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A WEB-BASED TECHNOLOGY FOR FOSTERING INITIATIVE
By

Su C. Jones

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Counseling, Educational Psychology, and Special Education

1998

ABSTRACT

A WEB-BASED TECHNOLOGY FOR FOSTERING INITIATIVE

By

Su C. Jones

Educators have been interested in promoting student agency in the classroom and
in encouraging students to take more action in their own learning. This research
addresses this issue by examining student initiative and focusing on how to foster this
kind of learning in the classroom. In this investigation, I developed The Mathematics
Chronicles (TMC) based on design principles for promoting initiative--public audience,
collaboration, and norms and practices. Final interviews with students, observations of
student interactions in TMC publishing and non-publishing classroom contexts, and
samples of student work from TMC and other classroom assignments were analyzed.
Results indicate that how students take initiative vary across publishing and non-
publishing contexts. In addition, collaboration, norms, and practices in TMC
environment were salient themes emerging from the data as significant in fostering
student initiative. Students’ perception of an audience, however, was the most powerful
aspect of TMC environment that led students to take action, make an effort, and

demonstrate volition in their learning.

For my loved ones, Mom, Dad, and Danny

iii

ACKNOWLEDGEMENTS

My experiences in developing this research have been a great chapter in my life
and have been encouraged and nurtured by many significant people. Dr. Ralph Putnam,
as my advisor, has played a key role in stimulating my thinking and interests, particularly
in shaping and completing my dissertation. He has shared his expertise in teaching and
learning in the classroom and has always been supportive of my goals and ambitions. He
has given much of his time and energy to assist me in developing my expertise and in
preparing me for my career. I will be forever thankful for Dr. Putnam’s genuine interest
and guidance throughout my endeavors at Michigan State University.

Mr. Watkins, the teacher I collaborated with for most of my studies, has shared
invaluable experiences as a teacher that steered my interests and philosophy on
education. I could not have completed my work without his contributions and am
thankful for his support.

I also want to thank my dissertation committee members, Dr. P. David Pearson,
Dr. Yong Zhao, and Dr. Dan Chazan. They have shared insights and expertise that have
been signiﬁcant in fostering my research skills and dissertation ideas.

Making it through graduate school has been a learning experience that I could not
have had without my friends, Becky Wai-Ling Packard and Valerie Worthington. They
have often volunteered their time to listen to my interests and to experience all of the
wonderful and excruciating moments in graduate school. I also want to thank Michael
Zsoldos for listening to my new ideas, frustrations, questions, and triumphs. I want to

thank them for their true friendships.

iv

Finally, I want to thank Dr. Evelyn Oka for inviting me to Michigan State

University.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................... ix

LIST OF FIGURES .................................................................................. x

INTRODUCTION ..................................................................................... 1
CHAPTER 1

UNDERSTANDING INITIATIVE ................................................................. 4

Encouraging Student Agency ............................................................... 4

The Role of Educational Technology ...................................................... 6

Focusing on Student Initiative in Leaming .............................................. 10

Engineering a Technology to Augment Student Initiative ........................... 13

Norms and Practices of the Community ....................................... 14

Collaboration ...................................................................... 15

Publishing to a Public Audience ................................................ 16

What Is Initiative and Howls It Supported? ........................................... 17
CHAPTER 2

METHOD ............................................................................................ 19

The Mathematics Chronicles, an Interpretation of the Design Principles .......... 19

Norms and Practices of the Community ........................................ 20

Collaboration ....................................................................... 22

Publishing to a Public Audience ................................................. 23

Why the Web? ............................................................................... 23

A Closer Look at The Mathematics Chronicles ........................................ 25

The Context. ................................................................................ 31

Nonpublishing Activities ......................................................... 33

TMC Environment ................................................................ 34

Publishing on The Mathematics Chronicles ................................... 35

Data Sources ................................................................................. 36
CHAPTER 3

FINAL INTERVIEWS ............................................................................. 39

Question 1: Do you think that publishing journals on the web is different from

writing in your math journals? How? ................................................ 40

Question 2: Do you think what you write is different on the web versus what you

write in your individual journal? How? ............................................ 42

Question 3: Would you rather write in your journal or on the web? ................ 43

Question 4: You know how we have a new student in this class or choose one of
your friends who is not in this class. How would you explain what the web
journal is all about to him or her? ................................................... 44

Question 5: What is the purpose? ........................................................ 45

vi

Question 6: What are important things to remember while publishing? Things that
you shouldn’t do? ...................................................................... 46
Question 7: Now I am going to ask you about how you like publishing or how you
ﬁnd it interesting, whether you like publishing or ﬁnd it interesting. So what
do you think about publishing? ....................................................... 48
Question 8: How do you like publishing compared to other activities? ............ 49
Question 9: Do you think publishing is different from other activities that we do
in class (makes it fun)? How?
(Probe) Next year, we are thinking about having publishing but trying to
understand whether we should have it. Do you think we should have it? Why
should I have publishing? or what is it about publishing that you like?

(Probe) How is it fun? What makes it fun? ......................................... 50

Question 10: Do you think that publishing on the web leads you to do math or

makes you want to do math more? Why? .......................................... 52

Question 11: Have written in journals before? Published before? Read other

student’s journal entries on TMC? ................................................... 53

Summary .................................................................................... 54
CHAPTER 4

OBSERVATIONS .................................................................................. 55

Developing a Rubric for Initiative ....................................................... 56

Snapshots of the 6 Targeted Episodes ................................................... 57

Understanding the Specific Aspects of Mathematical Learning ..................... 66

Mathematical Sense-making ..................................................... 66

Collaboration ....................................................................... 77

Other Management Tasks ......................................................... 85

Trends in How the Initiative Is Characterized Across the Contexts ................. 88

Summary ..................................................................................... 91
CHAPTER 5

COMPARING STUDENTS’ WORK ON TMC AND IN INDIVIDUAL JOURNALS..92

Developing a Rubric ....................................................................... 93

Comparing Entries from TMC and Individual Journals .............................. 94

Understanding the Components of TMC and Individual Journal Entries ........... 95

Problems From Class or Homework Assignments ............................ 97

Main Idea for Current Lesson .................................................. 100

Strategies for Solving Problems ................................................ 102

Example Solutions to Problems ................................................ 105

Explanations of Ideas, Strategies, or Example Solutions .................. 107

Summary .................................................................................... 1 13
CHAPTER 6

WEAVING AN UNDERSTANDING OF INITIATIVE .................................... 114

What Does Initiative Look Like? ....................................................... 115

Initiative in Interactions in Publishing and Nonpublishing Contexts. . ...115

Initiative in the Written Work .................................................. 121

vii

Extending the Understanding of Initiative and Its Relationship to Prior

Research ...................................................................... 121

Types of Initiative Vary Across Contexts .................................... 124

How TMC Environment Supported Student Initiative ..................... 128

Perception of an Audience ............................................. 128

Collaboration ............................................................ 134

The Chain Linking the Themes: Norms and Practices ............ 136

Summary .................................................................. 138

CHAPTER 7

CONCLUSIONS .................................................................................. 139
Implications ................................................................................ 139
Limitations and Future Directions ...................................................... 140

The Impact of TMC in Mr. Watkins’ Classroom ..................................... 142
APPENDICES ..................................................................................... 144
Appendix A Interview Protocol ......................................................... 145
Appendix B Chronological Order of Student Responses ........................... 146
Appendix C Responses According to Questions ...................................... 168
Appendix D Audience ..................................................................... 184
Appendix E Teaching and Learning .................................................. 186
Appendix F Ownership .................................................................. 188
Appendix G Collaboration ............................................................... 189
REFERENCES ..................................................................................... 190

viii

LIST OF TABLES

Table 1. Distribution of Responses on Writing in The Mathematics Chronicles Versus
Individual Journals as Same or Different .................................................... 40

Table 2. Distribution of Responses on Same or Difference of Content Between The
Mathematics Chronicles Versus Individual Journals ....................................... 42

Table 3. Distribution of Students’ Preference of Writing in Individual Journals or on The

Mathematics Chronicles ....................................................................... 43
Table 4. Distribution of Responses on Purpose of The Mathematics Chronicles ......... 46
Table 5. Distribution of Preference to Publish or Participate in Other Activities ......... 49

Table 6. Distribution of Responses on Whether Publishing is Same or Different From
Other Classroom Activities .................................................................... 51

Table 7. Distribution of Responses on Whether Publishing Encourages Doing More
Mathematics ..................................................................................... 52

Table 8. Distribution of Experience with Journals or Publishing and Whether Have Read

Others’ Work ..................................................................................... 53
Table 9. Different Types of Student Initiative in Classroom Interactions ................... 57
Table 10. Snapshots of Nonpublishing Cases ................................................... 58
Table 11. Snapshots of Publishing Cases ....................................................... 63
Table 12. Frequency of Initiative Across Cases ................................................ 89
Table 13. Percentage of Journal Entries Including Each Component ....................... 94

ix

LIST OF FIGURES

Figure 1. Login page for users to enter the site with a usemame and password ........... 26

Figure 2. Editing page for students to add entries to The Mathematics Chronicles. . ......27

Figure 3. Outline Of The Mathematics Chronicles into units and investigations ........... 27
Figure 4. Investigations included main ideas, solution strategies, and reﬂections. . . ...28
Figure 5. Response page for students to modify or comment on entries .................... 29
Figure 6. Interface enables visitors to e-mail comments or feedback to authors .......... 30
Figure 7. Shanon and Mandy’s responses ....................................................... 60
Figure 8. Tara’s responses ......................................................................... 61
Figure 9. Bali’s responses ......................................................................... 61
Figure 10. Ande’s responses ...................................................................... 62
Figure 11. Tinse’s responses ...................................................................... 62
Figure 12. Comparing fractions with pie graphs ............................................... 65
Figure 13. Jeff’s entry includes a general statement about the day’s activities ............. 96
Figure 14. Tara’s entry includes a general statement of the day’s activities ................ 96

Figure 15. Shanon and Mandy’s entry including problems from class or homework
assignments ...................................................................................... 98

Figure 16. Mandy’s entry including problems from class or homework assignments. . ..99
Figure 17. Canton and Justin’s entry including main idea for current lesson. . . . . . . . .....101
Figure 18. Lin and Camm’s entry including main idea for current lesson ................. 101
Figure 19. Tinse and Camm’s entry including strategies for solving problems. . . . ......103
Figure 20. Pie graph representing 25% and 75% ............................................. 103

Figure 21. Hazel’s entry including strategies for solving problems ....................... 105

Figure 22. Shanon and Mandy’s entry including example solution ........................ 106

Figure 23. Lin’s entry including example solution ........................................... 108
Figure 24. Dana’s entry including explanation ................................................ 110
Figure 25. Lynton and Tinse’s response to previously written entry ...................... 112

Figure 26. Bali’s entry providing a solution and explanation for how to add fractions
when the denominators are different when the denominators are different, and
Lynton, Tinse, and Justin’s response ........................................................ 112

xi

INTRODUCTION

It was a mathematics classroom at Gayle Middle School, a Midwestern
urban setting. Sixth graders sat in groups, as Mr. Watkins (pseudonyms have been used
for all participants and the school) had decided, to allow for more collaboration on
activities. For some, this was an incentive to participate in their mathematics learning
and make efforts to engage in small and whole group discussion. There were days,
however, when Mr. Watkins wondered when students would take more responsibility for
their learning and demonstrate some initiative. At times, students glanced away hoping
the teacher would not call on them to explain their solution strategies because they failed
to complete their assignment the night before. On other occasions, Mr. Watkins left the
day with disappointment because only one student offered her explanations without any
objections or alternative ideas from other classmates. He looked forward to the instances
when students would determine their assignments, develop different ways of thinking
about mathematics, and challenge peers’ arguments. Although some students provided
their own different justiﬁcations, Mr. Watkins expected more from his learners. “My
goal is to have the students rely on themselves instead of always coming to me for how to
solve a problem,” remarked Mr. Watkins. Indeed, as an Observer in this classroom, I
noticed events when students took on a role in their own and others’ learning as well as
moments when they did not seem to realize that learning is held to be an active process
rather than a passive one.

This issue of how students might be more active in their learning and how it could
be supported is not unique to this classroom, but is a salient theme in the broader

educational community. In this study, I ventured to address this problem of student

initiative in learning and how technology can be a medium for fostering initiative by
trying to understand how students take initiative and what factors might support it. I
developed a Web-based collaborative journal according to design principles for fostering
initiative as a context for understanding initiative, but also examined students’ work and
interactions in activities outside of this technology as a comparison and broader
exploration of the issue. The Web-based journal and activities outside of this journal were
anchored by the Connected Mathematics Curriculum (CMP) (Lappan, Fey, Fitzgerald,
Friel, & Phillips, 1996a), an innovative mathematics curriculum especially designed to
encourage students’ development of connecting mathematical ideas and thinking about
mathematics in different ways. Participants were from a middle-school mathematics
classroom and took part in this study for most of the school year.

In Chapter 1, I deﬁne initiative and review previous research literature on issues
related to student initiative. I also present the design principles that led to the
development of a Web-based collaborative journal, The Mathematics Chronicles (TMC),
to provide an understanding of the underlying assumptions of the development of the
environment. Finally, I introduce specific research questions for the study.

Chapter 2, the method of this study, includes discussion of the technology, the
context and participants, and data sources. A short description of TMC and explanation
of how the design principles were interpreted in the development of this technology is
given, as well as a justification for why this technology was built on the Web. Following
a more detailed look at TMC and the classroom participating in this study is an
explication of how TMC was introduced and how students generally participated in

writing in TMC. A description of regular classroom activities is also given to illustrate

students’ typical mathematical experience outside of TMC activities. Data sources are
then provided.

Chapters 3 through 5 examine the data. The fourth chapter reviews final
interviews conducted with students. The ﬁfth chapter targets student interactions in and
out of TMC environment, and the sixth chapter reveals a comparison of students’ work
on TMC and work in their own journals. Whereas the fourth chapter illuminates what
factors may foster student initiative in TMC environment, the ﬁfth and sixth chapters
illustrate how students took initiative.

Chapter 6 highlights themes across the interviews, observations, and journal
work. Data from these sources complement each other to produce a spectrum of themes
that provide an overall picture of the initiative occurring in this research and factors
promoting the initiative. In Chapter 7, I examine the implications and limitations of this

study, and I make my conclusions in Chapter 8.

CHAPTER 1

UNDERSTANDING INITIATIVE

In traditional mathematics schooling, there is an assumption that students
should acquire a body of knowledge from an authority such as the teacher (Schoenfeld,
1989). Studies indicate, however, that people Often do not use and learn mathematics in
the standard forms traditionally taught in school. Rather, learners may develop different
ways of thinking and using mathematics in out-of-school contexts (e.g., Millroy, 1992;
Nunes, Carraher, & Schliemann, 1993; Resnick, 1987). In recognizing this discontinuity,
reform efforts have emphasized the importance of teaching our students so that, instead
of “learning mathematics as accumulating facts and procedures,” (NCTM, 1991, p. 2),
they can construct their own meanings through examining, representing, transforming,
solving, applying, proving, and communicating (National Research Council, 1989).
Implicit in this goal is that educators and researchers should foster learners who take
more agency in their learning of mathematics. The agenda, then, is not simply to teach
students bodies of knowledge, but to prepare them to approach new mathematical

situations by formulating their own ways of thinking and solving problems.

Encouraging Student Agency
Researchers have been studying the broad issue of how to encourage students to
take more agency in their learning from a multitude of angles. Some researchers and
educators have been investigating innovative instruction that may support students in
developing their own ways of thinking about mathematics. In contrast to teaching

students a preconceived set of mathematical representations (Nesher, 1989), Ball (1992)

demonstrates how students and teachers can jointly construct mathematical
representations. In this kind of setting, students take a more central role in their own
learning by contributing their ways of thinking about mathematical ideas. Also moving
away from a traditional classroom where students rely on an authority rather than
themselves for mathematical reasoning, Lampert (1987) brings the students’ ideas to the
fore to “sanction students’ intuitive use of mathematical principles as legitimate” (p.
339). Lampert argues that highlighting students’ thinking supports students in deciding
for themselves whether something is mathematically reasonable. Likewise, Schoenfeld
(1989) illustrates how he deﬂects the teacher’s authority by asking students what they
think and “developing in the class the critical sense of mathematical argument that leads
it, as a community, to accept or reject on appropriate mathematical grounds the proposals
made by class members” (p. 362). i

Cobb, Yackel, and Wood ( 1992) also address the concern of how students
actively learn, emphasizing the role of socio-mathematical norms in encouraging students
to work cooperatively in mathematical meaning-making. In these classroom
communities, students are acculturated to the norms of formulating different
mathematical arguments and justifications. Indeed, research results suggest that students
from these experimental classrooms are more intellectually autonomous (Yackel, Cobb,
& Wood, 1991) and tend to believe that it is more important to “ﬁnd their own or
different ways to solve problems, rather than conforming to the method shown by the
teacher” (Wood & Sellers, 1996, p. 351). Instead of turning to the teacher as the source of
mathematical knowledge, students are expected to take a more central role in the

development of their knowledge by generating their own arguments and explanations.

The Role of Educational Technology

In addition to developing instructional methods, researchers have used new
technologies as tools for promoting students’ agency in their learning.

Success in cognitive functions such as reasoning, remembering, and

perceiving is understood as an achievement of a system, with

contributions of the individuals who participate, along with tools and

artifacts. (Greeno et al., 1996, p. 20)

Along similar lines, researchers have developed technology to represent (Pea, 1993) and
augment students’ own thinking. In this sense, technology is used to support students’
role in the system.

The Geometric Supposer (Schwartz, 1993) is a visual software package designed
to enable students to take a significant role in their knowledge development. Instead Of
presenting a problem for the student to solve, an underlying assumption of The Geometric
Supposer is that the learner will create her own mathematical problems through
formulating conjectures. By proving and generalizing these conjectures from specific
cases, the learner generates new understandings (Schwartz, 1993).

Whereas The Geometric Supposer enables students to take more agency in their
learning by supporting students’ development of their own problems to explore, The
Jasper Series, a collection of interactive laserdiscs, presents mathematical problems
embedded in complex, realistic settings (The Cognition and Technology Group at
Vanderbilt [CTGV], 1991, 1992). Technology in this setting is used as a rich medium for
conveying information and posing problems for the students. A central principle of this

series, however, is that students should synthesize the appropriate information from the

laserdiscs or other resources to formulate their own solutions. In this sense, The Jasper

Series continues to promote researchers’ goals of supporting students’ generation of their
own solution methods.

Whereas The Jasper Series has been designed as a source of information students
can search and review in problem solving, the Kids as Global Scientists Project (Songer,
1996) illustrates how students use the Internet, as a resource to research and explain
meteorological problems and topics that students themselves generated. In this setting,
the teacher’s role as the sole knowledge resource is diminished as students assume
greater responsibility by pursuing their own inquiries using the Internet as a resource
(Songer, 1996). Songer’s study illuminates how students take agency in their learning, as
they become experts who collaborate with other learners from remote sites.

Prior to Songer’s work, Brown (1992) proposed collaborative learning that also
capitalized on distributed expertise in intentional learning environments where students
are encouraged to “engage in self-reﬂective learning and critical inquiry” (p. 149). “In
this kind of community, each learner has ‘ownership’ of certain forms of expertise but no
one has it all. Responsible members of the community share the expertise they have or
take responsibility for finding out about needed knowledge” (Brown et al., 1993, p. 199).
As in Songer’s (1996) study, students became experts on various topics as they used the
computer to communicate and share their knowledge. However, students in Brown et
al.’s (1993) communities of learners developed collaborative software that organized
their knowledge as a hierarchical system of topics, themes, and list of resources. As in
Songer’s study, the computer enabled students to communicate, but here was specifically
used as a tool for intentional reﬂection, learning, and collaboration (Brown et al., 1993)

that resulted in a tangible product reﬂecting students’ knowledge development. Despite

the differences between Songer’s and Brown et al.’s specific goals, both researchers were
promoting the students’ role in their own learning.

Scardamalia and Bereiter (1994) have also explored ways to foster intentional
learners with the development of computer-supported intentional learning environments
(CSILE). In this context, learning is the primary goal rather than solely an incidental
outcome of students’ activities (Bereiter & Scardamalia, 1989). Students pose their own
questions and collaborate in the development of knowledge surrounding these questions.
A primary goal of this environment is to support students’ intentional learning by
requiring them, instead of the teacher or a computer, to carry out the diagnostic and
decision processes when encountering problems (Scardamalia & Bereiter, 1994). Central
to this idea of intentional learning is that students take more agency by not only being
able to ask questions in the classroom, but by “growing into a question-asking role”
(Scardamalia & Bereiter, 1991, p. 51).

Just as CSILE affords students’ choices in learning by allowing them to pose the
questions to pursue, TELE-Web, a Web-based literacy environment I have been
collaborating with others to develop, supports students in making decisions in their
reading and writing activities (Jones, Ferdig, & Zhao, 1998; Zhao, Englert, Jone’s, Chen,
& Ferdig, 1998). TELE-Web is a complex, interactive collaborative environment
designed to support students’ learning to read and write through a variety of activities.
Unlike other technologies that promote students’ agency in their learning, TELE-Web is
a dynamic web environment especially designed to delegate more responsibility to
students by allowing them to choose their audience, individualize their exercises, use or

ignore prompts, and create their own topics for discussion. Capitalizing on recently

developed Web-based technology, TELE-Web is another innovation that represents the

creativity and exploration of researchers who are striving towards developing a

community of learners.

In sum, an enduring theme in education has been how to promote student agency

in learning. In addressing this issue, researchers have suggested instructional and

technological solutions, but vary in the role they envision for students. Researchers have

presented numerous ways that students can participate in their learning to enhance their

agency:1

jointly constructing mathematical representations (Ball, 1992),

relying on themselves for reasonability of their mathematical thinking
(Lampert, 1987; Schoenfeld, 1989),

formulating their own mathematical arguments and explanations,
(Cobb et al., 1992; Wood & Sellers, 1996),

generating their own solutions for solving problems, rather than
conforming to the teachers’ methods (CGTV, 1991),

becoming experts who have gained “ownership” of their knowledge
through researching resources and collaboration (Brown et al., 1993;
Songer, 1996),

constructing and pursuing their own and other students’ questions
versus those asked by the teacher or another authority (Scardamalia &
Bereiter, 1994), and

taking more responsibility in their learning by choosing audience,
individualizing exercises, using prompts, and creating their own topics
(Jones, 1997a)

 

1These points are not mutually exclusive; references indicate ideas emphasized in
particular writings.

Although substantial research has examined instruction and technology that can
support students’ agency in their learning, this issue continues to be a prevalent theme in
education. Previous research I have conducted suggests that students, at times, generate
their own ways of completing mathematical activities, but reveals that students still have
difﬁculty with relying on themselves in approaching and making sense of mathematical
problems. In one study (Jones, 1998), I examined student-teacher and student-student
interactions in a 6‘h-grade mathematics classroom to investigate what the kinds of
interactions that seemed to lead students to take more responsibility for their learning.
Results of this work indicate that teachers’ questioning created a social expectation for

students to rely on themselves to generate mathematical ideas or make process decisions.

Focusing on Student Initiative in Learning
Encouraging students to take agency in their learning is a common thread running
through much research, but various researchers hold forth different conceptualizations of
the kind of learner they are promoting:

° learners who can rely on themselves in making sense of mathematical
problems (Lampert, 1987; Schoenfeld, 1989),

0 intellectually autonomous learners - students who are self-goveming
(Kamii, 1994) and think for themselves in mathematical knowledge
development (Cobb et al., 1992; Wood & Sellers,1996; Yackel et al.,
1991), .

0 problem-posers who generate their own conjectures to pursue
(Schwartz, 1993),

0 learners who can solve complex mathematical problems (CT GV,
1992)

10

° learners who develop their expertise through collaboration with other
learners and use Internet resources to conduct investigations of
genuine questions, rather than using the teacher as the primary
knowledge source (Songer, 1996), and

° intentional learners whose goal is to learn (Brown et al., 1993;
Scardamalia & Bereiter, 1994).

My research presents another perspective, students taking initiative in their
learning, which continues the examination of the broader issue of how to enhance
students’ agency. I chose the term initiative because it connotes ownership, volition,
agency, and action on the part of the student. Previous researchers have focused much of
their attention on particular kinds of interactions in mathematics learning. My research
builds upon and extends this work because initiative encompasses all of these kinds of
interactions, and emphasizes students’ action in their learning.

Not the same as agency in learning, initiative is one way that a leamer’s agency is
promoted. For example, students’ agency in learning mathematics with The Jasper
Series is supported because they are encouraged to generate their own solutions, rather
than being given a solution method to follow. One student may take initiative by
experimenting with different solution methods, whereas another student may wait for his
partner to ﬁnd the right strategies and then ask him what his solution is for the problem.
Both students are given opportunities for agency in their learning, but how they act upon
this opportunity determines whether they are indeed taking initiative.

In addition, student initiative emphasizes different aspects of a student’s learning
than those studied by other researchers. Cobb et al. (1992) and Yackel et al. (1991) stress
the importance of developing intellectually autonomous learners who take responsibility

for their learning. In their work, they foreground the issue of learners’ development of

11

arguments and explanations and construction of taken-as-shared meanings. I suggest that
students in these settings were encouraged to take initiative in mathematical sense-
making and problem solving. My examination, however, also highlights students’
initiative in other domains of mathematics learning - making decisions about what
resources to use and what procedures to follow in completing a task. Although Cobb et
al. (1992) and Yackel et a1. (1991) may also support initiative in these domains, I
foreground them to broaden the definition of student initiative and of mathematics
learning.

Student initiative in learning includes attempting to make sense of mathematical
problems (Lampert, 1987 ; Schoenfeld, 1989), but is not necessarily intentional in the
sense of the student explicitly holding learning as their primary goal (Brown et al., 1993;
Scardamalia & Bereiter, 1994). Intention to learn may be an incidental outcome, rather
than a conscious goal for a person taking initiative in her learning. The student’s
principal objective may not be to learn, but to simply present her own ideas. Initiative is
better characterized by a leamer’s volition in mathematical thinking and decision-making
in an activity that may not necessarily include learning as the explicit goal. Rather than
simply accepting the knowledge transmitted by the teacher or waiting for the teacher’s
decision about the one right answer or way to complete a problem, students taking
initiative in their learning, as I have defined it, are also making decisions and acting upon
them. This perspective, illustrated by students who make an effort or willfully act, is the
focus of this research. Although students asking for the teacher’s help in solving

problems or completing a task may be taking agency in their learning or intending to

12

learn, initiative is justiﬁed as taking a step beyond first turning to the teacher for
direction.

