely filtered out (Moses, 1993).

NAEP data showed that less than half Of the students from impoverished

urban SChC
not take at
develop a <
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urban schools takes more than one year Of algebra. As many as one in five do
not take any algebra at all (Silver 1997). Moses (1993) stated that we need to
develop a consensus around the right to learn algebra. He noted further in
Jetter (1993) that access to algebra is an important issue for a new civil-
rights movement for minorities. Moses argued that students, who do not take
algebra, left high school disadvantaged, and this is a situation that can no
longer be tolerated. Algebra for all students should be a top priority in

education.

Algebra for All

It can no longer be tolerated that some students are restricted in their
future by lack of success in an algebra course. Algebraic reasoning is a very
powerful tool for students to develop not only for the leverage it brings them
in society, but because it is one of the most powerful intellectual tools
civilization has developed. TO make the tools of algebraic understanding
available to students, “algebra for all” is becoming a matter of educational
policy in many states. (Olson, 1994).

N CTM commissioned several documents to represent a new vision Of
teaching algebra to all students. In one of their landmark publications, the
authors of the NCTM Curriculum and Evaluation Standards (1989) envision
a curriculum where algebra is expected for all students. The Standards

establish as a goal that students become mathematically powerful problem

solvers: p2
to help stt
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setting ad
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solvers; part of this power lies in the accessibility of algebraic understanding
to help students reason. In 1994, N CTM created the Algebra Working Group.
Their charge was to clarify the vision Of “algebra for all” through the K-12
setting advocated in earlier pieces of writing. The group did this by
illustrating and elaborating this goal with examples, practical ideas, and
promising practices to help educators raise questions about changing algebra
instruction. They published “A Framework for Constructing a Vision Of
Algebra” in 1997 that summarizes the group’s work. In the document, the
authors put forth diflefing perspectives on algebra, argue for algebra as more
than a course, and illustrate their view Of algebra in a K-12 setting.

In another NCTM publication, the authors of Algebra for Everyone
(1990) argue for the need to teach algebra to all students beginning in the
elementary curriculum, continuing through the middle grades, and
expanding in high school. These writers recognize that mandating the
traditional algebra described earlier in this chapter in ninth or even eighth
grade is unlikely to be successful at achieving mathematical equal
opportunity (Silver, 1997). In fact, many argue that requirements of this sort
might actually have an opposite effect and push more students away from
algebra. Worse yet, forcing students into an inappropriate traditional algebra
course might reinforce widely held destructive notions that algebra is only for
a select few students (Silver, 1997). Mandates to require traditional algebra

might have the ill-effect of accelerating students out of mathematics, rather

than ope
stresses
ahersch
uneas
the Ame
It is
but the e

careful r

Mos
manipul.
1993: Slt
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than opening gates (Prevost, 1985). The National Research Council (1989)
stresses in Everybody Counts that since algebra is required for opportunities
after school, all students should study a “meaningful algebra.” Rather than
serve as a filter, Everybody Counts argues that algebra should be a pump in
the American education pipeline and help students pass through the gate.

It is not only access to algebra that reformers suggest needs revision,
but the approaches towards algebra that is Offered to students needs a very

careful reexamination.

A Standards-Based Algebra

Most advocates of “algebra for all” do not assume the traditional symbol
manipulation algebra taught in the standard ways (Fouche, 1997; Kaput,
1993; Steen, 1992; Silver, 1997; and Chazan, 1994). When educators accept
the challenge Of algebra for all, they also support a change in the focus Of
both the teaching and content of algebra (Steen, 1992; Wheeler, 1989).
Chambers (1994) nicely summarizes this: “Algebra for all is the right goal at
the right time. We just need to get the right algebra.” (page 85).

Educators are trying to find this “right algebra.” One idea that cuts
across many interpretations is to consider a “strands” approach to the
learning Of algebra (Kaput, 1993; and Steen, 1990).

I suggest a strands organization, where major ideas weave
through many grade levels, frequently interweaving with one

anoth
natur
gener
basec
usua?
gradt

expei
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ideas and
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another to create a rich fabric, but one that has a direction, a

natural flow, from the wide watershed concrete experience to

generalizations and abstractions, fiom informal and language-

based representations to more formal representations. Filters are

usually built using layers, while strands provide a natural flow,

gradually drawing into mathematics even more diverse

experiences. (Kaput, 1993, page 34).

A strands approach assumes that students continually develop algebraic
ideas and algebraic reasoning across different grades and multiple courses.
The content can be developed in much more depth by continuing to build on
ideas introduced earlier. Students can also benefit from the increased
relevance of applications as algebra is used in more meaningful integrated
contexts.

Davis (1993) adds that less risk for students might be an additional
benefit to this K-12 approach. Rather than having only one or two
opportunities to learn algebra, the ideas and reasoning should span over
several years. Instead of filtering students out Of algebra, strands allow for
multiple Opportunities for students to have access, drawing in more students.
This approach afi'ords diversity in student learning over the entire K-12
curriculum where each student weaves a unique tapestry.

This longitudinal K-12 approach to algebra does not simply allow more
Opportunity to cover the traditional curriculum. It requires careful
consideration of what really makes meaningful content to be taught across
the K-12 span. Every topic placed in the curriculum should have significant

value to students’ learning Of mathematics (Thorpe, 1989). The American

mathematics curriculum is often criticized as being an “inch deep and a mile

10

wide." Cri‘
that each
The entire
reevaluat
through t'
overcrowc
(Barbeau
curriculu
respond t
The

be curric
documen

students

look quji

Course. I

when it

Fir

 

A Currie:

 

wide.” Critics imply that so many topics must be covered in the curriculum
that each only receives a cursory mention rather than a deep exploration.
The entire algebra and preparatory algebra curriculum needs to be
reevaluated for meaningful content starting in kindergarten and continuing
through twelfth grade. One problem with the current curriculum is that it is
overcrowded (U siskin, 1995). Fewer topics should be covered more in depth
(Barbeau, 1991; and Steen, 1993). In the following section I discuss how a
curriculum based on the NCTM Curriculum and Evaluation Standards could
respond to the charge of integrating algebra for all students.

The N CTM Curriculum and Evaluation Standards are not intended to
be curriculum materials implemented directly in a classroom. Rather, the
document represents a vision of teaching meaningful mathematics to all
students. An algebra curriculum in response to the NCTM Standards would
look quite difi'erent than the traditional Algebra I or eighth grade algebra
course. It would vary in who is taught, what is taught, where it is taught,
when it is taught, and how it is taught.

First, consider who is taught. Algebra for all students is a primary goal.
A curriculum supporting the Standards suggests heterogeneously grouped
classes for all students. It would not promote removing the top performers or
excluding those having difficulties. Rather, with the inclusion of all students,
they could suggest extensions for students who are prepared and Offer

additional support to others as needed.

11

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of numercw
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The next concern: what is included in the curriculum. The standard fare
of numerous symbol manipulations without a context would be minimized.
The focus would shift towards understanding algebraic ideas and multiple
representations.

The when and where algebra is taught could be taken together. Algebra
is not a specific course or a single chapter in a text. It represents a way of
thinking and reasoning. The Standards would support a vision of algebra
integrated throughout the entire K-12 curriculum.

Finally how algebra is taught in a Standards-based curriculum would
also change. The teacher is no longer the sole deliverer of knowledge. Her role
is to pose challenging and engaging problems for the students to work and

investigate.

Summary

There certainly seems to be a need to reconsider the content and
methods used to teach traditional algebra. There is little evidence to support
a claim that all students would develop solid algebraic reasoning by following
the traditional approach to both content and teaching. Neither is there yet
evidence to suggest that all students engaged in a curriculum based on the

NCTM Curriculum and Evaluation Standards would be successful. If these

reform ideas are to be accepted, then research must address this issue.

12

This
is: What 6
In C I
algebra a
rapid cha
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This study begins to do this. The initial question that fiames the study
is: What do students learn about algebra in a Standards-based curriculum?

In Chapter 1, I discuss some Of the issues surrounding the discipline of
algebra and its implementation as a curriculum. Mathematics is in a state of
rapid change and as a result what traditionally constitutes algebraic
understanding is coming under fire. Educators are re-thinking wheat is in
the curriculum and Offer suggestions to afford access to all students.

I address the question of what is algebra in Chapter 2. Since there is not
a consensus by educators, researchers, or mathematicians, I do not attempt
to resolve the issue. Rather, I offer four organizing themes—functions and
relations, modeling, structure, and language and representations—presentsd
by the Algebra Working Group to cluster some Of the different perspectives.

The methodology of this study is rendered in Chapter 3. I describe the
reasons I made some Of the choices surrounding the content—patterns which
represent functions and generalizing with symbols from patterns Of data, the
curriculum—the Connected Mathematics Project, the site—Heartland Middle
School, and the data—students’ investigations on performance tasks,
recordings while they worked, and interviews after completing the tasks.

The data is presented in Chapters 4 and 5. In the first of the two
chapters I carefully consider what students have done with each task. I
describe their solutions by considering the three sources Of data, written

responses, recordings while they worked, and interviews after they completed

13

each t2
commc

Ir
limitat
have (I

algebr:

 

 

each task. In the next chapter I look across students and tasks to recognize
common strategies and interesting aspects in their investigations.

In the final chapter, Chapter 6, I suggest some Of the implications and
limitations of this study. I summarize that the students who participated
have demonstrated a very solid understanding of a very important piece of

algebra.

l4

In
student
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CHAPTER 2

What is Algebra?

"A mathematician like a painter or a poet, is a maker of patterns. "
--Godfrey Harold Hardy

Introduction

In the previous chapter I raise the question about what it means for
students to understand algebra. Before students’ understanding Of algebra is
considered, I need to further explore the question “What is algebra?” “Solving
equations with variables” is a typical response to the question. Even among
educators, researchers, and mathematicians there is not a consensus to this
question. In this chapter I describe some of the differing perspectives in the
literature surrounding algebra and briefly discussing some Of the
implications for classrooms. I use as a framework the organizing themes
presented by the Algebra Working Group—algebra as the study of (a)
functions and relations, (b) modeling, (c) structure, and ((1) language and
representations.

After this overview of the discipline I narrow my discussion Of algebra to
describe the aspect which is the focus of this study—symbolically

generalizing from patterns of data. In chapter three I describe the tasks used

15

in the 51

(I)

discus
algebra.

student.

Set
that em
many it:
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Kaput ]

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in the study and how they related to this area of algebra. I continue to
discuss the implications for school mathematics from this perspective of
algebra. Finally, I describe a sorting scheme to classify the patterns that

students in this study encountered.
Some Perspectives on Algebra

Several researchers and groups Offer varying perspectives of algebra
that emphasize different aspects. Kaput (1995) explains that even though
many use the term “algebra” there is not an encompassing discipline, no
monolith which describes it. He provides a slightly different review on
algebraic thinking. He categorizes five aspects Of algebra:

. as generalizing and formalizing;

. as manipulations of formal Objects;

. as the study Of structure;

. as the study Of functions; and

. as the study Of languages.

Kaput is careful not to classify these as five disjoint categories, but rather as
“loosely spun and richly interwoven.” They represent guidelines to start
thinking of the different perspectives of algebra.

Kaput’s aspects are very similar, but not a direct match, to the themes

Ofi'ered by the Algebra Working Group (Burrill, 1995; and Phillips, 1995). The

16

Algebra '

four key

The don
functior
Fey and
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Algebra Working Group (1997) organizes the various perspectives around
four key concepts:
. functions and relations—where functions underlie all big ideas in
algebra,
. modeling—where finding ways to represent situations with
mathematical relations or models is key to algebra,
. structure—where algebra is conceptualized as generalized
arithmetic, and
. language and representation—where communicating ideas through
the syntax of the representations is the focus Of algebra.
The dominance of each theme varies with researcher. Some support the
function approach to algebra (Kieran, Boileau, & Garancon, 1996, Heid, 1996;
Fey and Good, 1985; Schwartz and Yerulshalmy, 1991; and Chazan, 1993),
while others seem to suggest a structure approach (Kieran, 1989). Just as the
emphasis changes by researcher, the focus has shifted with time. In the 60’s,
the “new math” movement was based primarily on structure, while the “back
to basics” movement that followed relied on thinking of algebra as a
language. The more current movement with the integration Of graphing
calculators has shifted the focus to functions and relations, and models
(Algebra Working Group, 1997). Although the trend has shifted through time,

researchers continue to advocate particular themes.

17

Altl
by Kaput
differenc
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SOFT out i
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Although there is much overlap between classification themes suggested
by Kaput (1995) and the Algebra Working Group (1997), there are also
differences. Their organizations overlap with the function and relation,
structure, and language themes. While the Algebra Working Group seems to
sort out modeling, Kaput does not. Instead he teases out generalizing and
formalizing, and the manipulation Of formal Objects as two separate aspects.

In the following sections I elaborate on the Algebra Working Group’s

organizing themes and consider curricular issues surrounding each.

Functions and Relations

Both the Algebra Working Group (1997) and Kaput (1995) consider the
basic study Of functions as one of the primary perspectives on algebra.
Researchers whose work falls into the function and relations theme view the
function as the central Object of study (Chazan, 1993; Fey and Good 1985;
Yerulshalmy and Schwartz, 1991; Thorpe, 1989; and Confi'ey, 1994). Fey
(1989) defines functions as relations where output variables depend on input
variables. Some would emphasize the rate of change between the variables in
the functions. An example of this dependence is how at a given rate the time
(input) it takes for a trip determines the distance traveled (output).

The NCTM Curriculum and Evaluation Standards (1989) advocate an
approach to algebra that focuses on functions and function-related ideas.

With the increasing access to computers and the technology of calculators,

18

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the concentration on functions in the algebra curriculum has become a more
reasonable part of study for students. Functions can be introduced to
students using different representations such as tables and graphs (Confrey,
1994). Schwartz and Yerulshalmy (1992) suggest one such approach where
the concept of a function is introduced much earlier than in the traditional
secondary school curriculum. Students explore specially designed software
that relates both the symbolic and graphical representation Of functions. Both
representations are important to help students understand the concepts of
functions and variables (Schwartz and Yerulshalmy, ????). Kaput (1989)
reminds that students need actually experiences with three different
representations—table, graphs, and symbols.

Chazan (1993) also suggests a curriculum where the function is the core
Object Of study. He clarifies his view when he describes an equation as a
comparison Of two functions. For example, the equation 3x — 2 = x + 5 is
identified as a question about two functions, f(x) and g(x), where f(x) = 3x — 2
and g(x) = x + 5. The equation asks: when is f(x) equal to g(x)? or, for what
values Of x will this produce the same output in both functions? (Chazan, in
press)

Chazan asserts several advantages to using this functional approach
over the traditional algebra approach. Students are Offered an alternative
from traditional symbolic manipulation to solve equations. Rather than being

limited to a single method in their solutions, students could potentially have

19

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three reasonable strategies to solve equations. They might apply operations
to create equivalent equations. Using the equation above:

3x — 2 = x + 5 Start with f(x) = g(x),

2x — 2 = 5 Subtract x fi‘om both f(x) and g(x),
2x = 7 Add 2 to both Of the new functions,
x = 3% Finally, divide both by 2.

Or students could try a ‘guess and test’ strategy to solve the equation and
find that x = g is a solution. Or they might consider a graphical representation
to find solutions. In this graphical approach, students could plot each

function, f(x) and g(x), on a coordinate graph and search for a point of
. . . . . 1
intersection. See Figure 1 where both hnes meet at the pOint where x = 32

and is a solution to the equation.

 

 

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f(x)=3x-2
------- g(x)=x+5

 

 

 

A :

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Figure l: Graphical Representation to Find Solutions
Each of these strategies could be considered an approach following this

functional view.

20

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Modeling

There are common ideas between the functions and relations theme and
the modeling theme. Kaput (1995) notes that most models have functions as
their core. Frudenthal (1983) would argue that modeling is the primary
reason to study algebra. In this version Of algebra, students start with some
situation and their goal is to mathematize it—find a mathematical relation
that models the phenomena. These mathematical models are represented
with equations, graphs, or tables (Kaput, 1995). The modeling perspective is
grounded more in data. It is finding mathematical relations that adequately
fit the data and that can be used to make reasonable predictions. An example
illustrates finding a model that will yield the weight Of an object if you know
how much a spring has been stretched. Students could collect the data
represented in Table 1 below.

Table 1: Spring Data

 

 

 

 

 

 

 

Length Weight
(cml (g)
6.5 0
7.3 100
7 .5 200
8.5 500
12.2 1000

 

 

 

Then use it to find the weight of a rock that stretches the spring a length Of

10 cm (see Figure 2).

21

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graph. 3
estimate

to fit the

 

 

7. :[Length = 10 cm

  

Figure 2: Spring Illustration
Part of their solution could involve making a graph, fitting a line on the
graph, and making a prediction. The graph in Figure 3 can be used to
estimate the weight of this rock to be about 700 grams. (I need to find a way

to fit the line on the graph).

 

 

 

 

 

 

 

 

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Figure 3: Spring Graph
Structure
Another way to consider algebra is to characterize the structure of the
system of algebra (Algebra Working Group, 1997; and Usiskin, 1988). An
example of this is algebra considered as generalized arithmetic (Thompson
and Thompson, 1993; Kieran, 1991; Peck and Jencks, 1988; and Sfard, 1995).

Peck and Jencks (1988) describe this as follows.

22

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of ar

is lea
numl
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CO

 

 

Algebra is a generalization Of arithmetic. It should arise logically and
naturally as a consequence of children’s decisions about how arithmetic
works. If arithmetic becomes completely sensible to children and becomes
a tool for their thinking, the decisions which make algebra sensible flow
naturally from it. (page 85).

They infer that algebra can arise quite naturally from a solid understanding
of arithmetic.

Educators whose work falls into this class support the view that algebra
is learned through studying and generalizing the properties of the real
number system. Algebraic assertions can be made about the numbers based
on what was generalized fiom arithmetic. This can be illustrated with the
distributive property. Before studying the general case, consider this

rectangular array representation example Of 3 x (4 + 5) in Figure 4.

4 5
A A
\f
I
3 I 3x(4+5)=3x(9)=27

Figure 4: Distributive Property of 3 x (4 + 5)

 

 

 

 

 

 

 

 

 

 

 

 

 

These 27 squares can be represented in another, equivalent way. This can
also be shown with (3 x 4) + (3 x 5) in Figure 5.

4 5
A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 3{ (3x4)+(3x5)=(12)+(15)=27

 

 

Figure 5: Distributive Property of (3 x 4) + (3 x 5)

23

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specific
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Ki
make tl
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Studying the structure of the distributive property involves moving fi'0m the
specific case of 3 x (4 + 5) = (3 x 4) + (3 x 5) to the general case of
ax(b+c)=(axb)+(axc).

Kieran (1985) asserts that students are able to learn algebra when they
make the transition from the arithmetic approach, finding the values of the
unknowns using simple Operations, to the algebraic approach, making use of
the structure of the system when the equation is the key Object of study.
Employing the algebraic approach means more than surface and structural
operations. It is about being able to compare expressions without evaluating
them directly. This can be illustrated with the equation x — 10 = 27. Students
using simple operators might think, “What minus ten is twenty-seven? Thirty
minus ten is twenty, forty minus ten is thirty, thirty-seven minus ten is
twenty-seven. SO, the answer is thirty-seven.” Whereas, students making use
of the structure might think, “Since I subtract ten from X, I know that I need
ten more than twenty-seven, or thirty-seven.”

The first approach is case dependent. If students tried to solve
x - 17 = 41 they would likely start this problem over, “What minus seventeen
is forty-one? Fifty minus seventeen is thirty-three, sixty minus seventeen is
forty-three, fifty-eight minus seventeen is forty-one. SO, the answer is fifty-
eight.” Students who used the structure could apply the same strategy in this
second equation, “Since I subtract seventeen from X, I know that I need

seventeen more than forty-one, or fifty-eight.”

24

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case A
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Kieran (1989) claims that this understanding of the structure is where
many students confront dificulty learning algebra. She illustrates this with a
case of students in a typical classroom. The equality between the left- and
right-hand sides of an equation is a foundation in traditional algebra
instruction. Students are Often taught to solve equations by doing the same
thing to both sides of the equal sign. When students do not see the equation
as this balance, but instead see the right-hand side as the answer, they
encounter problems. These students do not fully see the structure of the
equations and often have difficulty solving equations in this manner. Filloy
and Rojano (1989) write that these students who do not make the transition
have an arithmetic notion of equality. Kieran suggests that confionting these
issues that deal with structure earlier in their algebra classrooms will help

students gain a better understanding of algebra.

Language and Representation

In the final theme presented by the Algebra Working Group (1997) the
important characteristic of algebra is the language that is used to
communicate. Algebra is sometimes considered the “language of arithmetic”
(Kline, 1972; and Usiskin, 1988). Educators whose work falls into this theme
write that algebra can be thought Of as a language with syntax to be learned
that communicates mathematical ideas (Booth, 1989; and Bell and Malone,

1993).

25

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languag"
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In algebra there are many types Of representations that are used in the
language to communicate ideas. Many people’s first thoughts of algebra
relate to only the symbols present in a typical ninth grade algebra course.
This is very limiting. Considering only symbols in algebra would be like
restricting one’s writing to include only nouns. In both instances a much
more complete picture can be Obtained by using multiple representations, or
parts of speech. Just as a more informative story can be told when including
additional parts Of speech (e.g., nouns, verbs, adverbs, adjectives, and
prepositions) a more complete algebraic solution might include additional
representations (e.g., symbols, tables, words, graphs, and diagrams). One
aspect Of learning algebra means using one or more Of these representations
to communicate algebraic thoughts.

Learning this language is more complex than students think (Bell and
Malone, 1993). Students typically possess a very simplistic view of the
language Of algebra (Booth, 1989). They may not recognize the intricate
connections between different representations. Stacey and MacGregor, (1997)
caution that under this approach there is a new grammatical structure for
students to learn; the rules in the language Of algebra are not the same as the
rules in ordinary language. Some words that have multiple meanings in the
English language have very precise mathematical definitions, such as the

” u

words “product , and”, and “or”. Some operations that seem similar are not.

26

For exa
frustrat

Ar
what tl
what it
a “feel"
regardi
used, a
symbol
Symbol
acknov
that st'

(Hersh

 

 

 

For example, 2(x + y) = 2x + 2y is true, but many algebra teachers have been
frustrated when students incorrectly conclude that (x + y)2 = x2 + yz.

Arcavi (1994) warns that learning algebra is more than just learning
what the symbols mean. Rather, Obtaining “symbol sense” is at the heart Of
what it means to really know algebra. To him, “symbol sense” means having
a “feel” for symbols, or an “accurate appreciation, understanding, or instinct
regarding symbols.” (pg. 28). This involves knowing when symbols should be
used, when to use a different representation with symbols, when to abandon
symbols, how to manipulate and read symbolic expressions, and how different
symbols play different roles in difi'erent contexts. Some researchers
acknowledge that symbols should be introduced earlier in the curriculum SO
that students can more fully appreciate their power and learn the language

(Hershkowitz and Arcavi, 1990).

Patterns and the Study of Algebra

Studying patterns and finding generalizations cuts across all of the
themes discussed in the previous section. Students might emphasize
functions when they study the rate of change in a table to find a function to
represent the pattern. They could focus on the data collected in the search for

a model to predict additional values. They might study the data and

27

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(hflere:

P:
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generalize from the structure of the pattern. Or, they could use several
different representations to help generalize.

Patterns comprise a vital component Of the discipline of mathematics.
Interesting patterns arise in all areas of mathematics. Mathematicians can
use patterns of a sequence Of growing shapes in geometry to describe
characteristics Of the nth figure, patterns of simpler cases in probability to
explain a more complex probability, or patterns of numeric data in a table in
algebra to generalize the nth term in a sequence, or even patterns fi'0m a
continuous graph on a coordinate grid. The primary emphasis comes from the
function, relation, and modeling approach when students write
generalizations Of functions. They Often use the structure of algebra to help
write these generalizations, and Of course, they need knowledge Of the
language to do all of this.

Some researchers describe the discipline of mathematics as the “science
of patterns” (Hoffman, 1989; Steen 1988; American Association for the
Advancement Of Science, 1989; and Schoenfeld, 1989). In Land and Becher,
(1997) Van de Walle states:

The world is full of order and pattern: in nature, in art, in buildings, and
in music. Pattern and order are found in commerce, science, medicine,
manufacturing, and sociology. Mathematics discovers this order and uses
it in a multitude of fascinating ways... (page 301).

Steen (1988) writes,
Mathematics is the science of patterns. The mathematician seeks
patterns in number, in space, in science, in computers, and in

imagination. Mathematical theories explain the relations among
patterns; functions and maps, Operators and morphisms bind one type of

28

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Patter:

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pattern to another to yield lasting mathematical structures. Applications
of mathematics use these patterns to ‘explain’ and predict natural
phenomena that fit the patterns. Patterns suggest other patterns, Often
yielding patterns of patterns. In this way mathematics follows its own
logic, beginning with patterns from science and completing the portrait by
adding all patterns that derive from the initial ones. (page 616).

In the next sections I further illustrate the influence on mathematics
education of this view of patterns in mathematics. First, I consider the
implications for school mathematics, and then I describe a scheme to classify

the patterns students might study.

Patterns in School Mathematics

In school mathematics it is essential for students’ experiences to include
the exploration of patterns. The ability to recognize and describe patterns lies
at the foundation of mathematical science (Smist and Barkman, 1996). One
way this is useful is to illustrate an idea; mathematical conjectures Often
become more clear to Observers by examining patterns (Toumasis, 1994).
Some properties of positive integers can be more apparent through the
exploration Of patterns. There is much that can be inferred by studying the
way the numbers increase. An example is illustrated with the pattern Of

square numbers in Figure 6.

3 1 3 5 1 3 5 7
1 1+3=4 1+3+5=9 1+3+5+7=16

Figure 6: Square Number Pattern

29

that

nun

gm
DNA
and
pre

pat

TDV
f0“

and

By studying the dot representations Of square numbers, it becomes clearer
that all square numbers can be written as the sum of consecutive Odd
numbers,4=1+3,9=1+3+5,16=1+3+5+7,etc.

In addition to being a general foundation in mathematics, using patterns
to investigate relationships can specifically help students develop their
algebraic thinking (NCTM, 1989; 1994; Silver, 1997; Ferrini-Mundy, et al.,
1997; Phillips, et al., 1991; and Phillips, 1993). The authors of NCTM’S
Curriculum and Evaluation Standards (1989) and Kieran (1994) both
support the view that the inductive study of patterns should represent the
ground work for students’ initial learning Of algebra. Inductive thinking
involves students studying a small number of specific terms fi°om a pattern
and then making more general statements based on their explorations. These
preliminary experiences with generalizations arise through investigating
patterns (Curcio and Schwartz, 1997). Students might eventually use
symbolic notation to write their generalizations. Although those who do
consider verbal generalizations but do not formalize their expressions with
symbols are still involved in algebraic thinking (Lins, 1990).

There are other essential components to algebraic thinking based on
investigating patterns besides generalizations. These include exploring and
formalizing patterns, conjecturing about the patterns identified, verbalizing
relationships between and among elements in patterns, extending patterns,

and eventually representing the relationships using symbolic notation

3O

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(Silver, 1997; Kenney and Silver, 1997; and Sierpinska, 1992). Some
examples of these activities can be illustrated using the dot and sum
representations of square numbers in Figure 6 above. Students might
verbally describe or conjecture about several patterns. Students could
formalize the pattern by noticing that the square numbers can be written as
the sum of Odd numbers. Students might conjecture that all square numbers
could be written as the sum of consecutive Odd numbers. This relationship
might be further clarified between elements when students recognize that to
find the next square number, one can add the next Odd number to the
previous square numbers. This idea is extended in Table 2 below to illustrate

finding the first ten square numbers.

Table 2: Table of Square Numbers

 

 

 

 

 

 

 

 

 

 

 

 

N Sequence of Sum of Previous N“ Square

Odd Numbers Square Number and Number
(2N — 1) Next Odd Number

1 1 1 1

2 3 1+3=4 4

3 5 4+5=9 9

4 7 9+7=16 16

5 9 16+9=25 25

6 11 25+11=36 36

7 13 36+13=49 49

8 15 49+15=64 64

9 17 64+17=81 81

10 19 81+19=100 100

 

 

 

 

 

Finally, students could symbolize their generalization by writing the
hypothesis that to find the nth square number add the first n Odd numbers:

1 + 3 + 5 + + (2n-1) or simplify this with n2.

31

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Patterns that are appropriate for middle grade students to study have
regularity and predictability. These two characteristics of patterns make
explicit some assumptions about the behavior of the patterns advocated in
the reforms (Heaton, 1994). A general purpose of searching for patterns is to
use the information gained through the investigation to make predictions
about later terms in the sequence. Making predictions is only relevant when
a pattern maintains aspects of regularity since it is the regularity that leads
to the predictability of the pattern's behavior. Students need some reason to
believe that this regularity will be maintained. The only way this occurs with
reasonable certainty is in some context. (The issue of context is discussed
later in this chapter.) There is not a guarantee that a pattern continues based
solely on the table of numbers. Heaton (1994) adds further clarification.
"...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" (p. 149).

Investigating patterns is one foundation to learning functions for
students. Functions are an important concept for students to learn in
mathematics; yet they often have difficulty understanding the principles of
functions such as the notion of independent and dependent variable (Artigue,
1992; Eisenberg, 1992; and Sierpinska, 1992). In the NCTM Curriculum and

Evaluation Standards (1989) the authors emphasize the significance of

32

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studying patterns to support the learning of functions through the entire K-
12 curriculum. In the section for the middle grades, the authors write:

The theme of patterns and functions is woven throughout the 5-8
standards. It begins in K4, is extended and made more central in 5-8,
and reaches maturity with a natural extension to symbolic representation
and supporting concepts, such as domain and range, in grades 9—12 (page
98).

In the elementary standards these goals are formalized in Standard 13:
Patterns and Relations, extended for the middle grades in Standard 8:
Pattern and Functions and Standard .9: Algebra, and pushed further at the
secondary level in Standard 5: Algebra and Standard 6': Functions.

Several educators suggest ways to implement this approach to functions
through studying patterns at both the elementary and the middle grades
levels (Curcio, 1997; Austin and Thompson, 1997; and Herbert and Brown,
1997). Chappell (1997) suggests some pre-symbol experiences for students
that relate to algebra at the elementary levels where students verbally
describe patterns in a “guess my rule” game. She reminds that it is algebraic
thinking and not formal algebra that should receive the emphasis at this
level. Pegg and Redden (1990) describe a seventh grade course in South
Wales, Australia where algebraic ideas are introduced through studying
numbers patterns in data without introducing the early use of the
manipulation of symbols.

A goal at the middle grades is to describe a generalization. However, the
transformation to symbolic notation is not necessary for a student to initially

recognize and generalize the pattern (Lee and Wheeler, 1987). Students may

33

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verbally describe a pattern, but not be able to write a symbolic rule from the
numeric pattern in the data (Lee, 1987). This ability to generalize with words
represents an important initial part of algebraic reasoning. An example of
this can be illustrated in a problem involving phone charges. Suppose a
phone company charges a $1.50 connection fee for a phone call and an
additional $0.25 for each minute, or a fraction of a minute. Students might

represent the pattern in Table 3 as follows.

Table 3: Cost of Phone Call

 

 

 

 

 

 

Number of Cost in
Minutes Dollars
0 1.50
1 1.75
2 2.00
3 2.25

 

 

 

While completing the table they could use words to describe how the cost of
the phone call is increasing with time, but not be able to write a
generalization using symbolic notation.

Coming up with a symbolic representation to generalize a pattern often
proves challenging for students. It is not a trivial transition for students to
move from recognizing a pattern to writing an algebraic rule (MacGregor and
Stacey, 1993; Pegg and Redden, 1990; and English, 1995). Lee and Wheeler,
(1987) report that students can often formulate appropriate generalizations
without using algebraic symbolic notation. MacGregor and Stacey (1993)

identify four critical steps students must cross in order to move from

34

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to:

hiera
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recognizing a pattern in a function table to writing an algebraic rule with x as
the independent and y as the dependent variables. Students should to be able
to:

. articulate the relationship to find numerical values,

. look beyond a recurrence pattern to find one that links the two
variables,

. know the syntax of algebra, and

. know what can and can not be said with algebra (page 187).

While MacGregor and Stacey did not necessarily present these steps as a
hierarchy, I consider the order presented above. Prior to writing a symbolic
rule, students should recognize a pattern; this could be with a verbal
description, or by extending the pattern.

In many cases there are a number of different symbolic representations
that students can write. In this study, I am interested in how students write
symbolic generalizations where the dependent variable is expressed as a
function of the independent variable. Although this is not the only way to
represent functions, it is useful. To write a representation in this form,
students must look beyond a recurrence pattern.

To write symbolic rules students need to know the syntax. If they do not
know this language, writing a generalization can be nearly impossible. But,
not knowing the syntax is different than the fourth of MacGregor and
Stacey’s steps. Knowing what can be said with algebra might be considered a
more sophisticated step than the other three. To know this students would

need‘a more complete knowledge of the discipline. Students might confuse

35

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and believe that something can not be said, when actually they just do not

know the syntax to write a representation.

A Classification Scheme

There are two important, yet different, aspects in a classification of
problems involving patterns. Phillips, et al. (1991) along with other research
helps to distinguish these (Steen, 1990; 1988; Golos, 1981; Boles, 1990;
Heaton, 1994; and Algebra Working Group, 1997). The first is the content or
function type of the specific pattern; the second is the context in which the

problem is situated.

Content Class.

The first classification is based on the mathematical content, or function
type, of the pattern. Four functions suggested for a Standards-based middle
grades reform curriculum are linear, exponential, polynomial, and inverse
functions (N CTM, 1989). Although this list is not intended to be
representative of functions, it represents some of the typical functions
covered in the middle grades.

Linear functions are those which have a constant rate of change. They
typically are one of the first patterns students learn to recognize. Students
might observe these patterns by studying the linear graphs, or by recognizing

a constant difi‘erence in a table of data. Students might encounter these as

36

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either continuous or discrete functions. In some instances they might notice
that a pattern increases (or decreases) in a table by the same constant
number. For example, a problem might ask students to study the number of
perimeter dots of the following dot representations of squares with length

greater than or equal to two. The first four figures are illustrated in Figure 7

El
EB El. 0 0
El [El [:1 [E
4x1=4 4x2=8 4x3=12 4x4=16
Figure 7: Perimeter Dots on Squares

below.

[E3]

Students might recognize that this is a linear function by studying the
figures, tables, or a graph. In Figure 7 above, the perimeter dots for each
figure of the pattern are grouped into four boxes. For each figure in the
pattern, each box has one less dot than the length of the side. The number of
perimeter dots can then be represented as 4x(n-1) where n is the length of the
side. This form of the rule often tells students that this is a linear function.
The linearity of this pattern could also be observed by studying Table 4.

Table 4: Perimeter Dots on Squares

 

 

 

 

 

 

Length of Perimeter Rate of
Side Dots Change
2 4
3 8 8 — 4 = 4
4 12 12 - 8 = 4
5 16 16 — 12 = 4

 

 

 

 

The constant rate of change can be found by subtracting subsequent terms:

8 - 4 = 4, 12 — 8 = 4, etc. Since these differences are all 4, this must be a

37

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linear function. Finally, a plot of the data on a graph supplies information
about the pattern because the data forms a straight line, it is a linear

function (See Figure 8).

 

 

 

 

 

‘31
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g . ‘. l
E 15 T i l 'i l i i 5
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a 5 ~ , is; 7 — ——»-—~—— —— —
E 0 i i I i g - l
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‘ Length ofSide

 

t_¥_ __,__, _,,,_.

Figure 8: Linear Graph

 

A second function appropriate to the middle grades is the exponential
pattern. Students typically experience exponential functions as either growth
or decay patterns. One way to recognize this pattern is to note the
multiplicative growth (or decay) factor in a numeric table of data (See
Table 5).

Table 5: Exponential Table

 

 

 

 

 

 

 

 

Growth
x y : 4'3(x—1) Factor
' 1 4
2 12 12 + 4 = 3
3 36 36 + 12 = 3
4 108 108 + 36 = 3

 

 

This could be presented to students as a problem where students find their
allowance after 10 weeks under the following plan. They receive one cent the
first week and then on subsequent weeks, they double their previous week’s

allowance. They would receive one cent the first week, two cents the second

38

weel

data

 

week, four cents the third week, eight cents the fourth week, and so on. The
data can be organized in Table 6 below.

Table 6: Allowance Table

 

 

 

 

 

 

 

 

 

 

 

 

Week Number Allowance Growth Factor

1 $0.01

2 0.02 0.02 + 0.01 = 2
3 0.04 0.04 + 0.02 = 2
4 0.08 0.08 + 0.04 = 2
5 0.16 0.16 + 0.08 = 2
6 0.32 0.32 + 0.16 = 2
7 0.64 0.64 + 0.32 = 2
8 1.28 1.28 + 0.64 = 2
9 2.56 2.56 + 1.28 = 2
10 5.12 5.12 + 2.56 = 2

 

 

 

 

They could determine that this problem was exponential since the next term
could be found by multiplying by the growth factor of 2. The pattern can be
represented with a rule where A is the allowance for the wth week:

A = 0.01x2(w'1). The graph of an exponential also takes a different form as is

 

 

illustrated in Figure 9 below.
$6.00 T "T ,— , TE“
$5.00 T l " T“T ~~ T -_ T -.- . _¥_;,_fi_ f—“T_‘_0
0 ~ 1.: l 3t l L !___.'__
§$4'°°i“wl
B $3.00 T ' i i ' ' 1T
0 l l . T T
l ‘ ! ' i T T + T
$0.00 +4 +4 4, 9 ‘2 . l.-. 4
0 l 2 3 4 5 6 7 8 9 10
WeekNumber

 

 

 

Figure 9: Allowance Graph

39

In
emerit
y = x3!
polync

difiert

diflert

 

 

 

 

In addition, students could study polynomials. They often first
experience polynomial patterns as the less complex power functions: y = x2,
y = :03, y = x4, and so on. But, more generally these patterns include
polynomial functions that can all be recognized by finding constant
differences in numeric tables of data. In Tables 7 and 8 below, the constant
differences are found for y = x2 and y = x3 respectively.

Table 7: y = x2 Constant Differences1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x y _—_-_ x2 181: 2nd
Difference Difference
0 0
1 1 l — 0 = 1
2 4 4 — l = 3 3 - 1 = 2
3 9 9 — 4 = 5 5 — 3 = 2
4 16 16 - 9 = 7 7 - 5 = 2
5 25 25—16=9 9—7=2
6 36 36—25=11 11—9=2
Table 8: y = x3 Constant Differences
x y = x3 lot 2nd 3rd
Difference Difference Difference
0 0
1 1 1 — 0 = 1
2 8 8 — 1 = 7 7 — 1 = 6
3 27 27—8=19 19—7=12 12—6=6
4 64 64—27=37 37—19=18 18-12=6
5 125 125—64:61 61—37=24 24-18=6
6 216 216-125:91 91-61=30 30—24=6

 

 

 

 

 

 

 

In each case the first difference is not constant. In all quadratics the second

difi'erence is constant. In all cubics the third difference is constant. This can

 

1 Recall that the difi'erences are related to the derivatives.

40

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be extended to show that the fourth difference is constant in a quartic, but
middle school students do not typically explore polynomials to greater
degrees. Functions of this general polynomial type are typically more diffith
for students to generalize using symbols when they are other than the power
functions. In a typical problem, they might be asked to find the total number

of squares in the pattern illustrated in Figure 10.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ED

Figure 1 Figure 2 Figure 3
lx2=2 2x3=6 3x4=12

 

 

Figure 10: Polynomial Pattern of Figures

The number of squares follows a quadratic pattern. This could be recognized
in either the rule or the table. A symbolic rule for the total number of squares
can be written as T = n(n+1) where n is the figure number. This could be
recognized as a quadratic since it is a quantity of n multiplied by a quantity
of n. The pattern can also be extended in Table 9 to find that the second
difi'erence is constant, which also signifies that it is a quadratic pattern.

Table 9: y = Constant Differences in Total Squares Table

 

 

 

 

 

 

. Figure Total 18‘ 2nd
Number Squares Difference Difference
1 2
2 6 6 — 2 = 4
3 12 12 - 6 = 6 6 - 4 = 2
4 20 20—12=8 8-6=2
5 30 30—20=10 10-8=2

 

 

 

 

 

 

41

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A fourth function that students might study in the middle grades is the
. . . 1
inverse function. These are often in the form of ;. Students may first
encounter this function when they study distance-rate-time problems, d = rt,

d

and solve for the rate or time. These functions then would be of the form r = “t—

d . .
or t = :. Students could explore how changing the rate to walk 8 miles affects
the time walked as represented in a table, rule, or graph. The rule for this

8
case would be t = 7, while the table and graph are represented below as

Table 10 and Figure 11.

Table 10: Walking Times Table

 

 

 

 

 

 

 

 

 

 

Walking Rate Time
(mph) (in hours)
1 8
2 4
4 2
6 1.33
8 1
10 0.8
12 0.67
14 0.57
16 0.5

 

 

 

 

42

Cont

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T 0246810121416
Walking Rate (mph)

 

 

Figure 11: Walking Times Graph

Context Class.

The content is not the only important aspect of a pattern. The Algebra
Working Group (1997) reminds that any mathematics must be about
something. Algebra cannot be learned without some kind of context.
“Students build concepts and develop ways to think in pursuit of activities
that engage them in different contextual settings; such settings help students
make sense of the algebra they are studying (Algebra Working Group, page
9).” The group identifies five contextual settings: growth and change, size and
shape, data and uncertainty, number, and patterns. These are based on the
settings Steen (1990) presents: dimension, quantity, shape, uncertainty, and
change. Problems that involve studying expanding populations involve
growth and change. They might include linear or exponential growth. A key
idea in these type of problems is the relationship between how the change in
one variable affects the other. Size and shape problems are geometric in
nature. Students could study polygons and investigate which shapes could be

used to tile a surface. Data and uncertainty can be considered as data and

43

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able
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chance, the ideas in statistics and probability. Initially students can delve
into ways to explore the representations of data and the probability of certain
outcomes. Numbers, or quantities, are fundamental to school mathematics.
The problems students can perform with the availability of powerful,
inexpensive calculators are no longer routine calculations. They need to be
able to reason about the numbers and quantities. Patterns are the main
context for the focus of this study. Problems with numeric patterns can be
found in tables of data, or they can be recognized in a series of shapes.

Any content can be placed in a different context. For example, the
exponential problem that asks students to find their allowance after ten
weeks maintains the same mathematical content as a problem regarding a
certain bacteria that doubles every hour. Strong problem solvers are able to

investigate problems in all difierent contexts.

Summary

Educators, policy makers, researchers, and mathematicians have not
reached a consensus on what should dominate the study of algebra. Nor do
they agree on a main focus. The Algebra Working Group (1997) suggests a
vision of algebra with multiple organizing themes: functions and relations,
modeling, structure, and language and representations. In the study of

functions and relations the rate of change between variables is emphasized.

44

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The modeling theme focuses on writing mathematical models to represent
data. Supporters of the structure theme believe that the understanding of the
structure of the number system lies at the root of understanding algebra.
While the language and representations theme seems to suggest that
understanding and communicating algebra ideas are key.

The Algebra Working Group and Kaput (1995) both remind that any one
of these themes or aspects to the exclusion of the others is not sufficient to
represent school algebra. Instead, aspects of each are important to develop a
rich knowledge and understanding of algebra.

The study of generalizations of mathematical patterns is one area in the
discipline where four of these themes can be emphasized at different times.
Investigating patterns is suggested in the reforms as valuable mathematics

and worthwhile for students to study.

45

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CHAPTER 3

The Study

Introduction

What is algebra? Although, I did not resolve it in the last chapter, I
presented some different perspectives in this discipline. These views helped
to define the study and influenced some of the choices I made to investigate
the question of what students know and understand about algebra after three
years in a Standards-based curriculum. These decisions focused my
exploration of the research question. This chapter represents an opportunity
for the reader to both learn how I investigated the question, and why I made
specific choices. There were four initial selections I made that shaped this
study into its current form.

1. The content—patterns which represent functions and generalizing

with symbols from patterns of data.

2. The curriculum—the Connected Mathematics Praject.

3. The site—Heartland Middle School.

4. The data—students’ responses to performance tasks, recordings

while students worked, and interviews after they completed each

task.

46

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Having made these choices, with these parameters, the question evolved to,
“What do eighth grade students know about writing symbolic generalizations
from patterns which can be represented with functions, after three years in

the Connected Mathematics Project curriculum?”

The Content

I demonstrated in the previous chapter that studying patterns renders
one fundamental aspect of algebra. Although this represented the initial slice
I made to study the algebraic thinking of middle grade students, the content
was not sufiiciently defined for this study. I looked further at my interests
and the foundational aspects of algebraic understanding. Once students
studied and recognized patterns, I wondered what tools they had to help
them represent the patterns. The study evolved to investigate the content of
how students represented patterns with symbols. I saw this as one of the

fundamental aspects of algebraic understanding.

The Curriculum

A second decision I made involved the curriculum. I selected the

Connected Mathematics Project (CMP) to study for six main reasons.1 The

 

1 The CMP was funded by the National Science Foundation (NSF) to develop complete
curriculum materials for grades 6-8. After the NCTM Curriculum and Evaluation Standards

47

curr

tron

mat

t9 ac

 

 

first was the stance taken towards algebra. The CMP takes an approach
towards algebra different from traditional algebra curriculum. In CMP,
algebra is a strand woven throughout the curriculum. It was important to me
that the curriculum I studied took this stance since I was curious about the
strands approach.

The second was the position towards instruction. The developers of the
curriculum acknowledge that it is not possible to separate what is taught
from how it is taught. Both are important to the students’ understanding of
mathematics. The authors of CMP support an investigative approach to the

teaching of mathematics.2 I saw this approach as different than traditional

 

(1989) were published NSF supported several projects with 5-year grants to create materials
at the elementary, middle, and secondary levels that would support the teaching and
learning envisioned in the NCTM Standards. The goal of the CMP was “to develop a
complete mathematics curriculum with teacher support materials for the Middle Grades, 6,
7, and 8.” (page v, GTK CMP). Glenda Lappan, William Fitzgerald, and Elizabeth Phillips of
Michigan State University, James Fey of the University of Maryland, and Susan Friel of the
University of North Carolina are the principal investigators of the project.

2 Extensive teaching materials are available to support the implementation of CMP. The
CMP materials for teachers are organized around an instructional model that supports
problem-centered teaching. The model considers three phases of instruction: launch, explore,
and summarize.

Initially, during the launch, the teacher sets the context of the problem. She launches the
problem for the class to begin their investigation. This is the time the teacher could introduce
new ideas, clarify definitions, review old concepts, or connect the problem to previous work
done by the students.

The exploration phase allows time for students to “dig in” and investigate the problem.
They typically might do this with a partner or in a small group, but at times work
individually or even as a whole class, depending on the problem. The teacher’s role during
this phase of instruction is to observe students while they work, offer prompts to students
who are not making progress, redirect students as needed, and suggest extensions as
students complete solutions.

The final phase of instruction calls on the teacher to aid students summarize their work.
This is often done as a whole class when some students could share their strategies of the
investigation. The teacher helps students to deepen their understanding of the mathematics
in the problem by noting similarities or differences in students work or pushing students to
consider an extension.

48

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algebra instruction and was interested how students could learn in this
environment.

The third was the use of rich problems in the curriculum. The CMP is
organized around interesting problem settings. Students are presented with
worthwhile mathematics tasks to explore.3 This approach to problems
supports my use of tasks to evaluate student understanding. I will discuss
the tasks in further detail in the section surrounding the data later in this
chapter.

The fourth was a belief that all students can learn algebra. The authors
maintain a commitment towards teaching all children mathematics; as a
result they acknowledge that all students can learn a meaningful algebra.
The CMP takes the stand that all students can thrive in a heterogeneously
grouped classroom. The top performing students' mathematical
understanding is deepened when they consider ways to justify their solutions.
The teacher may also choose to suggest an extension to the problem that
continues to challenge the child. Students who are low performers are in an
environment where they are expected to learn valuable mathematics. The

materials meet the daunting task. of engaging while challenging all students.

 

3 The NCTM Professional Teaching Standards (1991) describe the importance of posing
worthwhile mathematical tasks for students to solve. They argue that the problems should
be based on sound mathematics, perceptions of students experiences, and knowledge of
diverse ways that students learn mathematics. The authors of CMP recognize the influence
of the tasks chosen by the teachers on students learning of mathematics. “There is no other
decision that teachers make that has a greater impact on students’ opportunity to learn and
on their perceptions about what mathematics is than the selection or creation of the tasks
with which the teacher engages the students in studying mathematics.” (Lappan, G., et. al.,
1996, page 40).

49

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The authors state, “The Connected Mathematics Project assumes that when
all students are held to the same high expectations and given a chance to
explore rich problems, all students can succeed in mathematics” (Lappan, G.,
et. al., 1996, p. 80). This vision supports a societal goal of educational
opportunities for all, as stated in the N CTM Curriculum and Evaluation
Standards.

The fifth was the importance of connections. The curriculum is called the
Connected Mathematics Project; this title alone is evidence that the authors
feel connections are vital for students to make. CMP is based on a foundation
that supports all students’ learning of mathematics by connecting it with
other areas of students’ learning and interests. Connections in the curriculum
are made between what students are learning in mathematics, different
areas of mathematics, ideas from other school subjects, and the world outside
of their mathematics classroom. The problem settings for each investigation
where students explore mathematics present many of the opportunities for
the connections. I felt it was important that for students to see the overall
picture of mathematics they should recognize how mathematical ideas are
related.

The sixth was my familiarity with the curriculum. I worked with the
authors of the CMP during the development phase. I helped with revisions to
both teacher and student editions in primarily the algebra strand. As a

result, I had the advantage of knowing the curriculum thoroughly. This

50

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insight helped me see that the CMP represented a curriculum that paralleled

many of my ideas regarding the teaching and learning of algebra.

CMP Algebra

In the section above I describe my interest in, CMPs stance towards
algebra for all students. Recall in Chapter 1, I reference Chambers (1994)
when she states, “Algebra for all is the right goal at the right time. We just
need to get the right algebra.” (p. 85). While I do not pursue CMP as
necessarily the “right algebra” in this study, I do see the curriculum as an
approach to algebra and a set of goals for the learners of algebra that grows
out of the NCTM Curriculum and Evaluation Standards. In this section I

describe this curriculum’s approach to algebra.

Algebra Goals

The developers of CMP believe that every child can learn mathematics,
and specifically every child can gain a meaningful understanding of algebra
(Lappan, G., et. al., 1996). This means that the authors do not expect a
portion of students to be skimmed off and placed in an advanced mathematics
track. Rather, they believe all students together can be challenged and be
successful in this curriculum. One of the first things recognized as different
fi'om the typical pre-Algebra or Algebra I in 8th grade course is that the

algebra is not isolated in one course or grade level. Although there are

51

specific units focused on algebra, algebraic ideas are woven throughout the

entire curriculum. This is in parallel with Kaput’s suggestion for a “strands”

approach to algebra.

The primary goal of the authors of CMP is to help students “reason and
communicate proficiently in mathematics” (Connected Mathematics Project).
The strand specific algebra goals for students who complete three years in
the CMP are that most students should be able to:

. Recognize situations in which important problems and decisions involve
relations among quantitative variables-one variable changing over time or
several variables changing in response to each other.