Student initiative can appear in various domains of mathematics learning. I
suggest it can be demonstrated in:

' mathematical sense-making,

' participating in group processes, and

0 management tasks
Student initiative in learning does not include:

0 waiting for the teacher or peer for prompts to complete an activity, or

0 asking the teacher for the one way to approach or solve a mathematical
problem

Although technology may afford different kinds of learning, little exploration of
how to design technologies to support student initiative in mathematics learning has taken
place. My research agenda extends this line of inquiry. Like Brown (1992), I “attempt to
engineer” an innovative technology designed to support student initiative and at the same
time study this tool. I also examine the salient factors of this environment that may or
may not foster initiative in mathematics learning to understand how technology may be

designed to support and enhance these kinds of interactions.

Engineering a Technology to Augment Student Initiative
Engineering an environment that fosters initiative in the classroom requires

thoughtful consideration of a sundry of theoretical premises. I draw from three principles

13

to design a technology to foster initiative:

1. Researchers have argued that norms and practices significantly shape how
learners participate in the classroom community (e.g., Ball, 1992; Cobb et al., 1992) as
well as in other settings (Lave, 1988; Millroy, 1992; Nunes, Schliemann & Carraher,
1993). Thus, norms and practices of the classroom community that encourage student
initiative in learning should be established.

2. Opportunities for students to offer their own thinking arise when they
encounter conﬂicting ideas of other learners (Leikin & Zaslavsky, 1997, Yackel et al.,
1991); therefore students should be encouraged to collaborate.

3. Students who present their work to a public forum (Lampert, 1987) and to a
wider audience than the teacher (Cohen & Riel, 1989) tend to be more explicit and
generate their own thinking. As a result, a technology, which affords publishing to an
audience outside the classroom, as well as the learners in the classroom, should be

developed to afford student initiative in elaborating their thinking and ideas.

Norms and Practices of the Community

Participating “in social practices plays a crucial role in all aspects of a student’s
learning in and out of school” (Greeno, Collins, & Resnick, 1996, p. 20). Brown (1993)
attempted to create communities of learners in which students and teachers felt individual
responsibility in communal sharing of their knowledge. In these communities, each
person was viewed as an expert, and the norms and practices entailed developing one’s
expertise and sharing of knowledge with other learners. With the support of the teachers’
modeling, students assumed roles as experts and were acculturated to the normsand

practices of these communities of distributed expertise.

14

Contrasting with Brown’s communities, which were speciﬁcally designed to
foster cooperation, classrooms in which the norms and practices were inextricably
mathematical activities have been engineered (Yackel et al., 1991; Wood & Sellers,
1996). Students in these classrooms were enculturated into developing their own ways of
thinking about mathematics, offering peers’ help in making sense of their solution
methods, and providing arguments and explanations. The teacher did not reward or
punish students for failing to participate in these ways of learning. Rather, she guided the
establishment of the community practices by initiating a discussion of the expectations
during group interactions and of spontaneous cases when students were or were not
attuned to the practices. Indeed, students not only indicated the importance of these
practices, but also emulated them in cooperative learning environments (Yackel et al.,

1991)

Collaboration

Collaboration has been the core framework for innovative instructional practices
that afford student initiative in learning in many different settings. Leikin and Zaslavsky
(1997) report that students who were collaborating had more opportunities to offer their
ideas to help others. Likewise, Yackel et al. (1991) argue that in the collaborative
environments fostered in their classrooms,

opportunities arise naturally for [students] to verbalize their thinking,

explain or justify their solutions, and ask for clarifications. Further,

attempts to resolve conﬂicts led to both the opportunity to reconceptualize

a problem and thus construct a framework for another solution method,

and the opportunity to analyze an erroneous solution method and provide a
clarifying explanation. (p. 401)

15

Collaboration afforded student initiative in these classrooms by providing
students appropriate contexts to Offer their ideas and feedback. In addition, encounters
with peer disagreement encouraged them to reforrnulate and clarify their thinking. A
student who explains his thinking to a partner presenting an erroneous answer must
extend his “own conceptualization of the problem to try to infer how the partner might
have been trying to make sense of the problem" (Yackel et al., 1991, p. 402).
Collaborative work not only allows students to offer their own thinking, but also provides
grounds for generating new arguments and explanations. Therefore, collaboration affords
student initiative in learning by providing opportunities for students to present and
generate their thinking.

More recently, technology has taken a central role in maintaining and extending
collaboration in learning environments in which students choose to pursue their own
questions or area of expertise. CSILE (Scardamalia & Bereiter, 1994) has been used to
support students’ collaboration, but has afforded student initiative by enabling students to
propose questions to pursue and jointly construct ideas in answering these inquiries. In
addition, Brown et al. (1993), Songer (1996), and Jones (1998) have illustrated how the
Internet can be used to enable students to collaborate with others in cultivating their
learning. Technology in these contexts has augmented students’ initiative in learning by
allowing students to both choose from a wider audience with more diverse expertise and

to contribute their own knowledge.

Publishing to a Public Audience
Publishing to a public audience, the third design principle, enhances initiative in

learning in several ways. As discussed, students can publish their work for their own

16

classmates’ reﬂection and review. However, one potent factor is that the audience
potentially extends beyond the classroom. Rather than completing their assignments or
doing mathematics only for the teachers’ review and approval, students can publish to a
wider audience of other children who can learn from the journal, other teachers who can
generate ideas about how they can teach, and parents who can understand how students
are learning. Publishing to a wider audience may lead students to articulate their ideas
better, thus enhance their writing. Cohen and Riel (1989) illustrate that learners writing
to a real audience who were interested in more than grading students’ work wrote more
ﬂuent, better organized compositions with ideas more clearly stated and supported. With
the perception that the audience would not have the knowledge base pertaining to their
composition topic, students writing to distant audiences seemed to elaborate on their
ideas more. Publishing to a public audience, in this sense, affords student initiative in
learning because through attempting to clarify ideas, students are more likely to develop
and make their thinking explicit as they present their work. Although the students in this
study were not necessarily writing on mathematics, its implications may apply to a

mathematical context as well.

What Is Initiative and How Is It Supported?

This study investigates student initiative through several strands. First, I develop
and study an innovative publishing environment on the Web, The Mathematics
Chronicles (TMC), which is based on these principles for fostering student initiative in
learning mathematics. I also examine the kinds of initiative students demonstrate in and

out of the publishing environment and which activities tend to support these kinds Of

17

initiative. My research questions are:
1. What kinds of initiative does TMC foster?

2. What factors of TMC publishing support initiative?
3. What kinds of initiative do nonpublishing activities support?

18

CHAPTER 2

METHOD

I have discussed three principles that motivated the design of a technology
I developed to study initiative. In this chapter, I begin with a detailed overview of this
technology and how illustrates these principles. I continue with a more detailed
description of the Mr. Watkins’s 6th grade classroom that participated in this work, and
how this setting met the needs of this research. Finally, I include the instruction around

the technology environment, and data analyzed for this study.

The Mathematics Chronicles, an Interpretation of the Design Principles
The Mathematics Chronicles (TMC), a Web-based interactive publishing

environment was designed to foster initiative. TMC serves as a collaborative journal
organized to be used in conjunction with the Connected Mathematics Project (CMP)
curriculum (Lappan et al., 1996a). The CMP curriculum consists of unit topics divided
into investigations focusing on subtopics. This Web-based journal supports students’
documentation of their major mathematical ideas and solution strategies in studying the
investigations of each unit and is structured to enable students to write entries—main
ideas, solution strategies, or reﬂections—for each of the investigations. TMC is
organized in this fashion to align it with the curriculum, but more importantly, to provide
a loosely structured and cohesive journal for easy navigation that allows ﬂexibility in the
content students develop. A more detailed overview of TMC will be provided later.
First, I discuss how TMC illustrates the three principles: norms and practices of the

community, collaboration, and publishing to public audience.

19

Norms and Practices of the Community

An underlying principle of TMC is that norms and “practices of a community
provide facilitating and inhibiting patterns that organize the group’s activities and the
participation of individuals who are attuned to those regularities” (Greeno et al., 1996, p.
20). The following norms and practices around TMC publishing were promoted to foster
student initiative:

1. Students are the authors. The Web-based Mathematics Chronicles is a product
of the students’ collaboration; therefore, the students are viewed as the authors. Since
students, rather than the teacher, take the role as the source of knowledge, they take on
greater responsibility for generating their own thinking and ways to solve problems. By
viewing themselves as the experts who are sharing their knowledge, students are more
likely to research the mathematics they would like to present to the audience (Songer,
1996). In this sense, with the understanding that they are the creators of The Chronicles,
I expect students to use this opportunity to take initiative in developing The Chronicles.

2. Students should take the initiative. The Mathematics Chronicles should
represent the diversity of students’ own mathematical learning; therefore, they are
expected to take the initiative in generating arguments, explanations, solution strategies,
and questions they would like to pursue, as well as deciding how to complete and publish
their work. Because the typical school rewards of grades is not given for publishing on
The Chronicles, students’ initiative in the development of their materials for publication
should not reﬂect students’ goal to attain a grade. Yackel et al. (1991) illustrates how

making expectations for classroom norms explicit can shape how students learn in the

20

classroom. Likewise, I suggest that presenting the expectations for student initiative
supports students in beginning to emulate this kind of learning.

3. Students need to collaborate. Students are expected to collaborate with peers
in developing and reﬁning questions, arguments, explanations, reﬂections, and solution
strategies that they would like to publish to The Web-based Mathematics Chronicle and
in making decisions as they complete their work. In collaborative settings, students tend
to offer their own thinking (Leikin & Zaslavsky, 1997; Yackel et al., 1991), and in this
sense, may enhance student initiative. I will detail the benefits of collaboration as I
explain its importance in the design of The Web-based Mathematics Chronicles below.

4. Students should use strategies for promoting collaboration and student
initiative. As advisors and authors, students are given the following questions to keep in
mind:

Is the work an example of the main idea of the investigation?
Is there evidence for what is being argued?

Can the work be changed to make it clearer?

Are there other ways to explain what is being argued?

Are there other suggestions for providing better evidence?

Collaboration and student initiative are supported through providing students with
questions that give them a sense of the kind of initiative in learning that is promoted in
the classroom.

5. TMC should represent diverse authorship. Students are asked and encouraged
to volunteer to publish their work. TMC is presented as a shared product that should
illustrate the diverse and valued knowledge of all the students.

6. Students should think about their audience when writing for TMC. Some

people may be reading the journal because they are interested in what the students are

21

learning (e.g., parents, teachers). Other readers may also learn from their work (e.g.,
students in and outside the class, teachers). Cohen and Riel’s (1989) study suggests that
students who believed that their audience did not have an understanding of their work
tended to write more highly rated papers that were more explicit and detailed than those
written for only the teacher. With the perception of a diverse audience who may or may
not already have a prior knowledge about the students’ work, students are expected to
take more initiative in clarifying their thinking and seeking resources that will support the

development of their publication.

Collaboration

Another important design principle of TMC is collaboration. Students are
expected to collaborate in pairs when developing work for the Web-based journal. As
advisors who encourage and support other learners, students can assist each other in
developing entries of an investigation for publication. Because students are expected to
negotiate their mathematical sense-making through collaboration, their initiative in
learning is fostered as they are given the opportunity to generate their thinking and offer
their suggestions to partners (Cobb, Wood, & Yackel, 1990).

TMC also promotes student initiative as a collaborative tool, a common space for
learning. On one level, students may use their own work or those of their partners to
initiate discussion of mathematical ideas or questions. On another level, however, TMC
can be used as a record or artifact of all students’ learning to reﬂect upon in formulating
thinking. As a public space for discussion, students can review other learners’ Work to
question, build upon, or seed their own learning (Brown, 1993). Initiative in thinking

may take the form of annotations to other students’ published work, offering of ideas

22

related to examples from the journal during whole class discussion, or providing
suggestions during one-to-one interactions with an author. In this respect, TMC is both a

means for and an Object of discussion.

Publishing to a Public Audience

Presenting a mathematical journal to a public audience can also support students’
initiative. I argue that in this Web-based publishing environment, students can take more
initiative as they reformulate their questions and ideas to better “teach” or illustrate to the
audience the mathematics they are studying. As important, another beneﬁt of TMC is
that it is a publication of the students’ thinking. The presentation of the students’ work
does not only give the learners’ thinking more equal weight to the teachers’ ideas, but
sanctions students’ intuitive use of mathematical principles as legitimate in a public
forum (Lampert, 1987). Since the students are given authority over how TMC entries is
developed and the thinking that they want to make public, they may demonstrate more

initiative in deciding what to publish and in the process of developing their publications.

Why the Web?

Publications can take many forms: newspapers, paper portfolios, a CD-ROM of
the collection of students’ work, and so forth. However, I decided to use a Web-based
environment because of the following unique features:

1. The Web is a multimedia tool. Unlike a text-based document, The Web-based
Mathematics Chronicles allows students to capture images, sounds, and videos to present
their work. Because students may have difﬁculty expressing their ideas solely through

text, multimedia options may enable students to communicate their thinking more easily

23

(CT GV, 1991). Difficulties in reading and writing may create obstacles for some
children trying to understand mathematical text. Multimedia, however, may support and
extend students’ mathematical experiences by enabling them to capture and reify their
learning. In addition, students who do not offer their mathematical ideas because of their
difficulty with reading or writing may take initiative in presenting their thinking when
provided with other ways of expressing themselves.

2. Publications on the Web are immediately distributed. Immediate distribution
is another important benefit of the Web. Instead of waiting to have their work processed
through a printing press for a newspaper or replicated as software to be distributed,
students can publish their work instantly. Publications can be posted daily, so the
audience can view the students’ work on an on-going basis. More importantly, students
can receive responses from peers and maintain a conversation about their mathematical
work. As with electronic mail, students can read and critique other learners’ work in
their spare time (i.e., after other class work is completed, during study hall, at home).
Ultimately, the goal is that because the Web serves as a publication that can be accessed
in and out of class, students can write mathematics entries outside of the regular class
period.

3. The Web allows for a common forum. As a collaborative product, The
Chronicles serves as a common forum for students’ presentations. Unlike individual
journals, this Web-based publication is a collection of students’ ideas and work. With the
ability to read and comment on other students’ work more immediately than with other
publications (e.g., newspapers, CD-ROMs), children can share their ideas with anyone in

the classroom. These conversations are documented as a reference, but presented as a

24

public record, so a child can collaborate with one, two or more students who are not
typically working together.

4. The Web saves time and money. Publishing work as CD-ROMs, newsletters,
or booklets may be less expensive than producing Web sites if a school does not have
Internet access or Web space. With Internet access and Web space, however, publishing
to the Web saves time because students can immediately post their work instead of
through the process of collecting samples, compiling them, and then submitting them for
print. If a classroom has already incurred the costs of a computer and Internet access, a
Web publication is the least expensive. Moreover, a wider audience can view the work
without the further costs that might be incurred in producing CD-ROMs or paper

publications.

A Closer Look at The Mathematics Chronicles

The Mathematics Chronicles (TMC) is an interactive Web-based journal that the
students create and develop. TMC is organized in parallel with the structure of the CMP
curriculum (Lappan et al., 1996a), as units and investigations within those units. Upon
entering the site (see Figure 1), the viewer was required to login with a usemame and
password. Students were given login names and chose their own passwords to access the
site as authors to ensure security of their own work. Visitors were advised Of a usemame
and password to use for viewing student work.

Students (authors) and the viewers (audience outside the classroom) had separate
interfaces. After logging in, students could create a new publication or read the journal
(see Figure 2). When editing, students needed to determine the unit, investigation, type

of entry, content of the publication, and whether to include any annotations (graphics,

25

sound ﬁles, or video clips). Annotations were organized on the school hard drive as well
as the Web server. In addition, students could choose to “read it from school.” This
option allowed them to view work that may have included multimedia with a shorter wait
time because these ﬁles were on the hard drive they were using at school and allowed
them to upload them more quickly. Otherwise, they could view it through the “journal
from home” option. Students were then provided a list of two units, Bits and Pieces I and
Bits and Pieces II, in addition to the investigations within those units (see Figure 3).
Authors were able to click on an investigation that they chose to read that was organized
according to type of publication: main idea, solution strategy, reﬂection (see Figure 4).
When reading publications, students click on response and provide the response as in
Figure 5.

The audience outside the classroom was not able to edit publications, but could
read student work. After logging in, they could choose from the list of units and

investigations as in Figure 3. Their interface, shown in Figure 6, enabled these viewers

 

the Ma thematics

A( hronicles

 

Figure 1. Login page for users to enter the site with a usemame and password.

26

 

Unit: r') Bits and Pieces II

 

0;) Using Percent;
\ .1 More About Percent:
I:_)EstiI'I'IIiting with Fractions and Decimals

 

 

InvestIgation. L} Adding and Subtracting Fractions
(_} Finding Areas and Other Products
': 2' Computing with Decimals
‘:_ : Main idea
What kind of r L" ' ' 'k' 2; IE)“
C :IFietlection

 

 

Please submit your work

 

 

..l i:

J

 

 

 

 

 

Other annotations [e g ,grophickfs wdioclips videoelips] "iorName of the File:
mug-nu: u.‘ t'he Investigation (e. 9., lber. ipg]

 

Title of annotation [Name of the lile
r——-—-— r-~—‘

 

I Submit |

Figure 2. Editing page for students to add entries to The Mathematics Chronicles.

 

 

 

 

The Mathematics Chronicles

Bits and Pieces I
' Fund-Raising Fractions
0 Comparing Fractions
' Cooking with Fractions
' From Fractions to Decimals
' Moving Between Fractions and Decimals
' Out of One Hundred

Bits and Pieces II
° Using Percents
- More About Percents
- Estimating with Fractions and Decimals
' Adding and Subtracting Fractions
' Finding Areas and Other Products

' Computing with Decimals

 

 

Figure 3. Outline of The Mathematics Chronicles into units and investigations.

27

Bits and Pieces I
Comparing Fractions

 

Main Ideas 1

 

 

 

 

 

 

 

Which is larger?
By Student A
Date, Time
Compare
Responses
Add a response
Solution Strategies ]
Benchmarks
By Student B
Date, Time

Problem 1 part a- You make a fraction strip labled the fractions 3/ 12 and 7/12. Then you
see what fraction is closer to one then the other. Then you see that the fraction 7/ 12 is
larger.

fraction

 

Responses

Add a response

 

 

 

Reﬂections I

 

Benchmarks
By Student C
Date, Time

Some numbers are closer to 0 and some are closer to 1. If a fraction is between 0 and 1/2
then it is closer to 0. For example, 3/12. Because 6 is half of 12 and 3 is less than 6. You
have 6/12. 6/12 is 1/2. 3/12 is closer to 0 because 3/ 12 is less than 6/12. It is closer to 0
than a half.

 

Responses

 

Add a response

 

 

Figure 4. Investigations included main ideas, solution strategies, and reﬂections.

28

. Visa-{313:3} I 2i,';3i.}i..2: (“at

Pleese submit your comments

 

 

 

 

 

 

 

Other ennotetions [e .g .,grephics, eudioclips, videociips] “tor Name of the File:
remember to begin the name with the number ot the investigetion (e .g ., tberbg]

litie of ennotetion | "me of the tile

i

 

 

 

 

Figure 5. Response page for students to modify or comment on entries.

29

Bits and Pieces I
Comparing Fractions

 

Main Ideas 1

 

 

 

 

 

 

 

Which is larger?
By Student A
Date, Time
Compare
Responses
Would you like to respond?
Solution Strategies I
Benchmarks
By Student B
Date, Time

Problem 1 part a- You make a fraction strip labled the fractions 3/ 12 and 7/12. Then you
see what fraction is closer to one then the other. Then you see that the fraction 7/12 is
larger.

fraction

 

Responses

Would you like to respond?

 

 

 

Reflections j

 

Benchmarks
By Student C
Date, Time

Some numbers are closer to 0 and some are closer to 1. If a fraction is between 0 and 1/2
then it is closer to 0. For example, 3/12. Because 6 is half of 12 and 3 is less than 6. You
have 6/12. 6/12 is 1/2. 3/12 is closer to 0 because 3/12 is less than 6/12. It is closer to 0
than a half.

 

Responses

 

Would you like to respond?

 

 

Figure 6. Interface enables visitors to e-mail comments or feedback to authors.

30

to read and respond to publications as well through electronic mail. The mail was

directed to me so that it could be filtered before presentation to the students.

The Context

Teachers’ knowledge and beliefs about learning heavily inﬂuences their
classroom instruction (Borko & Putnam, 1994). Therefore, a teacher’s perception of the
importance of student initiative in learning would inﬂuence how any proposed
instructional method or technology is integrated. If I conducted this study with a teacher
who did not support students’ initiative in learning, he or she may not have implemented
the use of TMC as I had designed it. As a result, I would be unable to attribute a lack of
student initiative to the technology or to the teacher’s belief that students’ initiative in
learning was not important. Therefore, it was important to conduct this research with a
teacher who wanted to encourage students’ initiative in learning. Mr. Watkins had this
goal.

This classroom site also had students who took initiative in their learning at times,
but continued to struggle with this kind of learning overall. Conducting research in this
site allowed me to understand what kinds of interactions were supporting student
initiative. Because the students in this classroom were not consistently taking initiative in
their learning already, I was able to study the impact of the Web-based publishing
environment. In addition, since students were sometimes taking initiative in learning, I
could better formulate how the technology and interactions around the technology might
be designed to support and enhance student initiative. An alternative would have been to
study a classroom in which the teacher was already implementing instructional methods

that successfully support initiative. In such a classroom, I might have learned more about

31

factors I could augment with technology. I might not, however, have found sufficient
variance in student initiative between the context of TMC and other activities to be able
to gauge the inﬂuence of TMC on initiative.

I have been collaborating with Mr. Watkins, a middle-school teacher who was
indeed interested in fostering student initiative in learning. He was looking for resources,
aids, and teaching techniques that could help him encourage learners to “rely on
themselves and their own resources” when encountering mathematical problems instead
of depending on the teacher to answer questions (Personal communication, September,
19, 1996). Mr. Watkin’s classroom was optimal for my research in many respects. The
teacher already valued learning in which students thought and decided for themselves and
was very excited about learning and integrating new technology and instruction which
would help him encourage student initiative in learning (Personal communication,
October, 28, 1997). Informal observations of one of Mr. Watkins’s 6th-grade classes and
discussions with him conﬁrmed that these students, at times, demonstrated initiative in
their learning and at others had difficulty doing so. In this sense, I could study more
closely how students took initiative as well as how TMC might foster initiative.

Mr. Watkins’s classroom, set in a Midwestern urban school, included 22 students,
17 who consented to participate in this study. Pseudonyms were given for the teacher
and students. This classroom had two computers for publishing TMC entries.

Students studied from the CMP curriculum (Lappan et al., 1996a). The CMP
curriculum is organized as units and investigations within those units. The units
represent domains of mathematics (e.g., fractions and decimals, geometry, probability)

and investigations represent particular lessons within those domains (e.g., relationship

32

between fractions and decimals, determining probability through experiments). The
CMP curriculum was designed to foster students’ development of connecting
mathematical ideas across domains within mathematics. Mathematical concepts are
presented through problems in realistic contexts ranging from considering discounts in a
store to determining how much of an ingredient to include in a brownie recipe. In
contrast to more didactic instruction, the CMP curriculum encourages students to develop
their own strategies for addressing problems and emphasizes the importance of diversity
in students’ thinking. Because the CMP curriculum is supportive of students’
development of thinking about mathematics in different ways, it was appropriate for

supporting the goals of this study.

Nonpublishing Activities

Class activities outside of TMC environment, nonpublishing activities, included
individual, whole group, and small group work. Typical instruction included whole
group discussion of an investigation or problems taken from the CMP curriculum, with
students presenting their questions, ideas, and strategies. During these whole class
activities, Mr. Watkins facilitated discussion on new or studied topics and led
examinations of homework or class assigned problems. Students were asked to talk
about their ideas and strategies for solving problems and to provide written explanations
or drawings on the overhead. These kinds of interactions were encouraged by Mr.
Watkins’ solicitation of students’ ideas and asking students whether they agreed or
disagreed with the ideas to highlight their authority in mathematics learning. Students
also worked in small groups or individually to complete assignments or review completed

work. Each investigation ended with a reﬂections exercise during which students wrote

33

general summaries of what they learned from the investigations and a Unit Project which

was the culmination of the ideas of the unit.

TMC Environment

Introduction to The Mathematics Chronicles. Mr. Watkins introduced TMC to
his students at the beginning of this study by outlining the discussed norms and practices
in his own words. The following were the norms and practices Mr. Watkins presented to
the class:

1. Authors are Students. The Web-based Mathematics Chronicles is a product of
your collaboration. YOU are the authors and creators of this journal.

2. Collaboration. You are expected to collaborate in small groups on the work
that you would like to publish. You will also be acting as advisors to each other.

3. Student Initiative. The Chronicles will represent the diversity of your
mathematical learning. You WILL NOT receive a grade for publishing. Therefore, you
are expected to take the lead in generating arguments, explanations, solution strategies,
questions, and deciding how to complete and publish your work.

4. Strategies for Taking Initiative and Collaboration. As Advisors and Authors,
keep the following questions in mind:

0 Is the work an example of the investigation?

0 Is there evidence for what you’re saying?

' Can you change anything to make it clearer?

' Are there any other ways that you can explain it?

0 Are there other suggestions for providing better evidence?

5. Diverse Authorship. You will be encouraged to publish your work. The

Chronicles is a shared product that should show the variety of thinking that we have here.

34

6. Audience analysis. The Chronicles will reﬂect all of our various mathematical
experiences. Whoever reads this journal might be interested in what you are learning or
in learning something new themselves. Keep your audience in mind as you write.

On the ﬁrst day TMC was presented, a whole-class discussion reviewed these
norms and practices to begin establishing the publishing activity. Students were
encouraged to contribute their explanations of each practice to involve them in
constructing a common understanding of how children can promote each practice during
publishing. For example, Mr. Watkins suggested what it meant to collaborate and for
students to take initiative in accomplishing their task, but asked for students’ ideas on
what they, themselves, thought about these norms and practices. This negotiation helped
us get a sense of whether students understood the norms and practices but also generated
students’ thinking about what each practice promotes. These norms and practices were
posted in the classroom so students could review it while publishing, and students were
also given their own copy of these norms and practices to keep in their notebooks for
reference. In addition, I reminded students of these norms and practices when students
published on TMC for the first time.

Publishing on The Mathematics Chronicles. Because students requested the
opportunity to publish on almost a daily basis, we kept a record of who had published to
ensure that all students had the opportunity to publish. Students chose partners to work
with while publishing. For the first month of this study, only one group was chosen to
publish each day, so I could provide enough support to establish norms and practices
around the publishing environment and assist students with technical problems (e.g.,

typing, logging in). The authors of the day sat at the computer and were also required to

35

participate and be responsible for material discussed and completed during regular
(nonpublishing) class discussion. Beginning the second month of the study, a laptop was
also used for students to create publications. Students could publish on TMC throughout
the 6 months of this study.