. Use numerical tables, graphs, symbolic expressions, and verbal
descriptions to describe and predict the patterns of change in variables.

. Recognize (in various representational forms) the patterns of change
associated with linear, exponential, and quadratic functions.

. Use numeric, graphic, and symbolic strategies to solve common problems
involving linear, exponential, and quadratic functions. (Lappan, G. T., et.
a1, 1996, pg. 22)

This list of goals looks quite different than what students in a
traditional Algebra I course would be able to do. Some of the areas that are
emphasized in traditional curricula that are not part of the CMP are: an
emphasis on multiplying and factoring polynomials, operating on algebraic
fractions, simplifying radicals, operating on non-linear polynomials, and
completing the square (Lappan, G., et. al., 1996, p. 28). By eliminating these
types of exercises, more time can be spent developing a solid underpinning in

algebraic reasoning. Some of the ideas included in CMP that traditional

curricula do not include are:

52

. Emphasis on variables and the representations of the relation
between variables in words, numeric tables, graphs, and symbolic
statements.

. Focus, on the rate of change between two variables, not only linear.

. Development of functional point of view and applications.

- Emphasis on modeling

. Earlier introduction of exponential growth and decay

. Development of alternative strategies for answering questions about
algebraic expressions and equations (Lappan, G., et. al., 1996, p. 28).

It is apparent that students following this curriculum will have different

experiences than students taking a traditional Algebra I course. There is less

emphasis on manipulating symbols and more of a focus on understanding the

relationship between variables.

Organization of the Algebra Strand

The curriculum consists of eight units at each of the three grade levels.4
Each unit has a primary strand (content goal) as the focus of mathematical
content, but all units make connections to the other strands throughout.5 (See
Appendix A for a complete list of units.)

The six units with algebra as a primary strand are listed in Table 11 by

suggested grade level.

 

4 Each unit of the CMP curriculum is divided into four to seven investigations that are built
on big mathematical problems that students solve. Some problem situations are real; some
are whimsical, while others are pure mathematical investigations.

5 The materials were developed to be used in the order suggested in Appendix A, although
other paths through the curriculum may reasonably be followed, based on local
circumstances. The authors caution, that some adaptations may need to be made when

53

Table 11: CMP Algebra Units

Grade Seven Algebra Units Grade Eight Algebra Units

Variables and Patterns: Thinking with Mathematical Models:
Introducing Algebra Representing Mathematical Relationships
Moving Straight Ahead: Growing, Growing, Growing...:
Linear Relationships Exponential Relationships

Frogs, Fleas, and Painted Cubes:

Quadratic Relationships

Say It with Symbols:

Algebraic Reasoning

The first unit with a central focus of algebra, Variables and Patterns, is
designed to start the seventh grade year. This unit builds on student’s prior
experiences and introduces them informally at first to the notion of variables
and representations of relationships. These ideas of representations are
revisited with a concentration on linear relationships in the next algebra
unit, Moving Straight Ahead. In the following year, four of the eight units at
grade eight have algebra as their primary mathematical strand. The first,
Thinking with Mathematical Models, introduces students more formally to
functions and modeling. Growing, Growing, Growing..., examines
exponential growth and decay in tables, graphs, and simple symbolic forms.
Frogs, Fleas, and Painted Cubes, focus on quadratic growth and functions.
The last algebra unit, Say It with Symbols, students investigate equivalent

symbolic expressions and solving linear equations symbolically.

 

following alternate routes since most units build on student understanding developed in
prior units.

54

The six units in the algebra strand do not represent the full extent of
algebra in CMP. Consistent with the philosophy of connecting and weaving
the mathematical strands, ideas from algebraic reasoning are present in all
24 of the units. Prior to working more formally with algebraic symbols
students thoroughly investigate relationships in verbal, tabular, and
sometimes graphic forms. When algebraic symbols are introduced, they are
presented as a natural extension and representation of the ideas explored. By
the end of grade eight, it is expected that students will have a deep
understanding of the meaning of symbols and the relationships in tabular,
graphic, and verbal forms.

Recall that both the Algebra Working Group (1997) and Kaput (1995)
remind that a complete algebra experience for students involves all of the
organizing themes—functions and relations, modeling, structure, and
language and representationsThis is CMPs intent. Although the overall
focus is on the functions and relations theme throughout the algebra strand,
the other themes are present in various algebra units. A big idea in
functions and relations is rate of change. Students study rate of change in
linear patterns in Moving Straight Ahead, in exponential patterns in
Growing, Growing, Growing..., and in quadratics and cubics in Frogs, Fleas,
and Painted Cubes. Students explore models in Thinking with Mathematical

Models. In Say It with Symbols they study the structure of algebra. And, in

55

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Variables and Patterns they spend a lot of time studying difi'erent
representations of relations.

The authors of CMP acknowledge the importance of studying patterns.
They state that “(o)bservations of patterns and relationships lie at the heart
of acquiring deep understanding in mathematics...Students solve problems
and in so doing they observe patterns and relationships; they conjecture, test,
discuss, verbalize, and generalize these patterns and relationships.” (Lappan,
G., et. al., 1996, p. 1). Studying data to determine a pattern is a big part of
the algebra curriculum of CMP. It is also something reasonable for middle
grade students to do in their classes as a basis for algebraic understanding
(N CTM, 1989). This is expressed in two of the CMP algebra goals: use
numerical tables, graphs, symbolic expressions, and verbal descriptions to
describe and predict the patterns of change in variables, and recognize (in
various representational forms) the patterns of change associated with linear,
exponential, and quadratic functions. Students are expected to search for
patterns and relationships and then express their conclusions verbally and

eventually symbolically.

The Site

Another choice I made determined the site to conduct the study. Since

CMP was still in development when I conducted the study, I sought a location

56

that was piloting the materials, was committed to the implementation of
CMP, and would allow my research. I selected Heartland. This site interested
me because they had demonstrated a record of their commitment to the CMP

curriculum and because of their proximity to the university.

Heartland

Heartland is a small, rural community with a population of around
4,000 residents. A strong German Catholic heritage is often credited for the
hard work ethics of the community. This ideal is often carried over into the
schools. Community members respect the teachers and it is expected that
children follow their parents’ lead in this regard. The town had been a fairly
closed community with many small family farms and blue collar automotive
workers, but it is changing to more of a bedroom community as residents
commute to jobs in other nearby urban areas.

Heartland Middle School, the one public middle school in the
community, averages around 450 students in grades six through eight. There
is a K-12 parochial school that has about 40 students per grade in the
community. These schools together account for all of the students in the
middle grades. Heartland Middle School has been involved in mathematics
staff development for a number of years. Heartland is only about 30 minutes
away from Michigan State University; this proximity to the university helped

the school actively participate in the early reform movement.

57

Heartland Middle School has achieved academic awards in different
areas. Lamar Alexander, at the time United States Secretary of Education,
granted the school's mathematics department the "A+ for Breaking the Mold"
award for the 1992-93 academic year. The social studies department also
excels. They won many awards in the National American Express Geography

Competition.

Commitment to Reform

There are a number of sources to credit for the achievement in this
school: willing teachers, supportive administrators, quality materials, and an
informed mathematics teacher leader. Of primary importance is the notion
that this reform evolved over a number of years; it has not happened with a
“quick fix.”

Table 12 charts the reform projects at Heartland Middle School.

Table 12: Heartland Middle School Reform Table

 

Time Reform Project

1983 — 1985 Piloted MGMP
1987 — 1990 Participated in a “County Project”
1991 — 1996 Piloted CMP

 

 

 

 

 

 

 

In 1983, Heartland was chosen as a pilot site for the Middle Grades

Mathematics Project (MGMP).6 The middle school teachers from Heartland

 

3 MGMP is a set of five separate stand-alone units on equivalent fractions, factors and
multiples, perimeter and area, probability, and spatial visualization. They were written by

58

participated in workshops given by the authors of MGMP as part of the
piloting agreement. In 1987, Emily Clark, a local leader in the school,
initiated a countywide project.7 This began a four-year project to help the
mathematics teachers become more reflective about their practice. In 1991
CMP selected Heartland Middle School as a pilot site. Heartland seemed a
reasonable choice due to its closeness to the university, the stafi' development
that had already occurred with the teachers, and the disposition towards
reform by the administrators. The school agreed to teach the curriculum to
all of the students in heterogeneously grouped middle grade classes. When
the piloting of CMP began, teachers participated in regular workshops during
both the summer months and the school years to familiarize them with the

CMP philosophy, instructional model, and materials.

Evaluating the Reform Efforts: MEAP

Schools are evaluated in Michigan based on their students’ performance
on the Michigan Educational Assessment of Progress (MEAP) test. Students
score within one of three ranges: satisfactory, moderate, or low. In 1991, early
in the reform story, only 44% of the students in Heartland scored in the
satisfactory range. (See Figure 12 for the data from 1991 to 1995). Only four

years later, 78.8% of the students scored in the satisfactory range. This is a

 

several of the same authors as CMP. They were not intended to represent an entire
curriculum, but rather as replacement units for these topics.

59

huge increase! But this does not just show that students are moving up from
the moderate range, they are moving out of the low range. The number of
students scoring low decreased from 20% in 1991 down to only 6.8% in 1995.
Teachers and administrators were pleased. They felt this demonstrated that
students enrolled in a curriculum that focuses on problem solving continue to
do well on a more traditional standardized test while learning more powerful

mathematics.

5 % of students scoring in
satisfactory category

15 I% of students scoring in
.6 .4 68 low category

I j 1
1992 1993 1994 1995
Years

Percent of Students

 

 

 

 

E if "”“ffigne 12: Heartland Mm; an.
I credit two major factors for the improvement in the middle school
mathematics performance at Heartland. First was the availability of quality
materials, and second was the support from the administration.8

The closeness to the university afford teachers the opportunity to have

access to resources—both people and quality materials. The materials that

 

7 Emily convinced her superintendent along with the other five school districts in the county
to collectively pool their money to provide staff development for the 25 middle school

60

were used, first MGMP and later CMP, represented a model for the teachers
of what it means to teach differently.

The final key aspect that was vital to the reform equation was the role of
the administration. John Roberts, the middle school principal, has
demonstrated openness, vision, and trust. He was willing to listen and allow
Emily to pursue her goals for the district. He readily approved the time for
the teachers to spend in their coaching sessions. I recall our conversation
when I sought permission to use Heartland Middle School as a site to conduct
my study; with his approval he added that he trusted his teachers as
professionals to make the best decision for the students. He further stated
that if Evelyn Howard, the classroom teacher whose students I wanted to

participate, agreed then he would also support my choice.

Implementation of CMP

The CMP was implemented in the entire sixth grade at Heartland
Middle School in the fall of 1992. The seventh grade was added the following
year in 1993 as the materials were developed. Finally in the fall of 1994 the
eighth grade materials were available and the entire curriculum was
implemented in all three middle grades.

As the CMP was implemented, all students had this curriculum in their

mathematics classes. Students at Heartland are not pulled out for gifted and

 

mathematics teachers of the county rather than just the four teachers in their school.
5 Conversation with Emily Clark on November 24, 1997.

61

talented sessions, eighth grade algebra classes, or special education classes.
The classes are heterogeneously grouped. The school typically has between 10
and 15% of their students identified as special education students. All of
these are mainstreamed into regular classrooms. This percentage has risen to
as high as 18% at times. Evelyn Howard, the classroom teacher, has had as
many as one third of her students in one class labeled as special education.
There is a very minute minority population in this predominantly white
community. There are typically only five minority students in Heartland

Middle School in any given year.

The Students

All of the students participating were in their third year of CMP
mathematics. They represented the second cohort of students at Heartland
Middle School to have the entire CMP curriculum, sixth grade in 1993-94,
seventh grade in 1994-1995, and eighth grade in 1995-1996. They all had
Evelyn Howard as their eighth grade teacher, but came from two different
classes. In the seventh grade students had one of two mathematics teachers,
Evelyn or Jim Johnson. As sixth graders, these students had one of two
different teachers.

I introduced this study to two of Evelyn’s eighth grade mathematics
classes and described my interest in understanding how students in a CMP

classroom investigated problems with algebra. I sought permission from all

62

students in the class to participate in this study and explained the time
commitment. I wanted to study the performance of students who found
varying degrees of success with the program, not just the top performers. I
asked Evelyn to identify students who would give me a range of achievement,
but would be able to participate in conversations with me about their sense
making. Of the volunteers whose parents had signed permission slips, Evelyn
and I together selected ten students to participate.

I decided that students would work with a partner because some of the
tasks were challenging and I felt that the students would be more successful
if they worked in pairs. I decided to videotape each pair while they worked on
their tasks, I saw the additional conversation occurring between partners an
additional benefit. Students worked with a partner fi'om their class on
selected problems. They worked with the same partner for all four tasks. I
assigned pairs randomly with the intention of having a diverse blend of
groups. I had two groups of male pairs, one group of female partners, and two
groups with a male and a female student working together. The partners are

listed in Table 13 below.

Table 13: List of Students

Students with Partners
Zachary and Todd

Ben and Joe
Anna and Katrina
Julie and Dan
Sara and Ryan

63

The Data

Once I had determined the content, the curriculum, and setting, my final
decision involved the data to collect. I had spent a number of years studying
assessment with the Balanced Assessment Project (BA).9 While at BA I had
the opportunity to explore what represents a quality task, and to analyze
students’ written responses to tasks. This helped me recognize that often
some of what students did while working on a task did not make its way to
the written record. This understanding helped me to determine the data that
I collected in this study. I determined that assessment tasks would be part of
my data. I developed and revised four performance tasks based on the
algebra content described earlier for the students to complete.

I collected three sources of data in this study.

. Students’ written responses to the performance tasks,

. Video recordings of the small group work, and

. Follow-up interviews.

Data Collection
I felt that each of the three sources of data could offer different evidence

about these students’ understanding.

 

9 Balanced Assessment was funded by the National Science Foundation in 1991. Their goal
was to develop a comprehensive range of performance assessment tasks in mathematics, and
assemble them into balanced packages at grades 4, 8, 10, and 12.

64

Written responses. This is valuable data since typically classroom
teachers make decisions about student understanding and instruction based
on this written record. I was interested in how these students made sense of
these tasks. I felt that some of this would be revealed to me in their written
responses.

Tape-recordings (igroup work. I also thought that a lot of what students
did could potentially be lost if I limited my data to their written responses
alone. So while students worked with their partners, I either videotaped or
audiotaped them. (Most were videotaped, only one pair was audiotaped for
two of the tasks). I had hoped that these recordings would allow me insight
into the students’ conversations while they worked. I could see a little more of
what these students did in their solutions, when they were stuck on a
problem, or changed strategies.

Follow-up interviews. I wanted the opportunity to ask the students
questions about their solutions. I conducted follow-up interviews with each
pair of students after they completed each task, all of which were video-
recorded. I did not review the recording of the students’ work prior to the
interview, but I returned the students’ written responses and allowed them
some time to review their work before we began the interview. When they
were ready I used a protocol that I wrote while designing the study for some
of my initial questions. The general protocol I used to conduct each interview

is included in Appendix B. I realized during the interview that additional

65

questions were needed to clarify a response or probe a little further. For
example, after Zachary and Todd replied that they made tables in all of the
problems, I asked them to further describe their strategies for constructing
them. Some of the additional questions I asked: When did you use the table?
Okay, what does that tell you? How did you construct tables? How many of
these did you have to draw before you could actually continued the table?
What does that tell you that the second difi’erence is one? Okay, what does
that mean?

The data was collected over a three week period. Several pairs of
students left their mathematics class each day to help me with this study.
The students completed a task and then participated in the interview before
completing the next task. The students spent an average of 23 minutes to
complete each task and 19 minutes for the interviews. The approximate times
each pair spent investigating the tasks and participating in the interviews
are given in Appendix C.

Although all students completed the tasks in the same order—Borders,
Cutting, Dominoes, and then Toothpicks—they were not all investigating the
same problem at the same time because I had only two video-cameras to
record their work. Not all interviews were done immediately after students
completed their work. At times several days passed after a student completed
a task before I conducted the interview. Since I had two ‘pairs working at a

time, I sometimes had to have one pair wait to complete their follow-up

66

interviews. On some days, students were absent so I did not ask one student

from the pair participate.

Task Descriptions

The four tasks in this study were chosen based on their mathematical
content, context, and the students’ experiences in their mathematics
classrooms. Students were allowed as much time as needed to complete the
tasks and had access to calculators. (See Appendix C to see how long each
pair worked on each task.) If the students did not complete a problem during
their class time, then I allowed them additional time in our next meeting to
finish. The tasks are similar in that they all ask students to study some
regularity, make predictions for future values, and then generalize about
what they have found out. The tasks are dissimilar in the patterns they
represent—linear, quadratic, and exponential—and the context in which each

of them are set. A copy of each task is included in Appendix D.

Borders
First I asked the students to complete Borders. They were asked to write
generalizations for both the number of blue tiles in the center and the white

tiles on the outside border for any figure n in the pattern given in Figure 13.

67

 

 

 

 

 

 

 

 

 

 

 

 

 

ilk

Figure 1 Figure 2 Figure 3

 

 

 

 

Figure 13: Borders Graphic

I thought that the two patterns—one quadratic and one linear—would
be straight forward for the students to investigate. The number of blue center
tiles increased in the basic quadratic pattern of n2, while the number of white
border tiles grew in the linear pattern of 4n + 4.

I perceived this as a good initial task since I felt that all of the students
could have some point of entry into the problem and therefore some success
with it. I thought that most students would complete the task and find a
symbolic generalization since the patterns were fairly basic and it was
similar to the types of problems they saw in the curriculum—collect and

organize data, and then write symbolic generalizations.

Cutting

After the students completed Borders, I asked them to explore Cutting.
Like Borders, Cutting also asked students to search for two generalizations,
but these patterns were not as familiar to the students. The first pattern

followed an exponential growth pattern, 2", while the second represented
exponential decay, §2%. Although, I administered the task with the expectation

that most students would not find symbolic generalizations for these

68

patterns, I still considered this a worthwhile problem for the students to
explore for three main reasons. First, I was interested to find if the students
used any language that would allow them to verbally describe a
generalization with a recursive pattern. For example, they might have stated
that the number of sheets was always double the number of sheets for the out
before. Second, I wondered what strategies they would rely on to solve
unfamiliar problems—would they call upon a comfortable process or search
for something new? Third, I was curious to see how students from a
Standards-based curriculum would struggle with a generalization for a rule
that was unfamiliar to them—for what length of time would they pursue the
problem, at what point they became fi'ustrated with the problem, how they
handled their frustration, and how they coped with an unfamiliar problem.

Since this exponential pattern has a base of two, I felt students could
conduct a meaningful exploration of the problem prior to writing a symbolic
rule. I predicted that they could work with the doubling in the problem prior
to their formal introduction of exponential symbolism. I discovered after the
pairs of students finished the task that some did recall doing similar types of
problems in earlier CMP units or their seventh grade science class. I found it
interesting that some students were able to make this connection.

I felt that the generalizations for the rules in this task would prove to be
more challenging for these students. Since, these students had yet to

complete the CMP unit on exponentials, Growing, Growing, and Growing..., I

69

saw this task as more challenging. They were scheduled to begin this unit

during their final weeks of school, shortly after the completion of this study.

Dominoes

The third problem students completed was Dominoes. The final question
asked students to find the total number of domino faces possible with from
zero to n dots.

I predicted that the students might observe and extend the pattern in

the data for a specific case, but would probably have difficulty writing a rule
for the nth case, g(n + 2)(n + 1).

I saw this as a problem that would challenge all of the students, but for
different reasons than with Cutting. Unlike the exponential unit, Growing,
Growing, and Growing..., all of the students in my study had completed the
CMP unit on quadratics, Frogs, Fleas, and Painted Cubes. The students had
some experiences in their mathematics class classifying patterns as
quadratics, but they did not have a systematic way to write generalizations
based on data for a quadratic relationship. This was not a basic quadratic

pattern that I felt the students would find easily.

Toothpicks
Next, the students explored Toothpicks which involved two number

patterns based on stair-step shape figures in Figure 14 below.

70

 

a ._: 1F

Fig. 1 Fig. 2 Fig.8 Fig.4

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 14: Toothpicks Figures

Students explored both the number of toothpicks in the perimeter, and the
total number of toothpicks in the figures. This task possesses a similar
structure to Borders; one pattern is linear and one pattern is quadratic in
both tasks.

The pattern of the perimeters of the Toothpick shapes is linear, 4n, while
the total number of toothpicks is quadratic, n(n + 3). I saw the pattern for the
quadratic total number of toothpicks, as a little more difficult for the students

than the quadratic pattern in Borders, n2, but not as challenging as the rule
. . 1 .
1n Dominoes, ’(n + 2)(n + 1). I was interested to see what tools the students

would use to write a rule for this pattern.
I felt that most students could find these generalizations, unlike
Dominoes. The linear perimeter pattern was more basic and the students

spent time in the curriculum studying factored.

Task Solutions
One of the strengths of the four tasks the students completed was the
potential for multiple solution strategies. This opportunity for a variety of

approaches is one aspect that makes a worthwhile mathematical task

71

according to the N CTM Professional Teaching Standards (1991). One possible
solution is explored in Appendix E for each of the tasks. This section is not
intended to be representative of the students’ work, rather ofier one possible
solution to each task for the reader.

In my solutions I make assumptions that these patterns do continue
infinitely in either the linear, quadratic, or exponential pattern I represented.
I realize that given any finite amount of data these patterns could reasonably
be extended in an infinite number of ways. The students also seem to make
this assumption, but it is important to note that these regular extensions of

the patterns are not the only possible completions.

Data Analysis

In my initial analysis I used only students’ written responses. I grouped
the responses by student pairs. I first studied all tasks completed by one pair
of students and looked for whether they used common strategies across
several tasks. I repeated this process with the other four pairs focusing on
each pair and searching for similarities across their solutions. After I
collected notes regarding what the students did by pairs, I looked across the
five pairs, looking for common approaches. I formulated tentative hunches of
what I saw in the students’ work. One of the strategies fairly common across
tasks and across pairs was the construction of a table and then the search for

a pattern in the table. I noted after studying the written work that:

72

. Tables seemed to be a starting point for students when they encountered
unfamiliar problem situations.

. Tables seemed to serve as a means to systematically generate and
organize data.

. Students were able to answer questions about specific cases.

. Students used tables to study the data in a search for patterns by
looking at constant differences.

Students’ written work was not suficient to inform me about the
students’ understanding. It did two things for me. First, the written
responses offered confirming evidence for some of my hunches, and second,
this work raised some questions for me about students’ understanding. I still
had questions about how the students used their tables, and I was puzzled in
the instances where students did not construct tables.

I looked at other sources of data to support my assertions. In the
videotapes while students worked I saw further evidence that confirmed that
constructing tables was a reliable strategy for these students. When several
pairs independently voiced their uncertainty, one of the pair usually
suggested making a table. With this lead, the all made progress with the
tasks. During the interview the students elaborated further that the tables

gave them a lot of information and usually worked in their problems. This

represents an example of how I found confirming evidence and triangulated

the data (Bogdan and Biklen, 1982).

73

Summary

This chapter represents a narration of the choices that shaped this study
to investigate the question of what eighth grade students know after three
years in the Connected Mathematics Project about writing symbolic
generalizations for patterns which can be represented with functions.

The next two chapters offer data and some analysis for the reader to see
how the question was addressed. First, in chapter four, I thoroughly describe
what each pair of students have done in their investigation for each of the
tasks. All three sources of data—written responses, recordings while working
and recordings of the interviews—are used to support this. Next, in chapter
five, I step back and look across students and across tasks to describe two
common strategies that were used in most of the solutions, making tables

and studying the shapes.

74

CHAPTER 4

Digging Deep—

A Careful Look at Student Investigations

Introduction

In this chapter, I describe students’ solutions in four performance tasks.
I rely on three different sources of data—students’ written responses,
recordings while students worked, and interview recordings after students
completed each task—to help me understand what the students have done. I
describe each pairs’ investigation separately in each task.

Most students classified the four tasks I administered to them as
primarily algebra because they searched for symbolic generalizations. A few
classified some of the tasks as mostly geometric because of the shapes that
were involved.

The students’ interpretations of the problems, algebraic or geometric,
influenced their solution strategy. When the students considered a problem
algebraic, they constructed a table of numeric data to study the pattern. In
some of the less complex, linear cases, the students recognized a pattern in
the numeric data prior to organizing it in a table. When they saw a problem

as geometric, they studied the changing shapes to describe the pattern.

75

The dominant strategy used across students and tasks was to construct
a table of data. Students who made tables followed some or all of these steps.
They used tables to (a) record or re-present data, (b) extend given data, (c)
find specific cases, ((1) study patterns, (9) write rules, and (f) verify rules.

Students followed an alternative strategy when they saw tasks as
geometric. They did not make a numeric table of data, instead they focused
on the changing shapes and found generalizations based on how the figures
grew. In a small number of cases, students made pictorial representations to
study the patterns. These students often found generalizations based on
sketches.

Regardless of the strategy used in the investigation, the students
demonstrated understanding about the patterns in all cases, even when they
did not write a symbolic generalization. The students demonstrated this at
times by describing the patterns verbally. In other cases some made
connections between the patterns of numeric data in a table to the shapes of

the graphical representations.
Zachary and Todd
Students often made and studied tables to help them solve situations

involving patterns. Zachary and Todd constructed tables for all four tasks.

The pair successfully found rules for all patterns.

76

Borders

These students described Borders as an algebra problem since it dealt
primarily with “equations and variables.” This interpretation led them to
solve the problem by creating and studying tables as discussed during our
interview.

Todd: We could have just looked at pictures. First, I did and we saw
that it got taller and wider, but it was kind of hard to make
an equation based just on that.

Zachary: We just tried to find patterns in the table to see how much it
changes by.

Todd: Yeah, it was easier seeing the numbers.

Using a Table

Zachary and Todd began working on Borders by counting the tiles in the
drawings and making marks on the figures as they counted. Then, they made
tables to record their data. Next, they generated new data by observing the
pattern of change in the numbers they had recorded in their table. In the case
of the blue tiles, they extended the table to Figure 10; in the case of the white
tiles they extended the table to Figure 8. After they had extended their tables
for both blue and white tiles, they used the data in their table to read the

number of tiles needed for the specific case of the fourth figure.

77

 

 

     

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Figure 16: Todd's work on Borders

The pair studied the patterns in the data table to extend their table and
to determine if a relationship was linear or non-linear. They quickly noted
the constant increase in the number of white tiles while working on the task.

Todd: After the first one is eight. Wait, we can start at zero, the

y-intercept is four.

Zachary: It increases by four each time.

Todd: So, for each blue1 square, it’s four times...

Once they recognized this change of four they attempted to write a
generalization.

The pattern for the blue tiles did not result in a linear relationship, so
the pair examined the data to see if and when they would get a constant
difference. Zachary noticed, “The increase increases by two each time, so
three, five, seven, nine. One times one is one, two times two is four, three
times three is nine.” In an interview I asked them about this.

AK: When did you stop looking for the pattern?

Todd: When the differences were the same.

AK: When was that?

Zachary: The increase increases by two each time.

Todd: I stopped about here [points to the table around four].

Using this they found the rule of x2 for the number of blue tiles.

 

1 Todd says “blue” but he refers to the pattern in the white tiles.

80

Zachary and Todd found the linear rule for the white tiles by using the
constant rate of change they calculated from the table and extending the
pattern back to find the number of white tiles for figure 0.

Todd: It would be what?

Zachary: Four X plus four?

Todd: That [four] would be the intercept, even though we don’t have

zero [in our table].
In an interview they explained further.

AK: What did you get for your equations?

Todd: Four X plus four.

AK: How is this four [the X-coefficient], how is how it changes

shown in your work?

Todd: Right here [pointing to the increase in the table].

AK: What does that last four show?

Todd: That’s the y-intercept and it means...

Zachary: ...where it starts.

Todd: Yeah. This figure over here [pointing to the left of figure I]

wouldn’t have any blue squares only four white squares.

Zachary: For the four corners.

Todd: That would be figure zero.

81'

In this complete explanation they elaborated on what each four in their rule
of 4x + 4 represented, describing how it related back to the geometric context

in which the problem was posed.

Cutting

Although Zachary and Todd recalled completing a problem similar to
this in their seventh grade science class, they did not remember the process
they used to solve the problem. They were initially unsure how to start this
problem, but decided to make a table again, influenced by the demand of the

task to find an algebraic rule.

Using a Table

The pair began by immediately writing the number of sheets in each
stack above the figures on their paper and completed the pattern up to 64
sheets in a stack after six cuts (see Zachary’s calculations above and to the
right of the four figures). They soon recognized that a table would be useful
as evidenced in their conversation while they worked.

Zachary: So, one, two, four...

Todd: Squared?

Zachary: ...times two, so eight.

Todd: Then, sixteen, then thirty-two? [pause] I don’t know.

Zachary: Do you want to try to make a table?

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Figure 18: Todd’s work on Cutting

Each student proceeded to flip his paper over and recorded the data in a
table for one to 10 cuts. Before they generalized their pattern, they used their
calculators to continue doubling the pattern and found 1,048,576 sheets in a
stack for the specific case of twenty cuts.

This exponential pattern was puzzling to the pair. They readily found
the recursive doubling pattern to find the number of sheets after each
successive cut by studying the pattern, but they had difficulty writing a co-
variational rule for the pattern. Part of the difficulty was that this situation
did not produce a constant difference. They continued to search out to the
sixth difference. In an interview I asked them about what they had tried.

AK: How did you fill out your table?

Zachary: Times two, times two, times two.

Todd: It was easy filling out the table.

AK: Why did it take so much longer to answer the question after
you filled in the table?

Todd: Not a real obvious problem. I tried finding the first difference,
second difference, and third difference and there really wasn’t
any other pattern besides it doubling.

AK: Why did you find the difi‘erences?

Zachary: To try to find the equation, that works for the one in the last

problem [Borders].

85

They knew that constant differences helped them write algebraic rules, but
this problem did not lend itself to this strategy.

However, they recalled doing a similar problem in their science class the

year before. They explained during our interview.

Todd: We were trying to find the time for bacteria to touch the
moon. How many times it duplicated. Looking at the
equation, it was almost the same.

AK: Did that help you to solve this problem?

Todd: I wouldn’t have known how to use the y-x key.2

Even with this recognition they struggled for some time before they finally
came up with the rule. They followed a “guess and adjust” strategy recording
(C — 1), 02, and (C — 1)2 x (C — 1). After about twenty minutes, Zachary
recognized something that led to the rule.

Zachary: It’s whatever number that is, the number of two times. I’m
not sure though how to write that in an equation.

Todd: It’s that right there.

Zachary: Two X [and writes 21‘].

Once the students had written their rule of y = 23‘, they tested it by trying a
specific case. In our interview, they “proved” that their rule was appropriate

by demonstrating how it worked for the values in their table.

 

2 They had everyday access to the 77- 30 Challenger in both their seventh and eighth grade
mathematics classrooms. These calculators used a y" to represent the exponential function or
(Footnote continued on next page.)

86

This problem proved to be challenging for Zachary and Todd. What was
interesting was they way in which they persevered in their investigation.
Even when I told them they could stop and we could discuss the rule
together, the pair seemed to believe that they should be able to find an
equation given any problem. They continued their investigation until they

had completed the problem and found a symbolic generalization.

Dominoes

Zachary and Todd made tables again in Dominoes. But, before they did,
they sketched representations of the dominoes to generate their data. First,
they drew the dominoes with zero to two white spots. Todd drew nine
possibilities (0-0, 0-1, 0-2, 1-0, 1-1, 1-2, 2-0, 2-1, and 2-2), decided that the 0-1
and 1-0 dominoes were the same, and crossed off the doubles. Rather than re-
draw the dominoes with up to two spots again in the set of dominoes with up
to three spots, Zachary drew the four additional dominoes to arrive at a total
of ten dominoes with up to three white spots. Zachary then started to sketch
the dominoes with up to six spots, but quickly abandoned that strategy to try

an alternative approach.

 

“power key”. Students also had access to the TI-82 graphing calculators in their classroom. I
supplied both calculators for the students to use as needed while they completed the tasks.

87

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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89

Using a Table

With the preliminary sketches made, they recorded and organized the
data into a table to complete the task. They studied the pattern of the data in
the table and then extended their table up to the number of dominoes
possible with one to 10 white spots.

Zachary first recognized that once he had the dominoes with from zero to
two white spots, he could find the ones with up to three spots by adding four
additional dominoes (0-3, 1-3, 2-3, and 3-3). He saw that the pattern would
continue; to find the dominoes with zero to four spots he needed to add five
more dominoes. They repeated this pattern to find the specific case that there
were 28 dominoes with from zero to six white spots.

This iterative pattern did not help them write the symbolic rule. It was
when they organized the data into a table and studied constant difl'erences
that they found something in the pattern that could help with the rule. This
was noticed while they completed the task.

Zachary: There is kind of a pattern, like that changes, three, then four.
Remember? So that would be fifteen, the next would be
twenty-one. Is that what we got?

They used the 28 dominoes they found earlier for zero to six to confirm the
pattern of difi‘erences.

Zachary and Todd collected a lot of information about the pattern prior

to writing their symbolic rule. During the interview they explained that since

90

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the first difl'erence increased by a constant amount of one, they knew the

pattern was quadratic.

AK:

Todd:

Todd:

Todd:

Todd:

How many of these did you have to draw before you could
actually continue the table?

Probably the first three, after it went up by three, then four,
then five.

Then that was enough for you to say [what]?

The second difi'erence was one.

What does that tell you, that the second difference is one?
That, uhh, it’s a quadratic.

Okay, what does that mean?

Umm, the graph makes a parabola and X is multiplied by

itself or a quantity of X is multiplied by itself.

They found one as the constant term of the rule by extending their table

back to find that there was one domino possible with up to zero spots. They

seemed to try to fit their understanding of constant terms in linear situations

to constant terms in quadratic situations. During their investigation they

tried potential rules that were quadratic and had a constant term of one.

Zachary: The starting point would be one, so zero zero would be one.

Todd:

So, its got to be something N plus one. N squared divided by

two plus one? N squared divided by three plus one?

Zachary: Maybe minus one? N [pause] N minus two times N minus?

Todd:

You have to add one.

91

These trials did not result in an appropriate symbolic rule.

The pair considered that they might need to divide by two because of the
way they counted the dominoes and crossed off the doubles. They explained
during our interview.

AK: Okay, what was your strategy for this one?

Todd: First we tried to make a chart to find all possible ways...

AK: Okay, and then what did you do?

Todd: Took out doubles.

AK: How did that help you with the problem?

Todd: Kind of found out that in the equation, that we will probably

needed to divide our 11 by two and then add one.

They used all of this information to make some trial guesses for a rule.
Zachary started to consider the consecutive integers when he noticed for the
10 dominoes with zero to three white spots, or “1 + 2 + 3 + 4.” After some
calculations, he arrived at (n + 2) + 2 x n, but realized his answers were off,
so he adjusted that expression and wrote his final rule.

Todd: Did you get it?

Zachary: Yea, I think I might, [11 plus two] divided by two times N.

That does not work.
Todd: That work?
Zachary: It doesn’t work for that, but it gets the answer before it, so

[pause] you have to add N plus one. [pause] Let’s try that out.

92

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ruleof(n+1)-:—2xn+n+1.

Toothpicks
Zachary and Todd also saw this as an algebraic task. They continued
with their numeric strategy of making and studying tables to address the

questions in this task.

Using a Table

The pair began the problem by counting the toothpicks and recording
these numbers above the shapes. They extended the pattern by drawing a
sketch of figure five. They recorded the perimeter toothpicks they counted in
the table. The pair used pattern of the increase of four they studied in the
table to complete it and find 24 toothpicks for the perimeter of the specific
case for figure 6.3

Even though their strategy of looking at differences was not apparent in
their written work for Toothpicks they demonstrated that they recognized

when the first difference was constant by referring to the increase when

 

3 Question 2 asked about the total number of toothpicks of figure six, while Zachary and Todd
recorded 24, the number of toothpicks in the perimeter of figure 6. I believe that this pair
would have likely been able to find the total number for figure 6 based on the table they
made and the pattern they noted for the total number of toothpicks.

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Figure 22: Todd's work on Toothpicks

95

counting the perimeter. Todd noted, “It goes up by four,” while studying the
numbers he recorded above the figures.

To find the total number of todthpicks, a quadratic pattern, the pair
repeated their process of counting the toothpicks, recording the data,
observing the increase, and extending the table. They stopped searching for
patterns early in their solutions and continued the pattern they saw. Todd
explained this during our interview.

AK: When do you know to stop looking for a pattern and start to

generalize [to find the total number of toothpicks]?

Todd: We filled out five and six [in the table].

Alter Zachary and Todd determined that the perimeter of the toothpicks
was a linear pattern, they extended their tables back to zero. In their tables
they found corresponding values for Figure 0, I was curious how they thought
about this, so I asked them during the interview.

AK: How did you find zero?

Zachary: If it goes up by four, then it goes down by four.

AK: Does that make sense with the figures?

Zachary: Yes.

Although it was not clear how this made sense based on the pattern or
figures, in the pattern of the data (0,0) was reasonable. They used the
y-intercept of zero and the increase of four to write a rule of 4n for the total

number of toothpicks.

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to Zachary. He studied the table and wrote a rule that seemed to work.
Zachary explained to Todd what he saw.

Zachary: So, N times N plus three.

Todd: Yeah. So how did you get the answer?

Zachary: I was looking to see what it was. It is kind of a coincidence
that I found it. Like five times two equals ten, so then five is
three more than two. Then I did the next one, six times three.

Todd: So you looked at the pattern in the table?

Zachary: Yeah.

Although they do not directly state it, Zachary began his trials with a
rule in the factored form of “n times itself or a quantity of itself.” They knew
that the pattern was quadratic since they found a constant second difference.
This helped them to make their initial guess that worked for the pattern of

data.

Ben and Joe

During each interview I asked Ben and Joe what they felt was the

mathematics in the problem. The pair saw the four tasks as algebra, or a

blend of both algebra and geometry. They claimed that all of the tasks

involved algebra because they were asked to find rules fi'om patterns in the

97

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tables. They added that two of the tasks also included geometry since they
involved changing shapes. The strategy they used to study the patterns—
numeric or geometric—seemed to determine their view of the mathematics in
the problem.

The pair constructed tables in three out of the four tasks. In the problem
where they did not make a table, they used a geometric approach to
investigate the problem. In all situations, Ben and Joe successfully wrote

symbolic rules.

Borders

In Borders, Ben described the task during our interview as “algebra a
lot, maybe a little geometry. They made tables to investigate the problem,
but verified their solution using a geometric approach.
Using a Table

Ben and Joe first counted the white border tiles and then recorded their
data into a table. They constructed tables with data for figures 0 through 10.

The pattern they observed helped them to extend the table to find the specific

cases.
AK: When do you know to stop looking for a pattern and start to
generalize?
Ben: For one b, [we] just saw a pattern once we got to four.

From the table, they formalized the pattern.

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Figure 23: Ben and Joe’s work on Borders

99

AK: 'What did you use the table for?

Joe: To find the pattern. Pattern made it easier to find the rule.

AK: Did you find the table before the rules?

Ben: For one a, we found the equation first. For one b we found

table first. L times four plus four.4
Joe described the constant increase of the first difi'erence in his response to 1
a, when he wrote, “Figure 1 to Figure 2 is +3, +5, +7, +9...” (See Joe’s work
under 1 a).

In this problem the pair wrote the quadratic rule, L2, for the blue
interior tiles quickly. The linear white tile pattern also proved
straightforward to generalize for the pair of students. They wrote L x 4 + 4.
The increase of 4 in the data was represented as the coefficient of L. They
extended the tables back to find that 4 corresponded to a value of O for n and

used this 4 as the constant term.

Geometric Approach
Although this pair used a numeric approach to investigate this problem,
when asked to justify their results, they switched to a geometric approach.
AK: How would you justify your solution?

Joe: Sides are always the same length so it is squared.

 

4 l. a) If the pattern of blue squares with white borders continues, how many blue tiles are
needed to build the 4th square? the nth square? Show how you figured this out.
(Footnote continued on next page.)

100

AK: In one b, what is L?

Ben: It looks like the length of blue tiles was L. That way there are
four on each side of the blue tiles, so that is L times four plus
the four corners.

They pointed to figure 3 and traced lines to illustrate their thinking as

illustrated in Figure 24.

 

L
Figure 24: Ben’s Borders diagram

Cutting

This was an unfamiliar problem for Ben and Joe since, “we had not
worked on anything with an X power before.” When I conducted my study the
class had not yet completed the CMP unit on exponential growth, but during
our interview these two students recalled a similar problem from the
previous year.

AK: Have you ever done this problem before?

Joe: In science class last year when we stacked things up.

Ben: In biology class we were multiplying things.

 

b) If the pattern of blue squares with white borders continues, how many white tiles
are needed to build the 4th square? the nth square? Show how you figured this out.

101

 

 

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103

Joe: We talked about cells doubling to reach the moon.
They referenced a problem from their seventh grade science class that had a
similar mathematical structure of exponential growth.

They saw Cutting as “(a)lgebra in writing a rule and table. Just algebra.
It doesn’t seem to have geometry.” Making and studying tables was a
strategy that they eventually used.

AK: What strategy did you use for this problem?

Ben: Guess and check

Joe: Made a table.

Ben: Yeah, and from the table we got the equation.

Using a Table
Initially Ben and Joe recorded their data in an incomplete table and

made the table shown in Figure 27

 

2. a) How many sheets of paper thick would the paper pile be after 4
cuts? 5 cuts? 10 cuts? 20 cuts? n cuts? Explain how you figured this

out. IO 20
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They soon recognized their error and corrected it on a fresh sheet of paper.
Ben: It wouldn’t be sixty-four, for ten. Do you have an eraser or a
sheet of paper [to me]? It doubles from five to six, not five to
ten. It wouldn’t be sixty-four it would be something else.

Joe: Yeah, so double it.

104

 

They created new tables and recorded data that ranged fiom 1 to 10 cuts,
including all numbers in between. They found that the number of sheets in a
stack after 10 cuts was 1,024 and not the 64 they originally wrote.

The pair completed the tables based on the doubling pattern they
observed in the data. They continued the pattern to correctly find the specific
case of 1,048,576 sheets in the stack after 20 cuts. Next, they searched for a
rule by first examining the difl'erences. They tried several iterations of
differences but did not find a constant.

The symbolic rule eluded Ben and Joe for some time. They found the
doubling factor for the growth problem and the halving factor in the decay
problem but had difficulty translating this into a symbolic rule.

Ben: I don’t see where the two is. For the first, second cuts, its
double, the number of cuts is two sheets and four is double
two, for three you get eight sheets.

Once Ben and Joe described the pattern in an iterative form, they searched
for a closed form of the rule. Joe began to guess.

Joe: Okay, X times Y times two?

Ben: Times? No, you can’t have X and Y in the same. You are
trying to find Y. This right here is Y [points to sheets thick].
You don’t know these, you are trying to get these [sheets
thick] from X [cuts].

Ben seemed to have a clear understanding of how the equation involved X

(the number of cuts) and Y (the number of sheets thick) as independent and

105

dependent variables. They eventually wrote N = 2". Ben credits, “I was just

thinking and it popped into my head.” It remains a puzzle how they found
their rule. The pair did not articulate any further how they found it, nor is

there evidence in any of the videotape to suggest anything else.

Dominoes
Ben and Joe saw Dominoes as an algebraic task since they were asked to
find a rule. Making and studying a table was the strategy they relied on for

algebraic tasks.

Using a Table

Their investigation began with the pair sketching dominoes. They
counted the number sketched and recorded the data in a table. They followed
the familiar strategy of recognizing a pattern, generating new data,
answering specific cases, and classifying the pattern.

They recognized the pattern of increase in the first difl'erence and used
that to help them complete the table.

AK: When did you know when to stop looking for a pattern?

Ben: We filled in the table from one, two, and three and then

looked for constant difi'erences.

AK: For zero to six spots? How did you find that one?

106

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108

Figure 29: Joe's work on Dominoes

Ben: We did the table, we saw it went up by two, three, four, five.
So we put one there [extended the first difference back] and
for 0 [dominoes with up to zero spots] it would be one then for
the third one we put ten, then added five, for fifteen. Add one
every time. So I found the second difference of plus one.

After the pair recognized the pattern, they then used that to find the 28
dominoes with from zero to six spots.

During the interview they explained how the constant difference helped
them write the equation.

AK: Okay, let me ask you something different now. When you
found that the second difference was one, what did that tell
you about the equation?

Ben: I knew the n had to be multiplied by something.

They started to describe a factored form of a quadratic.

As an initial guess, they seemed to connect the numbers in the table to
the sum of consecutive integers. They partitioned the 10 dominoes they drew
for zero to three white spots into sections of four dominoes, three dominoes,
two dominoes, and one domino in their written work (See Ben’s work beneath
1 b). These 10 dominoes could be represented by the sum of consecutive
integers, 4 + 3 + 2 + 1. They continued and represented the number of
dominoes with from zero to four white spots as the sum of 5 consecutive

integers (5 + 4 + 3 + 2 + 1) on the back of Ben’s paper. After they wrote this

109

sum, it prompted them to use N+1 x N + (N-l) as their initial guess. Ben
adjusted this guess and tried N+2, N+3, N+4, as different variations.

It was unclear what the pair tried for their rules after these guesses, but
they eventually wrote y = (N+ l)(0.5N+1). They verified their rule with the
zero to three, zero to four, and zero to six cases by finding the 10, 15, and 28

dominoes respectively. (See the back of Ben and Joe’s written responses.)

Toothpicks
Ben and Joe used a different approach in Toothpicks. They did not
construct any tables as they did for the algebra tasks. Instead, they classified

it as geometric and studied the pattern of the changing shapes.

Geometric Approach
The pair used the geometric patterns of change in the figures to help

write rules. They explained their approach in our interview.

AK: Do you want to describe your strategy you used for this
problem?

Ben: For one, we...

Joe: ...we drew pictures.

Ben: We added on to the first one, we drew extra steps and we did

that for number two also.

Joe: We added on to the pictures.

110

 

 

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111

AK: What did you mean you added on?

Ben: We drew the extra steps
Ben and Joe drew additional toothpicks on the steps of figure four to
represent the next two figures in the pattern (See Ben’s Figure 4 in his work).

Ben and Joe’s original rules of Y: (N-N) + (Nx2) for the perimeter and

Y: (NN) + (Nx2) + (N- 1) - N for the total number of toothpicks were not
correct. They recognized their error during the interview, revised their rules
and wrote correct generalizations.

Although their initial rules were incorrect, the reasoning they ofi‘ered

during our interview to find them was reasonable.

Ben: Umm, for four, we knew that each one had n times n to get
the two straight sides [the base and right side], then Joe came
up with this [the step portion on the left side] being two n.
Because you’ve got n here [vertical steps] and n here
[horizontal steps], so that you get two 11.

AK: Do you want...?

Joe: Yeah, 11 times n would be this times this part of the perimeter
[points to the base and right side]. Then n times two [the
steps portion] plus this perimeter is two, and right here is two
times two would be four. And it works over here too [See Joe’s

marks on Figure 3 in his work].

112

The pair had mistakenly used n x n (rather than n + n) to represent the
number of toothpicks in the base and on the right hand side. After they
recognized their mistake they correctly adjusted their rule.
Ben: We counted this one. Oh, for the n times 11, it was this times
this. For two n and there’s an n and two 11, three 11 there and
four n. I think it would actually be four 11.

AK: Why’s that?

Ben: Because you’ve got one 11 here, two 11, three n there, and four 11
there. It works the same here for one, two, three, so it would
be Y is four 11.

The pair found the total number of toothpicks by considering two
separate categories: the perimeter toothpicks and the interior toothpicks.
They revised and wrote Y = 4N + (N-1) -N for the total number of toothpicks.

It was unclear how they verified that (N -1) -N would always result in the
interior toothpicks. Ben demonstrated on his paper how he could show 2
groups of 1 for figure 1, 3 groups of 2 for figure 2, and 4 groups of 3 for
Figure 3 (See Ben’s work for Figures 1-3). Even though this will be
numerically true, it was unclear how to continue this pattern and mark 5
groups of 4 in Figure 4, or the general case of n+1 groups of n in Figure 11, nor
was it evident how this pair saw this aspect of their generalization

continuing.

113

Julie and Dan

This pair of students used two different approaches in their
investigation of the four tasks. Their strategy for each problem focused on
their interpretation of the mathematical content. Although they thought that
all of the tasks involved some algebra, they saw Borders and Toothpicks as
primarily geometry.

When Julie and Dan felt the main mathematics of a problem was
algebra, they constructed tables of numeric data to initiate their
investigation. For the tasks they interpreted as geometric they used a
strategy based on studying the shapes.

The pair found correct rules in Borders and Toothpicks. They found a
rule in Dominoes for Case 1 where they counted 1-0 and 0-1 as two distinct
dominoes, but were unsuccessful in Case 2 when they counted them as one
unique domino. They were unable to write a rule for the exponential patterns

in Cutting.

Borders

The pair seemed to find their rules in Borders by observing geometric
characteristics of the figures in the task. Their solution primarily focused on
geometry, but Julie also used the numeric values to search for a pattern in

part of the problem.

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Figure 31: Julie and Dan’s work on Borders

115

Geometric Approach

Julie and Dan found their rule of n2 for the blue center tiles by studying
the shape of the figures. Dan supported his rule and wrote, “For the Nth
square the answer is N2 because one side squared equals the area of the
square.” They recognized that the blue tiles formed a square with dimensions
of n, where n was the figure number. The pair described their strategy during
our interview.

AK: How did you decide what strategy to use?

Julie: Kind of looked at a pattern for these and that [points to
Figure 2, Figure 1, and Figure 3], when two by two, one by
one, three by three, so we figured four by four and got sixteen.

They observed the pattern in the shapes, extended it to answer questions
about specific cases, and then finally wrote the generalization.

Julie and Dan continued with this strategy to find the generalization of
the white border tiles. Dan’s written work again offered evidence to support
their rule. “If you take one side and multiplied it by four, youd [sic] have
everything accept [sic] the corners, that’s why I added four.” They further
explained their strategy during an interview.

AK: How would you justify your solution for the white tiles in the

equation?

Julie: Yeah, four N plus four?

AK: Mm, hmm. Could you prove it to me?

116

Julie: See, this [traces the side of the blue square] would be N, four
of these, then four corners.

Dan: You always have four corners left over.

Numeric Approach

Julie searched for a numeric pattern in at least one instance while
solving Borders, as illustrated in her written work where she counted the
number of white tiles and wrote the numbers, 8, 12, and 16, above the
figures. While she worked on the task she noticed an increase of four in her
data, “No, there’s [pause] you need to find out how many borders. Each
increases by four.” She further noted the increase in her written work where
she supported her rule of 4n + 4 by claiming, “So for n square, it would be
four n plus four. We found that the amount of tiles increases by four each
amount.” In her justification, she seemed to consider the numeric increase of

four in her statement rather than the geometric change.

Cutting
Even though Julie and Dan did not find symbolic rules, they
demonstrated understanding of the pattern and correctly responded to all of

the specific cases.

117

 

 

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Using a Table

The pair made and studied tables to investigate the task.

AK: How did you decide what strategy to use?

Dan: We tried to figure out simple ones, the first few. We tried to
make a table on back. We tried guess and check.

Julie: We found a pattern for them, but we couldn’t find the rule.