Students were expected to publish work they thought demonstrated what they
were learning in the investigations or annotations to their peers’ work. They could write
text, scan work, and create audio clips or video clips for publication on TMC. Because
establishing norms and practices is an ongoing activity (Yackel et al., 1991), students
were reminded of them when appropriate. For example, if a student asked me what they

should publish, students were reminded to ask their partner for help.

Data Sources

My general research goal is to understand how technology can support initiative.
To address this issue, I examined what kinds of initiative students take and what factors
of contexts support initiative. Data sources included observations of class activities in
and outside of TMC environment (audiotapes, videotapes, and field notes), samples of
student work, recordings (audiotapes and ﬁeld notes) of discussions with Mr. Watkins,
and ﬁnal interviews.

Observations included videotapes, audiotapes, field notes, and samples of student
work. Students who were observed were given a tape recorder and were asked to keep
the recorder with them during class activities (publishing and nonpublishing). The
videocamera was focused on as many of the observed students as possible and was used
primarily to capture the whole class discussions to set the context for an episode. Samples

of student work completed during these activities were also collected to provide a better

36

understanding of the object of discussion and students’ products. Observations of
nonpublishing activities included small group, whole group, and individual work.

In the ﬁrst few weeks of the project, observations of all students within the
nonpublishing context were collected. I videotaped and audiotaped 2-6 students a day
until all of the students were observed. In addition, videotapes and audiotapes of the
introduction to the norms and practices of TMC and students’ first experience with TMC
were recorded. Since I attended this class on almost a daily basis, these data were
collected throughout the 6 months of the study. Data included at least 3 observations of
each student participating in both publishing and nonpublishing activities in addition to
other data collected incidentally throughout the study. However, data analyzed for this
research was taken from end of the year observations of the students in each context
because it was at this point that students had experience in both contexts.

Other student work included TMC entries and each student’s individual journal
entries spanning the course of this project. These data were used to compare how
students took initiative in writing in TMC and their individual journals. TMC entries
were publications from the Web and individual journal entries were copied from
students’ spiral-bound notebooks. All 47 TMC entries were analyzed for this work and
represented publications made throughout the course of the project. To compare with
TMC entries, 44 of students’ individual journal were analyzed and were selected to also
represent work over the course of the project. To make my selections, I determined the
first and last dates of TMC entries and the date that fell in the middle of the entries to
represent key dates of the beginning, middle, and end of the project. Selected individual

journal entries were those closest to the key dates. On a few occasions, students

37

discontinued writing in their individual journals before the end of the project, so entries
representing end dates were not selected from these journals. Recordings of
conversations with Mr. Watkins were used to complement and explain observations and
student work. These data were used to understand in what ways students took initiative.

In addition, ﬁnal interviews with students were conducted at the end of this study.
The interviews were used to capture why students may or may not take initiative,
particularly in TMC environment. Questions targeted the students’ perceptions of TMC
and their comparisons of this environment to other mathematics activities (see Appendix
A). Topics included:

' how students believed writing on TMC was similar and different from
writing in their own journals or in class activities,

0 why students would rather publish on TMC or write in their journal,

0 why students would rather publish on TMC or participate in other
activities,

0 how students would describe TMC and its purpose,

0 why students found TMC interesting,

' why TMC might support them in doing mathematics more,
0 if students read other students’ TMC entries,

' if students have written in journals before, and

if students had published on the Web before this project.
Each student was interviewed separately in another room. Interviews were

audiotaped and transcribed for analysis.

38

CHAPTER 3

FINAL INTERVIEWS

In the following three chapters, I present data for how students took
initiative and themes that may explain how initiative was supported in the various
contexts. Chapters 5 and 6 reveal how students take initiative in publishing or
nonpublishing contexts and in writing in TMC or in their individual journals. I begin
with students’ ﬁnal interviews to present emerging themes that allude to why students
may take initiative. I then move on to Chapters 5 and 6 to continue with how students
took initiative to understand how students’ perceptions reﬂected in their interviews were
realized in student interactions in activities and in their written work.

The ﬁnal interview (Appendix A) was conducted during the last week of the
project with 16 of the 17 students. One student was absent during this period. In this
chapter, the description of students’ responses is structured around the interview as it was
generally conducted. The interviews with particular students, however, may have
departed from this structure as was appropriate at the time. When students’ thinking was
not apparent or clear during the interview, I asked prompting questions. At other times,
students had already addressed an issue, so I moved on to a different question. In this
chapter, I discuss the student responses and highlight themes with speciﬁc examples
representing these themes. A transcript of each student’s responses can be found in
Appendix B; a list of students’ responses organized by question appears in Appendix C.

The goal of the interview was to reveal:

0 how students would describe The Mathematics Chronicles (TMC),

0 students’ perceptions of its purpose,

39

0 whether and why they prefer the TMC over other mathematics
activities, and

0 how the Web journal compares with other activities.

Exploration of these issues delves into themes that, in the following chapters, will
be compared with and complemented by data from student interactions in and outside of
the TMC environment and by a comparison of student work from TMC and individual
journals. Here, ﬁnal interviews provide a window into students’ understanding and
beliefs about TMC to begin the examination of how TMC may support initiative.
Question 1: Do you think that publishing journals on the web is different from writing
in your math journals? How?

Table 1 illustrates the distribution of student responses to Question 1. Overall, the
majority of the students (75%) indicated that writing on the TMC was indeed different
from writing in their own journal.

Table 1. Distribution of Responses on Writing in The Mathematics Chronicles Versus
Individual Journals as Same or Different

 

 

n %_..__.,
Different 12 75
Same and different 2 12.5
Same 2 12.5

 

In addition, students gave several reasons for the difference:
1. The Web journal seemed to be different from individual journals for some

students because they perceived an audience. People besides the author or teacher could

4O

read students’ work because it was on the Web:

when you publish in your journal it’s just for you, but when you publish
on the Internet, it’s for everybody else. [Mandy, ﬁnal interview]

in the math journal, the only people who see it are you (Jones) and Mr.
Watkins, but on the Web, other kids who are having problems or

whatever, they can look on our sheet. [Camm, ﬁnal interview]

2. Other students enjoyed publishing their journals on the Web more than writing

in their own journals. Students repeatedly expressed that it was “ n ”.°

It’s more fun because you get to type on the computer, you are on the
Internet and put what you want to publish. [Lynton, ﬁnal interview]

I think because when you get an experience with typing and working on
the computer. It’s more fun. You get to learn how to publish things and
it’s more exciting. We can look at it from home and look over it. It seems
better than doing it with paper and pencil. [Justin, final interview]

3. Students also compared the difference in the content of the two kinds of
journals. They indicated that in their own journals, they write what they did in class, but
on the Web, provided more detailed explanations of the mathematics they were learning:

Yeah, because when you are on the computer, you talk about how you do

it and how you get the answer, but in your journal, you write what did you

do in class. Like, in class we did 3/4 and 1/5 and we learned how to do

times. On the web, it’s about the same thing, but you talk to other people

and tell them how to do it. [Ande, final interview]

because everybody can read it. You write all the steps and stuff. You

tell the solution strategy. In the journal, you just write what you did that

day and an example. [Jake, final interview]

The following are the responses that 2 (12.5%) students gave indicating that the
TMC was both similar to and different from their individual journals:

Sort of different because I write more in my journal, but not as long. But it

might be the same because I don’t write much in my journal, either. [Jeff,
final interview]

41

It’s different because. Well, it’s the same. I write about like what we did in
class, homework assignments in my journal, well, I don’t write homework
assignments on the web. It’s sort of both, same and different. [Kyndi,

final interview]

Finally, the following are the responses of the 2 (12.5%) students who suggested

that they were basically the same:

No. I like the web better. You don’t have to have a pencil, so you can just
publish. [Ganton, final interview]

Not really because it is just writing on the computer instead of the paper.

[Tinse, final interview]
Question 2: Do you think what you write is different on the web versus what you write
in your individual journal ? How?

As Table 2 summarizes, for Question 2, 43.75% of the 16 students indicated that
the content of their TMC entries was different from that of their own journals, whereas
56.25% indicated they were the same.

Table 2. Distribution of Responses on Same or Difference of Content Between The
Mathematics Chronicles Versus Individual Journals

 

 

n %
Different 7 43.75
Same 9 56.25

 

The general trend among those who reported a difference in content between the
two journals was that students perceived a wider audience on the Web and felt they
should explain it so everybody could understand it:

Yeah. You go into more detail on the Web. People explain it better so
other people can understand it. [Kyndi, final interview]

42

Indeed, students distinguish how they would write when the readers are other than
themselves:

Sometimes [what I write in my journal is different from the web] because

it’s more for myself, so you only have to explain it so you understand. [In

the web journal,] you have to write it so everybody can understand it.

[Jeff, final interview]

Those suggesting that their writing on the Web was not different often simply
reported that “No, they aren’t different.” Some students, however, were more specific,

explaining that they wrote strategies, examples, problems, or what they did in class:

No. I write things like strategies and problems I am doing. [Mandy, ﬁnal
interview]

A little bit of the same thing because you have to put examples and what
you did in class. [Camm, ﬁnal interview]

One student suggested that there was no difference between what she wrote on the
Web and what she wrote in her own journal “because when I write something in my

journal, I usually write that on the web.”

Question 3: Would you rather write in your journal or on the web?
For Question 3, Table 3 shows that a majority (87.5%) of the students would

rather write on the Web than in their own journals.

Table 3. Distribution of Students’ Preference of Writing in Individual Journals or on The
Mathematics Chronicles

 

n 1%
Web 14 87.5
no matter 2 12.5

 

43

Students preferred the Web journal because it enabled learning, teaching the
audience, sharing of ideas, collaboration, and typing:

On the web because it helps other kids learn new things and it’s fun
typing. [Mandy, final interview]

On the web because it just gives you a chance to show your ideas and
thoughts. [Haze], final interview]

On the web because I don't really write in my journal. I don’t really know
what to put in there, but on the web, there’s somebody else who can help
you word it right or whatever when we are publishing. [Camm, ﬁnal
interview]

On the web because of typing, telling how to do it, and publishing. [Ande,
ﬁnal interview]

T wo students had no preference for the Web journal or their own journal:

It doesn’t really matter. [Jeff, final interview]

Either one’s fine. I like to do them both. [Shanon, final interview]
Question 4: You know how we have a new student in this class or choose one of your
friends who is not in this class. How would you explain what the web journal is all
about to him or her?

Several key themes emerged from students’ responses to Question 4. For some,
the TMC was a place where students were able to choose what to write and to talk about

what they were learning:

It’s fun because you get to type and pick what you want to write about.
[Lynton, ﬁnal interview]

Every once in a while someone gets chosen to go and write about what
we’re doing in class, you can choose a subject. Like, how to ﬁnd the sums
of adding fractions and publish what you wrote. [J ake, final interview]

When you publish, you write what you learned from one of the lessons
and show how you did it. [Tinse, final interview]

44

In this sense, students seemed to distinguish between simply presenting problems
or homework on the Web and actually writing about mathematics.

In contrast, some students were more detailed about what to write on the
Web and suggested including “solutions, how to do it, ideas and examples:”

You have to explain what you did in class and a solution for how to do it
and give them a couple of examples. [Ande, ﬁnal interview]

You write what you did in class and how to do it, the instructions to do
that paper or problem and write down the ideas. [Lin, final interview]

Other students emphasized the importance of the audience of the Web journal:

You pick something, people and they publish it so other people can look at
it.... To help other people on the Internet learn. [Tara, final interview]

On the web, you have to explain to other people what you are doing and
how the problems work. [Shanon, ﬁnal interview]

It seems that writing to a remote audience outside the classroom is a recurrent

theme in students’ perception of the TMC.

Question 5: What is the purpose?

As presented in Table 4, 87.5% of the students explained that the purpose of the
TMC was to share ideas with other people and 12.5% believed that the purpose was to
use it as a future reference. Students did not only perceive an audience for their work,
but thought it should be used to help others learn:

to help other kids, like maybe in the X school district or all over, to help

them if they don’t know or if they are doing the same thing or if they want
to see how other people are doing it. [Camm, final interview]

45

Table 4. Distribution of Responses on Purpose of The Mathematics Chronicles

 

 

n %
Share ideas with others 14 87.5
Use as reference 2 12.5

 

As part of the purpose of sharing their work, students also thought the
audience could read and respond to their entries. Along this line, people could
also act as a sounding board for students’ thoughts:

I think to share your ideas. People respond to it. It’s not really bad if they

respond to it because you know that they are reading your stuff and

listening to what you have to say. [Hazel, final interview]

Other students valued the Web journal as a reference they could review
when having difﬁculty with future work:

If you are stuck on something, you can go and look back. If you forget

how to do something you can look back on your notes. [Ganton, ﬁnal

interview]
In addition, implying that the TMC is a supportive tool for collaboration, students
suggested they could better understand their mathematics through other people’s
ideas:

To remember things and if someone else published and you didn’t

understand something, you can go to their publication and read it and
understand it. [Ande, ﬁnal interview]

Question 6: What are important things to remember while publishing? Things that you
shouldn ’t do?

There were three main themes from responses to Question 6. Some students

simply gave a general idea for what to write about on the TMC. They could take

46

something from their homework:

You could do whatever homework you did and explain how you did it.

Like your homework, you can scan it through and write what goes with it

and you can cut out stuff that doesn’t. [Lynton, ﬁnal interview]

And “if you find something interesting, you can publish it.”

Second, because students perceived there was an audience, they believed authors
needed to write their entries properly by checking for grammatical errors and writing
mathematical ideas that made sense:

Everybody can see it, your audience you are writing to. Use proper

spelling, punctuation, all the editing stuff and make sure your information

is correct. If it isn’t, everybody is going to get messed up. [Abraham, ﬁnal

interview]

Shouldn’t tell them the wrong way to do a problem. Remember that

everybody is going to read it. Make sure you have the right answer before

typing it up. You shouldn’t publish the wrong answer or the wrong way to
write it. [Jake, ﬁnal interview]

It seemed that the students’ work might used as an authority on mathematical
ideas, so authors needed to be especially careful with what they wrote. Other people
would be reading their entries and possibly learning from their ideas, so students needed
to make sure their publications would not “tell them the wrong way.”

Finally, students not only emphasized the importance of writing grammatically
correct or mathematically sound ideas, but also highlighted the significance of writing
how they arrived at a solution. In publishing, students need

to do it step by step. Like if you can say that these all go together, you

have to say 2 and 2/5 plus this is what. Instead of just writing down what

everything is altogether, go step by step and tell what you are doing. You
should go step by step and add it together. [Jeff, final interview]

47

Similarly,

remember to title your paper and tell what you are doing, to show it and
don’t just put some answers down. Make sure you show how to do
it....Don’t just put the answers, but make sure you tell what the questions
are and what you did. [Lin, ﬁnal interview]

When solving a problem as an entry, students valued the importance of
including the problem as well as details about how to solve it. In this sense, one

role of the journal was to explain to readers how to do mathematics.

Question 7: Now I am going to ask you about how you like publishing or how you ﬁnd
it interesting, whether you like publishing or ﬁnd it interesting. So what do you think
about publishing?

All but one said they did ﬁnd the Web journal interesting, and there were several
trends in students’ responses for Question 7. They thought the Web journal was
interesting to students because it enabled them to learn and to teach. Writing about their
mathematics and collaborating with others enhanced some students’ understanding:

It’s fun. I think that you should have it next year to help out the kids
because this has helped me a lot in understanding the computer and math
because I was a bad student. When I explain it on the computer, I know
exactly what I am saying. That helps me out with my math. When we
have partners, like smart people helping us, you get it because once you
read it, you say, oh, now I get it. [Lynton, final interview]

Students could also use the TMC as a reference when they do not know “how to do
something”

I think it’s a good way to learn how to do different things and learn how
to, for example, if you didn’t know how to do something, you can go back
and learn all different kinds of things. If you didn’t know how to do
something and somebody published it and you were looking around on the
computer and found something you didn’t know how to do and you read it
and they explained it really carefully, you can learn how to do it. [Ande,
ﬁnal interview]

48

Others liked the TMC because it was a medium that allowed them to support the
audience’s learning as well:

Publishing makes sense for people to do it, not like all those things that the

teacher makes you have to do. It’s something people want to do... because

it is fun and if you are going to help someone else and if other people are

going to help you. If there is someone else in another city trying to do a

problem, I can write this down and try to help them if they have a problem

with it. [Ande, final interview]

Only one student was not as interested as the others in writing her journal on the

Web because “at ﬁrst, it’s kind of interesting, but it gets boring after a little bit” [Kyndi,

ﬁnal interview].

Question 8: How do you like publishing compared to other activities?

As Table 5 indicates, 87.5% of the students responded that they would rather
publish compared to other mathematics activities. Like others, one student responded
that he would rather publish than take a test, but in his case, he would leave worksheets
as homework, so he can publish in class: i

I would rather publish than take a test or would publish and do the

worksheets at home. [Jake, final interview]

Table 5. Distribution of Preference to Publish or Participate in Other Activities

 

 

n %
prefer publishing 14 87.5
prefer other activities 1 6.25
no preference 1 6.25

 

49

For others, publishing was more engaging because they had more ownership over
the work:

Yeah I like publishing, sometimes in class, it is boring, but when you get

to publish, you can write down what you think about it, how you got the

problem. [Camm, final interview]

One student would rather participate in other activities, but could not explain

why:

I probably like the other things better. I don’t know why, I just do. [Kyndi, final
interview]

Finally, another person had no preference because publications were based on
other class activities:
It doesn’t matter because I usually publish what I do in math class. [Shanon, final

interview]

Question 9: Do you think publishing is different from other activities that we do in
class (makes it fun) ? How?

(Probe) Next year, we are thinking about having publishing but trying to

understand whether we should have it. Do you think we should have it? Why

should I have publishing? Or what is it about publishing that you like?

(Probe) How is it fun? What makes it fun?

Question 9 was used as a means for understanding why students would rather
publish than participate in other activities or why they thought it was more fun by
capturing how students thought publishing was different. As Table 6 indicates, most of
the students (87.5%) reported that publishing was different. For some of these students,
publishing is different from other math activities because the TMC allows for a
worldwide audience and also enables students’ to author their work:

Because you get to use the computer, you get to put what you say, so

everyone in the entire world can read it. You can put it in your own words,

but you have to make sure other people can understand it. [Jeff, final
interview]

50

Table 6.~ Distribution of Responses on Whether Publishing is Same or Different From
Other Classroom Activities

 

_ 11%
different 14 87.5
no difference 2 12.5

 

Others emphasized that publishing is different because although you may simply
write the answer for other mathematics activities, it is important to explain when
publishing:

Publishing is different because with a worksheet, you just put down the
answer, but with the publishing you have to explain your answer, how you
got it. It’s better to explain your answer than just right down the answer
because if you get the answer from somebody, you can’t explain it. But if
you publish, you have to explain it. [Ande, final interview]

Finally,

When you publish, you are not by yourself, and sometimes kids don’t do
the work because they have to sit by themselves or do it by themselves and
don’t understand it. But when you are publishing and you put down a
problem, you have a friend there who can help you solve the problem or a
friend who can help you think of a problem to put down. It is also fun '
because it lets you show other kids, not just those you are working with, it
shows other kids. [Camm, ﬁnal interview]

In this sense, publishing is unique and captures students’ interest because it supports
collaboration among the writers and between the writers and the readers.

Two students did not believe publishing was significantly different from other
activities because instead of simply “doing” the work, students were writing about what
they did:

Sort of because you write down what you did down in class, so it is sort of the

same, but you don’t just do it, you write about what you did. [Lin, ﬁnal
interview]

51

Not really. You just talk about what you are doing. When you publish, you talk
about the ditto sheet or what you did in class today. [Haze], ﬁnal interview]

Question 10: Do you think that publishing on the web leads you to do math or makes
you want to do math more? Why?

As Table 7 indicates, a majority of the students (75%) agreed that publishing on
the Web did make them want to do mathematics more. Through publishing, students
could develop new ways for approaching mathematics:

because when I publish I start writing something and think, oh my gosh,
it does really work and start using it. [Mandy, ﬁnal interview]

Table 7. Distribution of Responses on Whether Publishing Encourages Doing More

 

 

Mathematics
n %
Yes 12 75.0
N o 4 25.0

 

Similarly, while publishing, students are encouraged to think about what they are
writing. Through this experience, they can better understand the ideas:

Yeah, because it is fun. You get to learn more about what you are doing

that way. That makes you understand it more. Once you think about it

more to write it down, it makes you understand the problem more. [Lin,
ﬁnal interview]

52

In addition, it makes some students want to do mathematics more

because you want to learn stuff. When others login, they can see your

initials and decide to read yours. They can learn from them. I think it’s

cool. I get to see other solution strategies and I get more involved in math

because I learn more stuff everyday. And I get to write more stuff. After I

type stuff, I want to learn more about that subject and publish more on that

subject. [Abraham, final interview]
In this sense, students may “get more involved” because they learn from others and
become more curious about the topic they discuss in their own publications.

Of those who did not feel that publishing made them want to do mathematics
more, one explained that it was, however, a way for her to “take a break while you are
working.”

Question 11 : Have written in journals before? Published before? Read other student’s
journal entries on TMC?

Finally, Table 8 indicates what percentage of participants:

0 had written in mathematics journals before this class,

° had published on the Web before this class, and

0 had read other students’ journal entries on the Web.

Table 8. Distribution of Experience with Journals or Publishing and Whether Have Read
Others’ Work

 

Experience with journals yes 6 37.5
"0 10 62.5

 

 

Experience with publishing yes 0 0.0
no 16 100.0

Read others’ work yes 10 62.5
no 6 37.5

 

53

A large percentage of students had done journals before (62.5%) and generally
indicated that previous journals comprised class assignments or solution strategies
teachers recited to students. All of the students had not published before on the Web.

Finally, a majority of the students (62.5%) had read other people’s work.

Summary

Several recurrent themes emerged across the students’ responses. The perception
of a remote audience for the TMC was one signiﬁcant theme students discussed that also
shaped students’ idea of the Web journal. Believing their work would be shared with an
audience outside of the classroom, students emphasized the importance of explaining
how they did mathematics and of grammatical correctness to enable understanding. In
addition, publishing on the TMC was a forum for collaboration while writing entries or
through responding to each other’s work. Students claimed that through these
interactions and through the care they took in writing their ideas, they not only learned,
but also became more interested in mathematics learning. All of these points illuminated
by students in addition to their perception that they had authority over their work
published on the Web journal made the TMC an interesting activity that “makes sense for
people to do.”

To examine these themes in more depth, I present trends emerging from
observations in and out of the publishing environment (Chapter 4) and from students’
work done in the TMC and their individual journals (Chapter 5). Then, in Chapter 6, I
present an analysis that highlights overall themes emerging from these three sources of

data (interviews, observations, journals).

54

CHAPTER 4

STUDENT INITIATIVE IN PUBLISHING AND NONPUBLISHING CONTEXT S

Observations of student interactions were conducted throughout the
project and included field notes, videotapes, and audiotapes. I focused on the data for the
last few sessions (17 episodes) because it was at this point that students had experience in
both the publishing and nonpublishing contexts. These were used to examine students’
initiative in the context of publishing on The Mathematics Chronicles and of regular
(nonpublishing) activities during the final phase of the school year. The purpose of this
investigation was to understand, broadly, how students demonstrate initiative in a
mathematics classroom and, speciﬁcally, how initiative may vary across the publishing
and nonpublishing contexts.

Data collected in the ﬁnal phase of the project includes each of the students in
each context. For example, I observed Shanon in the publishing and the nonpublishing
context during the last few sessions. In these episodes, students worked in small and
whole groups.

Nonpublishing activities included whole class discussion of ideas, class and
homework assignments as well as small group work. During whole class discussion, Mr.
Watkins facilitated discussion by presenting or reviewing material and encouraging
students to clarify or justify their ideas and strategies. He asked students to explain how
they arrived at their solutions and to provide reasons for agreeing or disagreeing with
their peers’ arguments. In small groups, students collaborated on assignments and were

encouraged to support each other in developing ideas, strategies, and solutions.

55

At the beginning of class, students either volunteered or were asked to write on
TMC. In the publishing context, students worked with partners at the computers for the
entire class period and completed entries by the end of class. Homework assignments,
class discussion, or interests in previously studied concepts sparked students’ ideas for an
entry. Students worked together in dividing tasks, contributing ideas or criticisms, and
composing the entries. While publishing, students were also expected to participate in
the nonpublishing classroom activities, so they needed to attend and plan their time

wisely.

Developing a Rubric for Initiative

Developing a rubric for initiative was a multi-step process. Before coding
observations, 1 developed a draft rubric that included student interactions illustrating
initiative. While reviewing the data for each of the episodes, I modiﬁed this rubric by
coding for other interactions of initiative in mathematics learning. By analyzing each of
the episodes more closely for a third time, I collapsed or refined the categories for
initiative. Afterwards, I chose 3 episodes from the publishing and nonpublishing contexts
(total of 6) that represented the range of initiative demonstrated across the 17 episodes. I
will discuss these 6 cases in more depth later. After another thorough examination of
these selected episodes, I developed categories that could be more broadly defined as
different aspects of mathematics learning:

0 mathematical sense-making,

' collaboration or working together in small groups, and

' management.

56

Within these broad domains of mathematical sense-making, collaboration, and
management, there were several specific ways that students demonstrated initiative.
Table 9 illustrates the different types of initiative students demonstrated under the

broader categories of mathematics learning.

Table 9. Different Types of Student Initiative in Classroom Interactions

 

Mathematical sense-making:

0 deciding on a topic for the journal entry

0 deﬁning the problem or task at hand

0 discussing mathematical ideas or strategies (procedure or visual aid to solve
problem)

° generating solutions for problems

0 explaining mathematical idea or strategy (talking about idea or strategy to
reiterate them or to respond to partner’s idea or strategy or to a question)

 

 

Collaboration:
' ask partner for help in completing work
' challenge, question, or ask for clarification on propositions or solution strategies
' continue or modify work
0 edit for misspellings or revised the work so that it made sense

 

 

 

Management:
0 negotiate roles in task
0 decide upon resource to use (other than paper: calculator, overhead, ruler)

 

Before providing cases of what I credited as initiative for each specific domain, I

present the 6 cases in more depth.

Snapshots of the 6 Targeted Episodes
The 6 targeted cases included 3 publishing and 3 nonpublishing episodes. I
present snapshots of each episode to provide a clearer view of the students and the
activities involved in the cases. In Table 10, I present the nonpublishing cases, and in

Table 11, I present the publishing cases.

57

 

Table 10. Snapshots of Nonpublishing Cases

 

Context - Nonpublishing

 

Shanon and Mandy:

Mr. Watkins started this class period by assigning work from the investigation, Adding
and Subtracting Fractions. Students needed to add and subtract fractions of an
advertisement page to find the cost of the ads. They had the entire class period to work
with partners on the assignment. Mr. Watkins interrupted small group work on occasions
to discuss students’ questions or solutions.