They used repeated multiplication on their calculators and recorded
data in their table. As evidenced on Julie’s paper, they correctly found the
specific cases for the number of sheets of paper in a stack, the height of the
stack in inches, and the area of the pieces of paper. As Julie passed her
calculator over to Dan to see the display she said, “That’s how many after ten
cuts. Would you believe that?” She recorded 1,024 on her paper fiom the
calculator display. Using the procedures they devised, Julie and Dan
correctly answered all of the questions related to specific cases.

It was apparent by the marks on the tables that both students searched
for constant differences while investigating the task in an attempt to describe
the pattern.

Julie: What’s it [the differences] going up by? Each time it increases.
Maybe [pause] I have no idea how to do this. Plus two, plus
sixteen, plus thirty-two. Then plus two, plus four.

Dan: That’s what I mean.

Julie: It just keeps going out.

118

 

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120

They found up to the fifth difference, but it was not constant. Although they

recognized that the column of difi'erences kept repeating (2, 4, 8, 16, 32) this

did not seem to help them write the rule. Dan hypothesized that this pattern

would never produce a constant difference. Julie was not convinced of Dan’s

generalization.

Julie:

Dan:

I suppose if you make it really long, it will.

No, it won’t.

During the interview the pair clarified their purpose of searching for constant

differences.

Julie:
AK:
Julie:
AK:

Julie:

We tried to find if it was a quadratic.

What did you find?

It [the differences] kept going and increasing.
Will it ever stop?

No.

Although they did not find a constant difference, Julie felt they did find a

pattern in the data.

Julie:

AK:

Julie:

We found a pattern for them [the differences], but we couldn’t
find the rule.

Okay, what was the pattern you found?

It went up by, it like increased by itself, like okay. Like it you
had four, four cuts, sixteen. Five cuts would increase by

sixteen.

121

But, this description of a pattern of increase did not help the pair write the
symbolic rules.
Julie and Dan searched for a rule that would give them the number of
sheets of paper in a stack fi'om the number of cuts.
Julie: Want to get the next one?
Dan: I’m not sure. There has to be a rule...times a half? Find out
what the heck six has to do with sixty-four?
Julie: That’s what six has to do with sixty-four. It doesn’t make any
sense.
After we discussed the rule y = 2" for the number of sheets of paper in a stack
after n cuts, I asked them to reconsider the other questions.
AK: Take a second to see if you can figure out a rule for three a.5
Julie: It would be thirty two divided by two to the n, I think.
Julie and Dan possessed a great deal of understanding about the task that
was not first apparent by analyzing only their written responses. On first
glance one might assume that Dan’s contribution to the solution was minimal
since his written work appears sparse. This was not the case; he continued to
puzzle over the rule for the number of sheets in a stack, while Julie continued
with other specific cases. He was determined to find the rule, even after I

suggested that they could continue with the rest of the problem.

 

5 3. Suppose the original piece of cut paper has an area of 32 cm2.
a) What is the area of each piece formed after 2 cuts? 3 cuts? 10 cuts? n cuts? Show how
you figured this out.

122

Dominoes

Julie and Dan saw this task as algebra because of the rules, variables,
and symbols they used to explore the problem. They held two different
interpretations of Dominoes. In both situations it was evident that they used
primarily numeric solutions to investigate the problems.

Their two interpretations differed in the way they counted the 1-0 and
the 0-1 dominoes.

Table 14: Two Cases of Dominoes

 

 

 

 

 

Case 1: Case 2:
Count the 0-1 and 1-0 as two Count the 0-1 and 1-0
difi’erent dominoes. Therefore dominoes as the same. Using
four dominoes possible with this counting scheme, only
fiom zero to one white spot three domino faces possible
(0-0, 0-1, 1-0, and 1-1). with from zero to one white
spot (0-0, 0-1, and 1-1).

 

The labels Julie used on her tables in her written work to distinguish
the two patterns were unclear. She seemed to reverse her labels when she
called Case 1 (where the 0-1 and the 1-0 dominoes were classified as two
distinct dominoes) as “same,” and Case 2 (where the 0-1 and 1-0 dominoes are
counted as one domino) “different.” The pair successfully found a correct

symbolic generalization for Case 1 but was unable to write a rule for Case 2.

Using a Table
Julie explained their data collection: “The tree diagrams helped us with

like making our tables.” Both students organized their data with tables for

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Figure 35: Dan's work on Dominoes

125

each case. In Julie’s tables there was evidence that she searched for constant
difl'erences. In the interview the students clarified how the constant second
differences informed them regarding the pattern.

AK: When the dominoes aren’t counted as the same,6 how did you
fill in that table?

Julie: Well that we found a pattern, every single time, the first one
it went up by four, and then five, six, seven, eight, nine so we
found the pattern in the table, but not anything else.

AK: Does that pattern tell you anything?

Julie: It looked to me like a quadratic

AK: What does that mean that it is a quadratic?

Dan: I think that Ms. Howard told us that if something goes up by
two degrees she called it that it was possibly a quadratic.

They seemed to recall from their mathematics class that a constant second
difi'erence meant that the pattern was quadratic; it was unclear what this
meant to Julie and Dan.

They successfully found a quadratic rule by studying the table for

Case 1, (n + 1)2 where n was the maximum number of white spots.
AK: How did you find N plus one squared [rule for Case 1]?
Julie: Well, we know that four plus one is five, obviously, and then

like five times five is twenty five. Five plus one is six, six

 

6 This is using Julie’s labels, but it refers to Case 2.

126

times six. I don’t know. We just kind of saw the pattern in the
table.
This situation seemed to be a straight forward rule for them to find.
Although Julie and Dan did not find a generalization for Case 2, they
possessed some additional knowledge about the relationship. First, they
knew that the equation was quadratic while they worked.
Julie: This is a quadratic right?
Dan: I don’t know.
Julie: It is because it takes two degrees to get a constant.
Dan: Oh, yeah, but what is that going to help.
Julie: It helps with the equation. Umm, I don’t know its not like one
of the real obvious ones.
This seemed to inform them about a possible form of the rule. Although it
was still not evident what quadratic meant to this pair, they recorded
n(n + 3) and n(n — 1) as reasonable quadratic guesses for the pattern. It was
not clear why they tried these two expressions. They might have started with
them because they believed quadratics could be written in the factored form
of n(n + x). Besides being quadratic, they also knew that the rule had a
constant term of one. They explained.
Dan: For, zero it is one so, we have to find like some way to get one

fiom zero.

127

Julie: So, then you would have to add one. So, it has to be something
plus one.
They combined these and tried n2 — 2n + 1, but continued to be unsuccessful
in their search.
Additionally, they referenced the shape of the graph. Dan noted during
the interview that the graph held the shape of a parabola and sketched one

on his paper (See the lower left corner of Dan’s work).

Toothpicks

The pair easily found a generalization based on the numeric data for the
perimeter pattern prior to making a table for the total number of toothpicks.
Julie and Dan systematically collected data, searched for patterns, and wrote
symbolic rules for both patterns in this task. Making tables played a partial
role in their solutions. They only used tables to investigate the total number

of toothpicks pattern.

Numeric Approach

In the perimeter pattern the pair counted toothpicks, recorded the data
on the shapes, and readily recognized the increase of four. While they
worked, Julie described how she used that pattern to find the perimeter of
Figure 5, “They go up by four each time, so the perimeter of five would be

twenty.” Once they had the increase of four, they readily wrote, y = 4n.

128

Using a Table

While working, they easily found the pattern in the perimeter number of

toothpicks.

The increase?

Julie:

It’s four.

Dan:

Four...N.

Julie:

N equals the base.

Dan:

I got Y equals four N.

Julie:

Okay.

Dan:

 

 

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[Figure 37: Dan's work on Toothpicks

130

They further described how they found the rule in the interview.

AK:

Julie:

Julie:

Julie:

The perimeters, four, eight, twelve, and sixteen, you found
that by counting?

Yeah.

What was the pattern?

It increased by four each time.

How does that help you write the rule?

Because I know one times four, two times four, I just figured

N times four gives perimeter.

Once they collected the data for the total number of toothpicks they

recorded it in a table, searched for a rule, and easily recognized that the

second difference was constant.

AK:

Dan:

Julie:

What about total number of toothpicks? You sketched and
counted?

[nods]

Julie?

I used my pattern. First, I made a table and found the

amount of increase [second difference] by two, then I could

find fifty-four.

To verify how Julie extended the pattern in the table, Dan sketched figure 6

on the back of his paper.

Once they had the table complete, the rule came quickly to Julie as she

explained during our interview.

131

Julie:

Julie:

AK:

Julie:

What about total number of toothpicks?

I don’t know, I just like looked at my table and it [the rule]
just popped into my head. I guess I don’t know. I looked at it
[the table] and I tried it [the rule] and it was right.

How did you know it was right?

I like checked with my table with different variables and it
worked with all of them. It just kind of popped in here. I just
kind of tried it.

Just sort of tried some for this one then?

No that was the first one I got.

Julie’s first trial in their guess and adjust strategy yielded an appropriate

rule n(n + 3). Julie and Dan verified their rule by checking it against any

ordered pair in the table. When Dan saw that it yielded the appropriate

value, it convinced him that their rule was reasonable.

Sara and Ryan

According to Sara and Ryan, all of the tasks were algebra because they

“looked for patterns” and “made tables”. They added that Toothpicks also

involved some geometry since it involved shapes. In three out of their four

solutions they studied numeric data, but in only two of these three tasks did

they use tables. Additionally, in some cases they created graphs to help

132

explore the patterns. The pair only found rules in two out of the four tasks,
but they demonstrated significant understanding of the patterns in those

instances where they were unable to generalize symbolically.

Borders

Sara and Ryan used a geometric approach to investigate Borders by
studying the shapes of the figures and they way they changed. The pair made
an interesting choice of variables in their generalizations. The question asked
them to generalize about the nth square, where n was the figure number, but

Sara and Ryan considered the length and width as their variables.

Geometric Approach
The pair used the drawings of the figures in the task to find the
generalization for the number of blue interior tiles.
AK: What strategy did you use for this problem?
Ryan: What do you mean?
AK: When you sat down and I gave you this paper, what did you
do?
Ryan: Umm, we started with the perimeters and know we needed to
get minus two fiom the center area. So we kind of looked at

[pause] there’s two extra on every side, so we took two away,

133

 

 

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Figure 38: Sara and Ryan’s work on Borders

134

we go length minus two. To get the area of anything length
times width.

AK: Did you do this by the pictures?

Ryan: Yeah.

They wrote the rule, (L — 2)-(W— 2), for the number of blue interior tiles using
L and W as the dimensions of the figure. Using similar notation, Sara and
Ryan wrote an accurate rule to find the number of white tiles,

4(L - 2) + 4. In this expression they recognized the regularity of the squares,
and used one variable L to represent the length of all four sides. During our
interview they demonstrated their rule.

AK: What about one b?7

Ryan: L minus two times four plus four.

AK: How did you get that?

Sara: Oh, like we took away these corners, length minus two, so we
took away these two corners times four, so you’d be like
multiplying these ones [white tile sides minus the corners],
then you add the four corners.

Ryan: Yeah. Take away these areas [the four corners], then times by

four, then the four corners.

 

7 Question one b refers to the white border tiles. It states:
“If the pattern of blue squares with white borders continues, how many white tiles are
needed to build the 4th square? the nth square? Show how you figured this out.”

135

Ryan traced the drawing for Figure 3 in the task as illustrated in Figure 39.

L—2

  

L—2

 

 

 

 

L—2
Figure 39: Ryan’s tracing in Borders

 

He covered the four corners; each side had (L — 2) tiles remaining. Since there
were four sides, they multiplied L — 2 by 4. To find the total number of white
tiles they added the four corner tiles back and wrote 4(L — 2) + 4 as their rule.
Their geometric approach of studying characteristics of the shapes helped
them write their expressions.

Much was left unclear in the students’ written work about their choice of
variables. In the first pattern for the blue tiles they did not state that L and
W represented the dimensions of the figures, although it can be inferred fi'om
their written work. Nor, did they consider the specific case of a square where
L = W. In the rule they wrote for the number of white tiles, they specifically
used squares with length, L, for all four sides. Their rules were reasonable
with the dimensions of L and W, but the pair never related these lengths
back to the figure number; the students never clarified that the side of the
squares, L, was two more than the figure number, n. They explained in an
interview that they chose dimensions because that seemed the important
characteristics of each shape to them.

Sara’s written work offers some evidence of her understanding of a

specific case. When she wrote, “4(L - 2) + 4, 6 - 2 x 4 + 4 = 20 white tiles” she

136

recorded that the length of figure 4 was six (See Sara’s work under 1 b). Sara

successfully used their rule to support finding twenty white tiles. It still

remains unclear if they could adapt this to find the length of figure n for the

general case.

Cutting

Ryan and Sara approached Cutting by constructing tables. They

explained in an interview.

AK:

Ryan:

AK:

Sara:

Ryan:

AK:
Sara:
AK:

Sara:

Okay, umm, how did you decide what strategy to use?
Looked for patterns.

How did you do that?

Made a table.

Yeah.

What was an advantage to using the table?
To see a pattern.
What do you mean by that?

Get how much each of the variables increased by.

They felt confident using this approach when they were in an unfamiliar

problem context.

137

 

 

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Figure 40: Sara and Ryan’s work on Cutting

138

Using a Table

The pair seemed to understand the doubling nature in finding the
number of sheets in a stack after n cuts. They found answers for most of the

specific cases requested in the problem by doubling the previous number of

sheets.

Sara: Maybe it goes up by eight and then sixteen. Yeah, it does. Do
you see what I am saying?
Ryan: Yeah, sort of.
Sara: It’s just thirty-two. You just double it. Double that, it goes up
by sixty-four. But, I don’t know how to find the equation.
Sara and Ryan successfully found the number of sheets in a stack up to
10 cuts and recorded this data in a table. They used this data to find the
specific cases for the thickness of the stack. The pair worked hard to make
sense of the situation. When they started question 2 b3, Ryan suggested an
answer while Sara was not clear how he found this. She pushed him to clarify
his process for her until she finally understood what he intended. The
following transcript illustrates their discussion.

Ryan: Okay, do you have that number, like six percent, I think.

Sara: Six percent?

 

8 2. b) For ordinary copier paper it takes about 250 sheets to make a pile 1 inch high. How
thick (in inches) would a stack starting with one sheet of paper be after 4 cuts? 5
cuts? 10 cuts? n cuts? Explain how you figured this out.

139

Ryan:

Sara:

Ryan:

Sara:

Ryan:

Sara:

Ryan:

Sara:

Ryan:

Sara:

Ryan:

Sara:

Ryan:

Sara:

Ryan:

Yeah, okay, umm. Cause that’s how many sheets. You get
that, I guess it’s like a fraction.

Do you get a percent?

I don’t know. I don’t know, I just did a bunch of stuff, okay.
So if it’s six percent, what is it six percent of? Six percent of
one inch?

Help me out.

I’m doing it. [Pause] Okay, point oh six inches. Would you just
take one inch divided by that [points to 250 sheets].

I tried that, it didn’t work.

One inch divided by [pause] fifteen point six [250 sheets = 1
inch, Sara calculates 250 -:— 16 = 15.6]?

Explain how you figured it out.

I umm, isn’t there sixteen sheets [in a stack after four cuts],
this [250 sheets] is one inch. [Pause] And then divide that by
one inch [16 + 250]?

Yeah, that makes sense.

It does?

Yeah.

Point oh six four [pause] inches?

Yeah.

140

Not only did the pair note the doubling in the number of sheets in a
stack, they further described a doubling in the thickness of the stack.
Ryan: It [the thickness after five cuts] should have doubled, right?
Sara: Doubled? So, it would be like point one two eight [double the
thickness after four cuts]?
Ryan: We’ve got to figure out a rule.
They had clearly described both patterns with verbal recursive rules that the

pattern doubles, but the symbolic expressions continued to elude them.

Graphic Connection

Although the students were not asked to create graphs in the tasks,
Sara and Ryan considered the nature of the shape of the graph while
searching for the rules.

While the pair worked, Ryan described the growth and decay graphs,
“Whatever it is, it would take, this [the number of sheets in a stack pattern]
sort of rises, going in the positive direction and this [the area of the pieces] is
going in the negative direction [slopes down].” In their search for
generalizations, they studied all they knew about the pattern, including the
shape of the graphs. They created representations of the graphs, and tried to

use this to help find the generalizations. They could still not find the rules.

141

Dominoes

Sara and Ryan mainly considered the algebraic nature of Dominoes to
search for a symbolic rule. Sara described to me the path they followed
during their investigation, “I got this answer by drawing pictures, then I put
it in the table, and then I found a pattern.”

This pair did not find a symbolic rule. But they did demonstrate a deep

understanding of the pattern when they discussed the shape of the graph.

Using a Table

Prior to recording the data in a table Sara and Ryan made sketches and
counted the dominoes. This strategy of drawing or representing dominoes
helped them to find the number of dominoes for specific cases. They discussed
how to count the 0-1 and 1-0 dominoes and completed the task under the
assumption that they would count them as one unique domino. Sara
continued with her strategy of sketching dominoes to find the number of
dominoes with up to six white spots. She began to systematically sketch
dominoes (See the top three rows on the back of Sara’s work). First, she made
the doubles for zero through six, next she represented the dominoes that had
a one, then two, and she continued up to six. After sketching all of these she
realized that she had repeated some and she used an “X” to cross them off.
This sketch was incomplete since she had forgotten the six additional

dominoes that could be made with zero spots (0-1, 0-2, 0-3, 0-4, 0-5, and 0-6).

142

 

 

 

 

 

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Figure 42: Ryan’s work on Dominoes

144

She initially answered question 2 c, with the 22 dominoes she sketched and
later revised her response when she recognized a pattern in the table.

Ryan opted for an alternative representation for his dominoes.

AK: When did you decide to try a new strategy?

Sara: I drew them, he did something different.

Ryan: I first kind of thought what they were. I thought about it and

drew tally marks.

AK: So, you thought about it and had a mental picture?

Ryan: Yeah.

Ryan mentally represented the dominoes. After noting the 28 dominoes with
up to six white spots, he proceeded to use his “tally marks” representations to
count the 15 dominoes with up to four white spots. (See the lower right corner
on the front of Ryan’s work.)

After they represented the dominoes, they used a table to organize the
data. They sketched additional domino representations to complete their
tables. Sara drew the number of dominoes with up to zero, up to one, and
then up four white spots and inserted these numbers into her table.

The pair studied the data in the table by searching for constant
differences. They used the pattern of a constant second difference of one to
complete their table and concluded that there were 21 dominoes with up to
five white spots, and 28 with up to six white spots. Here Sara recognized that
she had incorrectly represented 22 dominoes for the number of dominoes with

up to six spots and changed the value in both her table and response to 28.

145

During the interview Sara and Ryan explained the ease with which they
changed the data. The pattern they found in the table was more convincing to
them than the sketches. Besides, Sara added, it was quite possible that she
missed some dominoes in her sketches.

Sara and Ryan did not find a symbolic rule for the number of dominoes.
They did suggest some reasonable ideas regarding the form of the equation,
based on their work. After they found the second difl'erence was constant,
Sara used (X + 1) (X - ) as a template for her initial guesses. Because the
second difference was constant, they knew the rule was a quadratic (that
meant that the rule would be X or a quantity of X times a quantity of X).
They continued with a guess and adjust strategy but never found an

appropriate symbolic generalization.

Graphic Connection
Ryan described a graph of the pattern in Dominoes during our interview.
Ryan: It forms a parabola in the graph.
AK: How do you know it forms a parabola in the graph?
Ryan: The way the table is written when you write it out.
AK: What about the table tells you that it is a parabola?
Ryan: It increases by more each time. See right here, it increases by

three, then it increases by four. Makes it curve up more.

146

They analyzed an aspect of the pattern (the shape of their graph) even
though the task did not pose questions about graphical representations of the
situations. Ryan offered a very reasonable description of a parabola based on
this analysis of the data in their table. As he said, “Makes it curve up more,”
his hand curved up to indicate the upward swing of the curve.

Not only did he know that the graph would be a parabola, he offered a
very reasonable description of the graph based on the data in the table. They
made a strong connection between two representations (tables and graphs) of

quadratic functions.

Toothpicks

Although Sara and Ryan did not make a formal table in their solution to
Toothpicks they studied the pattern in the numeric data. They followed a
similar strategy of collecting data, generating new data, studying patterns,

and writing the rules.

Numeric Approach

Sara and Ryan first counted the number of toothpicks in the perimeter
for each of the four figures given and wrote these values (4, 8, 12, and 16)
above each shape. When they began investigating the second pattern for the

total number of toothpicks they repeated their process and counted the total

147

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Figure 43: Sara and Ryan’s work on Toothpicks

148

number of toothpicks in the four figures and wrote these numbers (4, 10, 18,

and 28) above the perimeters.

Sara and Ryan quickly saw the pattern of increase in the perimeter

data.
AK: Can you describe your strategy for this problem?
Ryan: We tried finding a pattern.
AK: How did you try to find a pattern?
Sara: We figured out like the perimeter of these.
AK: You counted?
Sara: Yeah, it went up by four.
AK: Okay, did you use that to answer the question about five or
did you draw five?
Sara: Five, we just counted them.
Ryan: We just found the pattern goes up by four every time.
AK: How many of these did you have to find, to find the pattern?
Ryan: The first three, but we just kind of wrote in the last one.
AK: Like a check?
Ryan: Yeah.

The total number of toothpicks pattern did not come to the pair as

quickly, but they were still successful in their search. Once they had the total
number of toothpicks for the four figures (4, 10, 18, and 28) they wrote out
the first differences of 6, 8, and 10 above the numbers. They did not write the

second difference, but demonstrated their recognition that the second

149

difference was two when Sara verbally extended the first differences during
our interview, “It increased by six, then eight, then ten, so then we added
twelve and then fourteen.”

Sara and Ryan accurately found rules for both the perimeter and the
total number of toothpicks of the nth figure. Finding the perimeter rule was
straight forward for them once they saw the constant increase of four in the
data. Sara supported her rule of 4-N in her written response with, “We found
that the figures [sic] perimeter increased by 4 each time so we multiplied by 4
and it worked.”

It was not as clear how they arrived at their second rule of
[(x+1)(x+1)] + (x-l) for the total number of toothpicks. Sara and Ryan both
wrote that they “guess and checked.” Their discussion during our interview
suggested that they knew a form the equation might take and used that to
inform their guesses.

Sara: We saw how much they increased by.

Ryan: Yeah. We just started guessing and we knew it formed a

parabola.

AK: How did you know it formed a parabola?

Ryan: Because we sort of plotted it.

AK: Okay, what did that tell you when you knew it formed a

parabola?

Ryan: In the equation X had to multiplied by another form of X.

150

The constant increase they observed in the first differences of 6, 8, 10, 12, 14
and so on, helped describe the shape of the graph to the pair. The shape of
the graph determined the form of the equation. Once they knew the form of
the equation, they embarked on their guess and check strategy.
AK: How did you guess and check that one?
Sara: We knew it would be a form of X times itself or a form of X,
and then I don’t know.
AK: Do you remember?
Ryan: It was boom. There it was. I just tried something, I just tried
the plus one and it worked.
Their guess and check resulted in a rule that satisfied the data. Although
they seemed to interpret their finding a rule for the pattern to be luck, their
initial guesses were guided by some fundamental understanding about the
pattern. They used a reasonable guess based on what they knew of the data

and were then successful in finding a rule.

Anna and Katrina

This pair of students claimed that all of these tasks involved algebra
since their goal in all was to write symbolic rules. They described algebra
during an interview.

AK: What do you think algebra is?

151

Katrina: Taking math to a different level, using numbers and symbols

in the rules

Anna: I think using tables and graphs to get rules using data to find

patterns.

Although, they considered all of the tasks to be algebra their solutions
varied. In Cutting and Toothpicks they constructed tables, in Borders they
followed a geometric approach, and in Dominoes they considered a pictorial
strategy. They found adequate rules for the patterns in three out of the four

tasks.

Borders
Anna and Katrina wrote their symbolic rules for the patterns in Borders
based on the shapes of the figures, a geometric approach. They found the

rules early in their solutions and used these rules to generate specific cases.

Geometric Approach

Although their written work offered minimal insight into their approach
to finding the rules, the videotape provided evidence of their process. Anna
and Katrina readily described the rules b = n x n and w = (n + 2)~2 + (n + n)
in words shortly after they began the task.

Anna: So, for this, the fourth square is like N times N. So, for the

number of blue tiles it would be N times N. So, four times four

152

 

 

 

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Figure 44: Anna and Katrina’s work on Borders

153

is sixteen [pause]. For, white tiles, N plus one, or N plus two,
plus N plus two plus N plus N.
While Anna discussed her generalization with Katrina, she motioned along
the lines sketched in Figure 44 below to the corresponding sides of figure 3 to
illustrate the pieces of the rule.

n+2 n+2

 

11
Figure 45: Anna’s figure 3 in Borders

When pushed to describe the strategy they used to complete this problem,
Katrina claimed that is was a simple problem, “we just looked for a pattern
in our heads.”

The pair found the number of blue and white tiles in figure 4 by using
the rules they generated. Anna described how she found the 20 white tiles for
figure 4 using her rule, “In this case N is three, and it goes, you add two on
each end [pause]. Okay, so in this case N is four, so N plus two is six, is
twelve, plus four plus four is twenty.” While stating her explanation she
wrote, “6 + 6 + 4 + 4 = 20 white tiles.”

Once they found the 16 blue and 20 white tiles for figure 4 using the
rules, they verified this for themselves by sketching figure 4 and counting
tiles. Not until this confirmation were they satisfied with their work.

During the interview Anna and Katrina again justified their results

with a specific case.

154

AK: Please justify your solution in one a9, how did you get N times
N?

Anna: N times N is the area of the blue squares, Like for figure
three, three blue squares across, three here, so three times
three.