Shanon and Mandy began this episode by updating their individual journals in their
notebooks, and soon after began completing their assignment. In this episode, these two
were able to address a and b:

1. A local magazine sells advertising space. It charges advertisers according to
the fraction of a page their ad will ﬁll.

a. For page 20 in the magazine, advertisers have purchased 1/8 of the page
and 1/16 of the page. What fraction of the page will be used for ads?
What fraction of the page will remain for other uses? Explain your
reasoning.

b. The Cool Sub Shop is having its grand opening and has purchased several
ads. They buy three 1/4 page ads, four 1/8 page ads, and ten 1/16 page
ads. What is the total amount of space that they have bought? Explain
your reasoning.

c. The magazine wants to make $160 for each page of advertising sold.
What might the magazine charge for each size ad if ads can be any of the
following sizes: 1/32, 1/16, 1/8, 1/4, 1/2, or a whole page? Explain your
reasoning.

d. Using the pricing scheme you developed in part c, what would the bill for
the Cool Sub Shop be for the ads they purchased in part b? Explain your
reasoning. (Lappan, Fey, Fitzgerald, Friel, & Phillips, 1996b, p. 49)

The two worked together by asking for each other’s help, offering ideas for solving the
problems, and providing different ways for solving some of the problems. They drew
pictures to represent fractions and also converted fractions into equivalents or converted
improper fractions to mixed fractions to compare them. Through these interactions, the
pair made some errors that led to explanations and development of each student’s own
solution to the problems. See Figure 7 for Shanon and Mandy’s completed work.

 

Bali and Tara:

Bali and Tara also participated in the activities deﬁned by Mr. Watkins discussed
above, but they addressed questions a-d. In solving the problems, the two also drew
pictures to represent fractions or ﬁnd equivalent fractions. They spent a great amount of
their energy on explaining their strategies. In addition, conﬂicts between the two
students’ ideas and strategies led them to question, challenge, and explain the ideas and
strategies. During these conﬂicts, Bali asked Tara to copy her answers, but Tara refused
and individualized her work by providing her own solutions and explanations. See
Figures 8 and 9 for Tara’s and Bali’s completed works.

 

58

 

Table 10 (cont’d)

 

Context - Nonpublishing

 

 

Tinse and Ande:

Mr. Watkins began this class period by having one of the students read the
introduction for the investigation, Finding Areas and Other Products. Then, students
split into small groups to work on the following questions:

Paulo and Paula are tending the brownie booth at the school fair. All evening
long they have run into interesting situations in which they have to find
fractional parts of other fractions. Let’s look at some of the problems. Paulo and
Paula had to solve while they were selling brownies.

The brownies are baked in square pans, and they are sold as fractional parts of a
pan. A whole pan of brownies costs $24 dollars. The cost of any fractional part
of a pan is that fraction of $24.

A. One pan of brownies was 2/3 full. Mr. Sims bought 1/2 of what was in the
pan. What fraction of a full pan did Mr. Sims buy? How much did he
pay?

B. Paulo’s aunt Serena asked to buy 3A of what was left in another pan. The
pan was half full. How much of a whole pan did Aunt Serena buy? How
much did she pay? (Lappan et al., 1996b, pp. 54-55)

While working together, Tinse and Ande provided strategies and solutions to
problems and completed this assignment. After giving students some time to discuss
these problems, Mr. Watkins called on a few students to talk about their solutions or to
present their work on the overhead. After this whole group discussion, students. were
given an assignment to work in class and finish up at home if needed. Tinse and Ande
worked on the following problem for the rest of the class period:

The school fair was almost over. Paulo and Paula wanted to sell all the remaining
brownies in a hurry, so they decided to offer a discount of 20% on all sales. They
had 2 1/4 pans of brownies left. Remember, they originally sold a pan of
brownies for $24.
Mr. Vargas offered to buy half of all that they had left.

A. How much will Mr. Vargas purchase? (Lappan et al., 1996b, p. 57)

In solving this problem, Tinse had difficulty remembering how to find the percentage
of an amount and requested help from the Mr. Watkins and me. Her requests for help
were deﬂected as suggestions to ask her peers. During these interactions, Ande
attempted several times to help Tinse and said to her partner that she was trying “to teach
her.” The remaining part of this episode involved figuring out how to find 20% of $24.
See Figurele and 11 for Ande’s and Tinse’s works.

 

59

 

 

 

   
 
 

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Figure 7. Shanon and Mandy’s responses

60

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Figure 9. Bali’s responses

61

 

 

 

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Figure 11. Tinse’s responses

62

Table 11. Snapshots of Publishing Cases

 

Context - Publishing

 

Tara and Ande:

Mr. Watkins began class by asking students to provide their algorithms for multiplying
fractions. After a few students offered their algorithms for discussion, Mr. Watkins
divided students into small groups to work on an assignment taken from the book. T and
B discussed what they would publish for their TMC entry and developed an algorithm for
multiplying fractions and provided an example. Each student shared in the typing and
development of the entry. Their collaboration around the entry was like a coordinated
dance between the students illustrated as continuations and modiﬁcations of each other’s
contribution. The following is their ﬁnished entry:

 

Making an algorithm

We were in math class and our teacher told us to ﬁnd an algorithm. First of all we had to
know what an algorithm was. An algorithm is a plan or series of steps. Then we chose
3/8 x 3/4. Then we need to know how to times fractions. First you multiply the
numerator (the to number of a fraction) by the other numerator. Then you multiply the
denominator (the bottom number of a fraction) by the other denominator. We got 9/32. ‘
Kyndi, Lin, and Hazel:

It had been a couple of days since students were in their mathematics class, so Mr.
Watkins gave students time to review their homework assignment with their partners
before handing it to him. Kyndi, Lin, and Hazel completed their work and wanted to
publish. The group threw out a couple ideas for an entry and settled on a problem from
the homework assignment. Most of their time was spent on editing and modifying their
work so that it made sense. They also provided solutions and corrected errors they made
during this episode to conclude with the following entry:

 

Finding the fraction cost of an advertisement page

This section is about adding and subtracting fractions. The problem asks you to use this
lpricing scheme to figure out how much it would cost for a person to buy: 3/4, 4/8, and
10/16 of a page. The prices are:

1 pg = $160

1/4 pg = $40

US pg = $20

1/ 16 pg = $10

We figured out this problem by multiplying: 3x$40=$120 (=3/4),
4x$20=$80 (=4/8), 10x$10=100 (=10/ 16). Then we added
$120+$80+$100=$300

So this person spent $300 on advertisements.

 

 

 

63

Table 11 (cont’d)

 

Context - Publishing

 

 

Bali and Camrn:

Mr. Watkins reviewed a check-up with the students and during this discussion, called
on students to provide how they arrived at their solutions. Afterwards, students were
given time to discuss mathematical reﬂections on Finding Areas and Other Products.

Bali and Camrn offered several different topics for their TMC entry and began the entry
several times before settling on writing about comparing fractions. They drew pie graphs
as a strategy to compare fractions and included their sketches as annotations to their
written work:

Comparing Fractions
We drew pie graphs to figure out which fractions were greater, less than, or equal. We

had 3/5 and 7/8, 12/15 and 3/4, and 5/8 and 10/16.
See Figure 12 for the pie graphs Bali and Camm drew to compare these fractions.

 

64

 

Understanding the Speciﬁc Aspects of Mathematical Learning

To understand how student initiative may be demonstrated in the mathematics
classroom, I analyzed the speciﬁc ways that student took initiative. The following
provide cases of what I credited as initiative for each specific domain. The unit for
coding initiative was an utterance unless otherwise indicated. Since excerpts may include
several different kinds of initiative, I highlight in bold those speciﬁc parts of the
examples that illustrate the type of initiative under discussion. The interactions I
excluded as initiative were those that students demonstrated at the teacher’s prompting or

leading.

Mathematical S ease-making

I suggest that mathematical sense-making comprised several kinds of interactions.
Students ﬁrst needed to deﬁne the problem or task at hand and/or decide on a topic for
their journal entry. In addition, to work problems or compose journal entries, students
often discussed mathematical ideas or strategies, and generated solutions. Learners also
provided explanations of mathematical ideas or strategies. I considered all of these
events as part of mathematical sense-making.

Deciding on a topic for a journal entry. While writing journal entries, students
needed to ﬁrst decide on what topic to write. This activity essentially begins
mathematical sense-making because it becomes the subject of discussion between
partners and in student work. I coded these instances as initiative in this domain when
students decided for themselves what the topic of discussion would be for their journal

entry.

66

Case of Shanon and Mandy (nonpublishing). Deciding on a topic for a journal
entry might entail figuring out what generally occurred on particular dates as it did for
Shanon and Mandy. In their case, Shanon and Mandy used class time to update their
individual journals with statements about particular activities they accomplished for
certain dates:

Shanon: Did you get any more dates? Did you get any of these? [Asking Mandy
whether she has completed journal entries for dates that Shanon has blank]

Mandy: Did we take a practice test? This was a test and this was a practice test.
Shanon: We had Spring Break, so when did we get back?
Mandy: On the 7th.

Shanon: Mhmm, 13th? [Replaying events to decipher what activities occurred on
which dates]

Mandy: Yeah.

Shanon: Whatever. 13th. We took the practice test.

Mandy: No, for the XXY [statewide test].

[They both write in their journals]

In this scenario, Shanon and Mandy, along with the rest of the students, were
given an assignment to work on in groups, but these two ﬁrst used some time tocatch up
on the dates they had missed for their individual journals. They discussed generally what
activities they did for each of the missing entries.

Case of Bali and Camrn (publishing). During publishing, students sometimes
began their TMC entry with an idea of what to discuss for their publication. On other
occasions, however, defining the topic unfolded as in the case of Bali and Camrn. These

two students generated several ideas for their entry and finally settled on one. Camm

67

began writing an entry and erased it to initiate another that Bali continued and modiﬁed.
After some editing, Camm decided to write their mathematical reflections2 for the
investigation on “Finding the fraction of fractions or whole numbers.” Again, Bali
decided that she wanted to publish something different and illustrated how to solve the
following problem:

While traveling in Mexico, Samantha found some beautiful ceramic tiles.

The tiles are square, 6 1/2 inches on each side. Samantha wants to buy

enough tiles to cover the floor of her sun room. The sun room is also

square, 108 inches on each side. How many tiles does Samantha need?

(Lappan et al., 1996b, p. 62)

By the end of this episode, however, Bali had decided to write on how they compared the

following fractions and decided whether one was greater than, less than, or equal to the

other:
3/5 __ 7/8
12/ 15 _ 3/4
5/8 _ 10/16

In this setting, Bali and Camrn had suggested several topics for their journal entry
and ﬁnally settled on the last idea. I considered cases like this, with students
demonstrating an effort to deﬁne and redeﬁne the topic, to be initiative.

Summary. Students, sometimes, were quite thoughtful in deciding on a topic for
their journal entries as Bali and Camm demonstrated or, as Shanon and Mandy
illustrated, simply remembered what activity the class completed for the day. In either

case, it was during these interactions when students, themselves, were struggling with

 

2Reflections were student’s overall summaries of mathematical ideas and solution
strategies they learned during an investigation

68

formulating the topic of discussion for their journal that I highlight this kind of initiative
in the different episodes.

Deﬁnition of the problem or task at hand. Similar to choosing a topic for journal
entries, students demonstrated initiative by defining the problem or task at hand for
mathematical activities other than journal writing because this deﬁning illustrated
students’ active participation in addressing mathematical problems. In this case, learners
exempliﬁed effort in undertaking a task by asking other learners for clariﬁcation of
problems assigned.

Case of Shanon and Mandy (nonpublishing). During nonpublishing activities,
the teacher often defined the activities for the day. However, it was the responsibility of
the students to be aware of the problems or assignments they needed to complete. In the
following, Mr. Watkins had begun class and Shanon and Mandy were trying to determine
what the task was:

Shanon: What are we supposed to do, right now?

Mandy: The heck if I know.

Shanon: What are we supposed to do? [turns around to IT to ask]

Case ofAnde and Tara (publishing). In the case of Ande and Tara, the class had
recently been given an assignment, but Ande apparently did not hear what it was and
asked her partner:

Ande: What’d you say? What do we have to do?

Tara: 5-13.

Ande: We have to do 5-13.

69

Summary. In scenarios like these, students may have missed the assignment when
it was given or were not paying attention at the time. I assert, however, that these
students took initiative in defining the task at hand by demonstrating an effort in
determining the assignment. It was in these kinds of instances when students attempted
to begin the activities themselves without needing the teacher or researcher to remind
them to complete their work that I credited students with taking initiative in this domain.

Discussing mathematical ideas or strategies. After establishing a topic or
pinpointing the problem at hand, students may have discussed mathematical ideas or
strategies. Through these kinds of interactions, I argue students took initiative by
attempting to understand how to solve the problem.

Case of Tinse and Ande (nonpublishing). Tinse and Ande had the following
problem:

The brownies are baked in square pans, and they are sold as fractional

parts of a pan. A whole pan of brownies costs $24. The cost of any

fractional part of a pan is that fraction of $24.

One pan of brownies was 2/3 full. Mr. Sims bought 1/2 of what was in the

pan. What fraction of a full pan did Mr. Sims buy? How much did he

pay? (Lappan et al., 1996b, p. 57)

Tinse and Ande attempt to solve the problem:
Tinse: Ok, we divide this [a pan of brownies] into 2/3.
Ande: We have to do [problem] A ﬁrst.

Tinse: That’s what I’m doing. I’m dividing it. Divide it into 2/3.

Ande: Oh, ok.
Tinse: In Mr. Sims.

Ande: That’s not 2/3. What do you mean empty? [Tinse had shaded in 2/3 of the
pan and indicated the remaining was empty because there was 2/3 left]

70

Tinse: It’s empty, see cause 2/3.

Ande: Yeah and then.

Tinse: You divide it into 3 and 2 parts of it.
Ande: Ok, that’s 2/3 of it, right? 1,2,3, right?
Tinse: That’s in thirds.

Ande: Right, but that’s not empty.

Tinse: Yes, it is. It says that one pan of the brownies was 2/3 full.
See?

Ms. Sims buys 1/2 of what’s left in the pan. Oh, that’s easy. We can divide
this into 6ths instead.

Ande: (gasps)

Tinse: Oh, we don’t need this into 6ths. Ok, he buys half of 2/3. So that’s
this. Mr. Sims buys this, right? [points to 1/3 portion]

Ande: Mr. Smith?

Tinse: Mr. Sims. Right, because he said he bought 1/2 of what was in there.
Right?

In this setting, Tinse first suggested a strategy of drawing a diagram of a pan and
dividing the pan into 2/3, and second, to ﬁnd 1/2 of this 2/3 portion, she suggested a
strategy of dividing the whole pan into sixths. Instead, Tinse concluded that “we don’t
need this into 6ths” when she realized that 1/2 of the 2/3 left in the pan is obviously one
out of the two pieces, which is 1/3. In this case, she tried a few different strategies to
solve the problem.

Case of Bali and Camrn (publishing). While publishing on the TMC, students

drew pictures as a solution strategy for solving a problem as well. As previously

71

mentioned, Bali and Camrn compared the following fractions:

3/5 _ 7/8
12/15 _ 3/4
5/8 _ 10/16

They scanned in Figure 12 as part of their journal entry and stated the following:

Bali: “We drew pie graphs to figure out”

Camm: “To figure out if”

Bali: “which fractions were bigger or smaller.”

Camrn: “which fractions were greater than, less than, or equal to.”

This interaction along with Figure 12 illustrates the strategy Bali and Camrn used
to compare the fractions. They drew pie graphs for each fraction and compared them to
decide whether one pie had more pieces shaded or both pies looked like they had the
same amount of pieces shaded. This comparison was used to decide which fraction was
greater or whether they were equal.

Summary. During the various episodes, students drew pictures or used algorithms
as strategies for solving problems, as in the mentioned cases. Students also demonstrated
thoughtfulness about mathematical concepts such as what a fraction, decimal, percentage
represents. I coded strategies and mathematical ideas under the same category because
students’ statements could be ambiguous characterized by either strategies or ideas. For
example, in the case of Tinse and Ande, the strategy of dividing the pan into 6ths to
determine 1/2 of 2/3 of the pan could also be interpreted as a mathematical idea. It was
for these events when students attempted to solve the problem by offering a strategy or

mathematical idea in the discussion that I credited students with this type of initiative.

72

Generating solutions for problems. Students not only discussed mathematical
ideas or strategies, but also provided example solutions. Coding for students’ solution of
problems was at times intermingled with ideas or strategies, but since mathematical ideas
or strategies may not have been mentioned alongside solutions, generating a solution was
coded separately.

Case of Tinse and Ande (nonpublishing). Tinse and Ande illustrated how students
may use several strategies until they solve a problem. In this case, the two attempted to
solve the following:

A whole pan costs $24. Paulo’s aunt Serena asked to buy 3/4 of what was

left in another pan. The pan was half full. How much of a whole pan did

Aunt Serena buy? How much did she pay? (Lappan et al., 1996a, p. 55)

Tinse and Ande had figured out that Serena was buying a total of 3/8 of the pan
and determine the cost of this portion by taking this fraction of the cost of the whole pan:

Tinse: 3/8.

Ande: Ok, she buys 3/8.

Tinse: And how much does she pay?

Ande: You have to take 3/8.

Tinse: Ok, so 24 divided by 8 is 3 dollars. 3 dollars. It’s 3 dollars. Wait it’s
not 3 dollars. It’s 3 dollars for each.

Ande: It’s 3 dollars for each.

Tinse: 3 dollars is for 1/8. But 3/8 is 9 dollars. She’s only buying three.
In this episode, Tinse and Ande used a strategy to figure out how much 3/8 of the

pan would cost. First they figured out that 1/8 of the pan would equal 24 divided by 8 or
$3. With this information, they concluded that 3/8 is 3 times this amount, or $9, because

3/8 is 3 of the 1/8 portions. In this case, the focus was to solve the problem, so they

73

determined how much of the pan was bought by Serena and solved the problem of what
this portion of the pan would cost.

Case of Tara and Ande (publishing). At times, students also generated solutions
using an example to illustrate a mathematical idea without explicating the strategy or to
simply solve a problem. In the following, Tara and Ande published a journal entry about
how to multiply fractions and provide an example:

Ande: For example, 3/8. 3/8.

Tara: 3/8 times.

Ande: No and.

Tara: Put the multiply sign.

Ande: Times 3/4 would be. 9. 9.

Tara: 9 over 32.

In this case, Tara solved the problem of 3/8 times 3/4 and came up with 9/32. The
goal in offering this solution was to explicate an example of how to multiply fractions.

Summary. As in these cases, students demonstrated initiative by taking it upon
themselves to contribute to the mathematical sense-making by providing solutions
without the leading of the teacher or researcher. They applied ideas or strategies to
present examples of how to solve a particular problem.

Explaining mathematical idea, strategy solution, or suggestion. As in the
presented episodes, students were credited with taking initiative in explaining their ideas,
strategies, solution, or suggestions when they offered the explanations without the teacher
or researcher’s request. I considered explanations as descriptions about the ideas or

strategies in words that were provided when challenged or questioned by their partners’

74

or as a summary given at the end of a presentation of ideas, strategies, solution, or other
suggestion.

Case of Tara and Bali (nonpublishing). Students often not only offered
mathematical ideas, strategies, solutions, or suggestions, but also provided explanations.
For example, Tara and Bali compared their explanations of what the total of 1/8, 1/4, and
1/16 of an advertisement page is in the following:

Tara: So, it would be 7/16.

Bali: Yeah. 7, because 1/8 is 2 pieces and 1/4 is 4 and l/l6 is 7. 7/16

because 1/4 is a quarter of a page which on the 16 side is 4 squares so

we added 4 squares.

Tara: I put down because 1/8 is 2/16 and 1/4 is 4/16 and add 1/16 and
you get 7/16.

Bali: Ok.

Tara: Ok.

In this excerpt, Bali and Tara had discussed strategies for solving the problem and
at this point were reading their ﬁnal explanations by explicating the strategies and ideas
used to arrive at their solution of 7/16. Bali explained that 1/8, 1/4, and 1/16, altogether
is 7/16 because 1/8 is 2, 1/4 is 4, and 2 plus 4 plus 1 is 7 or 7/16. She also explained that
she found 1/4 is 4/ 16 because she drew two grids on her paper (one in sixteenths and one
in fourths) and found that 1/4 of the fourths grid equals 4 pieces of the sixteenths. Tara
provided a slightly different explanation that 1/8 is 2/ 16, 1/4 is 4/16, and adding them
along with the 1/16 totals 7/16.

Case of Kyndi, Lin, and Hazel (publishing). Students also provided explanations
of editing remarks. In the following episode, Kyndi, Lin, and Hazel tried to determine

the total cost of a group of advertisement pages and needed to include the information

75

about the pricing scheme for different fractions of a page given in a different problem.
Hazel typed how much different fractions of a page cost with the help of her partners, and
Kyndi suggested they exclude the irrelevant information and explained why:

Lin: How about 1 whole?

Hazel: Equals $160.

Kyndi: Why are you doing one page?

Hazel: Because that is what it says.

Kyndi: Oh, you’re writing down all the prices?

Hazel: Yeah.

Kyndi: You could just put 1/4 equals. You don’t have to give the rest

of the prices that you don’t use. You could just put 4/8, sixteenth.

[Makes an editing suggestion and explains]

Lin: Yeah, that would work.

Kyndi: Instead of doing all that. We don’t really need that
information. [Continues to explain the suggestion]

Hazel: Equals how much?

Kyndi suggested that they not include all of the prices because in their example,
they only discussed the cost of some of the fractions. In this sense, Kyndi provided an
editing suggestion and explained that they did not need to include other facts that would
not be used in solving the problem.

Summary. Students provided explanations to compare responses to problems as
in the case of Bali and Tara or as a response to partners’ challenges or questions as in the
case of Kyndi, Hazel, and Lin. Taking initiative in this domain was demonstrated in
either of these scenarios as students exerted effort in contributing and clarifying each

other’s understanding. The distinguishing feature between the explanations and the

76

offering of ideas, strategies, solutions, or suggestions is that with explanations, students

talk about how they generated them.

Collaboration

During small and whole group discussion, students collaborated and supported
each other in several ways. They asked partners for help in completing work. Likewise,
students took initiative in challenging, questioning or asking for clariﬁcation of partners’
propositions or strategies. Also in assisting others, students added to or modiﬁed
students’ work or ideas. While publishing on the TMC, students were often concerned
with grammatical errors and edited collaborative entries for misspellings or revised them
so they made sense.

Asks partner for help in completing work. In mathematics learning, students
may have difficulty making sense of a problem or solving it. Instead of looking to the
teacher, who is typically viewed as the authority in the classroom, students may turn to
peers for help. Along these lines, students take initiative by making an effort to employ
their partner’s assistance in solving or making sense of a problem.

Case of Shanon and Mandy (nonpublishing). Shanon demonstrated how students
might solicit the help of their partners when confused about the problem. She read the
following problem:

A local magazine sells advertising space. It charges advertisers according

to the fraction of a page their ad will fill.

For page 20 in the magazine, advertisers have purchased 1/8 of the page

and 1/ 16 of the page. What fraction of the page will be used for ads?

What fraction of the page will remain for other uses? Explain your
reasoning. (Lappan et al., 1996b, p.49)

77

Afterwards, Shanon asked her partner, Mandy, for help because she did not understand
the problem or how to solve it:

Shanon: 20, and they are advertising, which is 1/8 [reading the problem].

(idle talk about grandma)

Shanon: I don’t get it. Help me.

Mandy: Listen to the problem. Listen to the question. “A local magazine

sells advertising space. It charges advertising according to the fraction of

a page”

Shanon: Yes.

Mandy: “their ad will”

Shanon: Yes.

Mandy: “fill. For page 20 in the magazine, advertisers have purchased 1/8
of the page and 1/ 16 of the page. What fraction.”

Shanon: It’s 2/16.

Mandy: Yes. No, it’s 2/8. 2/8.

Shanon: It’s 2/ 16 equal to 1/8, like I said before.

Mandy: No, it doesn’t equal to 1/8.

Shanon: 2, it’s divided by 2.

Mandy: Yeah, 2/8. I was right. It is 2/8. Ok, explain your reasoning.

Shanon commented, “I don’t get it. Help me,” as a request for some assistance
from her peer in deciding how to approach solving the problem. Instead of ignoring the
problem or avoiding her work, she took initiative by using her resources, her partner in
this case, to make sense of how to tackle the problem. Mandy reread the question with
her partner and the two began to peel away at possible solutions. Thus, Shanon

demonstrated an effort to solve the problem by soliciting the help of her partner.

78

Excluded from this synopsis is the girls’ continued discussion about how to solve the
problem that concluded with a sensible solution.

Case of Tara and Ande (publishing). On occasions, students also asked their
partners how to revise their work, particularly While publishing on the TMC. In the
following excerpt, Ande and Tara wove together their entry about how to multiply
fractions:

Ande: Times [Multiply]. Let me do it now.

Tara: Hold on.

Ande: And.

Tara: And then you get the answer.

(Ande typing)

Ande: And then you get the.

Ande: What do you want me to put after the answer?

Tara: For example.

Ande: For example, 3/8. 3/8.

Tara: 3/8 times.

Collaboratively composing this entry, Tara and Ande, each provided their
additions. After a pause, Ande asked her partner, “What do you want me to put after the
answer?” in an effort to invite Tara’s contributions.

Summary. Although asking partners for help is demonstrated differently in the
two settings, both pairs initiated or reinitiated steps towards approaching the task at hand

by looking to their peers to make sense of the problem. In the process they also made an

79

effort to involve their partners in the problem—solving. I considered this type of action to
be initiative because students were looking to their peers rather than teachers, who are
often thought to have the “right answer.” By requesting the aid of partners, students are
taking initiative in involving their partners in more collaborative problem-solving and
using other resources instead of turning directly to the teacher ﬁrst.

Challenging, questioning, or asking for clariﬁcation on propositions or
strategies. In response to partner’s arguments or strategies, students often challenged,
questioned or asked for clarification of the ideas and strategies. I considered these kinds
of events to be initiative because students were taking some action in trying to understand
their partners’ propositions and strategies.

Case of Tinse and Ande (nonpublishing). In the following episode, Tinse and
Ande needed to decide how to find out how much brownies would cost if they were 20%
off, considering they cost $24 per pan:

Ande: Ok, take 24.

(Idle talk)

Ande: Then you take that.

Tinse: What are you doing? I don’t remember that.

Ande: I am trying to teach you. And then you take 24 and divide it by 4.

Tinse: Why do you divide it by 4?

??: I don’t know. Why do you divide it by 4, Ande?

Ande: Ok, listen 20% of 24 dollars, right?

Tinse: Yes.

80

In this setting, Ande attempted to provide a strategy for solving the problem by
taking the 24 and dividing it by 4. Realizing that this did not make sense to her, Tinse
questioned Ande’s strategy, “Why do you divide it by 4?” Tinse was not accepting
Ande’s argument, but took initiative in making sense of the problem by demanding
further explanation.