AK: How about one b?10 N plus two times two plus N plus N?

Anna: N plus two is like taking three and adding these two corners,
so that whole length would be five, then you times it by two,
because you have another one up here. What you have left is
N, then three, then N or another three.

Although Anna specifically used figure 3 for her support, her description
could be generalized for the nth figure. It was unclear based on the evidence if

Anna and Katrina saw this further generalization.

Cutting
This was an unfamiliar task to Anna and Katrina. They did not recall
completing it or a similar problem, but because Cutting asked the students to

write rules, they felt it was an algebra task. When faced with a new

 

9 1. a) If the pattern of blue squares with white borders continues, how many blue tiles are
needed to build the 4th square? the nth square? Show how you figured this out.

10 1. b) If the pattern of blue squares with white borders continues, how many white tiles
are needed to build the 4th square? the nth square? Show how you figured this out.

155

 

 

 

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Figure 46: Anna and Katrina’s work on Cutting

156

situation, Anna and Katrina made and studied tables to investigate the

pattern.

Using a Table

The first step Anna and Katrina followed was to collect their data and

record that in a table.

Anna:

Katrina:

Anna:

Katrina:

Anna:

Katrina:

Anna:

Katrina:

Anna:

Katrina:

After one cut, two pieces, right?

Yeah.

After two, you out these pieces in half. You have four.

Uh huh.

Then you take those four and cut them.

I don’t know its either going to be eight or six.

Okay, let’s think about it. If you have four pieces of paper in
your hand just regular size and then in half, it would be
[pause] eight.

Eight.

So, would it be eight?

It would have to be. I guess because if you rip them in half it

would be eight papers.

They considered the context of the problem and used that knowledge to help

them collect their data.

157

They found the data for up to 20 cuts by using calculators. They
recorded the data into a table and then used that representation to find the
specific cases for which the problem asked.

Anna: For ten it would be . . . one thousand twenty four.

Katrina: All right, twenty? [Long pause]. Is that what you got? One oh

four eight five seven six?

Anna: Yup.

Next, they tried to find the number of sheets in a stack after 10 cuts.
They studied the table and searched for patterns. They calculated the first
difi'erence in the table, but that did not seem to help them describe the
pattern.

Although Anna and Katrina did not find the symbolic rules for the
exponential patterns it was evident during our interview that they
recognized that they needed the rules for the number of sheets in a stack to
find the thickness of a stack in a later problem.

AK: What about b?11 What was your strategy?

Anna: We had to take the number of stacks and divide by two

hundred fifty. Our table helped us, so we needed to know for
four, so we took sixteen stacks and divided that two hundred

fifty and we got point oh six four.

 

11 2. b) For ordinary copier paper it takes about 250 sheets to make a pile 1 inch high. How
thick (in inches) would a stack starting with one sheet of paper be after 4 cuts? 5
cuts? 10 cuts? n cuts? Explain how you figured this out.

158

AK: All but the N?

Anna: We needed to know the equation for a, but we never figured it
out.12

AK: Katrina?

Katrina: I don’t think we did it.

Anna: I wrote how you could do it. [Pause] Four cuts, gives you
sixteen stacks of paper, so sixteen divided by two fifty, so
point oh six four of an inch. So, for any number of cuts , it
would be stacks divided by two fifty.

They generalized that they could find the thickness of the stack when they

were given the number of sheets in a stack.

Dominoes

Anna and Katrina saw this problem as algebra and probability. Algebra,
“because you need to write a rule” and probability, “to get different ways of
charting out all possibilities.” Their approach was to represent the dominoes

with tree diagrams and find a pattern in their sketches to generalize.

 

12 2. a) How many sheets of paper thick would the paper pile be after 4 cuts? 5 cuts? 10 cuts?
20 cuts? 11 cuts? Explain how you figured this out.

159

 

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Figure 47: Anna and Katrina’s work on Dominoes

160

Pictorial Representation

The pair had to first determine how to count the 0-1 and 1-0 dominoes.
This question arose early during their investigation. They decided 0-1 and
1-0 were two distinct dominoes and completed the task with this assumption
(Julie and Dan’s Case 1).

These two students made tree diagrams to represent the dominoes after
they sketched out the nine dominoes with up to two white spots. Making tree
diagrams was a strategy they recalled from the probability they studied in
their mathematics class. From the tree diagrams, they saw how to write the
rules of (n + l)-(n + 1).

Anna: Zero to three white spots it would be sixteen possibilities

right?

Katrina: Uh huh.

Anna: All you do is you add, through, add one to the total number,
so, for six it would be seven times seven, so it would be forty-
nine.

Katrina: How did you get that?

Anna: You do like, when you draw the diagrams you do zero, then
one, two, then three, up to three. When you include zero, its’
like one more, so you count its like one two three four.

Katrina: All right, I understand.

161

Katrina demonstrated her understanding further in her written explanation
for question three when she wrote, “because when you have like 0-3 spots you
add 1 on which represents 0.” She illustrated this with a diagram to clarify

(See Katrina’s work under question 3).

Toothpicks

This was a straight forward task for Anna and Katrina. They readily
recognized the linear pattern for the perimeter toothpicks in a table. It took a
little more time but they also found the quadratic pattern for the total

number of toothpicks in another table.

Using a Table
The pair made tables to investigate the problem and to collect the data
for the specific case in the first pattern.
Anna: Let’s do a table to see if there is a pattern [in the perimeter
pattern]. Be four.
Katrina: Then twelve [is the perimeter in Figure 3].
Anna: One-two-three-four-five-six-seven-eight [counts perimeter in
figure 2].
When they investigated the second pattern for the total number of

toothpicks, Anna again suggested a table.

162

 

 

      

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Figure 48: Anna and Katrina’s work on Toothpicks

163

Anna: Okay, let’s do a table for this one too [the total number of
toothpicks]. Figure one there’s one, two, three, four. This is for
perimeter and this is for toothpicks. Okay.

Once they decided to use a table they collected data by counting the
toothpicks. Anna described a shortcut for counting the total number of
toothpicks.

Anna: One, two, three, four, five, six, seven, eight, nine, nine, yup
eighteen [is the perimeter for figure 3]. All it is if you find all
of the toothpicks that go like this way [lie in the horizontal
direction] and you double ‘em, because its like going one, two,
nine, and then instead of counting all of the one like this [in
the vertical direction] you can just turn it and it would be the
ones that go across. Do you know what I am saying?

Katrina: Ohhh, okay.

Anna: For four: one, two, three, four, five, six, seven, eight, nine, ten,
eleven, twelve, thirteen, fourteen, twenty-eight.

They proceeded to use this shortcut to collect the remainder of the data.

With the data recorded in the table, they studied it in search of a

pattern. They quickly noticed the increase of four in the table and wrote the
rule 4N.

Anna: Sixteen? So, its...

Katrina: increases by four each time?

164

Anna: The figure times four?

Katrina: Yup.

They studied their second table, for the total number of toothpicks, and
found the first difi'erences. Katrina continued to find the constant second
difference of two, but it was unclear how and if they used that information.

Anna: Okay, fi'om here to here, six, eight, ten, twelve, fourteen.
What are you doing?

Katrina: And then we can do this again, they each increase by two.

Anna: Okay, so how can we do that two times X plus six. No, that
doesn’t work, then umm. ..

Once they had this pattern the rule for the total number of toothpicks was
not immediately evident.

Anna: I totally forgot how to write equations fi'om tables. Is it
something like a X squared plus b...?

Katrina: Umm. I remember that for.

Anna: Plus b X? So [pause] no wait [pause] I totally forgot how to do
equations when they are like this, where you have to know
the previous ones. That’s why we had problems with that one
[Cutting]. Right?

Katrina: Mm, hmm. That’s paper stacks.

Anna eventually suggested that the equation be of the form x2 + b or x2 + bx.

It was unclear what guided her to select these families of quadratics. Rather

165

than basing her use of a quadratic on the constant second difference, she

seemed to focus on the iterative nature of the problem. But she was

successful finding the rule, n2 + 3n, using this starting point.

Anna:

Katrina:

Anna:

Katrina:

Anna:

Katrina:

Anna:

Katrina:

Anna:

Oh my gosh.

Huh?

I got it. I got it, hold on. Three times four is twelve. Okay
remember twelve.

Right.

I got it. It’s N times N plus three times N. Don’t ask me how I
got it. It was just.

Guess and check?

So, three times three is nine plus three oops, times three is
nine.

Nine plus nine.

Eighteen. Okay.

Katrina: Good job.

A key to Anna finding their rule was starting with the form a:2 + bx. She did

not articulate a reason for her selection during their work or during our

interview. But, something about the pattern seemed to guide her to a

quadratic that in turn helped the pair find a rule.

166

Summary

Making and studying tables represents a reliable strategy for these
students when studying patterns in data. There were 31 instances of
students investigating patterns in this study.13 The students constructed
tables in over 60% of the instances (19 out of 31). The range of values the
students used for their independent variables in the tables is listed in
Table 15. Of the tables that were made, half ranged from O to 10 or 1 to 10.
Only one table extended further. The less complex generalizations are
represented in the first four rows of the table (Borders—linear, Toothpicks—
linear, Borders—quadratic, and Dominoes—Case 1). The more challenging
problems are listed in the last three rows (Dominoes—Case 2, Toothpicks—
quadratic, and Cutting).

Ten out of the 12 instances where students did not make and study a
table were in the less complex cases. Six of these were from students
studying Borders, a fairly basic task where students immediately recognized
the patterns. These were both linear and quadratic patterns. Three of these
were the linear pattern in Toothpicks. The only remaining situation where

the students did not use a table was when Anna and Katrina completed

 

‘3 Borders and Toothpicks each had two patterns (one linear and one quadratic). Cutting and
Dominoes both had only one pattern, but Julie and Dan studied two patterns in Dominoes.
The 6 student pairs each studied 6 patterns or 30 total, plus Julie and Dan’s additional
pattern from Dominoes, gives a final total of 31.

167

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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168

Dominoes. They solved this using the Case 1 assumption,” making it a fairly
straightforward pattern to generalize with symbols.

In the 14 more complex instances only two did not involve tables.
Students used tables in 85% of the more complex instances. In the two
instances where they did not use tables students followed a geometric
approach to solve Toothpicks. In Cutting, which was an unfamiliar task for
the students all constructed tables.

There were only five cases where the students did not find symbolic
rules. In all of these instances they constructed tables in their solutions.
Table 16 lists the symbolic rules found by students.

In the following chapter I step back and look for common ideas across
students and across tasks. I also consider some of the difl'erences that make

some of the solutions unique.

 

‘4 In the Case 1 assumption, the 1-0 and 0-1 dominoes are counted as two unique dominoes.

169

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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CHAPTER 5
Stepping Back—

An Analysis of What These Students Know

Introduction

In the previous chapter I describe what each of the pairs of students did
with each of the individual tasks. Now I consider a different framework and
review the data as a collection and describe common threads across students
and across tasks.

All of the students hold some strategies to help them formalize
generalizations. In the majority of situations, students constructed tables of
data. They studied patterns of data and regularly sought out constant
differences to help write rules. In some instances, the students used visual
models and studied changing shapes in the patterns. A few students went
beyond the demands of the task and considered the graphs of the data to help
them think about the pattern.

Another common idea I saw across students was the ability to make
connections. These students made connections between the representations of
the patterns—tables, symbols, and graphs. They also made connections

between the tasks and recognized similarities in the mathematical structure

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of the problems. Finally, some students made connections between these
tasks and other mathematics that they recalled.

In the final section, I write a bit about the students’ disposition towards
mathematics and mathematics problems. They all seemed frustrated and

surprised when they had difficulty with a problem.

Strategies to Formalize Generalizations

These students demonstrated that there was much that they knew about
formalizing generalizations in specific situations. This was evident in the
work they completed and in the interviews in which they participated. All
student pairs made reasonable progress in each of the tasks; all were able to
draw some strategies that afforded them access into the tasks.

Two aspects of the tasks determined the strategy students used to
investigate it. First was the mathematics they saw in the problems, and
second, the familiarity of the problems. When the students classified the
problem as primarily an algebra task, they constructed a table of data to (a)
record or re-present data, (b) extend given data, (c) find specific cases, (d)
study patterns, (9) write rules, and (f) verify rules. If they considered it a
geometry task, they used a geometric approach to investigate and studied the
changing shapes. They considered a single figure and related the figure
number to the measure of interest, then used the next figure to confirm their

assertion. In familiar situations the students had several strategies they used

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in their investigations. In unfamiliar situations they studied tables as an
entry point into the problem.

The students had several difi'erent strategies they used to investigate
the situations. Most used tables, others followed a geometric approach, while

a few considered graphic representations.

Numeric Approach: Generalizing From a Table of Data

Constructing and studying tables was the dominant strategy used by
students in these four tasks. In over 60% (19 of 31) of the situations fiom this
study, students made tables of numeric data to help with their investigations.
Of the 31 situations, there were only five where the students did not
successfully find symbolic generalizations. In all of these five cases, the
students constructed tables to investigate the patterns.

The students saw making tables as a reliable strategy to investigate
challenging problems. When the students realized that a problem was
demanding, I heard many conversations where one student suggested trying
to make a table after they were unsure how to proceed. In the only problem
that was an unfamiliar content, Cutting, all five pairs of students constructed
tables to analyze the data. When the students were uncertain how to
approach a problem, making a table afforded them a reasonable entry point

to begin their investigation. After the students constructed tables of data,

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they usually found the constant differences and used this information to help
them write their rules.

Once students re-presented their data in a table, they usually did not
relate what they found back to the context of the problem. Most studied the
numeric data in the table and did not verify that their solution with the
numeric data was reasonable in the setting of the problems. In a few
instances when students related their solution back to the context they
extended their tables back to include 0 for the independent variable and

referenced figure 0 in their pattern.

Finding Constant Differences

The students were quite facile at finding constant differences. In most of
the tables they wrote the differences they found. They used this information
to describe and extend the pattern and then write the rules.

Linear functions. Writing a linear rule was fairly routine for the
students in the study. When they found a constant first difference, they knew
that the pattern was linear. The students often wrote linear rules in the form
of y = mx + b. They easily recognized the rate of change in the table as the
coefiicient of x and proceeded to find the y-intercept. They found this by
either considering Figure 0 in the pattern of shapes, or by extending the

numeric pattern back in the table to include case 0. Some referred to this as

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the y-intercept, while others called it the constant term. All student pairs
easily found the linear rules in Borders and Toothpicks.

Quadratic Functions. Rules that corresponded to a quadratic function
proved to be more difficult for the students to find. A constant second
difference was a clue to the students that the pattern was quadratic.l A
quadratic meant to most of the students that it was a rule in the form of a
quantity of x (or x itself) times a quantity of x; their views represented the
factored forms (x + a)-(x + b) and x- (x + b). To others, it led to a specialized
expanded form of a quadratic x2 + bx or x2 + b.2 Students did not possess an
algorithm to write a quadratic rule. But, they knew some things about
quadratics that informed a conjecture and adjust strategy. They knew a
potential form of the rule, (x + a)-(x + b) or x-(x + b), that helped them write
an initial guess. From there they used a guess and check strategy to try to
find a rule in quadratic situations. They checked to see if their rule satisfied
the data; is so they were done, if not they adjusted the rule and tried again.

Some students tried to fit what they knew about linear cases to guide
their guesses. The most common idea was the notion of the constant term. In

the quadratic cases some students extended their tables back to include a

 

1 This is a reasonable conclusion to draw. When the x terms differ by one, the second
difi'erence is actually the second derivative. Recall that the second derivative of a quadratic is
a constant (non-zero) term.

2 These factored and expanded forms of the rules restricted students’ guesses. They
eliminated all quadratics with irrational roots from their trials. Some limited their guesses to
only integer roots.

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value of O for their independent variable and used the corresponding value
for the dependent variable as a constant term in their rule.

Students’ knowledge of quadratic situations was not as robust as it was
with linear situations. However, they had enough understanding of the
situation and a set of tools to conduct a reasonable investigation. They knew
how to determine if a pattern was quadratic fi'om a table of data. They knew
the form of a quadratic rule. Some recognized the factored form, while others
focused on an expanded form. They also knew things about the graphs of the
pattern, which I describe in later sections.

Exponential functions. Quadratics were not the only non-linear patterns
the students explored. I asked them to consider an exponential pattern.
When the students could not find a constant difference, they were unsure
what form the rule might take. Most followed a guess and check strategy
hoping they might find something that worked for their situation. Even
though they were in an unfamiliar situation, they had some strategies that

allowed them access into the task.

Formulating Rules

After these students constructed tables, it was not trivial for them to
find symbolic rules in non-linear situations. MacGregor and Stacey (1993)
found that when students had difficulties writing algebraic rules from

function tables there were several different steps along the way where

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students might lack the needed skills or knowledge. They identified four of
these steps that students must cross before they can make the transition to
symbolic generalizations with x as the independent variable and y as the
dependent variable. Students must be able to (a) articulate the relationship
to find numerical values, (b) look beyond a recurrence pattern to find one that
links the two variables, (c) know the syntax of algebra, and (d) know what
can and cannot be said with algebra.

Students who were successful finding symbolic rules demonstrated
competence in each of the four steps outlined. Students who did not find
symbolic rules stumbled in at least one of the steps. This is illustrated in the
following sections.

Articulating the relationship to find numeric values. All students found
numeric values for specific cases in the tasks. They generally did this in one
of two ways. In some situations they extended a recursive pattern, at other
times they continued the pattern of differences. Examples to illustrate these
different strategies can be pulled fiom two tasks, Cutting and Toothpicks. In
Cutting students were asked to find the number of sheets in the stack of
paper after 20 cuts. Anna and Katrina extended their table by doubling the
previous term to find the 1,048,57 6 sheets after 20 cuts; they used a recursive
pattern to complete this. After finishing Toothpicks, Julie clarified during our
interview how she used her understanding of the constant second difference

to find the total number of toothpicks in the sixth figure. “I used my pattern.

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First, I made a table and found the amount of increase by two [the second
difi’erence], then I could find fifty-four [for the sixth figure].”

The numeric patterns in the data represented very strong support for
the students. When they noticed that a value they found through sketching or
counting did not agree with the values predicted by the pattern, they quickly
abandoned their counting strategy in favor of the pattern.

When the rules were straight forward to write, as in the linear case,
there were a few instances where students wrote their rules first and used
this articulated relationship to find the specific cases. This process seemed to
be the exception; students only did this in simpler linear cases like
Toothpicks where the pattern was 4n. Most students readily saw the increase
of 4 after they found the perimeter for two only cases. They noticed that the
perimeters were also multiples of 4 and wrote the rule

Looking beyond a recurrence pattern. This is the step that determined
for most students whether they were able to write the symbolic rules. In
nearly 84% (26 out of 31) of the situations, students were successful in
writing rules. Three pairs of students did not generalize with symbols for the
exponential case in Cutting and two pairs could not find the rules for a
difficult quadratic situation in Dominoes. In all five of these cases the
students had difficulty expressing a relationship that linked the two

variables.

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Cutting proved to be quite a challenging task for all five pairs of
students. The three pairs who did not write a rule, all described the doubling
in the tables, and used that doubling pattern to find the specific cases. Yet,
they could not make the transition fi'om doubling to an exponential to link
the number of cuts to the number of sheets in a stack.

It was clear that some students knew they were searching for the
relationship that linked the two variables. Dan said while they worked,
“There has to be a rule [pause]. Times a half? Find out what the heck six has
to do with sixty four?”

Ben and Joe also discussed this link. Joe suggested “X times Y times
two?” as a possible rule. Ben responded “Times? No, you can’t have X and Y
in the same. You are trying to find Y. This right here is Y [points to sheets
thick]. You don’t know these, you are trying to get these [sheets thick] fiom X
[cuts].”

Both pairs demonstrated that they recognized the roles of the
independent and dependent variables in the generalizations. Dan did this
with a specific case of searching for a relationship between 6 and 64 while
Ben generalized the relationship using x and y. While Ben and Joe finally
wrote N = 2x to represent the number of sheets in a stack after x cuts, Julie
and Dan were not successful searching for a rule. One possible factor to
explain this distinction was the recognition of the structure of the problem.

Julie and Dan never recalled seeing a problem like this, while Brad and Joe

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recalled a similar problem fiom their seventh grade science class. The science
problem involved studying bacteria doubling. Although Brad and Joe
remembered the bacteria problem they added that they did not recall using x
as the exponent.

The second task where students did not write a rule was Dominoes. Two
pairs of students were unsuccessful with this rule; they both constructed
tables and used the patterns to extend the table, but neither found a rule.

The rule the students searched for was difiicult. The total number of

dominoes with up to n spots could be written in factored form as

in + 2)(n + 1) or én + l)(n + 1). Both of these are slightly different than the

factored form the students used to guide their search, (x + a)(x + b). Their
rules all included a coefficient of 1 for the x2 term.

All of the rules that the students wrote in all situations did correctly
link independent and dependent variables in their patterns. When students
had difficulty finding the rule, it was not that they did not know they needed
to make the link, rather they did not know how to do this. In Dominoes, all
students recognized that they were trying to find a rule that related the
maximum number of spots (independent variable) to the total number of
dominoes (dependent variable), some could just not find a rule.

Knowing the syntax. A student who could not find a symbolic rule
because of the syntax might have possessed a verbal generalization, but did

not know the syntax, or algebraic language, that could represent the rule

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with symbols. The lack of knowledge about the syntax seemed to prevent the
students fi'om writing their rules in Cutting. Since the students were
unfamiliar with the appropriate syntax needed, they had difficulties
verbalizing their generalizations in a way that could be represented with
algebraic language. In the interviews after they completed the tasks I
discussed the possible solutions. We talked about adding specific columns to
their tables, illustrated in Table 17, to help them think about a rule.

Table 17: Cutting Table with Factors of 2 Represented

 

 

 

 

 

 

 

Number Number of Sheets
Of Cuts in a Stack
1 2 2x1 2
2 4 2x2 2x2
3 8 2x4 2x2x2
4 16 2x8 2x2x2x2
10 1,024 2x512 2x2x2x2x2x2x2x2x2x2

 

 

 

 

 

 

 

The three pairs who did not write rules were not familiar with the
exponential notation for functions. Although they had worked numerically
with exponents, they were unsure whether they could use them in an
algebraic rule. The lack of familiarity with the syntax represented at least
one step where the students tripped in their solutions.

The pairs who successfully found rules recalled a similar problem from
their seventh grade science class, which involved doubling bacteria. Todd
stated that if had he not done the problem the previous year he would not

have known how to use the “y-x” key. It seemed that the understanding of the

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calculator syntax fi'om their science class helped the pair write their rule of
y = 2x for the number of sheets in a stack after x cuts.

The other task where some students encountered difficulties with a rule
was Dominoes. The students systematically observed the pattern of increase
in the differences fi'om the table. Since the rules did not fit the factored forms
of some students’ vision of quadratics, these students did not write rules in
this situation.3 They did not have the appropriate syntax, or language, to
represent the rule. The quadratic pattern in Toothpicks nicely fit their
assumed form, so that all pairs were successful. For the students who saw
the factored form it was n(n + 3). Anna and Katrina found n2 + 3n to fit their
general form of x2 + bx. It was unclear that Sara and Ryan saw how their
solution matched the factored form. They wrote [(x + l)(x + 1)] + x — 1, which
is an equivalent expression. Ryan started with (x + l)(x + 1), a factored form,
and recognized that he needed to adjust his rule and added (x — 1).

Knowing what can and cannot be said. This step might involve higher
order thinking than middle grade students’ algebraic understanding. Since
they do not have a complete view of the discipline they could confuse what
they don’t know the syntax for with something that cannot be said with

algebra.

 

3 Students looked for rules in the factored forms of x-x, x(x + b), or (x + a)(x + b). Some used
the expanded forms of x2 + b or x2 + bx.

18‘2

Geometric Approach: Generalizing about Changing Shapes

Part of being a mathematically powerful problem solver is for students
to have a number of strategies at their disposal to investigate problems. The
authors of the NCTM Curriculum and Evaluation Standards (1989) wrote
that instruction should increase its focus on having students use geometry to
investigate problems. A few students did successfully use geometric
approaches to solve the situations presented. Using these strategies, rather
than generating numeric data in a table, students observed all the patterns
spatially to recognize a generalization. When the students followed a
geometric approach, they were all successful seeing the geometric structure of
the pattern.

Four out of the five pairs used a geometric approach in the solution to
Borders. Julie and Dan, Sara and Ryan, and Anna and Katrina all used it to
launch their investigation while Ben and Joe followed a numeric approach in
their solution but verified their rule geometrically during the interview. The
students also followed a geometric approach for the quadratic pattern of blue
tiles in Borders, but they represented the pattern as n2 or n x n, without
much discussion. This pattern seemed trivial to these students. The
geometric interpretations of the patterns to find the white border tiles
allowed different ways to count. The rules the students wrote for white tiles

follow in Table 18.

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Table 18: Rules for White Tiles in Borders

 

 

 

 

 

 

 

 

Students Rule
Julie and Dan 4n + 4
Sara and Ryan 4(L — 2) + 4
Anna and Katrina w = (n + 2)-2 + (n + n)
Ben and Joe Lx4 + 4

 

The first two rules, 4n + 4 and 4(L — 2) + 4, could represent counting the sides
without including the corner tiles. The students knew that there were four of
these sides (n or (L — 2)), and then added the four corner tiles (+ 4). Anna and
Katrina’s rule represented counting the tiles slightly differently. They
counted the two long sides (n + 2), including the corners, and then added the
two short sides (n + n).

There were only two other situations where students studied the way
the pattern changed spatially to write their generalizations. Anna and
Katrina used a visual model in their solution of Dominoes. Ben and Joe
analyzed Toothpicks geometrically to find their rules.

Recall that Anna and Katrina explored the Case 1 (considering 1-0 and
0-1 as two distinct dominoes) interpretation of Dominoes. The girls relied on a
visual model or tree diagram to describe how they wrote (n+1)’. Katrina

wrote a nice explanation to visually support their rule (See Figure 49).