Case of Kyndi, Lin, and Hazel (publishing). While writing on TMC, students also
asked for justiﬁcation for what partners included in an entry. In the following setting,
Hazel was typing the cost of different fractions of a page and her partners questioned why
she was including some of the information:

Lin: Yeah, that would work.

Kyndi: Instead of doing all that. We don’t really need that information.

Hazel: Equals how much?

Lin: $80.

Kyndi: I’d just do fourths, H, and then the 8ths, and the 16ths.

Hazel: I know. I’m not doing all of them. I’m not doing all of them.

Lin: Then why are you doing the ones that we don’t need?

Kyndi: Why are you doing the half?

Lin: And the 1/3?

Hazel: You all are picky.

Although Hazel explained that, no, she was not including all of the fractions,
Kyndi asked, “Why are you doing the half?” and Lin followed with “And the 1/3?” This

setting exempliﬁes how students may challenge the work of their partners. It was along

81

these lines that the partners took initiative because they were not satisfied with Hazel’s
modiﬁcation of the journal entry and made an effort to clarify or strengthen the work.

Summary. I considered events like these to be initiative because students were
voluntarily seeking to make sense of their partner’s propositions or strategies by
challenging, questioning, or asking for clariﬁcation. At times, when students
demonstrated this kind of initiative, they also provided alternative explanations, solution
strategies, or missing information. Learners collaborated and pushed themselves and their
group towards a clearer understanding of the mathematics through this process.

Continue or modify work. Contributing to the improvement of partner’s work is
often an important part of collaboration because it not only represents students’
involvement, but also moves the task forward. In the analyzed episodes, students assisted
each other by continuing or revising partners’ ideas. It was during these events when
students took a step into shaping work by providing suggestions on how their partner’s
could elaborate ideas, strategies, or explanations that I coded their interactions as
continuing or revising work.

Case of Tara and Bali (nonpublishing). The case of Tara and Bali illustrates a
typical situation in which a student modified the work of a partner. Tara and Bali were
explaining how much of an advertisement page would be used and left over if US and
1/ 16 of the page was bought. The two had discussed how to solve the problem up to this
point and were reviewing each other’s responses to the question:

Tara: Ok, do you want to hear what I wrote?

Bali: Hold on.

82

Tara: Ok, 3/ 16 would be used for the ad and 13/ 16 will be left because 1/8

equals 2/16 and you add 1/16 to that you get 3/16 and there will be 13/16

left unused.

Bali: Mine is 13/16 because we divided a piece of paper into 16 squares

and ﬁlled in 1 because there is 1/16. Then we turned to the back and

divided it into 8 and ﬁlled in 1. I ﬁgured that 1 square on the 8ths side

equals 2 squares on the 16th side so I ﬁlled in 3 squares altogether on the

16th side. I came up with 13/16.

Tara: And there were 13/16 left because we came out with 3/16.

Bali: Yes, and there was 3/16.

Tara provided her explanation and as Bali proceeded to read hers, Tara interjected
with a suggestion of how Bali might modify her explanation, “And there were 13/ 16 left
because we came out with 3/16” to elaborate what Bali meant by “came up with 13/16.”
In this case, Tara demonstrated how students took initiative in this domain by offering an
explanation of Bali’s answer.

Case of Tara and Ande (publishing). While publishing, students more often
collaborated on one product than when working on the individual journals. As a result,
partners took turns adding to each other’s assertions. Consider the case of Ande and Tara.
Their example illustrates how students often continued or modified each other’s
statements:

Ande: Wait, wait.

[typing]

Ande: [reads the entry].

Tara: We found out that the algorithm.

[Ande types].

Ande: The algorithm.

83

Tara: No.

Ande: That’s what you told me to put. Then we chose to...

[typing]

Ande: We chose to put.

Tara: 3/8. Problems.

Ande: Fractions or problems? Fractions. 3/8.

As Ande typed, Tara added “We found out that the algorithm,” and Ande
continued with “We chose to put.” Tara added, “3/8” and Ande revised this with
“fractions 3/8...” In this scenario, the two were composing an entry explaining how to
multiply fractions. Each student took a role in contributing to the development of the
work, and through this process, the entry was transformed continually until each author
approved the final product.

Summary. Contributing to the revision or continuation of partners’ and
collaborative work is an important demonstration of exerting effort in students’ own and
others’ learning. Indeed in publishing and nonpublishing activities, students wove their
ideas into their partners’ or collaborative work. I suggest this willful action to shape a
partner’s or collaborative work demonstrates another way students take initiative in
collaboration.

Edit for misspellings or revised the work so that it made sense. Students also
edited work for misspellings and grammatical errors. In this way, learners addressed the
syntactical problems to ensure that the product made sense. Initiative in this domain is

demonstrated because students are making an effort to better develop the work. Since

84

there were no examples of this kind of initiative in the nonpublishing environment, I only
provide an example from the publishing context.

Case of Tara and Ande (publishing). Ande and Tara illustrate how they
collaborated in explaining that to multiply fractions you find the product of the
numerators and the product of the denominators. Up to this point, they had written that
“you multiply the numerator (the top number...)” and Tara added “to a fraction,” but
Ande edited the following, so it made grammatical sense:

Ande: Where am I at? A what.

Tara: To a fraction.

Ande: Of a fraction?

Ande continued the publication “you multiply the numerator (the top number”
with “of a fraction.” The final thought was “First you multiply the numerator (the top
number of a fraction) by the other numerator.” This event illustrates how students often
made grammatical changes to the entries. In addition, students made spelling corrections,
but in both cases, students took initiative in revising their work for publication.

Summary. In addition to modifying the work so that it made sense, students
attended to the structural features of their work. In this sense, it seems students valued
the appearance of their work. Student initiative was demonstrated as they made spelling
and grammatical corrections by voluntarily contributing their ideas for how to make the

entry more readable.

Other Management Tasks
Finally, students demonstrated the importance of management or organizational

tasks. Students needed to delegate roles to work together effectively. Also, deciding on

85

what kinds of resources to use illustrates that students took initiative in managing tools
that supported them in doing mathematics.

Negotiate roles in task. Negotiating roles was another important type of initiative
that exempliﬁed students taking some authority by deciding who would accomplish
which parts of an activity. Although students’ roles might be implicit, I explored ways
students made their roles explicit by discussing their roles in a task.

Case of Bali and Camrn (publishing). During publishing, students decided who
would type and which parts of the entry each would compose. For example, in one
episode, Bali and Camm had alternated in typing the publication, and Camrn asked her
partner, “Do you want me to type that?” in an effort to redefine the typing roles.

On other occasions, students divided parts of the task by deciding who would
solve which part of a problem. In either of these kinds of scenarios, students took
initiative by taking it upon themselves to manage students’ roles in the group.

Decide upon resource to use. Instead of asking the teacher or researcher how
they should approach solving a problem, students, themselves, often chose tools to
support their problem-solving. In the episodes analyzed, the resources excluded paper
and pencil and included calculators, the overhead, and information from a previous
solution.

Case of Tara and Bali (nonpublishing). Bali and Camrn needed to decide how
much of an advertisement page 1/8 plus 1/16 would equal. In the following, Bali decided
she wanted to use the overhead. Bali thought she could use transparencies to draw two
grids: an eighths and a sixteenths. She had planned to overlap the eighths grid with the

sixteenths to make sense of an equivalence from the sixteenths grid for the 1/8:

86

Bali: Mr. Watkins this is something we need the overhead for. Can we use
the overhead for this? Because I need the transparencies.

Mr. Watkins: For what?
Bali: For the overhead.

[gets up and leaves for a moment]
[comes back]

Ok, we got all this. [Begins to solve problem by drawing grids]

In this episode, Bali decided to use transparencies and the overhead as a resource for
comparing drawings of 1/8 on a sixteenths grid. She spontaneously thought of this
resource for helping her, thus taking initiative in solving the problem.

Case of Kyndi, Lin, and Hazel (publishing). In addition to using tangible tools
(e.g., calculators, rulers, blocks, transparencies), students also selected relevant solutions
for solving their problems. Kyndi, Lin, and Hazel were composing their entry and at the
same time deciding how they would solve the problem. Kyndi realized that they needed
to solve a different problem and use the information from the solution in order to solve
the current problem:

Kyndi: To ﬁgure out how much.

Hazel: A person to buy.

Lin: Cost a person to buy a certain parts of a magazine.

Kyndi: Wait, wait, wait, if we do D, we’re going to have to do C, kind

of, but not explain, cause it says, use the pricing scheme, so we have to

put the prices down, like one page is. So we have to put that
information down.

Hazel: 3/4, 4/8.

Lin: 4/8 and 10/ 16 of a page. The prices are.

87

Demonstrating an effort to contribute, Kyndi suggested another piece of the puzzle, or
resource, that would enable them to complete their entry.

Summary. Students may use abstract resources such as mathematical ideas or
strategies to solve problems. They may also use tangible tools such as calculators,
overheads, or information from previous problems to support their problem-solving.
Students were taking responsibility for coordinating how to solve a problem instead of
requesting the ideas of the teacher. In these events, students demonstrated initiative by
attempting to make some progress with solving a problem through employing tangible

I'CSOUI'CCS.

Trends in How the Initiative Is Characterized Across the Contexts

In the discussed cases, I have illustrated the different types of initiative that
students demonstrated in the nonpublishing and publishing context. These examples
provide a clearer picture of the initiative that students took. Table 12 presents how often
each type of initiative was coded in each case and the total number of times it occurred in
the nonpublishing and publishing context. Although there were not enough cases to
make a statistical comparison, this descriptive data can suggest some trends of initiative
that characterize the nonpublishing or publishing context.

Overall, the initiative that students took in the nonpublishing context could be
most characterized as

0 clarifying tasks and

0 offering or explaining mathematical ideas, strategies, solutions, or other
suggestions.

88

The initiative that students took in the publishing context could be most
characterized as

0 deciding on topic for journal entry,

0 asks partner for help in completing work,

0 continue or modify work,

0 edit for misspellings or revised the work so that it made sense,

0 negotiate roles in task, and

0 decide on resource to use (other than paper calculator, overhead, ruler)
Finally, challenging, questioning, or asking for clariﬁcation on propositions or solution

strategies seemed to be a common way for students to demonstrate their initiative in both

 

 

 

 

 

 

 

contexts.
Table 12. Frequency of Initiative Across Cases
Nonpublishing Cases Publishing Cases
Type of Shanon Bali Tinse Total Tara Kyndi Bali Total
Initiative Mandy Tara Ande Ande Lin Camrn
Hazel
Deciding on 2 0 0 2 1 7 4 12
topic for
journal entry
Deﬁnition of 1 2 5 8 1 0 0 1
problem or
task at hand
Discussing 9 17 12 38 2 2 1 5
mathematical
ideas or
solution
straEgies
Generating 4 7 4 15 2 4 2 8
solutions for
problems

 

 

 

 

 

 

 

 

 

89

 

Table 12 (cont’d)

 

Nonpublishing Cases

Publishing Cases

 

Type of
Initiative

Shanon
Mandy

Bali
Tara

Tinse
Ande

Total

Tara
Ande

Kyndi
Lin
Hazel

Bali
Camrn

Total

 

Explaining
mathematical
idea or
solution
strategy

9

18

5

32

2

4

 

Asks partner
for help in
completing
work

13

 

Challenging,
questioning,
or asking for
clariﬁcation
on
propositions
or solution
strategies

10

20

12

18

 

Continue or
modify work

29

39

13

81

 

Edit for
misspellings
or revised the
work so that
it made sense

10

18

28

 

Negotiate
roles in task

 

 

Decide on
resource to
use (other
than paper
calculator,
overhead,
ruler)

 

 

 

 

 

 

 

 

 

90

 

Summary

The analyzed episodes illustrate how students took initiative in different domains
and suggest that nonpublishing and publishing contexts afforded different kinds of
initiative. During nonpublishing activities, students tended to clarify tasks, discuss main
ideas, strategies, or explanations, and generate solutions. Perhaps these tendencies arise
from the nature of the nonpublishing context. First, students would be expected to clarify
tasks because they are given an assignment. Furthermore, when students tackle more
problems, they may have a greater opportunity to contribute mathematical ideas,
strategies, explanations, and solutions. Students have more of these opportunities during
nonpublishing activities because they work on more problems than during publishing.

Since TMC exists as a journal on the Web, it is not surprising that composing
journal entries seemed to characterize the initiative students took in this context. In the
publishing environment, students also tended to ask partners for help, continue or modify
work, edit for misspellings or revise the work so it made sense, negotiate roles in the task,
and decide on resources to use. Students’ initiative may have been characterized by these
types of interactions because, although they focused on fewer problems, they spent more
time explicating their entries, so they would make sense to readers. In addition, because
students were interacting around the completion of one piece of work versus separate
assignments, they tended to demonstrate initiative characterized as collaborative and
management-oriented. While publishing, students created a co-constructed product and
may have been more interested in sharing and shaping the work. In this sense, I argue
that, by interacting around one piece of work, students were more likely to collaborate on

the publication in an interconnected fashion.

91

CHAPTER 5

COMPARING STUDENTS’ WORK ON TMC AND IN INDIVIDUAL JOURNALS

I have presented interview and observation data to provide a sense of
students’ perception of the Web journal versus their individual journals and what kinds of
interactions illustrate students’ initiative in publishing and nonpublishing environments.
In this chapter, I analyze students’ TMC and individual journals to examine how initiative
is manifested in their work. This examination complements and elaborates on themes that
emerged from interviews and classroom observations. I also use this analysis of students’
journals to understand what kinds of initiative characterize the work students
accomplished in TMC or their individual journals.

As discussed earlier, TMC is a Web-based mathematics journal that is
dynamically developed as students contribute their entries and responses to others’
writings. Students usually worked with partners, but on a couple of occasions, wrote
entries by themselves. TMC’s organization is aligned with the Connected Mathematics
Project (CMP) curriculum (Lappan et al., 1996a), so it is arranged into investigations
(lessons) within units. Students were encouraged to write on the investigation they were
learning at the time, but otherwise could choose the topic of their entry. In addition,
students were not graded or given credit towards their grade for completing TMC entries.

Individual journals were also given to students to maintain. These journals were
blank notebooks that each student was issued at the beginning of the year. The teacher
credited students for writing in this journal daily and encouraged students to write

something every day, so students’ journals were organized chronologically. Classwork,

92

mathematical reﬂections, or general ideas, strategies, and explanations were the type of
entries the teacher expected. After reviewing TMC, Mr. Watkins made announcements
throughout the year that the kinds of entries they wrote on TMC should also be written in
their own journals.

Analyzed entries included all 47 student entries on TMC, and 44 entries from the
students’ own journals. Since the entries from the TMC spanned the course of the
project, I selected entries from individual work that also represented work over the course
of the project. To make my selections, I determined the first and last dates of the TMC
entries, as well as the date that fell in the middle of the entries, to represent key dates of
the beginning, middle, and end of the project. Then I selected individual journal entries
that were closest to the key dates for analysis. In a few cases, students discontinued
writing in their individual journals before the end of the project, so I did not select entries
for the end dates from these journals. I also excluded entries irrelevant to mathematical
activities (e.g., workshop on self-control) and chose the individual journal entry second

closest to the key dates for analysis.

Developing a Rubric
Before analyzing the journals, I drew from interview and observation themes to
develop a draft rubric of components that might be apparent in journals. After reviewing
the journals, I highlighted other components that seemed to occur in the entries to reﬁne
the ﬁnal rubric that included the following:
° general statements about the activities accomplished for the day,

° problems from class or homework assignments,

93

' main or mathematical idea for the current lessons,

0 strategies or techniques for solving problems,

0 example solutions to problems,

° explanations of ideas, strategies, or example solutions, and

0 responses to journal entries.

Comparing Entries from TMC and Individual Journals

Students took initiative in writing in TMC and their own journals by including
these different types of components. To understand the kind of initiative students took in
writing on TMC or in their individual journal, I used the rubric to review the journal
entries a third time and coded for the different types of components. I coded each entry
by indicating whether it included at least one instance of the component. I used this
coding system to understand what percentage of entries from each journal included the
component (see Table 13). I used these data to understand what kinds of components are
more likely to appear in each journal with an overall goal of investigating further how

students take initiative in writing on TMC or in writing in their individual journals.

Table 13. Percentage of Journal Entries Including Each Component

 

 

Component Individual Journal TMC
% %
general statement 75.0 0.0
class or homework problems 6.8 40.4
main or mathematical idea 0.0 42.6
strategies or techniques 13.6 61.7
example solutions 11.4 87.2
explanations 6.8 78.7
responses 0.0 8.5

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Understanding the Components of TMC and Individual Journal Entries
In the following data presentation, I provide an example of each component from
each journal (when available), description of the component, and summary of the

percentage of journals that included the component.

General Statements About Activities Accomplished for the Day

Students’ individual journal entries often consisted of only general statements
about the assignments they worked on for the class period and sometimes included
statements regarding their affect (e.g., it was fun, boring, etc.). I included general
statements in the rubric because of the high frequency of this component in individual
journals, so coded TMC entries for it as well. Seventy-five percent of individual journal
entries included only these general statements whereas none of TMC entries were limited
to such general statements. In this sense, it was more likely that students would write
these general statements as their entry in their own journal than on TMC.

Jeff’s case (individual). Most students seemed to use their own journals as a log
of activities for the day as in Jeff’s case (Figure 13). Jeff wrote, “reread 5.2 and ACE
questions, 1-5. Homework 5.3.” His work represents entries commonly written in
individual journals that only stated the problems completed or given for assignments.

Tara’s case (individual). On occasions, students also included general feelings
about the activities that they accomplished for the day. For example, Tara (Figure 14),
wrote, “Today we did pg 23, we had to do the 2.4 follow-up on page 23 for our

homework.” Tara also expressed that “I like working with groups of people and Mr.

95

Watkins lets us do that.” This entry illustrates how students mentioned whether they

enjoyed an activity in their individual journals.

/

Tffﬂa/W.’ rel/cowl S. 2 ano/

ce- quesfmi, I B. «a: homewor

"5.3

Figure 13. Jeff’s entry includes a general statement about the day’s activities.

 

 

 

 

 

 

 

 

 

 

 

Figure 14. Tara’s entry includes a general statement of the day’s activities.

96

Problems From Class or Homework Assignments

Sometimes students included in their individual and Web entries, examples of the
assignment they were completing. On occasions, students also wrote a short narrative
about the main idea, solution strategy, or explanation, but at times, only completed
problems in the journal as though it was a notebook for completed assignments. This
component appeared in 40.4% of TMC entries and was more likely to appear in TMC
than in students’ individual journals where it appeared in 6.8% of the cases.

Shannon and Mandy’s case (TMC). Shannon and Mandy’s case (Figure 15)
illustrates the kind of assigned problems students might include in the journals. In this
event, Mr. Watkins asked students to write mathematical reﬂections about the
investigation they recently studied, and Shanon and Mandy decided to publish this on
TMC. As indicated, the two students completed questions 2 and 3. Question 3 is actually
taken from the students’ book and included in the entry: “When you multiply 2 whole
numbers, the product is larger than the factors. Is the product of the 2 fractions larger
than the fractions?” Students were expected to answer these reﬂections questions,
providing some reasoning as evidence. Indeed, Shanon and Mandy answered the
question by reasoning, “No, because every time you multiply a fraction the number gets
smaller. This happens because when you break a fraction in parts it will decrease.”

Mandy’s case (individual). Students completed assignments in their individual
journals as well. As Mandy’s case (Figure 16) illustrates, however, students sometimes
only provided the answers to assigned problems. In this case, Mandy solved “Problems

5.1 and 5.2” in her journal, but unlike Shannon and Mandy’s entry above, did not include

97

 

Mathematical Reﬂection
By Shanon and Mandy

2.SOMETIMES WE SEE PROBLEMS LIKE 3/5 X 4/5 ALL YOU HAVE TO DO TO
SOLVE THIS PROBLEM IS MULTIPLY THE NUMERATORS 3X4=12 SO THAT IS
THE NEW NUMERATOR. THEN YOU HAVE TO FIND THE DENOMINATOR SO
YOU MULTIPLY THE 2 DENOMINATORS 5X5=25. SO YOUR DENOMINATOR IS
25. YOUR FRACTION NOW IS 12/25. IN OTHER CASES YOU WILL END UP
WITH ANSWERS LIKE 12/24. SINCE YOU CAN REDUCE THAT THEN YOUR
ANSWER WILL BE 1/2.

3. QUESTION: WHEN YOU MULTIPLY 2 WHOLE NUMBERS, THE PRODUCT IS
LARGER THAN THE FACTORS. IS THE PRODUCT OF THE 2 FRACTIONS
LARGER THAN THE FRACTIONS. ANSWER;REASONING: NO BECAUSE
EVERYTIME YOU MULTIPLY A FRACTION THE NUMBER GETS SMALLER.
THIS HAPPENS BACAUSEWHEN YOU BREAK A FRACTION IN PARTS IT WILL
DECREASE.

 

Responses

 

 

Would you like to respond?

Figure 15. Shanon and Mandy’s entry including problems from class or homework
assignments.

98

 

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Figure 16. Mandy’s entry including problems from class or homework assignments.

99

the context of the answer. For example, she wrote “Mr. Sims paid $8.00 and bought 1/3,”
but did not include the problem she solved. As a result, it is difficult to decide what
question she addressed. Mandy’s example represents other cases of student journal
entries that demonstrated the completion of assignments, but provided little explanation

of the context or reasoning behind the answers.

Main Idea for Current Lesson

I coded the two journals for main or mathematical ideas because some students
claimed in their interviews that they wrote them in their individual journals as well as
they did on TMC. Main ideas included general statements about the topic students
discussed (e.g., “over the last few weeks, we have been learning how to reduce fractions,
rename fractions as decimals or percentages”) as well as more speciﬁc mathematical
ideas, including definitions (e.g., definition of fraction), labeling of mathematical
expressions (e.g., top number of a fraction is the numerator), and mathematical
relationships (e.g., “we are putting fractions into decimals”). This component occurred in
42.6% of the TMC entries and none of the individual journals.

Ganton and Justin ’s case (TMC). On some occasions, students made general
statements about the mathematical ideas they were learning in TMC publications. Ganton
and Justin’s entry (Figure 17) is a short description of the topic students reviewed along
with a part of an assignment they completed in class. They wrote that the main idea they
explored for the week was “working on adding and subtracting fractions.”

Lin and Camrn’s case (TMC). Lin and Camrn’s entry (Figure 18), which

illustrates how students also wrote about more specific main ideas, depicts the authors’

100

 

Tupelo township
By Ganton and Justin

In this lesson we have been working on adding and subtracting fractions. Lapp owns 1/4
of the town. Bouk's area multiplied by 4 equals Lapp's total area. Bouck's area is 1/ 16 and
Lapp's area is 4/ 16 of the town.

 

Responses

In this lesson we have been working on adding and subtracting fractions. For example
Lapp owns 1/4 of the town. Bouk's area multiplied by 4 equal parts is equal to Lapp's
total area. Bouck's area is 1/ 16 and Lapp's area is 4/ 16 of the town. 4/16 is equal to 1/4 of
the town.

Lynton and Tinse

 

 

Would you like to resEond?

 

 

Figure 17. Ganton and Justin’s entry including main idea for current lesson.

 

Fractions into Decimals
By Lin and Camrn

We are putting fractions into decimals. For example, today in class, we had to ﬁgure out
the fractions of the vegetables in Justin's Garden. Then we had to take the fractions and
put them into decimals. We had to take the number of vegetables which was out of a 100
square meters. For example, on one of them, we had to figure the least amount of potatoes
we could have in the garden. And the other one, we had to find the highest amount of
potatoes. We still had to get all of the other vegetables in the garden. One example is
60/100 equals 0.60 To get this, we had to first count all of the vegetables out and write
them down as fractions and then put the fractions into decimals. To change the fraction
into decimals, we knew that the hundredths stands for two over on the right side of the
decimal.

 

Responses

 

 

Would you like to respond?

 

 

Figure 18. Lin and Camm’s entry including main idea for current lesson.

101

understanding of decimals. In this setting, students reviewed how to change fractions
into decimals. The main idea that Lin and Camrn wrote was “To change the fraction into
decimals, we knew that the hundredths stands for two over on the right side of the
decimal.” They considered this idea in concluding that “60/100 equals 0.60.” This entry
exempliﬁes how students wrote about main ideas by labeling mathematical expressions,

in this case a decimal, and translated that knowledge into relating fractions and decimals.

Strategies for Solving Problems

Again, I coded for strategies (drawings or algorithms) in journal entries because
students claimed they wrote them in both journals and to understand whether students
indeed took initiative in this domain. As students afﬁrmed in their interviews, strategies
were quite common in their TMC journal entries, but contrary to what some students
claimed, were rarely apparent in their individual journals. It was more likely to appear in
TMC where it was included in 61.7% of the entries than in individual journals where it
was included in 13.6% of the entries.

Tinse and Carnrn’s case (TMC). Tinse and Camm’s case exemplifies how
students used drawings as solution strategies. In this case, Tinse and Camm needed to
determine what percentage of Figure 19 was shaded. Figures 19 and 20 illustrate that the
students decided that considering the whole pie was 100%, they could “divide it into 4
pieces,” and discover that one piece was equal to the “eaten” (shaded) portion, that is
“25% or a quarter of the pie.” Their solution strategy was to sketch a picture that divided

the rest of the “circle graph,” so that each portion would equal the shaded piece. Tinse

102

 

Finding what one piece of a Pie Graph is
By Tinse and Cam

Circle graphs,or pie charts,is a kind of graph that shows how a whole (100%)is divided
into several categories. On the pie chart that we drew someone ate 25% of a pizza.75% is
left of the pizza. We found out that someone ate 25% of the pizza by thinking of the pizza
as 100% or as a whole.Then divided it into 4 pieces so each piece is 25%.So if 1 piece is
eaten then we know that someone ate 25% or a quarter of the pizza.

This is by:

Tinse and Camrn

.2;

 

Responses

 

 

Would you like to respond?

 

 

Figure 19. Tinse and Camm’s entry including strategies for solving problems.

      

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Figure 20. Pie graph representing 25% and 75%.

103

and Camrn concluded four of these pieces made a whole or 100% and decided that the
shaded portion was 1/4 or a quarter of the 100% which came to 25 %.