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Figure 49: Katrina’s Answer to Question 3 in Dominoes

Ben and Joe described how the individual pieces of the shapes in
Toothpicks were represented in the rule. They initially had made an error for
the perimeter toothpicks, but recognized their mistake and corrected it. After
they rewrote their rule for the perimeter, it was easy for them to revise the
total number of toothpicks based on their correction.

When students studied these problems using a geometric approach, they
considered the shape of the n"' figure of representation. They supplied visual
support of how the parts of the rule related to the pieces of figure n. This
keeps their rules related to the context of the problem. The students who
wrote 4n + 4 for the number of white border tiles knew that the constant

term of 4 represented the 4 corners in the shape.

Graphic Representations
Strong problem solvers often have multiple strategies to investigate
problems. Some students relied on a third strategy when their others did not

result in a rule. In some instances the students discussed the graphs of the

185

patterns as a way to explore the problem. They were not asked to consider
the graph, but some used this to aid in their investigation.

Sara and Ryan discussed graphs while investigating Cutting and
Dominoes. They did not find rules in either of these patterns, but considered
graphs in both.

Although the exponentials represented an unfamiliar task for these
students, Ryan offered a reasonable description of the shape of both the
growth and decay graphs, “Whatever it is, it would take, this [the number of
sheets in a stack pattern] sort of rises, going in the positive direction and this
[the area of the pieces] is going in the negative direction.”

Ben and Joe sketched similar increasing and decreasing graphs on their
papers to represent the patterns while they searched for a rule. There are
several cases where the students made some reasonable sketches to help
them think about the problems. Although, it is not clear how the students
used this to help them write rules, this demonstrated an understanding of

the pattern that may not otherwise be evident.

Connections

Making mathematical connections is an important idea for students.

One of the four common standards in the NCTM Curriculum and Evaluation

186

Standards (1989) is “Mathematical Connections.” A part of the middle grades

standards state,

STANDARD 4: MATHEMATICAL CONNECTIONS

In grades 5-8, the mathematics curriculum should include the

investigation of mathematical connections so that students can--

. see mathematics as an integrated whole;

. explore problems and describe results using graphical, numerical,
physical, algebraic, and verbal mathematical models or
representations;

. use a mathematical idea to further their understanding of other
mathematical ideas;

. apply mathematical thinking and modeling to solVe problems that
arise in other disciplines, such as art, music, psychology, science, and
business;

. value the role of mathematics in our culture and society. (page ??).

The curriculum that the students had used was called the Connected
Mathematics Project; this title emphasizes the importance of connections to
the authors.

Students demonstrated many difl‘erent connections in their solutions to

these problems. There was evidence of the first three bullets as stated in the
Standards. They made (a) connections in each pattern between different

representations, (b) connections between the four tasks and additional

problems, and (c) connections with other mathematics.

Connections between Representations
The students readily made connections between several different
representations of the patterns. I have already described the links between

tabular and symbolic forms in the earlier discussion surrounding constant

187

difi'erences. Students recognized that tables of data with constant first
differences were linear and tables of data with constant second differences
were quadratic. They were unsure when they could not find a constant
difference about the form of the equation. Several students offered evidence

connecting a third representation, connecting the graphs of the patterns.

Connections between Tasks

Some students connected the linear problems in the Borders and
Toothpicks as similar. They recognized the rule, tables, and graphs as
similar. Although all students did not write symbolic rules for Dominoes
some connected it to the other quadratics either through sketches in their
work (see Dan’s work for Dominoes) or discussions we had. In an exchange
during an interview Ryan explained the similarities in the graphs between
patterns in several tasks.

AK: Is this [Dominoes] similar to any of the other problems I have

asked you to do?

Ryan: The Toothpicks one, sort of

AK: How is it similar to Toothpicks?

Ryan: It forms a parabola in the graph.

AK: How do you know it forms a parabola in the graph?

Ryan: The way the table is written when you write it out.

AK: What about the table tells you that it is a parabola?

188

Ryan: It increases by more each time. See right here, it increases by

three, then it increases by four. Makes it curve up more.

AK: Was that the same as Toothpicks?
Ryan: Yes.
AK: Were there any other parabolas?

Ryan: This one, one a in Borders.4
Since they additionally found symbolic rules in Borders and Toothpicks they
extended their connections between three representations—tables, graphs,
and symbols.

The students completed a fifth task, Dot Pattern, which I do not analyze
here.5 Sara and Ryan again made connections between the quadratic
situations in Dominoes and Toothpicks with the quadratic pattern in Dot
Pattern. They recognized the similarities in the quadratic patterns in both

tabular and graphic form. When questioned regarding any similarities or

 

4 l. a) If the pattern of blue squares with white borders continues, how many blue tiles are
needed to build the 4th square? the nth square? Show how you figured this out.

5 Dot Pattern asked the students to create their own pattern based on this '

first figure and then write several statements based on their pattern. Sara “2m”,

and Ryan completed the pattern with the triangular numbers that formed Figure 50:

a quadratic pattern similar to the one they found in Dominoes. The 3, 6, Dot Pattern

10, 15, and 18 were the total number of dots for the first five figures

sketched below. The numbers between these numbers (3, 4, 5, and 6)

represented the first differences. The following written represents their written work.

Graphic

 

The following figure made with dots is the first figure in a pattern.
«5 3 5. 4 IO 5’ 15' c a!
o “......m 3...--- .. - -‘o .'.
/““""’ ... o ‘ . . o o 0 o 0.0..
. ~ . ‘ ‘ .Q‘. c g a 0 .....0'0. '0...’.'.
Figure]. 1‘ ""° "
. y L] s

 

 

 

aura 51: Dot Pattern Written Response

189

differences between the tasks, Ryan first noted that Toothpicks and Dominoes

were similar. His clarification was, “Similar in the table, the way it increases.

In Dominoes: eight, ten, twelve and fourteen. In Toothpicks: three, four, five,

six, and seven.” He recognized the constant increase in the first difference or

that the second difierence in both cases was a constant number.

They added to their earlier discussion of the similarity of the graphs but

described that all had the shape of parabolas because of the data fi‘om the

table.

AK:

Ryan:

Ryan:

Sara:

Ryan:

AK:

Ryan:

Were there any other patterns?

Forms a parabola, the number of dots.

How do you know that?

The way the pattern is.

They increase by three, then four, then five, then six.

How do you know that is a parabola?

If you were to plot it. It starts out shallow and starts to get
steeper and steeper until it is almost straight up and down.
How do you know?

It increases by more and more each time.

While saying this Ryan made the sketch in Figure 52 to support his claim.

I

Figure 52: Ryan’s sketch during Dot Pattern interview.

190

In solving the problems that the pair saw as algebra, Sara and Ryan
made comparisons across the three different representations—tables, graphs,
and equations—of the relationship. Although they did not always find
symbolic generalizations for the patterns, they demonstrated a deep
understanding of the pattern.

Other students sketched graphs in their written work; some discussed it
during their investigations or interviews. Although they were not specifically
asked to consider graphs, they extended their solutions to include this
representation. The following diagram, Figure 51, illustrates the connections

that several students made between tasks, and the representations in the

 

(quadratic)

tasks.
Cutting Borders 1:: T oothpicks
(exponential) (linear) .................................... (linear)
I
I
I
!
Bacteria Doubling
Pizza Pirate
(exponential) I
I
I
Handshakes : Toothpicks
I
I
I
I ..-
l
l-'

Legend:

Symbolic Connections
----- Tabular Connections
......................... Graphic Connections

 

 

Figure 53: Connections between Tasks

191

In addition, some students linked the mathematical structure of the
tasks with problems they had previously solved. While two pairs connected
Cutting to their seventh grade science problem, Anna and Katrina related it
to a problem fiom their seventh grade mathematics.

AK: What do you think this problem [Cutting] is about?

Anna: Umm, like taking things and cutting them in half and keep
cutting them in half. There was a pizza problem that we did
like this last year.6 When they came at night and took half of
the pizza and left half, and then half. They kept taking half.

AK: How was this similar?

Anna: Pretty much the whole thing, see how much you have.

The Pizza Pirate Problem the pair referenced had a discrete exponential
pattern. They recognized that both patterns looked at the way data doubled
and halved.

Both of these problems had similar mathematical structure. In the
science class, the students did use exponential notation to discuss the time it
took doubling bacteria to reach the moon. The students recalling the Pizza

Pirate Problem did not use exponential notation. The problem was posed to

 

3 This was Problem 4.3 from the Bits and Pieces II unit of CMP. A Pizza Pirate was raiding
the pizza in a fi'eezer that a class was saving for their party. On the first night he crept in
and ate half of the pizza. On the second night, he ate half of what was left. Each night after
that he ate half of the pizza that remained. Students investigated what fraction of the pizza
was left after so much time.

192

consider using fiactions. But, both problems were reasonable to use as
comparisons.

Another comparison students made connected with Dominoes. Although
these students did not see that task prior to my administration of it, several
pairs recognized it a similar structure to some of the earlier problems they
did in their mathematics class.

Both pairs, Zachary and Todd, and Anna and Katrina felt they had done
similar problems prior to this in their mathematics class. Anna and Katrina
felt it was somewhat like the “handslaps, high fives, or handshakes”
problems they completed because of the way they charted the possibilities.
Zachary and Todd noted other similarities.

AK: Have you ever done this problem before?

Todd: I don’t think so.

Zachary: I don’t remember.

AK: Did you do anything kind of like it?

Todd: Yeah.

AK: When was that?

Todd: It was like only about a week ago, in this book, it was how

many games would a team play if there were five teams and
every team played every other team

AK: How was it the same?

193

Todd: I remembered making a chart kind of like this, where there
were certain possibilities, certain ones that were doubled so
we crossed them out.

Later, during the interview, Zachary admitted that it seemed similar to
another problem, “I remember a problem like this except it did not have the
N plus one. In the book of Frogs 7, I think, with handshakes.”

In both of these tasks, Cutting and Dominoes, students connected the
problems to previous problems they had completed and used that information
to assist with completing the tasks. The tasks may be situated in very
different contexts, but the students looked beyond this to see the similarities
in the structure of these tasks. This is a very important problem solving skill

for these students to possess.

Connections with Other Mathematics

After each pair of students completed the tasks, I asked what
mathematics was involved in the problem. One student stated the connection
in the curriculum during an interview. Once they completed the final task, I
pushed further to find out what they felt were some of the different areas of
mathematics. Todd responded, “I don’t know. In CMP math the teacher
doesn’t say right now you are doing algebra, right now you are doing

geometry.” Todd did not want to parse out his mathematics into separate

 

7 He referred to the Frogs, Fleas, and Painted Cubes quadratic unit from CMP.

194

categories, he further determined that he saw mathematics as mostly related

and connected.

Disposition

There is more to learning mathematics than knowing the content. Again
I refer to the NCTM Curriculum and Evaluation Standards (1989) document
as a source to consider what educators envision. The authors wrote of helping
students develop a mathematical disposition,

Disposition refers not simply to attitudes but to a tendency to think

and to act in positive ways. Students' mathematical dispositions are

manifested in the way they approach tasks--whether with

confidence, willingness to explore alternatives, perseverance, and
interest-—and in their tendency to reflect on their own thinking

(page ??).

When I initially designed this study, I had not intended to look at
students’ disposition. Then, some evidence unfolded that pushed me to
consider their attitudes toward mathematical problem solving more carefully.

There is strong evidence that these students approached mathematical
tasks with positive attitudes. They used a number of strategies to investigate
the problems. When they were presented with an unfamiliar problem they
had some strategies to investigate the situations. They seemed confident that
they would be able to solve the problems and were disappointed when they

could not.

195

 

I was also surprised at how long the students persevered on a
challenging problem. It frustrated them when they could not finish a
problem, but they persisted. Julie was quite animated while working. During
their solution of Cutting, she bemoaned, “I hate not being able to do this.”
When they investigated Dominoes she exclaimed, “This problem is driving me
crazy!” while she mocked pulling at her hair. She expressed her fi'ustration,
yet always continued to explore the problems.

The students all worked at Cutting until I reassured them that it was
understandable that they might have a difficult time with the problem and
we could discuss a solution together. I think that they might have worked on
the problem longer had I given them the opportunity. Some students did
continue their investigation, even after my warning.

These students all demonstrated confidence in their mathematics
ability, willingness to explore alternatives, and perseverance. I felt they all

revealed a very positive disposition towards mathematics.

Summary

When these students who had spent three years in a reform curriculum

confi'onted a situation involving patterns in functions, they demonstrated

understanding in several areas of mathematics. Each of the situations the

student pairs investigated could be represented by a function—linear,

196

quadratic, or exponential—embedded in different contexts. These students
demonstrated a deep understanding about various algebraic functions. They
all had a solid understanding regarding linear functions. There seemed to be
a lot that they understood about quadratics, but occasionally had some
difficulties in pulling it together to write a generalization. When a pattern
did not fit one of these two categories they relied on several strategies to help
them investigate, but were unclear what to do for the generalization.

They all had reasonable strategies at their disposal to investigate .
various function types. Students made connections among the tasks they
solved for this study and with other problems they had previously
investigated. In all cases, they held dispositions towards mathematics that

encouraged them to persevere with the difficult problems.

197

CHAPTER 6

Implications and Limitations

Implications

There are two major findings in this study surrounding students’
understanding. First, students who had three years in the Connected
Mathematics Project curriculum demonstrated a rich understanding of a
significant piece of algebra. And second, teachers can learn much more about
students’ understanding in algebra by drawing on multiple sources of

evidence, and not relying solely on their written work.

A Rich Understanding of an Important Piece of Algebra

There is not consensus among educators regarding what constitutes
algebraic thinking. Some would take issue with my view on the discipline. I
do not claim that this study represents students’ complete algebraic
understanding, but it does demonstrate that these students have significant
understanding of an important aspect of algebra.

Although this study draws on a narrow slice of algebra, it represents a
very significant piece of the discipline. In Chapter Two, I write of the

importance that studying patterns plays in the foundation of algebraic

198

 

thinking. Studying and symbolically generalizing these patterns cut across
all of the organizing themes presented by the Algebra Working Group (1997).

NCTM published a discussion draft of the Principles and Standards for
School Mathematics in 1998 to build on the foundation of the three
Standards documents published earlier. In their overview, the authors write
the following algebra standard across grades K-12.

Standard 2: Patterns, Functions, and Algebra

Mathematics instructional programs should include attention to

patterns, functions, symbols, and models so that all students-

. understand various types of patterns and functional
relationships;

0 use symbolic forms to represent and analyze mathematical
situations and structures;

- use mathematical models and analyze change in both real and
abstract contexts (page 56).

The NCTM authors write further that,

Patterns, functions, and algebra encompass the systematic use of
symbols, algebraic characteristics of mathematical systems,
modeling of phenomena, and the mathematical study of change.
These notions are not only linked to one another, but also closely
linked to number and operations and to geometry. They are
essential to all areas of mathematics and form the basic language
in which mathematics is expressed. Ideas included within this
standard compose a major component of the school curriculum

(page 56). (Emphasis added)
This study addresses all three of the main components of the Patterns,
Functions, and Algebra Standard.

Studying patterns was the starting point for this study. Mathematics is
sometimes considered the “science of patterns,” and represents a

fundamental aspect of the discipline. In studying patterns the regularity one

199

notices can be used to predict other values. I narrowed my field of patterns to
only those that held the potential for representation as a function. This'kind
of work can be seen as a precursor to more formal investigations with
functions.

In addition to studying patterns, I asked students to generalize the
patterns they recognized with symbolic representations. The use of symbolic
notation is also a powerful idea in mathematics. Some say that “(s)ymbolic
representation of quantitative relationships lies at the heart of algebra”
(NCTM, 1998, page 58).

This study, focused on the study of patterns, also included mathematical
modeling. Once the students recognized their patterns, they were asked to
write symbolic generalizations that modeled the situations. “One of the most
powerful uses of mathematics is the mathematical modeling of phenomena
(N CTM, 1998, page 60).” They were given a situation and asked to write a
mathematical model to represent it.

The students in this study demonstrated their competence in algebra in
a number of ways. The Algebra Working Group (1997) and Kaput (1995) both
recognize that solid algebraic understanding involves experiences across all
organizing themes of algebra. The students demonstrated competence across
three of the Algebra Working Group ’3 themes: functions and relations,

modeling, and language and representations. I did not see evidence of their

200

understanding of the structure of algebra, but that was not the intent of the
tasks or this study.

While the ten students of this study demonstrated understanding across
the organizing themes, they met the ambitious algebra goals established by
the CMP authors. The students showed that they had more than a procedural
understanding of the algebra and were able to think deeply about the

mathematics involved in this collection of problems.

Understanding Functions and Relations

When approaching algebra through the functions and relations theme,
one of the main ideas is the focus on the rate of change. Making numeric
tables of data was the dominant strategy employed by all students in this
study to attend to the rate of change. After students made tables, a common
tool for analysis was to search for constant differences. When they found that
the first or second difference was constant this informed them about the
patterns. A constant first difference meant the pattern was linear, while a
constant second difference told them it was quadratic.

All five pairs of students considered the rate of change when they
constructed tables and found constant differences. The students not only
noted the differences, but also related the differences to graphs and described

various patterns of change. They described that the linear patterns increased

201

 

by the same amount while quadratic patterns, “increase by more each time.”1
These students recognized how the rates of change in the tables affect the
patterns. They used this knowledge, to inform their generalizations. They had
reasonable general forms for the symbolic expressions of linear and quadratic
patterns that they matched with the pattern.

The students in this study also demonstrated a very strong sense of
changing quantities in the situations. They recognized the important
quantities that changed and the dependence between the quantities. All
students drew reasonable conclusions about the quantities in all tasks. There
were several instances in the challenging tasks (the quadratic patterns in
Toothpicks and Dominoes, and the exponential patterns in Cutting) that
while the students searched for rules they worked hard to make sense of the
variables. There was evidence while they worked of how they clarified the
independent and dependent variables in the situation.

The students did not seem to struggle as much with the variables in the
less complex problems (the linear patterns in Borders and Toothpicks, and
the quadratic pattern in Borders). They easily selected variables that seemed
to fit the problem. In Borders, one pair used different variables (length and
width instead of figure number) than the others for their independent
variable, but with their interpretation they found a reasonable solution for

the problem.

 

1 Quote from Ryan after during our interview after he completed Dominoes.

202

Understanding Modeling

The students modeled the situations when they sought symbolic
generalizations to represent the patterns in the data. In all situations they
collected, organized, and studied their data in search of a rule. Some students
studied numeric data in a table, while others considered a geometric
representation. Once the students recognized a pattern in their data they
usually wrote their generalization. They often extended their table to find
additional values and then verified the additional data generated fiom the
table with their rule. In all instances the students looked for rules that
modeled their data, and helped them to predict other values.

In some instances, the students used a graph model to describe the data.
They discussed the shapes of the graphs in some tasks and used patterns in
the shapes to help predict other values. These students felt that studying the

graphs could be helpful to describe their patterns.

Understanding Language and Relationships

All students demonstrated competence with the language of several
different representations of the patterns. All constructed tables, all found
some symbolic rules, and some pairs considered graphical representations. In
all instances the students used and connected different representations.

The dominant representation the students used to study the patterns

was a numeric table. They recognized both linear and quadratic patterns in

203

this tabular representation by finding constant differences. Once they knew
the function type, linear or quadratic, these students drew reasonable
conclusions about the symbolic representations. The tasks of this study did
not pose specific questions that asked students to consider other
representations, but some students additionally discussed their
understanding of linear and quadratic patterns in graphic form and related
that to other representations.

There was not sufficient evidence from this study to evaluate all
students’ understanding of the graphical representations since I did not pose
questions that addressed this in any of the tasks. However, several students
did demonstrate competence with this representation.

The students were not as solid in exponentials, the third pattern, that I
asked them to investigate. Some pairs were unfamiliar with this pattern
since they had not yet studied the CMP exponential unit in their
mathematics class. It is important to note that although they could not
classify the exponential pattern, they did not try to classify it as linear or
quadratic when they observed the pattern of differences in the table. They
did not use linear or quadratic rules inappropriately. The students who

sketched a graph also recognized it as different than the other two patterns.

204

Understanding Deeply

Masingila (1998) writes about the difference between knowing some
mathematical idea procedurally and knowing it conceptually. She
acknowledges the importance of students’ understanding mathematical
concepts. If what these students learned through the CMP curriculum is to be
classified as meaningful mathematics, they needed to learn more than just a
new algorithm to generate symbolic rules. I believe that these students did
have a deep understanding of symbolically generalizing patterns from data.

If these students were to have only a procedural understanding, then
you could expect all of the students to arrive at similar generalizations by
following the procedure. In most of the less complex cases (linear patterns in
Borders and Toothpicks and the quadratic pattern in Borders) students did
arrive at expressions that appear similar, but in the more complex quadratic
cases of Toothpicks and Dominoes students found very different looking
equivalent expressions.

The students quickly wrote the symbolic rules in both linear cases in
Borders and Toothpicks and the quadratic case of Borders. It is apparent in
Table 19 below that all students found the same form of a symbolic
generalization for the linear case in Toothpicks. All pairs, except Sara and
Ryan, used the generalization of x2 or xxx for the Borders quadratic pattern;
Sara and Ryan found a slightly different expression based on their choice of

variables. Zachary and Todd, Ben and Joe, and Julie and Dan all found

205

similar looking rules in the linear case of Borders, while Sara and Ryan and
Anna and Katrina based their rules on geometric interpretations and found
something that looked slightly different.

Table 19: Symbolic Rules Generated for Simple Cases

 

 

 

 

 

 

 

 

 

Tasks
Students Borders Toothpicks Borders
Linear Linear Quadratic

Zachary and Todd 4x + 4 4N x2
Ben and Joe Lx4+4 Y= 4N L2
Julie and Dan 4n+4 y = 4n n2
Sara and Ryan 4(L-2)+4 4N (L-2)(W-2)
Anna and Katrina w = (n+2)-2+(n+n) 4N b = nxn

 

 

 

 

There was not much discussion between the pairs in the recordings
regarding their solutions for these patterns. I suggest that these students do
have a solid understanding in these cases, although their work might appear
to be somewhat procedural. Their ease of working with these patterns and
their connections with other representations during our discussions supports
this view. Additionally, their work with the more complex quadratic suggests
a solid understanding of these cases.

These students seemed to have a conceptual understanding of the more
complex cases also. Table 20 below lists the four equivalent, but different

expressions they wrote to represent the patterns found by the five pairs of

206

students for the symbolic generalizations for the quadratic patterns in
Toothpicks and Dominoes.2

Table 20: Symbolic Rules Generated for More Complex Cases

 

 

 

 

 

 

 

Students Tasks
Dominoes Toothpicks
Quadratic Quadratic
Zachary and Todd (n+ l)+2-n+n+1 n(n+3)
Ben and Joe y = (N+l)(0.5N+l) y = 4N + (N-1)N
Julie and Dan No rule y = n(n+3)
Sara and Ryan No rule [(x+1)(x+1)]+x-1
Anna and Katrina - n2 + 3n

 

 

 

 

 

In their solutions, three pairs made tables and studied constant differences;
Sara and Ryan studied the differences in numeric data not organized in a
table, while Ben and Joe studied the pattern of the Toothpicks in the
changing shapes to write the rules. Only two of the four pairs of students who
worked with this interpretation of Dominoes arrived at symbolic rules. These
two expressions looked quite different.

In all of these more complex cases the students had access to a number
of tools that helped them investigate the problem in a meaningful way. Since
their final rules did not look the same, in fact some appear quite different,

this supports the view that these students did not follow an algorithm to

 

2 I refer to the Case 2 interpretation of Dominoes, counting the 0-1 and 1-0 as one domino.

207

generate symbolic rules. They used their understanding of the situations to

generate rules.

Assessment

This study also offers supporting evidence surrounding the uses of
multiple forms of assessment. If teachers want to know students’ sense
making of algebraic ideas, then a range of forms of assessment is required. To
paint a picture of students’ understanding it requires administering high-
quality tasks, observing students while they work, and talking with students.

High-quality tasks have some or all of the following characteristics.
They must engage all students while challenging them at the same time.
They should allow for multiple ways to find a solution. They should capture
important mathematics. They should support student discussions. They
should require higher levels of thinking. They should have the potential for
students to make connections. The problems should be based on sound
mathematics, perceptions of students experiences, and knowledge of diverse
ways that students learn mathematics (NCTM, 1991).

It is not enough to consider only students’ written responses. In this
study, much of the students’ thinking was not recorded in their written work.
In some instances I saw that students understood much more than what they
recorded on their papers. Watching the videotapes of the pairs working gave
me some insight into this additional understanding. Together, written

responses and careful observations still left some aspects of their

208

understanding hidden. I learned much more by talking with students about
their understanding that was not evident in the other data sources. Much of
students’ understanding about the graphs became apparent in our
discussions. In their work with graphs, the students often connected the
pattern in the problem with the pattern for other problems and connected the
representations. I might not have seen what these students really understood
had I not collected this additional data.

A goal of assessment is to accurately represent what students
understand; good assessment strategies will aid teachers with this task. For
classroom teachers, this means that they need to be aware that the written
record may not tell everything about students’ understanding. When
evaluating students’ understanding teachers need to be diligent about
collecting multiple forms of assessment to more accurately represent this.
They need to be careful observers while students work. It may not be
practical for them to make video recordings of students working, but they can
make careful notes while observing students engaged in the tasks. They
should talk with students about what they have written and ask them to

clarify what they have done.

Limitations

In any study it is important to acknowledge the limitations. In Chapter

Three I offer support regarding the decisions I made for this study. In this

209

 

section I describe three limitations based on the design of this study. First,
my selection of the performance tasks I administered limited what I could
learn about the students’ understanding. Second, I did not study the
implementation of the curriculum, and so I can not dismiss the importance of
the teachers’ role in the success these students had with algebra. And, third,
I did not have a control group and so I am unable to draw any comparison

conclusions.