Hazel ’s case (individual). Students also used algorithms for strategies as
depicted in Figure 21. During the events surrounding the writing of this entry, students
were learning how to determine the equivalent percentage for a given fraction. Hazel
knew fractions based on hundredths could easily be converted to percentages because a
percent was a portion out of a hundred. Therefore, she developed a strategy for deciding
how to change fractions so that it was based on hundredths. She concluded that “if the
denominator... is not 100, you multiply” the denominator so that it is 100. Then, “you
take the number of how many (numerators) and you multiply by that number,” the factor
of 100 or the number you multiplied the denominator by, to get the numerator. The
converted numerator and denominator would presumably result in the equivalent fraction
“based on hundredths.” Hazel also added that if the denominator is greater, you-divide it
so that it is 100. In this case, the student did not provide a thorough explanation of the
strategy because it needs to be assumed that the numerator is also divided by the same
factor. The written explanation may not include all of the details of the strategy because
it was written for herself so that she can make sense of her notes. This entry clearly
indicates a procedure Hazel can use under different circumstances for deciding the

equivalent fraction based on hundredths or percentage for a given fraction.

104

 

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Figure 21. Hazel’s entry including strategies for solving problems.

Example Solutions to Problems

Students’ entries also included solutions or the answers to problems, and I
analyzed journal entries for this component to examine whether students took initiative in
this domain. I coded students’ work for solutions that employed ideas or strategies
students discussed to distinguish them from simple answers. In both of the following
settings, students may have solved a problem, but in the latter, only provided an answer.
Example solutions were more common in TMC entries (87.2% of the entries) than in
students’ individual journals (11.4% of the entries).

Shanon and Mandy’s case (TMC). When explaining a strategy, students often
included an example, as in the case of Shanon and Mandy (Figure 22). They provided a
strategy for finding the percentage for a given amount when the percentage is a multiple

of 10% (i.e., “10, 20, 30, 40, etc.”). In addition, they provided examples of how to use

105

the algorithm. For instance, to find 40% of $34.00, the students took “10% of
34.00=$3.40, then multiply $3.40 by 4,” and provided an example solution using the
strategy:

Example: 3.40

__x_ﬂ
$ 13.60

 

Finding Percents
By Shanon and Mandy

When finding percents that are multiples of 10; 10, 20, 30, 40, etc. first find 10 percent.
EXAMPLE: $ 34.00 10% of 34.00 = $3.40

If your looking for 40%, Then multiply $3.40 by 4; 2 if your looking for 20%, etc.
EXAMPLE:
3.40
)_r_4
$13.60
So 40% of $34.00 would be $13.60

 

Responses

 

 

Would you like to respond?

 

 

Figure 22. Shanon and Mandy’s entry including example solution.

106

Lin ’s case (individual). At times, students simply wrote about the main idea and
provided an example solution, as in Lin’s case (Figure 23). While writing this entry, Lin
wrote the class “corrected homework [and] took a test on multiplying decimal,” and also
provided “an example of a problem we did:”

3.07
x 2.06

4842

000
614
6.3242

In this case, Lin solved this problem to demonstrate how to multiply decimals.

Explanations of Ideas, Strategies, or Example Solutions

In their interviews, students claimed that it was important to “tell how to do it”
and “explain the problems and answers” in the Web journal. I analyzed TMC entries to
examine whether students indeed provided explanations of their ideas, solution strategies,
or solutions and also studied students’ own journal entries for explanations to compare
work from the two journals. Students were more likely to provide explanations in TMC
(78.7% of the entries) than in their individual journals (6.8% of the entries).

I distinguish explanations of ideas, solution strategies, or solutions from only
presenting ideas or solution strategies because to qualify as explanations, students needed
to be more explicit by writing about the ideas, solution strategies, or solutions in words.
For example, in Figure 17, Ganton and Justin stated the main idea of the investigation,
adding and subtracting fractions, but neither explained what adding and subtracting
meant mathematically nor how to do it. The following two cases provide a better sense of

the kinds of explanations students wrote.

107

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Figure 23. Lin’s entry including example solution.

108

 

Kyndi, Lin, and Hazel ’s case (TMC). Kyndi, Lin, and Hazel’s case provides an
example of how students explained their solutions. In this case, the students included a
problem from an assignment and determined the total cost a person incurred by buying
3/4, 4/8, and 10/ 16 fractions of advertisement pages. They provided all of the
information needed to solve the problem including the fractional costs of a page and
explained how they solved it, indicating,

We ﬁgured out this problem by multiplying: 3x$40=$120 (3/4),

4x$20=$80 (=4/8), 10x$10=100 (=10/16). Then we added

$120+$80+$100=$300. So this person spent $300 on advertisements.

In this scenario, the students explained how they solved the problem. For example, they
knew that 1/4 of a page cost $40, so to find the cost of 3/4 of a page, they calculated 3
times the cost of 1/4 of a page or $40. They continued to explain how they found out
what “this person spent” by writing “then we added $120+$80+$100=$300.”

Dana’s case (individual). Although rare in students’ individual journals, some of
the entries included explanations. Dana’s entry (Figure 24), similar to the case in Figure
23, provides an example of how students explained strategies. Unlike Lin, Dana did not
provide a solution, but explained how to find an equivalent fraction that is based on
hundredths for a given fraction. I consider cases like hers to be explanations because she
wrote in words “how to do it,” instead of simply writing an algorithm such as the
following:

A with B>100: ﬂ? =

B B/? =100
A with B<100: A x ? =
B B x ?=100

109

This example presents an algorithm with variable manipulation to find an
equivalent fraction that is based on hundredths, whereas the entries that I consider

explanations included some kind of description.

  

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111; 1.31. m 1111111111131

 

Figure 24. Dana’s entry including explanation.

Responses to Peer ’s Journal Entries

Writing responses to the entries of their peers was another aspect of the TMC
students enjoyed, as indicated in their interviews. At times, students read TMC from
home or school and tried responding to others’ work or added to entries that they wanted
to revise themselves. I coded TMC entries to determine whether students indeed
contributed responses to the Web journal. Only a small percentage, 8.5%, of TMC
entries were responses, but the responses written provide a window into how students can
view the TMC as a collaborative journal and an avenue for supporting each other’s work.

Lynton and Tinse’s case (TMC). In responding to their peers’ entries, students

may have simply elaborated on the publication to provide a better explanation as in the

110

case of Lynton and Tinse (Figure 25). They read Ganton and Justin’s entry, which
explained that the class was “working on adding and subtracting fractions.” Lynton and
Tinse, however, were not satisfied with the authors’ example. Ganton and Justin took an
example from their assignment and wrote that Lapp owns 1/4 of the town and that Bouk’s
area multiplied by 4 equals Lapp’s total area. They then concluded that Bouk’s area is
1/ 16 and Lapp’s is 4/16. Lynton and Tinse responded to the entry with “4/ 16 is equal to
1/4 of the town.” In this case, they provided a connection between the claim that Lapp
owns 1/4 of the town or 4/16 of the town by indicating that 4/16 and 1/4 are equal.
Lynton, Justin, and Tinse’s case (T MC). Whereas some students may have
elaborated on others’ ideas to provide a better explanation for Web entries, other students
provided completely new strategies that made more sense. For example, Figure 26
illustrates that Bali provided a solution and explanation for how to add fractions when the
denominators are different: “If it was 2/5 + 2/4, the answer would be 4/5 because when
you have 2 fractions with different denominators you always go with the bigger number,”
the denominator. Lynton, Tinse, and Justin responded by providing other strategies:
They purported that 2/5 + 2/4 would be 9/10 “because when you change it into decimals
2/4 would be .5 and 2/5 would be .4. Then you add it together and you end up with .9
which is the same at 9/ 10.” They also provided another strategy of solving the problem
through drawing pie graphs. After shading 2/4 of the whole (or half of the whole) plus
2/5, they discovered that 1/ 10 was left, so 9/ 10 was the shaded portion or the parts that
were added together. In this case, the group responded by not only providing an
explanation, but also challenging the ideas of the author and presenting evidence of their

mathematical ideas and solution strategies.

111

 

Tupelo township
By Ganton and Justin

In this lesson we have been working on adding and subtracting fractions. Lapp owns 1/4
of the town. Bouk's area multiplied by 4 equals Lapp's total area. Bouck's area is 1/16 and
[@p's area is 4/ 16 of the town.

 

Responses

In this lesson we have been working on adding and subtracting fractions. For example
Lapp owns 1/4 of the town. Bouk's area multiplied by 4 equal parts is equal to Lapp's
total area. Bouck's area is 1/16 and Lapp’s area is 4/16 of the town. 4/16 is equal to 1/4 of
the town.

Lynton and Tinse

 

 

Would you like to respond?

 

 

Figure 25. Lynton and Tinse’s response to previously written entry.

 

Adding fractions
By Bali

2/4+2/4= 4/4 because we took the numerator and added them together. We left the
denominator the same. So the answer is 4/4th's. If it was 2/5 + 2/4 the answer would be
4/5th's because when you have 2 fractions with different denominators you always go
with the bjgger number.

 

Responses

Today I am going to explan how to add fraction.2/4+2/4= 4/4 because we took the
numerator and added them together. We left the denominator the same. So the answer is
4/4th's or equivialant to 1 whole.. If it was 2/5 + 2/4 the answer would be 9/ 10th because
if you change it into decimals 2/4 would be .5 and 2/5 would be .4. Then you add it
together and you end up with .9 which is the same as 9/ 10th. You all so can do 2/4 plus
2/5 would equal 9/10. I got that because 1/4 plus 1/4 equals 2/4 which is half a whole
plus US plus US would equal 2/5. 2/4 plus 2/5 would leave 1/10 left so that is why I got
9/ 10.

Lynton, Tinse, and Justin

 

 

Would you like to respond? g

 

 

Figure 26. Bali’s entry providing a solution and explanation for how to add fractions
when the denominators are different, and Lynton, Tinse, and Justin’s
response.

112

Summary

This analysis indicates that individual journals were likely to include only general
statements about activities students accomplished for the day, whereas publications in
TMC illustrated more detailed entries. TMC were more likely to include main ideas,
strategies, and example solutions (student-generated and class or homework problems).
Students were also more likely in these entries to explain in words the ideas, strategies,
and solutions. In addition, as students indicated in their interviews, some students
responded to peers’ entries.

The range of components students included in their TMC and individual journal
entries provides another view of how students demonstrate initiative through their work.
Overall, students wrote more substantively rich and clearly written entries in TMC than in
their individual journals. Since it was more likely that students would include more of
the various components in TMC, entries written on the Web were more likely to be

elaborate and specific.

113

CHAPTER 6

WEAVIN G AN UNDERSTANDING OF INITIATIVE

A broad goal of education has been to prepare students for novel situations
beyond the everyday tasks of school. Researchers have argued that some learners may
have difficulty connecting their school knowledge with practices in the workplace or the
community, but may still develop their own ways of thinking or approaching problems in
out-of-school settings (Nunes et al., 1993; Resnick, 1987). Many have proposed ways of
supporting learners in taking a more central role in the development of their mathematics
thinking and learning (Cobb et al., 1992; Ball, 1992; Lampert, 1987; Schoenfeld, 1989) to
prepare them for authentic problem-solving situations.

In beginning this work, I conceptualized student initiative as another way to
describe, understand, and promote learners who direct and own their learning. I defined
initiative as comprising volition, ownership, action, and effort. I suggested that students
taking initiative make decisions about how to proceed with solving a problem and author
the ideas or ways to approach the problem. In this sense, learners have the executive
control and come to own their work because they are the creators of it. Instead of taking
a passive role of knowledge consumption, initiative calls for students who take an active
role by generating their own mathematical thinking. Through all of these activities,
students demonstrate initiative through their efforts to move forward in completing tasks
or solving problems.

A main agenda of this study was to move beyond this general conceptualization of

initiative to better understand specific ways students can demonstrate initiative. In

114

Chapters 3, 4, and 5, I presented results from students’ final interviews, observations of
student interactions in and out of the publishing context, and a comparison of students’
individual journals and TMC entries. In this chapter, I use these results to consider how
students took initiative in these contexts. Then, I highlight the salient themes emerging
across these data sources to provide an understanding of what aspects of TMC
environment afforded or hindered different ways students demonstrated initiative. Along
with examining these themes, I revisit the design principles and how they are related to
the initiative and factors promoting initiative. Throughout this discussion lies the
contribution of this work in establishing a richer sense of the conceptualization of

initiative.

What Does Initiative Look Like?

To provide a better understanding of initiative, I intended to go beyond my initial
deﬁnition to examine what initiative looked like in the classroom. In this sense, I wanted
to begin the work needed to enable educators to recognize initiative. This research
reveals initiative as reﬂected in students’ interactions and in the written products of their

efforts.

Initiative in Interactions in Publishing and Nonpublishing Contexts

Analysis of student interactions in the publishing and nonpublishing contexts
revealed that students take initiative in different ways. In addition, the kinds of initiative
demonstrated could be categorized into three broad domains of mathematics learning --
mathematical sense-making, collaboration, and management. The following review

highlights the particular ways that students indeed took initiative.

115

Mathematical sense-making. Mathematical sense-making entailed learners’
efforts to understand the question or problem confronting them, mathematical ideas, and
ways of approaching the problem. Just as a painter decides the subject of her work and
owns the complexity of each stroke for ﬁnishing the painting, students taking initiative in
mathematical sense-making come to an understanding of a problem and then navigate the
path towards solving that problem.

I suggest that to begin the process of mathematical sense-making, students first
decided on a topic for a journal entry. At times, students were quite thoughtful about
their journal entry topic by deciding on a subject; at other times, they simply reported
what activity the class completed. Although students differed in how thoughtful they
were in deciding on a topic, they showed initiative in all of these ways of deciding on a
topic. Students may have also begun a task by defining the problem. In these cases, they
may not have heard the assignment, but took initiative by attempting to determine the task
by asking others in the classroom. Initiative in these ways reflected student effort in
determining the task that also fueled their engagement in solving the problem.
Determining the journal topic, problem, or task created the boundary for discussion and
launched students into mathematical sense-making.

These kinds of events were described as initiative, in contrast to cases in which
students made these kinds of decisions by following the lead of the teacher or researcher.
Such contrasting scenarios might include students talking about off-task subjects and
failing to launch into mathematical sense-making until the teacher or researcher
approaches them to suggest several topics for a journal entry or to remind them of the

assignment. In these scenarios, although students may have chosen a topic or announced

116

the assignment, they did not take the first step, themselves, in making these decisions;
rather, they waited for the prompting of an adult.

Students also demonstrated initiative in mathematical sense-making by suggesting
mathematical ideas or strategies and generating solutions to problems. They proposed
concepts such as the relationship among fractions, decimals, and percentages; used
pictures or algorithms to solve problems; and applied ideas or strategies to present
examples of how to solve a particular problem. These actions constituted initiative
because students were trying to make sense of the concepts and how to use them in
solving mathematical problems. Students took this kind of initiative in response to peers’
requests for help or in tackling the various steps of a problem. The time they spent on
discovering the ideas and strategies that lead to solving a problem rather than simply
determining the answers highlights the effort and action-- the initiative that they took in
their learning.

Fostering initiative in this particular area is a key to promoting learners who are
the originators of ideas or strategies and who follow through with applying them to solve
problems. As encouraged by the NCI‘M Standards (1991) and the National Research
Council (1989), students moved beyond merely learning the facts and procedures and
began owning their meaning. Suggesting ideas, strategies, and solutions illustrate ways
students examined, transformed, applied, and communicated (National Research Council,
1989) their knowledge in solving problems. The knowledge was not simply transferred
from an authority; rather it was learned and transformed by the students. Therefore,
taking initiative in this realm requires knowledge producers leading the discussion rather

than echoers or followers of a teacher.

117

Finally, students took ownership in mathematical sense-making through offering
explanations of the mathematical ideas, strategies, solutions or editing suggestions by
expressing in their own words what these were. Students may suggest mathematical
ideas, strategies, solutions, or editing remarks, but the distinguishing feature between
suggestions and explanations is that students talk about how they generated them. For
example, a student may suggest the solution for the addition of 1/8, 1/4, and 1/16 as 7/16,
but also provide an explanation that it is 7/16 “because 1/8 is 2/16 and 1/4 is 4/16 and add
1/ 16 and you get 7/ 16” (Tara and Bali interaction). Some students provided explanations
to compare responses to problems, and others used them as a means to respond to their
peers’ challenges and questions. Initiative in this area led them towards representing,
proving, and communicating (National Research Council, 1989) what they thought or
learned.

Taking initiative in providing explanations is crucial to developing learners who
can reexamine how they think about problems and peers who can beneﬁt from others’
knowledge. Through offering explanations, learners reflect on their own thinking as they
articulate their thoughts. This process of communicating more than a simple statement of
a concept, procedure, or answer for a problem requires students to organize their thoughts
into an effective explanation. Therefore, students’ own ways for solving problems
(Wood & Sellers, 1996) are crystallized out of their thoughts and presentations. In
addition, peers can Ieam from the expertise or knowledge of their partners (Brown et al.,
1993) by trying out ideas with one another and interconnecting their contributions

towards one or several ways of thinking about a problem.

118

Students thus took initiative in mathematical sense-making through determining
the task and discussing or explaining their ideas, strategies, solutions, and revisions.

Collaboration. Collaborative interactions are another broad domain within which
students took initiative. In these events, learners directed their requests and support
towards their partners.

Students worked together and assisted each other by asking partners for help in
completing work rather than automatically turning to Mr. Watkins ﬁrst for direction. On
occasions, students who needed clarification of the problem or were at a standstill about
how to tackle a part of the problem asked peers how they might approach it. In these
events, by relying on their partners for support, students were attempting to initiate or
reinitiate steps towards approaching the task at hand. Initiative in this area illustrates the
effort and action students exert in continuing the path towards solving a problem in
contrast to events in which students asked partners for help only after the teacher or
researcher encouraged them to turn to their peers. Students were acting out the behaviors
without taking initiative by relying on the teacher or researcher rather than assuming the
control or making the effort to collaborate with peers.

In addition, students demonstrated that challenging, questioning, or asking for
clariﬁcation on propositions and strategies was important in understanding their
partners’ thinking and the problem they were solving. These interactions were sparked
by conﬂicts between students’ and peers’ thinking or confusion about partners’ ideas and
strategies instead of the requested direction of an adult. Through these interactions,

students’ initiative was demonstrated as they took an active role in eliciting their partners

thinking. This kind of initiative is signiﬁcant because it promotes an environment where

119

students push the group further in understanding what they are learning and in reaching a
resolution for the problem.

Finally, students worked together through continuing or modifying each other ’3
ideas and editing for grammatical or spelling errors. While writing TMC entries,
students supported their peers by elaborating on and revising the co-product. As their
partners typed or contributed additions, students chimed in with their suggestions. Events
such as these illustrate how learners took initiative by offering their assistance on their
own accord, rather than through the lead of an adult to revise their work, and in this
sense, established ownership for the work.

Management of tasks. Also important in mathematics learning was students’
management of tasks through negotiating roles or deciding on resources to use (e. g.,
calculator, overhead, ruler). While publishing a TMC entry, students decided what parts
of a problem each would work or who would type. In these cases, students took action by
making executive decisions to organize or plan, dividing tasks, and choosing what tools
to use. This kind of initiative is significant in establishing expectations for who will be
accountable for what part of a task.

Students generated mathematical ideas to use and apply in solving problems,
worked with their partners, and managed the roles or tools. By taking initiative in these
three broad domains, students demonstrated effort in solving a problem or task. Learners
not only showed action on their part in their interactions with partners, but also owned the

work as the generators of ideas, strategies, and solutions.

120

Initiative in the Written Work

Students’ written work in their individual journals and TMC entries provides
further evidence for various kinds of initiative. Some student entries consisted of only
general statements about activities accomplished for the day. For example, one entire
entry consisted of, “Problems 4.2 and 4.3”; another consisted of, “We watched a video on
mathematics, and it was fun.” On other occasions, students wrote more substantive and
detailed entries that included examples of assigned problems, main ideas, strategies,
example solutions, or explanations. Compared to entries that included only general
statements, the more extensive journal entries provided richer descriptions of the ideas
discussed, how students could use these ideas, examples applying the ideas, and students’
own words to clarify what they were trying to convey. These entries reﬂected student
initiative in their thinking, contributing of ideas, and making decisions about appropriate
information to include. This initiative is signiﬁcant because through these kinds of
actions, students are likely examining their thinking and advancing their learning.

Some students also showed initiative in collaborating by writing entries that were
responses to the work of others. In these cases, students offered a different way of
communicating the topic of discussion by providing more sensible ideas, strategies,
example solutions, and explanations. This kind of initiative reﬂects students’ active role

in shaping their peers’ work and extending the ideas in the publications.

Extending the Understanding of Initiative and Its Relationship to Prior Research
In introducing this study, I suggested that initiative connoted action, ownership,

volition, and effort. Analysis of students’ interactions and written work have shown

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student initiative to be reﬂected in multiple ways--all illustrating action on the part of the
students through their contributions, challenges, and requests. Students in this study
gained ownership as they, themselves, generated the decisions, ideas, and solutions to
problems, and contributed this knowledge through collaborative interactions or their
written work. Aligned with the goals of other researchers (e.g., Brown, 1993; CGTV,
1991), students came to own some of the knowledge and shared it among others and, in
the process, generated their own solution methods and made their own decisions.

A number of researchers have examined concepts that I would term initiative:

formulating or challenging different arguments and justifications (Cobb et al.,
1992; Wood & Sellers, 1996; Yackel et al., 1991),

making decisions (Lampert, 1987; Scardamalia & Bereiter, 1994),

developing own solutions (CTGV, 1992), and

f pursuing own topics (Schwartz, 1993) or deﬁning a task (Schoenfeld, 1989).
These researchers were focusing on particular issues and were promoting different ways
to enhance mathematical thinking and learning. Drawing from and building upon these
researchers’ ideas, I propose that initiative encompasses all of these kinds of interactions
and is a common thread running through them. Whereas these researchers emphasized
different ways of thinking and learning, the focus in this study is the underlying general
characteristic of a learner, initiative or action on the part of the student in taking the lead
in their thinking and learning.

Extending beyond previous research on how to enhance students’ role in their
learning, this work shows other ways that students take initiative. This study suggests that

student initiative plays out in other kinds of interactions and the work that students

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produce. It continues other researchers’ examination of how students can collaborate in
mathematics thinking and learning (Cobb et al., 1992; Wood & Sellers, 1996; Yackel et
al., 1991) by emphasizing the significance of asking partners for help and continuing or
revising each others’ ideas. Along the same line, though Schoenfeld (1989) mentions the
importance of metacognitive skills in problem-solving, this work suggests that initiative
in other management skills--negotiation of roles or determination of resources to use--
supports the organization and progress of a task. Finally, initiative was evident in the
care and effort of students as they created elaborate and sensible entries in TMC, which
implies that student initiative is demonstrated not only in their interactions, but in their
written work as well.

As important, this work highlights that students may demonstrate these kinds of
interactions without taking initiative. In one scenario, students may define their own
questions, generate their own ideas, strategies, or solutions, request or provide assistance
to their peers, and challenge peers’ thinking, all with the lead and prompting of a teacher.
These events play out as students continually turn to a teacher to provide scaffolding by
reminding students to think on their own and collaborate. This kind of scenario illustrates
students demonstrating the kinds of interactions that other researchers and I promote, but
not the taking of initiative. In this work, I have emphasized that initiative requires
students to take the first step. Learners, themselves, need to take the responsibility to lead
the discussion and thinking. I believe other researchers (e.g., Cobb et al., 1992; Lampert,
1987; Schwartz, 1993) would agree that they were also promoting initiative in this sense,

but my work emphasizes its role and link between their ideas.

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Students’ initiative in the discussed areas mirrored the effort they made in moving
the task forward or in attempting to understand the problem. In each of these events,
initiative led learners towards accomplishing tasks and engaging them in thinking about
the ideas around their work. As a result, initiative laid a path for students to take a more
self-directed and central role in their thinking and learning.

This work enriches our understanding of initiative by providing examples of how
students may take initiative in a mathematics classroom and by framing initiative in
relation to previous thinking about how to encourage students to take a more central role
in their learning. I suggest, however, that this study is only a first step towards
understanding initiative and how it is demonstrated. Student initiative may look different
in other contexts that vary, for example, by subject matter or classroom culture. In
addition, the kinds of interactions or other evidence for initiative may emerge from
studies that include other data sources (e.g., incidental interviews focusing on students’

perceptions of their role).

Types of Initiative Vary Across Contexts

Another focus of this study was how students’ initiative varied across the settings
in which they worked. In their interviews, students expressed an overall preference for
TMC publishing over other mathematics activities and writing in their own journals.
They said they enjoyed writing journal entries on the Web, thought it was interesting and
“made sense for people to do.” Many students requested opportunities to publish and
would even leave class assignments for homework so they could work on TMC. It was

clear that students valued publishing by recommending in their interviews that learners in

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other classes should have the opportunity to write on TMC as well. For some, teaching
the audience made publishing an enjoyable and worthwhile activity:

If you publish and let everybody see it, you feel happy because you taught

somebody how to do something that they didn’t know how to do. [Ande,

ﬁnal interview]
A further discussion of the kinds of initiative taken and comprehensive analysis of TMC
context reveal how this environment afforded or was limited to some forms of initiative.

Analysis of the student interactions suggests that the kinds of initiative students
took differed across the publishing and nonpublishing contexts. The initiative students
took in the publishing context included:

0 deﬁning a topic for TMC entry,

0 asking partners for help,

0 continuing or modifying work,

0 editing for misspellings or revising so the entry made sense,

0 negotiating roles in the task, and

0 deciding on resources to use.
These types of initiative were largely collaborative and management-oriented; students’
actions centered on supporting each other and organizing plans in order to work through a
task.

In contrast, the initiative that students took in the nonpublishing environment
included:

° clarifying tasks and

0 suggesting or explaining mathematical ideas, strategies, solutions.

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The kinds of initiative in the nonpublishing environment seemed in some ways
more focused on mathematical sense-making than did the initiative demonstrated in the
publishing environment. Initiative in the nonpublishing environment entailed narrowing
down the purpose of the task and examining the ideas or solutions.

Recognizing some differences between nonpublishing and publishing context
might inform why the initiative demonstrated in the contexts tended to be different as
well. First, the publishing environment may limit how often students took initiative in
discussing main ideas, strategies, solutions, or explanations, primarily because students
focused on fewer problems for the publication than in the nonpublishing environment.
As a result, because they worked on more problems, students tended to discuss and
explain ideas, strategies, or solutions in the nonpublishing context. In contrast, while
publishing, students spent more time explicating and revising thoughts to develop richer
entries for TMC rather than focusing on multiple problems.

In addition, student initiative may have been more collaborative and management-
oriented in the publishing than in the nonpublishing context because learners were co-
constructing single pieces of work rather than completing separate products.
Consequently, students needed to be more aware of sharing and making their own
contributions to shaping the work when publishing than during nonpublishing activities.
Although student initiative differed between the nonpublishing and publishing contexts,
the student actions in both settings were significant in moving the task forward and
supporting students’ reﬂection on the ideas and solutions.

Analysis of students’ entries in TMC and individual journals also revealed

differences in the kinds of initiative students took. Students tended to limit their

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individual journal entries to general statements about the activities for the day. These
entries usually consisted of a few words or a couple of sentences regarding what
assignment students were given or how students enjoyed the day’s activity. Thus,
students did not appear to take much ownership or initiative in the writing of their
individual journal entries.

In contrast, students’ TMC entries tended to be more elaborate and included:

° class or homework problems,

0 main ideas,

0 strategies,

0 example solutions, and

' explanations.

TMC entries were the products of student action and effort in creating a detailed, sensible
publication. Initiative in this respect is significant because it represents students’
involvement and willingness to compose a purposeful entry.

In one setting, students discussed main ideas, strategies, explanations, or
solutions, and in another, they wrote them in journal entries. When writing on TMC,
students believed they could “put what they want to publish” and choose a topic they
were learning in class. In this sense, students expressed ownership of TMC publications,
and indeed, took initiative in generating a subject for their entries, often making several
attempts before settling on an idea. It seems that students did not take as much initiative
in discussing main ideas, strategies, explanations, or solutions while publishing as in
nonpublishing activities; What they wrote in publications, however, included more of

these components than did their individual journals. Examination of the interactions

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suggests that students tended to take more initiative in mathematical sense-making in the
nonpublishing than publishing environments. Students’ written work, however, revealed
that they were taking initiative by thinking about the ideas, strategies, or solutions to
write about them. In this sense, interactions reﬂect student initiative in their actions

whereas written work reﬂects initiative in their thinking.

How TMC Environment Supported Student Initiative

In addition to these arguments for why initiative varied across contexts lies an
explanation that was illuminated through weaving overall themes from the interviews,
observations, and journal entries. Several themes emerged from data analyses to suggest
how initiative was fostered in the publishing context. The perception of an audience,
learning and teaching others, ownership, and collaboration emerged as underlying
themes for why students took initiative in TMC environment. In addition, these themes
illustrate the significance of the design principles used as the basis for the TMC
environment. In the following sections I discuss the significance of the themes and

reﬂect on their relationship to the design principles.

Perception of an Audience

Other people can read TMC. A key theme of the TMC environment was
students’ perception of an audience. Students emphasized repeatedly in their interviews
(see Appendix D for a list of relevant responses) that a unique feature of writing on the
Web is that “everybody can read it.” Authors perceived an audience beyond their
classroom and argued that people “outside the classroom, not just Mr. Watkins or Ms.

Jones,” could read the publications. However, since I do not have documentation of

128

anyone visiting the site, it is difﬁcult to determine whether students indeed had an
audience outside of the classroom. Regardless, students perceived an outside audience
and claimed the importance of the possible readers in motivating them to work with more
enthusiasm and initiative in authoring the publication than in other mathematics activities.

Students’ consideration of an audience was made evident when they highlighted
their goals of editing and revising work so it made sense to the reader. Authors
emphasized that since everybody could see it, students should “make sure [the]
information is correct” and “use proper spelling [and] punctuation.” Otherwise, “people
will get messed up.” The importance of writing sensible journal entries was a salient
theme in students’ interviews that explains why authors were so meticulous when taking
initiative in this area when publishing.

You can be the teacher and you can learn how to do it. The educational value of
TMC (see Appendix E for a list of relevant responses), a theme related to a perceived
audience, was highlighted in students’ interviews and helps to explain why students were
so careful with what they wrote in their entries. TMC was a medium by which authors
could teach “other people on the Internet [to] learn.” On the Web, students “could be the
teachers,” something they could not do when writing in their own journals where their
“teacher was the one reading it.” In this sense, students thought they could help. learners
outside the classroom from other districts as well as from a wider community. A remote
audience could learn how authors solved the problems, “if they are doing the same
thing;” could receive help on other topics they are learning “if they are having problems

with it;” or could “see if anybody has put anything they did not know.”

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In addition to “helping other people,” students stressed the role of TMC in their
own learning. The process of publishing extended their learning as they examined
various topics new and old to students and discovered strategies to realize “it does really
work and start using it,” or through writing and thinking about TMC entries, students
“understand it more.” Through writing entries, students had a chance to revisit and
reﬂect upon the points made in the publications and, as a result, learn more about the
topic.

Students also alluded to the role of TMC in their own learning by saying they
could use it as a reference or review it to learn new mathematics. “If someone else
published and [students] didn’t understand something, [they] can go to their publication
and read it and understand it.” When authors “explained it really carefully,” students
could review previous entries to resolve problems or questions they might have about
mathematics they did not know or “to see how other people learn math instead of only the
way you learn.”

Finally, students emphasized the importance of the reader’s option to respond to
their work. Through this interchange, authors could learn from the suggestions readers
made “to get better ideas of what they have been doing,” but to provide some feedback as
to how authors may revise their writing or ideas in the entry. Some students did take
initiative by using this capability to respond to peers’ work and offer alternative ways of
discussing an idea or solving a problem.

The students’ emphasis of the educational role of TMC entries provides a
rationale for why students valued the mathematical content of what they wrote. This

valuing was illustrated by students’ effort and beliefs about writing coherent and main

130

ideas as well as how to solve mathematical problems. In distinguishing TMC entries
from their individual journals, students reported that their individual journals mentioned
general thoughts or “explained the whole thing,” such as what the assignment was for the
day, and rarely provided examples. In TMC entries, in contrast, they explained that they
provided more details about what the class was doing for the day, a strategy for solving a
problem, a couple examples or solutions, and an explanation for “what we did and how
we go about it.” They wrote more elaborate explanations of problems, a “step-~by--step”
process of how to solve them, and examples illustrating “how to do it” in TMC than in
their individual journals. These themes from student interviews corroborate the finding
that learners took initiative in including main ideas, strategies, solutions, and explanations
in TMC entries.

These sentiments expressed in the interviews may explain why students
demonstrated more effort in editing and modifying their entries in TMC environment than
in nonpublishing activities. Attempting to polish the arguments they were trying to
convey to the reader, students were compelled to discuss with their partners ways they
could improve the wording and points they were trying to make. Learners perceived an
audience for their Web entries other than themselves, but also imagined the educational
role of their work in their own and others’ learning. As a result, TMC entries were more
likely to “go into more detail on the Web.” Apparently, students’ perception of an
audience led them to provide more explicit journal entries to clarify ideas, strategies,
solutions, and explanations.

In publishing, you can write something no one has. Ownership of work was

another salient theme in student interviews that may illuminate why students took

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initiative (see Appendix F for relevant responses). In addition to valuing the affordance
of the Web to share publications with a remote audience, learners emphasized the
importance of having the ﬂexibility to write what they wanted to publish and to express
ideas in their own words. Students were able “to write something no one else had [which]
made publishing more interesting.” Instead of having the “teacher tell [students] how to
do it,” students, themselves could “explain things” and write their “own ways” for doing
mathematics.

Perhaps because students felt this ownership, they assumed authority over their
ideas. This sense of ownership may have led students to take more pride in their work
and more responsibility for what they wrote on TMC than in their individual journals.
Although students could write their own thoughts in their individual journals, they
believed the only people reading it were themselves, “ the teacher, or Ms. Jones.” The
Web seemed to establish more ownership because “other people on the Internet” could
read it. In this sense, their work was presented to the world--as students may bring
something of their own to class for show and tell. They perceived TMC entries as their
own publications for an outside audience. These assumptions also help to illuminate why
students were so careful with the content of their Web journal entries and why they took
initiative in elaborating the entries in the publications more than in their individual
journals.

Design principle: audience. When I designed TMC, I envisioned an audience
would lead students towards examining and re-examining their work to explain what they
were learning and to teach others. In this study, students not only assumed an audience

beyond the teacher (Cohen & Riel, 1989), but their perception of a public forum

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(Lampert, 1987) resulted in greater efforts to publish sensible, educational journal entries
on TMC. Aligned with my assumptions, students claimed that work presented on TMC
was expected to teach others. However, a role of audience that I did not expect was that
students considered the process of writing on TMC and the ability to use the publications
as a reference as a ways to extend their own learning. They could learn new mathematics
and clarify questions or problems they might have. Assuming that the audience for TMC
included those in and outside the classroom as well as themselves, students believed that
their publications could make an impact on the learning of an audience as well as of their
own.

In addition, although students had individual notebooks for journal entries
especially for themselves, they only emphasized a sense of ownership of knowledge
(Brown et al., 1993; Songer, 1996) when describing their writing on TMC. Writing on
the Web was an avenue to present their way of thinking to a public forum. With this
perception of an audience, students tended to take initiative in creating a meaningful
publication for others to use and read. Speciﬁcally, the efforts they made in TMC
environment-~the collaboration, management of tasks, and production of detailed entries--
reﬂect students’ action and ambition. In this sense, a perception of an audience was a key
to fostering initiative.

When designing TMC, I anticipated that students might actually have an audience
that would encourage student initiative. However, this study indicates that a perception
or the possibility of an audience is very powerful in leading students to take initiative in
carefully developing and revising their work or collaborating with others to create

substantive publications. As a result, to encourage students to take more initiative,

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teachers can take advantage of the Web’s capabilities of making student work available to

the world as a tool for creating a sense of audience.

Collaboration

You have someone else to help you. The role of working with their partners in
engaging them in the publishing environment was also signiﬁcant. Students expressed in
their interviews that through collaboration, they could ask their partners for help and
work together in completing TMC publications (see Appendix G for list of relevant
responses). Indeed, while publishing, students collaborated on thinking of a topic for the
entry, understanding ideas, solving problems, and deciding how to write the publication.
“When [students] do it, it’s a group thing, compromise,... teamwork,” so they negotiated
what to publish and worked together to “word it right” and develop a sensible
publication. In addition, as knowledgeable others, partners could help students “get more
stuff” to understand ideas. Through this interchange, students liked working with
partners to solve problems, “to figure it out.” “Sometimes kids don’t do the work
because they have to sit by themselves or do it by themselves and don’t understand it.”
However, when students published, they had a “friend there to help solve the problem.”
Collaboration also afforded students the opportunity to discuss with others how to word
entries or sensible mathematics that they “think everybody will understand.”

While publishing, learners accomplished these goals through asking peers for
help, continuing or editing publications, and challenging each other’s ideas. In addition,
perhaps because students were completing a collaborative product and seemed to take

some ownership for the work, they delegated roles to divide the tasks and determine what

134

students could contribute to the publication. Since students each had a role in developing
a collaborative product, students took more responsibility for requesting peer help,
advising revisions, and challenging ideas. I suggest students developed a sense of
accountability to the group, so took initiative in supporting each other and creating a
quality publication.

Design principle: Collaboration. The second design principle, collaboration,
indeed was another prominent factor that encouraged student initiative. A major premise
of the role of collaboration was that students would encounter conﬂicts that would lead
them to challenge each other or explain their thinking (Leikin & Zaslavsky, 1997; Yackel
et al., 1991). While publishing, Mr. Watkins and I encouraged these kinds of
interactions. We recommended that students turn to their partners with questions they
had about their contributions or ask them for help on dilemmas they had in determining
the topic or task and in solving a problem. The significance of collaboration was
reﬂected in students’ engagement in these kinds of interactions. While working together,
students not only encountered conﬂicts they needed to resolve through explanations of
ideas, strategies, and solutions, but also valued the importance of requesting and offering
assistance.

Since students had an opportunity to collaborate, they were able to take initiative
in resolving conﬂicts as I had projected. An important contribution of this work in
deﬁning the role of collaboration is that by having partners to consult, students took more
initiative in requesting assistance: They had “a friend there to help them” in contrast to

“being by yourself” with your individual journal and “not [knowing] what to put.”

135

Therefore, this work indicates that students may not only offer their own thoughts, but

also ask their peers for suggestions.

The Chain Linking the Themes: Norms and Practices

The chain linking these factors of audience, learning and teaching, ownership, and
collaboration was the norms and practices established when students were introduced to
the environment. Whole class discussion and incidental reminders of the norms and
practices were given during the project. Main points of the norms and practices
established were that:

0 TMC was the students’ own product,

° collaboration was expected,

0 students needed to take initiative in generating arguments,
explanations, strategies, and questions,

' learners needed to employ strategies to promote collaboration and
initiative,

0 TMC needed to represent the diversity of students’ thinking, and

0 students needed to consider the audience in deciding how and what to
write.

I expected these norms and practices to set the expectations for how students should
interact and that they, themselves, should take the initiative in developing their work.
Overall, students’ interactions around TMC environment and the publications
they created indicate that they were attuned to the norms and practices. Moreover,
students emulated these norms and practices that ultimately led to their initiative in
various domains. As anticipated, their ownership of the publications led to students’

initiative in contributing ideas of how to revise and improve publications, but students

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also discussed their value and emphasis on collaboration as a key factor for enhancing
their interest and involvement in publishing. A common assertion students made was that
they took more initiative by working with each other and by making an effort to develop
a coherent, meaningful publication for “anyone on the Internet.” In this sense, students
highlighted the importance of working together and felt their contributions justified their
authorship.

Students’ initiative was also demonstrated as volition, willful action, and effort in
including ideas, strategies, solutions, and explanations in their TMC work. As a result,
norms and practices indeed seemed to frame the kinds of publications students wrote on
TMC. Finally, a perception of an audience seemed to be the most highlighted difference
of TMC and the reason why students took initiative. Students’ serious consideration of a
possible audience was reﬂected in their emphasis on teaching, learning, and ownership
and explains why they took initiative in developing carefully written, sensible TMC
entries.

This work reinforces the premise that a context has its’ own norms and practices
(Green et al., 1996) that creates the boundary for how students perceive a situation and
interact in the environment. In addition, it suggests that educators can design
environments based on norms and practices to promote particular kinds of learning
experiences (Brown et al., 1993). This work provides, however, only one interpretation
of how to weave design principles into an environment with the goal of promoting

initiative.

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Summary

Students demonstrated initiative in different ways and the kinds of initiative
varied across contexts. Final interviews with students provided the underlying rationale
for why they took initiative and revealed how TMC environment supported students’
actions in mathematics learning. Whereas analyses of observations and journals
represented the manifestations of initiative, students’ interviews illuminated what
principles educators should consider in fostering initiative. In TMC environment,
audience, teaching, learning, ownership, and collaboration were central themes
highlighted in students’ interviews that seemed to be the catalyst for initiative. The
signiﬁcance of these themes is that they reﬂect the design principles used as the
framework for engineering TMC environment indicating that educators can be innovative

in developing an environment to promote initiative.

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CHAPTER 7
CONCLUSIONS

Currently, one of the greatest challenges of educators is to develop techniques for
capturing the interest of learners and motivate them to actively participate and navigate
their own learning. This study offers new insights into the issue of initiative by providing
one example of a Web-based technology, TMC, that illustrates how an environment
based on design principles may be used to foster initiative. The nature of the Web
environment allows for a perceived audience because publications to the Web are
viewable to anyone who has access. This perception of an audience was powerful in
students’ feelings of ownership, learning, teaching, and collaboration. With the belief
that a student “in another school,” someone at home, or people “from all over the world”
could view their work, students could display their own ways of doing mathematics,
could learn from the work as well as teach others, and help each other in developing or
revising work. Results from this study indicate that a Web-based environment may be a
powerful tool in promoting the perception of an audience and that this sense of an

audience could be a very important factor enhancing students’ initiative.

Implications
This research contributes to a better understanding of how initiative is taken, but
more importantly, a practical benefit of this work is that it provides some ideas for how
educators can foster it. As suggested in this study, educators can establish norms and

practices that cultivate student initiative. Students can assume roles and become

139

acculturated to norms and practices through which students take initiative in their
learning.

In addition, the role of tools is interpreted and manifested through students’ use of
them. This study of TMC provides insight into what aspects of an interactive Web-based
environment can be 'used to support initiative and how students perceive this tool in their
own and others’ learning. A perception of an audience seems to be a signiﬁcant factor of
a Web-based environment. This work contributes to a wider body of research focusing
on how the Web can be a partner to educators in fostering initiative and implies that they
should capitalize on the perception of an audience created by Web-based publishing.

On one level, this research advances the understanding of initiative, but ultimately
contributes to the broader issue of promoting learners who develop their own thinking
(National Research Council, 1989). By nurturing learners who take initiative, we are
educating our students to take authority over new problems they might encounter. In the
workplace and community, we expect people to take it upon themselves to be resourceful
and innovative in their thinking to accomplish tasks and solve problems. Likewise, we
should expect students to emulate the same kind of responsibility in their own learning to

acculturate them to these kinds of environments.

Limitations and Future Directions
An examination of initiative in this study only begins the journey toward
deconstructing this phenomenon. Students’ perceptions of and interactions in TMC
environment compared to nonpublishing activities were explored. However, more

investigations into students’ perceptions of other activities paired with analyses of more

140

observations of students across varying contexts can help isolate features that might
foster initiative.

Students in this classroom were not able to write in TMC environment regularly
throughout this study because there were only two computers in the classroom. It is
difﬁcult, therefore, to conclude that TMC made an impact on student initiative because
students had limited experience in this environment. I do not attempt to make this claim,
but suggest that this research provides a window into understanding what initiative may
look like and the factors that may support the different types of initiative. This
intervention may have been a small part of the classroom, but given that this was an urban
school with few technological resources, the experiences students had with TMC were
unusual. As students expressed, TMC was different from other mathematical activities in
the classroom. Although TMC may not have reengineered the norms in the overall
classroom, it was a microenvironment in the classroom that resulted in different
experiences and interactions than other activities.

In addition, it seems students emulated the norms and practices that were
established during the introductory phases of TMC environment. Future research that
includes interviews and observations of students before the presentation of this kind of
environment might reveal whether students’ emulation of norms and practices indeed
resulted from preconceived notions about the environment or from the norms and
practices established.

Whereas this study focused on what kinds of initiative students may take in the
publishing environment and the factors that may foster them, it did not address whether

TMC environment enhanced students’ initiative over the year. Since students did not

141

have as many experiences publishing on the Web as they had in other activities in the
classroom, it is questionable whether TMC had any impact in enhancing students’
initiative over the year. Future work tracing the development of students’ initiative may
reveal whether the amount of experiences students have publishing inﬂuences the amount

of initiative students take.

The Impact of TMC in Mr. Watkins’ Classroom

After the year ended and a new one began, Mr. Watkins and I sat in his new
classroom waiting for his next class to begin. We returned to the previous year and
remembered the role TMC had in the classroom. Mr. Watkins held his goal of
developing learners who took more control of their learning by taking initiative in their
mathematics thinking and problem-solving. He recalled that students’ interactions
around TMC were different from when they were participating in other activities. When
observing students himself, he noticed they took a “different perspective, a different
approach” when they published on TMC. “They were more reﬂective, more careful with
the words they chose--not as much verbiage as when they wrote in their individual
journals.”

TMC seemed to hook in some of the students. They were “excited to write on the
Web” and tried to “incorporate ideas from discussions.” Others, however, seemed so
engrossed in the work that it seemed difficult for them to also engage in the discussion.
As a result, pedagogically incorporating TMC himself as he tries to orchestrate the many
other tasks as a teacher is a puzzle. Mr. Watkins is very interested in using TMC and

managing the environment himself despite this struggle. He thinks it is a “creative way to

142

get them to do out of the ordinary doldrums that forces some students to think about
mathematics differently and in a concise way.” More importantly, it “blends in well with
what we are doing in the classroom with the CMP.” In this sense, Mr. Watkins values the
role of TMC in mathematics learning and wants to employ this tool as a way to get the
students to take more initiative by hooking in his students and getting them to take more
action in their learning. Mr. Watkins’ sentiments illustrate that he believed TMC
changed the way students interacted in their learning and has potential in shaping other
students’ initiative. Along these lines, fostering students’ initiative continues to be an

issue affecting Mr. Watkins’ pedagogical decisions in his new classrooms.

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APPENDICES

144

APPENDIX A

INTERVIEW PROTOCOL

Topic: What is the journal all about?

1.

PPN

95"

Topic: Do they like publishing/is it motivating?

7.

10.

11.

Do you think that publishing journals on the web is different from writing in your
math journals? How?

Do you think what you write is different? How?

Would you rather write in your journal or on the web? Why?

You know how we have a new student in this class, Josh (or one of your friends)? 1
How would you explain what the web journal is all about to him or her?

What is the purpose? ’

What are important things to remember while publishing? Things that you shouldn’t
do?

 

Now, I am going to ask you about how you like publishing or how you ﬁnd it
interesting whether you like publishing or find it interesting. So, what do you think
about publishing?

How do you like publishing compared to other activities? Why/ why not?

Do you think publishing is different from other activities that we do in class (makes it
fun)? How?

Next year, we are thinking about having publishing but trying to understand whether
we should have it. Do you think we should have it? Why should I have publishing?
or what is it about publishing that you like?

How is it fun? What makes it fun?

Do you think that publishing on the web leads you to do math or makes you want to
do math more? Why?

Have you read other people’s work on the web journal? Have you published before?
Have you done journals before in math classes?

145

APPENDIX B

CHRONOLOGICAL ORDER OF STUDENT RESPONSES

 

Case

Responses

 

Jake

[Writing on the web journal is different from writing in the individual
journals] because everybody can read it. You write all the steps and stuff.
You tell the solution strategy. In the journal, you just write what you did
that day and an example.

 

[I’d rather publish on the web than write in my journal] because you get to
review it while you are typing it.

 

Every once in awhile someone gets chosen to go and write about what we’re
doing in class, you can choose a subject. Like, how to find the sums of
addirg fractions and publish what you wrote.

 

[The purpose is] to share what you’re learning with people who are reading
it.

 

Shouldn’t tell them the wrong way to do a problem. Remember that
everybody is going to read it. Make sure you have the right answer before
typing it up. You shouldn’t publish the wrong answer or the wrong way to
write it.

 

I think publishing is neat because we get to share ideas with others outside
this school.

 

I would rather publish than take a test or would publish and do the
worksheets at home.

 

[Publishing is different from other activities] because you are using a
computer and sharing it with everybody else, whoever looks at the Internet,
share what you did in class.

 

It makes learning math more interesting because you get to type on the
computer and write about what you are doing in class.

 

[Publishing is different from other activities,] like with worksheets, you
have to do that alone, but with publishing you get to go up and work with a
partner.

 

[It would be different if Mr. W. told you to write something instead of
writing your own,] because when you publish you can write something no
one else has. This makes it more interesting.

 

Because you are using the computer, sharing your ideas.

 

AH and I went on and saw whzipeople published during the last time

 

[never published before]

 

 

 

[never written in journals before]

 

146

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Jeff

[The web journal is sort of different from my journal] because I write more
in my journal, but not as long. But it might be the same because I don’t
write much in my ﬂumal, either.

 

Sometimes [what I write in my journal is different from the web] because
it’s more for myself, so you only have to explain it so you understand. [In
the web joumallyou have to write it so everybody can understand it.

 

It doesn’t really matter [if I publish or write in my journal].

 

[The purpose is] so other people know how to do it or know what you are
talking about.

 

[In publishing, you need] to do it step by step. Like if you can say that these
all go together, you have to say 2 and 2/5 plus this is what. Instead of just
writing down what everything is altogether, go step by step and tell what
you are doing. you should go step by step and add it together. Make sure
you are spelling it right and double check the answer to make sure you
know it is right.

 

It’s fun, but sometimes it takes away from class. It’s hard to type and pay
attention at the same time.

 

Yeah, [I’d rather publish than do other math activities] because I don’t like
doing the other things in math class. It’s fun.

 

[Publishing is different from other math activities] because you get to use
the computer, you get to put what you say, so everyone in the entire world
can read it. You can put it in your own words, but you have to make sure
other people can understand it.

 

Sometimes I want to publish and sometimes not. If we are having class
discussion, I’d rather publish. Sometimes I’d rather publish, like when I’m
tired.

 

[If the teacher picked what you had to write about instead of letting you
pick want you wanted to write,] I don’t think it would be done as well. If
you publish about what you want to publish, then you know what you are
going to say, and if you have to publish what you want us to publish, you
have to figure out what you are going to say and how to figure it out instead
of what we already had planned for it or what was already in our journal.

 

It [publishing] is different from how we did math because we used to just
have the blackboard and no journals. It makes it a little more interesting and
different because you get to work with someone else and both of the people
are tﬂing to figure it out. I like that better.

 

[Although I can do small group work during other math activities,] I’d
ratherjublish because I like it better.

 

 

 

[has not seen what others done] [has not published] [has not done journals]

 

147

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Lynton

[Publishing is] more fun because you get to type on the computer, you are
on the Internet and put what you want to publish.

 

Yeah, I’d rather write on the web because it isn’t right in front of you and
the web is not like that. I can’t explain why.

 

It’s fun because you get to type and pick what you want to write about.

 

The purpose is to explain to other people what you are doing. Like, people
can click on Otto and see what we’re doing.

 

You could do whatever homework you did and explain how you did it.
Like your homework, you can scan it through and write what goes with it
and you can cut out stuff that doesn’t. You shouldn’t mess around or scan
stuff that shouldn’t be scanned.

 

Yeah [the things I write on the web are different from what I write in my
journal], because you can write how you got your answer and ask questions
like, do you get it and stuff. You can be the teacher on the web. When it’s
just in your journal, you can’t do that kind of stuff because the teacher is
just reading it.

 

It’s fun. I think that you should have it next year to help out the kids
because this has helped me a lot in understanding the computer and math
because I was a bad student. I’m surprised I got into enriched math. When I
explain it on the computer, I know exactly what I am saying. That helps
me out with my math. When we have partners, like smart people helping
us, you get it because once you read it, you say, oh, now I get it.

 

[Working with others during publishing is not really different from
working others in other activities] When you work with other people on
other stuff, it’s easier because you get more stuff. It’s harder when you
work by yourself because you don’t get the stuff.

 

Yeah, it’s funner because of typing and looking through the stuff like what
you wrote. Like when you go through the stuff and see what goes with the
stuff. It’s fun going through and seeing what other people wrote. And you
have a password so you can put in your own thing.

 

 

 

Publishing is like doing math, but doing it on the computer and writing it
out instead of just reading it and doing it. You are writing it out. I think it
would be more boring if you just did your own journals because, on the
web, you get to work with a partner. When you look at other people’s
work, you can cut out the stuff that doesn’t go with it. If they put, ‘oh, I
forgot to mention I got an A.’ You can cut that out because that has
nothing to do with it.

 

 

148

 

3‘“ F." H 11.11.
L.

APPENDIX B (cont’d)

 

Case

Responses

 

Lynton

Once you do your math more, you get to publish more. If you don’t do
your math, you can’t publish because you don’t have anything.

 

Yeah [I’ve seen other people’s work.] I corrected some of it. Responding
in class.

 

[has not published]

 

 

 

[has not done journals]

 

 

149

 

WI“.
1

APPENDIX B (cont’d)

 

Case

Responses

 

Ganton

No, well, I like the web better. You don’t have to have a pencil, so you can
just publish.

 

No, [the things I write in my journal] is not really [different from on the
web].

 

It’s [Publishing is] like notes. At the end of the day, you just have to write
down what you did.

 

If you are stuck on something, you can go and look back. If you forget how
to do something you can look back on your notes. It could tell the teacher if
you are paying attention in class or not.