Tasks

My selection of tasks limits what I could learn about these students’
algebraic understanding. This study considers students’ symbolic
generalizations of data from specific patterns. This cut on algebra is not
intended to represent the entire discipline or evaluate students’ complete
algebraic understanding, rather to survey students’ reasoning in an aspect
that represents a part of the foundation of the discipline. I described earlier
in this chapter that this narrow cut of algebra is quite significant for students
to learn. But, all of the tasks involved whole numbers and one could
speculate that other tasks with rational numbers might have proved more

challenging.

210

Implementation

I recognize that the nature of the implementation of any curriculum is
vital to its success or failure. But, the implementation of the CMP curriculum
was not the focus of my study. Even though this was not part of my study, I
acknowledge that students coming out of the eighth grade are not just
prepared because of the curriculum. Quality curriculum is not sufficient for
success; it may be necessary, but not sufficient. There are a number of
interesting stories to be told surrounding Heartland. Some have already been
done regarding this site (Bouck, M. and Wilcox, S., 1996).

A key element to these students’ achievement is that they had quite
skilled teachers in grades 6 through 8. There is quite a bit of local knowledge
surrounding Evelyn Howard, the teacher these students all had in grade
eight, and some in grade seven. She is a very accomplished mathematics
teacher who is held in high regard in the community. I feel quite certain that
Evelyn’s skill teaching mathematics is a contributing factor to the success for
these students.

It should again be noted that this school had been seriously working on
professional development for teachers for a number of years and offered
considerable support for teachers. Many, including Evelyn had the
opportunity to work closely with the developers of the CMP curriculum as
part of the piloting agreement. This association helped the teachers refine

their teaching and better understand the curriculum.

211

Comparison

This study was not designed to compare the algebraic understanding of
students from the CMP curriculum with that of students who had more
traditional experiences in mathematics. Instead, my intent was to carefully
describe the potential for students with three years in the CMP curriculum to
develop students’ algebraic thinking. This study suggests that there is great
promise for CMP students to develop a solid understanding of symbolically
generalizing from patterns of data.

There have been other studies done as a comparison and found that
CMP students did significantly better on challenging open-ended response
tasks than non-CMP students. In addition, on a traditional, multiple-choice
test, CMP students made gains comparable to the other students (Hoover,
M., Zawojewski, J ., and Ridgway, J ., 1997). This means that the CMP
students do better on the open-ended items and do not do any worse on the

traditional items.

Summary

This study represents an analysis of the learning of students in one

curriculum, in one site, and of one important slice of algebra. Even with these

limitations, this study provides an opportunity to get a very good look into

these students’ reasoning about algebraic situations. It offers compelling

212

evidence that this Standards-based curriculum has great potential for
student learning in algebra. A curriculum that focuses on algebra as much
more than symbolic manipulation, that has taken functions, modeling, and
representations as key components can provide students with a solid
understanding in a fundamental area of algebra.

A study such as this begins to answer some questions, but it also raises
many more. There is certainly much more to be learned about students’
understanding regarding other areas of algebra with these curricula. An
additional set of questions important to study would look at the
implementation of a standards-based curriculum. The authors of the CMP
curriculum recognize the importance of good teaching. What does it take to
successfully implement a curriculum like CMP? What kinds of experiences do
teachers need to have to teach this curriculum? What do we know about
students’ knowledge prior to entering the CMP curriculum in the sixth grade?

All of these questions are worthy of future study.

213

APPENDICIES

214

APPENDIX A

List of Cbfl’ Units

The units are listed in the form: Title: Subtitle, Strand in Table 17.
Table 21: Order of Cllfl’ Units

Grade Six Units

1. Prime Time:
Factors and Multiples
Number

2. Data About Us:
Statistics
Prob. and Stats.

3. Shapes and Design:
2-D Geometry
Geom. and Meas.

4. Bits and Pieces I:
Understanding Rational
Numbers
Number

5. Covering and
Surrounding:

2-D Measurement
Geom. and Meas.,
and Number

6. How Likely Is It:
Probability
Prob. and Stats.

7. Bits and Pieces II:
Using Rational Numbers

Number

8. Ruins of Montarek:
Spatial Visualization
Geom. and Meas.

Grade Seven Unit§

1. Variables and
Patterns:
Introducing Algebra
Algebra

2. Stretching and
Shrinking: Similarity
Geom. and Meas.

3. Comparing and
Scaling: Ratio,
Proportion, and Percent
Geom. and Meas.,
and Number

4. Accentuate the
Negative: Integers
Number

5. Moving Straight
Ahead:
Linear Relationships
Algebra

6. Filling and
Wrapping:
3-D Measurement
Geom. and Meas.,
and Number

7. What Do You

Expect?: Probability and

Expected Value
Prob. and Stats.

8. Data Around Us:
Number Sense

Number, and
Prob. and Stats.

215

Grade Eight Units

. Thinking with

Mathematical Models:
Representing Mathematical
Relationships

Algebra

. Looking for

Pythagoras:

The Pythagorean Theorem
Geom. and Meas.,

and Number

. Growing, Growing,

Growing...:
Exponential Relationships
Algebra

. Frogs, Fleas, and

Painted Cubes:
Quadratic Relationships
Algebra

. Say It with Symbols:

Algebraic Reasoning
Algebra

. Hubcaps,

Kaleidoscopes, and
Mirrors: Symmetry and
Transformations

Geom. and Meas.,

and Algebra

. Samples and

Populations:
Data and Probability
Prob. and Stats.

. Clever Counting:

Combinatorics

Prob. and Stats.

10.

11.

12.

13.

APPENDIX B

Interview Questions

Have you ever done this problem before?

Have you ever done this type of problem before? If yes, explain.

What do you think this problem is about?

What do you think you are asked to do in this problem?

What mathematics do you think is involved in this problem?

How did you decide what strategy to use? What strategies did you use?
When did you decide to try a new strategy (if so)?

How did you decide what variables to use to solve this problem?

When do you know to stop looking for a pattern and start to generalize?
How would you justify your solution?

How would you verify your results?

Is this similar to any of the other problems I have asked you to do? In
what ways?

Did anything about this problem surprise you?

216

APPENDIX C

Time Spent by Students Working on Tasks and During Interviews

Table 22: Approximate Number of Minutes Spent on Each Task

 

 

 

 

 

 

 

 

 

 

 

 

Tasks
Students Borders Cutting Dominoes Toothpicks
Zachary and Todd 17 21 24 8
Ben and Joe 10 34 33 8
Julie and Dan 13 30 31 10
Sara and Ryan 15 27 41 44
Anna and Katrina 20 29 13 26

 

Table 23: Approximate Number of Minutes Spent on Each Interview

 

 

 

 

 

 

 

 

 

 

 

 

Tasks
Students Borders Cutting Dominoes Toothpicks
Zachary and Todd 16 13 22 20
Ben and Joe 18 29 30 17
Julie and Dan 10 24 30 25
Sara and Ryan 14 17 10 13
Anna and Katrina 5 19 22 18

 

217

 

 

APPENDIX D

218

Tasks

 

Name
Date

 

Borders .

 

These three squares have been made using blue tiles and then a border of
white tiles is put around the blue square.

 

 

 

 

 

 

Figure 1 Figure 2

 

1. a) If the pattern of blue squares with white borders continues, how
many blue tiles are needed to build the 4th square? the nth square?
Show how you figured this out.

b) If the pattern of blue squares with white borders continues, how
many white tiles are needed to build the 4th square? the nth
square? Show how you figured this out.

2. Suppose the blue tiles are arranged as rectangles of any length and
width.

a) How many blue tiles are needed for this? Show how you figured
this out.

b) How many white tiles are needed for this? Show how you figured
this out.

 

 

 

Figure 54: Borders Task

219

 

 

 

Name

 

 

 

Date
Cutungan CuttmgandCut ”p =

 

 

If you take a sheet of notebook paper and cut it in half and stack the
pieces and then cut in half again and stack. and then in half again and
stack, each cut gives smaller pieces but a thicker pile of paper.

 

 

 

 

 

 

At the start, before any cuts, there is one sheet of paper. After one cut,

stack the pieces. The stack is now 2 sheets thick. After 2 cuts and stacking,
the pile is 4 sheets thick.

1. Describe what happens after 3 cuts. How many pieces of paper do you
have in the pile?

2. a) How many sheets of paper thick would the paper pile be after 4

cuts? 5 cuts? 10 cuts? 20 cuts? n cuts? Explain how you figured this
out.

b) For ordinary copier paper it takes about 250 sheets to make a pile
1 inch high. How thick (in inches) would a stack starting with one

sheet of paper be after 4 cuts? 5 cuts? 10 cuts? n cuts? Explain how
you figured this out.

c) How many cuts would you need to get a pile that is 1 foot thick?

3. Suppose the original piece of cut paper has an area of 32 cm2.

a) What is the area of each piece formed after 2 cuts? 3 cuts? 10 cuts?
n cuts? Show how you figured this out.

b) After how many cuts would you get a piece that is 1 cm2.

 

 

 

 

 

Figure 55: Cutting Task

220

 

 

Name
Date

 

 

 

Dommes

Dominoes are spotted tiles used in a board game. A regulation domino tile
is a black rectangle, split into square halves. On each half of one side there
are from O to 6 white spots. The other side is blank. For example. three
different domino faces are shown below.

   

a) Sketch all possible domino faces if your set is made up of dominoes
with from O to 2 white spots.

...;
I

b) How many different domino faces are there in a set made up of
dominoes with from O to 3 white spots on each half of the domino?

2. a) How many domino faces are possible if a set is made up of
dominoes with from O to 2 white spots?

b) O to 3 white spots?
c) O to 6 white sp0ts?
d) O to n white sp0ts?

3. How do you know that what you wrote for the above is true?

 

 

Figure 56: Dominoes Task

221

 

Name

 

Date

 

 

Toothplcks:1;;_;;.;;g...

 

 

 

 

 

Figure 1 Figure 2 Figure 3 Figure 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l. Extending the pattern, what is the perimeter of Figure 5? Show or
explain how you figured this out.

2. How many toothpicks are needed to make Figure 6? Show or explain
how you figured this out.

3. How did you decide how each new figure in the sequence is made?

4. Write a formula that you could use to find the perimfler of any
Figure N. Tell what your variables represent. Explain how you figured
this out.

Write a formula that you could use to find the total number of
tOOthpicks needed to make of any Figure N. Tell what your variables
represent. Explain how you figured this out.

Ul

 

 

 

Figure 57: Toothpicks Task

222

APPENDIX E

Task Solutions

Borders

These three squares have been made using blue tiles and then a border of
white tiles is put around the blue square.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1 Figure 2 Figure 3
Figure 58: Borders Figures

1. a) If the pattern of blue squares with white borders continues, how many
blue tiles are needed to build the 4th square? the nth square? Show
how you figured this out.

The pattern of squares can be continued

to sketch figure four as in Figure 59. The

 

blue interior tiles can be counted from

Figure 59: Figure 4 in
that figure to find 16 blue squares. Borders solution
To find the number of blue tiles in the nth square, try to observe a

pattern in the data. First, count the interior blue tiles from the shapes in

figures 1, 2, and 3 and organize that information in Table 24.

Table 24: Figure Number and
Blue Tiles in Borders solution

 

 

 

 

 

 

Figure Blue
Number Tiles
1 1
2 4
3 9
4 16

 

 

 

The number of blue tiles is the figure number squared, or

Blue tiles = n2, where n is the figure number.

b) If the pattern of blue squares with white borders continues, how many
white tiles are needed to build the 4th square? the nth square? Show
how you figured this out.

To find the number of white tiles in figure four, use the sketch drawn

above to count. The fourth figure has 20 white squares.

Use a similar strategy of observing a pattern to find the number of white

tiles in the nth square. Count the perimeter in figures 1, 2, and 3 and

record that information in Table 25.

Table 25: Figure Number and
White Tiles in Borders solution

 

 

 

 

 

Figure White
Number Tiles
1 8
2 12
3 16
4 20

 

 

 

 

The number of white tiles is a linear pattern since there is a constant

rate of increase of four in the table. This can be used to extend the table

224

back to find the number of white tiles in figure 0 to be four. The rule can
be written as: White tiles = 4n + 4, where n is the figure number.
Suppose the blue tiles are arranged as rectangles of any length and
width.

a) How many blue tiles are needed for this? Show how you figured this
out.

To find the tiles in rectangular figures, consider the general rectangle

with dimensions L by W as illustrated in Figure 60.

W

 

L
Figure 60: General Rectangle in Borders

The number of blue squares would be the inside area. The dimensions of
the interior blue rectangle is (L - 2) by (W -2). So, the number of blue
tiles is:

Blue tiles = (L - 2)(W- 2)

b) How many white tiles are needed for this? Show how you figured this
out.

If the number of white tiles were written as 2L + 2W, the four corner
pieces are double counted, so four must be subtracted:

White tiles = 2L + 2W — 4

225

Cutting

If you take a sheet of notebook paper and cut it in half and stack the pieces and
then cut in half again and stack, and then in half again and stack, each cut
gives smaller pieces but a thicker pile of paper.

 

 

 

 

 

 

 

Figure 61: Cutting Graphic
At the start, before any cuts, there is one sheet of paper. After one cut, stack the
pieces. The stack is now 2 sheets thick. After 2 cuts and stacking, the pile is 4
sheets thick.

1. Describe what happens after 3 cuts. How many pieces of paper do you
have in the pile?

A table that gives the number of sheets of paper in a stack after each cut
could be used to organize and display the data. First, the data that is
given in the problem is put into Table 26.

Table 26: Cut Number and Sheets of Paper
in Solution of Cutting

 

 

 

 

 

Cut Sheets of
Number Paper
0 1
1 2
2 4
3 ?

 

 

 

 

It appears in the table that the data is doubling from the previous term.
The number of sheets after 1 cut is double the number of sheets after 0
cuts. 2 is double the previous term of 1. Likewise, the number of sheets
after 2 cuts is double the number of sheets after 1 cut, or 4 is double the

previous term of 2. So, to find the number of sheets after 3 cuts, it would

226

be double the number of sheets after 2 cuts, or 8 sheets of paper in the

stack after 3 cuts.

a) How many sheets of paper thick would the paper pile be after 4 cuts? 5
cuts? 10 cuts? 20 cuts? n cuts? Explain how you figured this out.

From here the pattern can be extended to find the number of sheets

after 4, 5, and 10 cuts. It seems rather cumbersome to complete the table

to find 20, so first search for a pattern in Table 27 to find n.

Table 27: Cut Number and Sheets of Paper in Cutting solution
up to 10 cuts

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cut Sheets of Cut Sheets of

Number Paper Number Paper

0 1 6 64

1 2 7 128

2 4 8 256

3 8 9 512

4 16 10 1024

5 32

 

 

 

 

To find each of the answers, multiply the previous term by 2, so the

table can be rewritten as Table 28:

227

Table 28: Cutting solution with exponential notation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cut Sheets of Doubling Exponential
Number Paper Notation Notation

0 1 1 20

1 2 1x2=2 2:21

2 4 2x2=4 2x2=22

3 8 4x2=8 2x2x2=23

4 16 8x2=16 2x2x2x2=24

5 32 16x2=32 2x2x2x2x2=25

6 64 32x2=64 2x...x2=26

7 128 64x2=128 2x...x2=27

8 256 128x2=256 2x...x2=28

9 512 256x2=512 2x...x2=29

10 1024 512x2= 1024 2x x2=210

 

 

The answer to the questions are displayed in Table 29:

Table 29: Cut Number and Sheets of Paper in
Cutting solution for 4, 5, 10, 20, and n cuts

 

 

 

 

 

 

 

 

 

Cut Sheets of
Number Paper
4 16
5 32
10 210 = 1,024
20 220: 1,048,576
n 2n

 

b) For ordinary copier paper it takes about 250 sheets to make a pile 1
inch high. How thick (in inches) would a stack starting with one sheet
of paper be after 4 cuts? 5 cuts? 10 cuts? It cuts? Explain how you
figured this out.

Since it takes 250 sheets to make 1 inch, after 4 cuts the paper stack

would be 16 + 250 = 0.064 inches. The answers to the specific cases

asked about in this question are displayed in Table 30.

228

Table 30: Cut Number and Inches Thick in
Cutting solution

 

 

 

 

 

Cut Inches
Number Thick
4 16 + 250 = 0.064
5 32 + 250 = 0.128
10 1,024 + 250 = 4.096
n 2" + 250

 

 

 

 

c) How many cuts would you need to get a pile that is 1 foot thick?

To have a stack one foot thick, that would be 12 inches, or

12 x 250 = 3,000 sheets thick. After 10 cuts it would be 1,024 sheets
thick. By examining the table, we see that after 11 cuts, it would be
double that or 2,048 sheets. 12 cuts would produce a stack 4,096 sheets
thick. So it would not be until 12 cuts that the stack was at least one

foot thick.

Suppose the original piece of cut paper has an area of 32 cm2.

a) What is the area of each piece formed after 2 cuts? 3 cuts? 10 cuts? It
cuts? Show how you figured this out.

The area after 2 cuts would be 32 + 4 = 8 cm2 since the sheet of paper is

cut into 4 equal pieces after 2 cuts. The answers to the specific cases

asked about in this question are displayed in Table 31.

Table 31: Cut Number and Area in Cutting solution

 

 

 

 

 

 

 

 

Cut Area
Number
2 32 + 4 = 8 cm2
3 32 + 8 = 4 cm2
10 32 + 1,024 = 0.03125 cm2
n 32 -:— 2"

 

229

b) After how many cuts would you get a piece that is 1 cm2.
It would be when the sheet of paper was cut into 32 equal pieces. By
examining the table earlier, we can see that this is after 5 cuts.

32+25=32+32=1cm2.

Dominoes

Students interpreted this problem in two different ways. The first,
CASE 1, was to count the 0-1 and 1-0 domino as one unique domino. In
CASE 2, students counted 0-1 and 1-0 as two distinct dominoes. Since both
solutions follow in my analysis of students’ work I present both alternatives
in this section.
Dominoes are spotted tiles used in a board game. A regulation domino tile is a
black rectangle, split into square halves. On each half of one side there are

from 0 to 6 white spots. The other side is blank. For example, three different
domino faces are shown below.

 

Figure 62: Dominoes Graphic
CASE 1: 0—1 and 1-0 as one unique domino

1. a) Sketch all possible domino faces if your set is made up of dominoes
with from 0 to 2 white spots.

The sketch of the dominoes with from 0 to 2 white spots is represented

in Figure 63.

230

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 63: Sketch of Dominoes with 0-2 Spots—Case l

b) How many different domino faces are there in a set made up of
dominoes with from 0 to 3 white spots on each half of the domino?

The dominoes with from O to 3 white spots would include the six
dominoes with from 0 to 2 (above) plus the following 4 dominoes in

Figure 64.

WW

 

 

 

EH

 

 

 

 

 

 

Figure 64: Additional Dominoes with 3 Spots

a) How many domino faces are possible if a set is made up of dominoes
with from 0 to 2 white spots?

Counting the dominoes drawn in question 1 a, above, there are 6

dominoes with from 0 to 2 spots.

b) 0 to 3 white spots?
To find the dominoes with 3 spots add the four dominoes sketched in
question 1 b, to the set of six dominoes with up to 2 white spots. There

are 10 dominoes possible with from 0 to 3 spots.

231

c) 0 to 6 white spots?

The dominoes possible with from O to 6 white spots would be 28. For
each set of dominoes add one more than the maximum number of spots
to the previous set of dominoes. To find the number of dominoes with up
to 3 spots add 4 for the dominoes 0-3, 1-3, 2-3, and 3-3. To find the
dominoes with up to 4 spots, add 5: 0-4, 1-4, 2-4, 3-4, and 4-4 to the 10
dominoes with up to 3 spots. This is illustrated in Table 32.

Table 32: Maximum Spots and Possible Dominoes

 

 

 

 

 

 

 

 

 

Maximum Dominoes
Spots Possible
2 6
3 6 + 4 = 10
4 10 + 5 = 15
5 15 + 6 = 21
6 21 + 7 = 28

 

d) 0 to n white spots?
To find the number of dominoes with up to n white spots, consider the
triangular array in Figure 65 sketched for up to 2 spots:
00
0-1 1-1
0-2 1-2 2-2
Figure 65: Triangular Array of Dominoes with up to 2 Spots

This array can be placed in the rectangle that has dimensions 3 by 4

illustrated in Figure 66.

232

 

 

 

0-0

 

0-1|1-1

 

0-2| 1-2

 

 

2-2

 

 

Figure 66: Triangular Array in 3 by 4 rectangle

The number of dominoes is one half the area of the rectangle,
2(3 x 4) =%(12) = 6 dominoes. This arrangement can be extended for any

n in a rectangle of dimensions (n+ 1) by (n+2) illustrated in Figure 67.

 

 

O-OI

 

0-1 1-1

 

 

 

 

 

0-4 .I

 

n-Il»

 

Figure 67: Triangular Array in (n+1) by (n+2) rectangle

The number of dominoes is still one half the area of the rectangle,

1/2 (n+ 1)(n+2)

This can be verified using the diagrams sketched above.

CASE 2: 0-1 and 1-0 counted as two distinct dominoes

1. a) Sketch all possible domino faces if your set is made up of dominoes

with from 0 to 2 white spots.

The sketch of the dominoes with from 0 to 2 white follows in Figure 68.

233

How do you know what you wrote above is true?

 

 

 

 

 

 

 

 

 

E

 

 

 

 

 

 

 

.’0 00M
L_|-_'l

 

 

 

 

 

 

 

Figure 68: Sketch of Dominoes with 0-2 Spots—Case 2

b) How many different domino faces are there in a set made up of
dominoes with from 0 to 3 white spots on each half of the domino?

The dominoes with from 0 to 3 white spots are sketched below in

Figure 69.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lflfllflfl-fll fl

llo'L
.0 DIM—OJ 0| 0

 

 

 

 

Figure 69: Sketch of Dominoes with 0-3 Spots—Case 2

2. a) How many domino faces are possible if a set is made up of dominoes
with from 0 to 2 white spots?

Counting the dominoes drawn in question 1 a, above, there are 9

dominoes with from 0 to 2 spots.

234

b) 0 to 3 white spots?

Count the 16 dominoes drawn in question 1 b.

c) 0 to 6 white spots?

There are 49 dominoes possible with from 0 to 6 white spots. For each
set of dominoes add one more than the maximum number of spots to
include the zero. Then this number should be squared. So, for 6, you
take 7-squared to get 49. This is illustrated in Table 33.

Table 33: Maximum Spots and Possible Dominoes
in Dominoes Case 2 solution

 

 

 

 

 

 

Maximum Dominoes
Spots Possible
2 32 = 9
3 42 = 16
4 52 = 25
5 62 = 36
6 72 = 49

 

 

 

 

d) 0 to n white spots?

Following the description above, this can be generalized for n, (n+1)2

How do you know what you wrote above is true?
We can show why you need to add one with the dominoes sketched with

0 to 2 white spots. You need to add one, to include the value for zero.

235

Toothpicks

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1 Figure 2 Figure 3 Figure 4
Figure 70: Toothpicks Graphic

1. Extending the pattern, what is the perimeter of Figure 5? Show or
explain how you figured this out.

A sketch of figure 5 is shown in Figure 71, The perimeter can be counted

from the sketch as 20 toothpicks.

 

Figure 71: Toothpicks Figure 5

2. How many toothpicks are needed to make Figure 6? Show or explain
how you figured this out.

Figure 6 is sketched in Figure 72, 54 total toothpicks can be counted.

 

Figure 72: Toothpicks Figure 6

236

Write a formula that you could use to find the perimeter of any Figure
N. Tell what your variables represent. Explain how you figured this out.

The perimeter toothpicks can be found from counting the first four
shapes and recording that data in Table 34.

Table 34: Figure and Perimeter in Toothpicks solution

 

 

 

 

 

 

Figure Perimeter
1 4
2 8
3 12
4 16
5 20

 

 

 

 

The table has a constant increase of 4, so this is a linear relationship.
The table can be extended back to find a perimeter of 0 for figure 0.

Perimeter = 4n, where n is the figure number.

Write a formula that you could use to find the total number of
toothpicks needed to make of any Figure N. Tell what your variables
represent. Explain how you figured this out.

Table 35 shows the total number of toothpicks.

Table 35: Figure and Total Toothpicks in Toothpicks solution

 

Figure Total
Toothpicks
4
10
18
28
40
54

 

 

 

 

 

 

 

 

 

65014:me

 

237

Figure three is used to demonstrate how to write the rule. First, count

all of the horizontal toothpicks that are marked on the shape below in

 

 

Figure73.
-/—1
7'1'2
I‘I‘I‘3
:iis

 

 

 

 

 

Figure 73: Horizontal Toothpicks
There are 1 + 2 + 3 + 3 toothpicks. Likewise the vertical toothpicks can

be counted. They are marked in Figure 74.

 

 

 

 

 

 

 

‘\ ‘x
r-——-
ik ‘\ ‘k
‘\ ‘N ‘\ ‘
l 2 3 3

Figure 7 4: Vertical Toothpicks

There are 1 + 2 + 3 + 3 vertical toothpicks. The total number of
toothpicks is the sum of the horizontal and vertical toothpicks or
(1+2+2+3) + (1+2+3+3) which is 18 toothpicks. This can be re-written as
2(3) + 2(1+2+3) and generalized with n: 2n + 2(1+2+...+n).
1 + 2 + 3 + + n can be rewritten by pairing off the sums:
_ n+1

n+1
l u u

1 + 2 + 3 + + n-1 + n, this is g(nfl)

 

So the total number of toothpicks can be rewritten 2n + 26 (n+ 1)). In

simplified terms the total number of toothpicks = n2 + 3n.

238

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239

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