 

If you ﬁnd something interesting, you can publish it.

 

The purpose is so that other people in other schools can see what you are
doing.

 

Don’t write stupid stuff or things you don’t know what you are talking
about. You should write about stuff you know about.

 

If you don’t have a pencil, you don’t have to write everything down. I like
looking at other people’s journals, what other people have been putting on
there and see if I want to put a response.

 

With other stuff, all you got to do is work. This is fun, though. ...On the
web, you don’t have to do tests or other assignments. You just got to write
down one example problem and tell what it is about.

 

Yeah, it would be different [if you told us to type on a speciﬁc topic instead
of letting us choose our own] because the stuff that you want to type, you’ll
know and the stuff you want us to type, we might not know about.

 

Yeah, because if you don’t know what you are doing, you’ll just be sitting
there writing about something you don’t know. If you don’t know what you
are going to do right then, you can think of something.

 

If you are publishing and others are doing the Classwork, the publishers
usually have to do it at home and turn it in. I don’t mind because you have
all night to do. I’d rather publish and do that at home. Except when I don’t
know what we are doing covering in class.

 

Yeah, [I’ve seen what others have written.] When I go on sometimes, me
and my partner make reﬂections on people’s work.

 

[has not published]

 

 

 

[has done journals] in 5th grade about the assignment done

 

 

150

‘ a 1 “1
..

 

APPENDIX B (cont’d)

 

Case

Responses

 

Tara

[The web is different] because it is more fun to write on the computer. [I’d
rather publish on the web] because on the web, other people can see your
work.

 

We explain more on the web than in our journals. On the web, we explain and
do step by step how to do one problem and in our journals, we explain the
whole thing. We explain what we did and how we go about it. We pick one
problem and explain it step by step. In our journals, we explain the whole
thing. Like what we did and some examples of what we did. On the web, we’d
explain how do the problem.

 

You pick something, people and they publish it so other people can look at it....
To help other people on the Internet learn.

 

To stay on task, to listen to Mr. Watkins too while publishing. You shouldn’t
talk about other things beside our assignment, not listen to Mr. Watkins.

 

I like publishing. It’s fun. It’s more fun than other activities. When you
publish on the computer, it’s just more fun than sitting and doing your
assignment.

 

When you are publishing on the computer, you give other people chances to
learn, other people on the Internet.

 

When you are publishing, you are talking about math problems, and it just
makes you want to do math more.

 

When you are publishing, it’s just more fun then sitting down and doing a class
assignment because you get to be on the computer.

 

No, [it isn’t the same if students just wrote their journal on the computer and
not the web] because on the web, other people can see what you do and it can
help them more. [This is an important] because other people need to learn too.

 

[has not seen what others have published]

 

[has not published]

 

 

 

[has not done journals]

 

151

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Ande

Yeah, [the web is different from my journal] because when you are on the
computer, you talk about how you do it and how you get the answer, but in
your journal, you write what did you do in class. Like, in class we did 3/4 and
1/5 and we learned how to do times. On the web, it’s about the same thing,
buLyou talk to other people and tell them how to do it.

 

[I’d rather publish than write in my journal because] of typing, telling how to
do it, and publishing.

 

No, [I don’t think what I write on the web is that different from what is in my
journal.]

 

You have to explain what you did in class and a solution for how to do it and
give them a couple of examples.

 

[The purpose is] to remember things and if someone else published and you
didn’t understand something, you can go to their publication and read it and
understand it.

 

[You need to remember] how to do it. Before you publish, you have to know
what you are explaining and how to do it. You know what to do when you
get on there and how to explain your answers. You shouldn’t play around on
it.

 

I think it’s a good way to learn how to do different things and learn how to,
for example, if you didn’t know how to do something, you can go back and
learn all different kinds of things. If you didn’t know how to do something
and somebody published it and you were looking around on the computer and
found something you didn’t know how to do and you read it and they
explained it really carefully, you can learn how to do it.

 

Yeah, it’s fun because it’s like telling people how to do other things they
don’t know how to do.

 

Publishing is different because with a worksheet, you just put down the
answer, but with the publishing you have to explain your answer, how you
got it. It’s better to explain your answer than just right down the answer
because if you get the answer from somebody, you can’t explain it. But if you
publish, you have to explain it.

 

 

 

Publishing helps other people and gives you a chance to do other things you
don’t know how to do. If you publish and let everybody see it, you feel happy
because you taught somebody how to do something that they didn’t know
how to do. If you didn’t have publishing, most people couldn’t learn as much
stuff because they couldn’t look through and see if anybody has put anything
they did not know.

 

152

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Ande

Yeah [I’d rather publish than do other activities] Both of them help you
understand how to do it, but publishing is funner.

 

No, [publishing does not make me want to do math more], not that much.

 

[has done responses with AH]

 

[has not published]

 

 

 

[has done journals] in 5th grade discussing how long we spent in class and
what we did. In an hour’s worth, we did a sheet and it had this on it.

 

 

153

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Mandy

[Publishing is sort of different from writing in the individual journals]
because when you publish in your journal it’s just for you, but when you
publish on the Internet, it’s for everybodyelse.

 

[I’d rather publish] on the web [than write in my journal] because it helps
other kids learn new things and it’s fun typing.

 

No, [the web journal and my journal is not different] I write things like
strategies and problems I am doing.

 

[Publishing is] like you explain things, explain problems, how to do special
techniques, and easy ways to do things, and you just write it, gist type it on.

 

[The purpose is] to help other people and let kids know what you are doing
at your school.

 

Make sure other people can understand it, make sure it makes sense. You
shouldn’t do bad things, like English when it is about math and something
that doesn’t make sense.

 

Publishing makes sense for people to do it, not like all those things that the
teacher makes you have to do. It’s something people want to do... because it
is fun and if you are going to help someone else and if other people are
going to help you. If there is someone else in another city trying to do a
problem, I can write this down and try to help them if they have a problem
with it.

 

I think kids would rather publish than do other things in class. It’s different
because it let’s you explain things instead of letting the teacher tell you how
to do it. You can put how you think is easy to do it your own way.

 

Yeah, [publishing does make me want to do math more] because when I
publish I start writing something and think, oh my gosh, it does really work
and start using it.

 

[has done journal writing, but just put problems in it.]

 

[has not seen other entries]

 

 

 

[has not published]

 

 

154

‘ 3117‘. .. It?! If

 

I! 3 ..RE o. l -‘PK‘I I—-

APPENDIX B (cont’d)

 

Case

Responses

 

Camrn

Yeah, [the web journal and my journal are different] because in the math
journal, the only people who see it are you (Jones) and Mr. Watkins, but on
the web, other kids who are having problems or whatever, they can look on
our sheet.

 

[I’d rather publish] on the web [than write in my journal] because I don't
really write in my journal. I don’t really know what to put in there, but on
the web, there’s somebody else who can help you word it right or whatever
when we are publishing.

 

[What you write on the web and in your own journal is] a little bit of the
same thing because you have to put examples and what you did in class.

 

I would tell him that you organize it so that we have different partner and
that when you publish, you just pay attention to what’s going on in class and
on the computer, you can type it down on the computer and it will go
through. You still have to type examples and how you got the answer or like
a problem you are doing or checking over.

 

[The purpose is] to help other kids, like maybe in the X school district or all
over, to help them if they don’t know or if they are doing the same thing or if
they want to see how other people are doing it.

 

Remember your problem, and how you got it, what strategies did you use.
You shouldn’t put the whole problem, but just part of the problem, and the
examples. You could put down your answer and how you got it.

 

Publishing is fun because in your journal, when you are by yourself, you
might not be able to think of what to publish, but when you have a partner,
you can ask them how to word it right or help you.

 

Yeah I like publishing, sometimes in class, it is boring, but when you get to
publish, you can write down what you think about it, how you got the
problem.

 

It’s different [from other math activities] because, in our classroom, the only
people who get to see what you are doing is in your class, but when you
publish, other kids get to see it too.

 

When you publish, you are not by yourself, and sometimes kids don’t do the
work because they have to sit by themselves or do it by themselves and don’t
understand it. But when you are publishing and you put down a problem,
you have a friend there who can help you solve the problem or a friend who
can help you think of a problem to put down. It is also fun because it lets

you show other kids, not just those you are working with, it shows other
kids.

 

 

 

No, it’s not that different [from publishing if you get to work with other
people in another activity.] It don’t make any difference because you still
have kids helping you.

 

155

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Camrn

Yeah, [it makes me want to math more] because if other kids like it from
wherever, then your teacher might like it too. If you are with your partner
and doing a problem and they like it too, the teacher might like it too.

 

[Publishing], it’s fun because it gives you a chance to explain to other kids,
how to do a problem or how you got the answer or examples of what you
did.

 

[has not done journal writing]

 

[has not published before]

 

 

 

[has not read other people’s work.]

 

 

156

El;

 

APPENDIX B (cont’d)

 

Case

Responses

 

Tinse

[Publishing is] not really [different from writing in your journal] because it
is just writing on the computer instead of the paper.

 

[ I’d rather publish] on the web [than write in my journal] because it is more
fun typing than writing it down.

 

When you publish, you write what you learned from one of the lessons and
show how you did it.

 

[The purpose is] so that other people can see what you learned and what you
do in math class.

 

Don’t put your name. Make sure the sentences are right and not irregular
sentences and make sure what you wrote is right and like how most people
do it and not wrong.

 

No, [what I write in my journal is not different from what I write on the
web.] Things I write are what you should learn and strategies you use.
Sometimes, I forget to do it in my journal.

 

 

It’s interesting to see how other people learn math instead of only the way
you learn.

 

Yeah, I’d rather publish than other math activities] because others are
boring. You get to work with the computers in publishing.

 

I like publishing on the web [versus just writing on the computer] better
because other people can see what you wrote. If you just typed it on
something, not everybody would see it.

 

Publishing is different because you use computers and in other activities we
don’t use the computer as much.

 

Yeah, because other people can learn how you learn and it’s fun.

 

Not really, [publishing doesn’t make me want to do math more.]

 

[has not done journal writing]

 

[has not published before.]

 

 

 

[has read and responded to other people’s work during publishing]

 

 

157

APPENDIX B (cont’d)

 

Case

Responses

 

Abraham

Yeah [publishing the journal on the web is different from writing in the
individual journals], because everybody else can see it and they don’t
have to ask you. Theycan just get on and see it.

 

[I’d rather write] on the web [than in my journal] because it helps with
your typing skills and also other people get to see it from all over.

 

[What I write in my journal is] not really [different from what I write on
the webJ I write different problems, different solution strategies.

 

I’d say just write what you did in class on the journal, like a problem we
did today. Solution strategy that he learned from Mr. Watkins.

 

[The purpose is] so that other people, in smaller grades can see what
we’re doing if they haven’t learned it yet, or older people to see what
they are doing.

 

Everybody can see it, your audience you are writing to. Use proper
spelling, punctuation, all the editing stuff and make sure your information
is correct. If it isn’t, everybost going to get messed up.

 

Shouldn’t type something off your brain, something you think you
learned at home from your book, something you haven’t talked about yet,
something you don’t know totally about yet.

 

I think publishing is interesting. Sometimes, I get on the computer at my
grandmother’s house and get on to see what other people are doing.

 

Yeah, [publishing is interesting] because you get to work with other
students that you know. You get to type and work together and to publish
something that everyone can see, that is bigger than just in the classroom,
so everybody that has a computer can see it too.

 

Publishing is different because you use computers, you write solution
strategies, not just talk about them, so you can remember them. It’s not
just problems, but solution strategies, ways to solve stuff and you can
make up your own problems, too.

 

Yeah, [publishing is interesting] because you can work others and talk
about what you think is correct and what you should write on there,
whatever you think everybody will understand.

 

 

 

Yeah, [it makes me want to do math more] because you want to learn
stuff. When others login, they can see your initials and decide to read
yours. They can learn from them. I think it’s cool. I get to see other
solution strategies and I get more involved in math because I learn more
stuff everyday. And I get to write more stuff. After I type stuff, I'want to
learn more about that subject and publish more on that subject.

 

 

158

 

APPENDIX B (cont’d)

 

Case

Responses

 

Abraham

[has done journal writing]: multiplication tables. If she talked about an
idea, we had to copy it down, so then she would say look back your
journal and use the solution strategies I told you to use yesterday.

 

He thinks his journal is different. The web is longer, has different ideas,
different vocabulary.

 

 

 

[has not published before]

 

 

159

._ rn-u—

 

'F..-_

APPENDIX B (cont’d)

 

Case

Responses

 

Bali

Yeah, [the web journal is different from writing in you own journal]
because when you write in your journal, it is really boring. When you write
on the screen, it is fun because you can type. There is a delete button, so if
you make a mistake, you can change it, but if you use your pencil.... You
can delete mistakes more easily.

 

I’d rather write on the web because it is kind of fast and more fun because
you get to see interesting things and see new stuff. With pen and pencil,
you get used to it and gets boring.

 

No, [what I write on the web is not different from what I write in my
journal]. I think it is pretty much the same, just faster on the computer. [I
publish things about] fractions, decimals, things like that. One time, I put
that it was really cold in here, but would not write that in the web journal.
On the web journal, I would write something like I was doing fractions and
decimals and stuff came up with this problem, give the answer and how I
got it.

 

I’d tell him that it’s like writing in your own journal. Just think of
something and type it, then ask you [Jones] if it is alright. You would put
things we are doing in class.

 

To not forget it, have it there and other people can read it. Be serious
about it. Pay attention in class before you publish, so you know what we
are talking about that day. You shouldn’t mess around with the computer
and try to focus on class.

 

I think it’s interesting because it’s a new place to explore. On the web,
you can go back and say that’s really cool, and that’s really cool and I’ve
never seen that before.

 

I think it’s funner than just sitting in class and just paying attention to Mr.
W

 

Yeah publishing is different. If you are just sitting there listening to Mr.
Watkins, it gets boring, but publishing is fun.

 

Yeah [I think you should have publishing next year.] I think it is
interesting because it gives kids a break from Mr. W. talking and it is a
new thing to learn.

 

 

 

She doesn’t read what people publish.

 

160

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Bali

No, I think it is a more interesting way of taking a break. Instead of saying
that you are going to put down your pencil and take a break, you can just
go to the computer and take a break while you are working.

 

[has not done journal writing]

 

[has not published before.]

 

 

 

Just typing and trying to pay attention to Mr. W. at the same time, so we
can still get a good participation grade and still have some fun on the
computer.

 

 

161

 

[.Iﬂl“ _l‘l .I. .4

APPENDIX B (cont’d)

 

Case

Responses

 

Shanon

Sometimes, [the web journal is different from your own journals] because
you have to explain it better on the Internet. People won’t understand what
you are saying. You have to make it more clear so they can understand it.

 

Either one’s ﬁne. I like to do them both [writing on the web and in my own
journal]

 

[What I write in my journal and on the web is] basically the same. I write if
the problem was hard or if I have to remember something to do it better
later. I write it down, so I can remember it.

 

On the web, you have to explain to other people what you are doing and
how the problems work.

 

[The purpose is] so that other people can see what we are doing and they
can learn some stuff too.

 

You have to explain it really well like how you do the problem, how you
get the answer, what to do. You shouldn’t write something you don’t
explain at all and you should try it on somebody else to see if they
understand it, so then other people will probably understand it too.

 

[Publishing is] fun because you get to show other people what you are
doing and so they can understand it too.

 

It doesn’t matter [whether I publish or do other math activities] because I
usually publish what I do in math class.

 

Yeah , [other math activities are different] like with the overhead, we draw
graphs and shades things in, and we don’t do that in publishing.

 

She doesn’t read what people publish.

 

Yeah, [publishing makes me want to do math more] because it’s easier to
explain and you’ll be able to explain more stuff.

 

[has not done journal writing]

 

 

 

[has not published before.]

 

 

162

 

APPENDIX B (cont’d)

 

Case

Responses

 

Yeah, [the web journal is different from writing in your own journal]
because other people can read them not just teachers, and they can login
from home to read it.

 

[I’d rather publish] on the web [than in my journal] because it is fun to
type and to have people segyour ideas.

 

[What I write in my journal is] a little bit [different than what I write on the
web] because you show what you’ve done, but you do the same thing in
your journals, so it’s not that very much different.

 

You write what you did in class and how to do it, the instructions to do
that paper or problem and write down the ideas.

 

[The purpose is] so that people can see what you did in class.

 

Remember to title your paper and tell what you are doing, to show it and
don’t just put some answers down. Make sure you show how to do
it....Don’t just put the answers, but make sure you tell what the questions
are and what you did.

 

 

I think it’s [publishing] fun because you get to write on the computer, put
your ideas, and what you did in class. It’s nice to let people know what
you did in class.

 

Yeah, [I’d rather publish than do other math activities] because you have
to do other activities , but you get to sit there, learn how to use the
computer, write about stuff you did in class, instead of just write down
things on paper.

 

[Other math activities are] sort of [the same as publishing] because you
write down what you did down in class, so it is sort of the same, but you
don’t just do it, you write about what you did.

 

Yeah, I’ll read others and write down responses. I’ve read what they’ve
put and put a response. I’ve done that a couple of times. I did it at home,
but it didn’t work very well because my brother wanted to get on and I
didn’t get to ﬁnish it.

 

 

 

Yeah, [publishing makes me want to do math more] because it is fun. You
get to learn more about what you are doing that way. That makes you
understand it more. Once you think about it more to write it down, it
makes you understand the problem more.

 

 

163

APPENDIX B (cont’d)

 

Case

Responses

 

[has done journal writing, but it was just problems and the answers]

 

[has not published before. ]

 

 

 

Yeah, [publishing makes me want to do math more] because if you don’ t
do the homework, you won t be able to do that because you don’ t know
what you are doing, so it helps the kids because it is fun to do that, so they
get to do that, so they should do the homework. It makes them think I
better do this if I want to write on the computer tomorrow.

 

164

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Hazel

Yeah, [the web journal is different from your own journal] because you
share your ideas when it’s on the web and more people can see what you
are doing in class.

 

[Compared to writing on my journal, I’d rather publish] on the web
because it just gives you a chance to show your ideas and thoughts.

 

[What I write on the web is] not really [different from what I write in my
journal] because when if I write something in my journal, I usually write
that on the web.

 

The web journal is a web site on the computer, you log into and have your
own password and you publish things you did in class, things that you talk
about in school.

 

[The purpose] I think is to share your ideas. People respond to it. It’s not
really bad if they respond to it because you know that they are reading
your stuff and listening to what you have to say.

 

Write in what you talked about in class. You should reread it to make sure
you put what you wanted to say. Just fill out all the information on the
screen. Things you talk about in school or you can have your paper
scanned. Don’t use foul language and don’t put your full name.

 

Yeah, [publishing is interesting] When you do it, its a group thing,
compromise and think of something to put on. It’s just teamwork when
you put it on there.

 

If it’s just dittos, I don’t mind, but like tests, I’d rather publish. It’s just
fun. I don’t really know why, but I just like doing it.

 

[Publishing is] not really [different from what you do in other math
activities] You just talk about what you are doing. When you publish,
you talk about the ditto sheet or what you did in class today.

 

I don’t know [why publishing is fun]. I think it is that you just basically
get to work with your friends.

 

It doesn’t really matter [whether I do other activities that allow me to work
my partner.] I just like hanging out with her, and we usually work on
things together.

 

Yeah, sometimes, ifI am in science, I think about things I would put on the
web. Usually the class after because I think about that I could have put that

 

on before instead of this or something... or what I may want to do.
[has not published before] '

 

[has not written in journals before]

 

 

 

[has read others and responded]

 

 

165

 

 

APPENDIX B (cont’d)

 

Case

Responses

 

Justin

I think [publishing on the web is different from writing in your journal]
because when you get an experience with typing and working on the
computer. It’s more fun. You get to learn how to publish things and it’s
more exciting. We can look at it from home and look over it. It seems
better than doing it than with paper and pencil.

 

[I’d rather writg] on the web [than write in my joumalj

 

I think [what I write in my journal is different from what I write on the
web] because I always make things longer when I type so I can explain it
better than when I do it in my journal because I know what I’m talking
about in my journal.

 

First of all, you have to learn the stuff, then type all that you know about
it... all the stuff we’ve been learning in class.

 

The purpose is to let other people know what we are doing in class.

 

Just type all that you know about the subject. You shouldn’t do anything
else you weren’t told to do, just do what you were supposed to do and not
anything else. Like don’t mess with things, just type your publication.

 

I think publishing is interesting because this is the only class you get to do
this kind of stuff, like typing your journals, type what you know about
what you did in a certain area.

 

I’d rather publish than any other subject in school except for gym and in
math. It’s fun typing in publishing. On the other hand, you don’t get to do
the other stuff. You don’t get to learn the other stuff when you are typing_._1

 

I think publishing is different from other activities. Usually in other
activities, you don’t get to type, you just write problems. Here you can
explain why you put something down, why or how you came up with the
answer.

 

You miss out on the other stuff sometimes. It’s just fun doing things on
the computer.

 

It wouldn’t always make a difference [whether I was just writing on the
computer versus publishing on the web]. Well, I like publishing on the
web a lot. You get to tell other people your opinions and they listen to it.

 

[He’s read entries from home.]

 

Yeah, [publishing makes you want to do math more because] it makes you
want to get to math class or for first class because you are eager to do it.

 

[has not published before]

 

 

 

 

[has not done journals before]

 

166

 

APPENDIX B (cont’d)

 

Case

Responses

 

Kyndi

It’s different because. Well, it’s the same. I write about like what we did in
class, homework assignments in my journal, well, I don’t write homework
assignments on the web. It’s sort of both, same and different.

 

Publishing is interesting because kids that don’t know how to do things get
a chance to learn about computers, not only computers, but the Internet.
People respond to their journals and may get better ideas of what they have
been doing. For example, I put a math problem, an example on the
Internet and Shanon corrected it because I made a couple of mistakes.

 

Yeah, [what I write in my journal is different than what I write on the
web.] You go into more detail on the web. People explain it better so
other people can understand it.

 

[I’d rather publish] on the web [than write in my journal] because I like it
better. I like using computers.

 

[What I write in my journal is] not really [different from what I write on
the web.] It has the same kind of main idea.

 

You just write about investigations you do and explain it to people like
how you got your answer and what you are doing in school.

 

The purpose is so that people can see what you are learning and so other
people can learn.

 

Make sure that people can understand what you write and choose your
words carefully. Don’t go to other web pages while publishing.

 

At first, it’s kind of interesting, but it gets boring after a little bit.

 

I probably like the other things [math activities] better. I don’t know why, I
just do.

 

has read what others have published and responded

 

No [publishing does not make me want to do math more.]

 

[has not done journals before]

 

 

 

[has not published before]

 

167

 

 

APPENDIX C

RESPONSES ACCORDING TO QUESTIONS

 

Question 1: Do you think that publishing journals on the web is different from
writing in your math journals? How?

 

1.

[Writing on the web journal is different from writing in the individual journals]
because everybody can read it. You write all the steps and stuff. You tell the
solution strategy. In the journal, you just write what you did that day and an
example.

 

[The web journal is sort of different from my journal] because I write more in
my journal, but not as long. But it might be the same because I don’t write
much in my journal, either.

 

[Publishing is] more fun because you get to type on the computer, you are on
the Internet and put what you want to publish.

 

Yeah, [the web is different from my journal] because when you are on the
computer, you talk about how you do it and how you get the answer, but in your
journal, you write what did you do in class. Like, in class we did 3/4 and 1/5
and we learned how to do times. On the web, it’s about the same thing, but you
talk to other people and tell them how to do it.

 

[Publishing is sort of different from writing in the individual journals] because
when you publish in your journal it’s just for you, but when you publish on the
Internet, it’s for everybody else.

 

Yeah, [the web journal and my journal are different] because in the math
journal, the only people who see it are you (Jones) and Mr. Watkins, but on the
web, other kids who are having problems or whatever, they can look on our
sheet

 

[Publishing is] not really [different from writing in your journal] because it is
just writing on the computer instead of the paper.

 

Yeah [publishing the journal on the web is different from writing in the
individual journals], because everybody else can see it and they don’t have to
ask you. They can just get on and see it.

 

Yeah, [the web journal is different from writing in you own journal] because
when you write in your journal, it is really boring. When you write on the
screen, it is fun because you can type. There is a delete button, so if you make a
mistake, you can change it, but if you use your pencil.... You can delete
mistakes more easily.

 

 

10. Sometimes, [the web journal is different from your own journals] because you

have to explain it better on the Internet. People won’t understand what you are
saying. You have to make it more clear so they can understand it.

 

 

168

_ 33.; 1.4 7!". 513-1

 

IFv-r“. . _

APPENDIX C (cont’d)

 

11. Yeah, [the web journal is different from writing in your own journal] because
other people can read them not just teachers, and they can login from home to
read it.

 

12. Yeah, [the web journal is different from your own journal] because you share
your ideas when it’s on the web and more people can see what you are doing in
class.

 

13. I think [publishing on the web is different from writing in your journal] because
when you get an experience with typing and working on the computer. It’s more
fun. You get to learn how to publish things and it’s more exciting. We can look
at it from home and look over it. It seems better than doing it than with paper
and pencil.

 

14. It’s different because. Well, it’s the same. I write about like what we did in
class, homework assignments in my journal, well, I don’t write homework
assignments on the web. It’s sort of both, same and different.

 

 

15. Yeah, [what I write in my journal is different than what I write on the web.]
You go into more detail on the web. People explain it better so other people
can understand it.

 

 

169

 

APPENDIX C (cont’d)

 

Question 2: Do you think what you write is different on the Web versus what you
write in your individual journal? How?

 

1.

Sometimes [what I write in my journal is different from the web] because it’s
more for myself, so you only have to explain it so you understand. [In the web
journal,] you have to write it so everybody can understand it.

 

Yeah [the things I write on the web are different from what I write in my
journal], because you can write how you got your answer and ask questions
like, do you get it and stuff. You can be the teacher on the web. When it’s just
in your journal, you can’t do that kind of stuff because the teacher is just
reading it.

 

9’

Nthhe things I write in my journal] is not really [different from on the web].

 

We explain more on the web than in our journals. On the web, we explain and
do step by step how to do one problem and in our journals, we explain the
whole thing. We explain what we did and how we go about it. We pick one
problem and explain it step by step. In our journals, we explain the whole
thing. Like what we did and some examples of what we did. On the web, we’d
explain how do the problem.

 

No, [I don’t think what I write on the web is that different from what is in my
journal]

 

No, [the web journal and my journal is not different] I write things like
strategies and problems I am doing.

 

[What you write on the web and in your own journal is] a little bit of the same
thing because you have to put examples and what you did in class.

 

No, [what I write in my journal is not different from what I write on the web.]
Things I write are what you should learn and strategies you use. Sometimes, I
forget to do it in my journal.

 

9.

[What I write in my journal is] not really [different from what I write on the
web.] I write different problems, different solution strategies.

 

10. He thinks his journal is different. The web is longer, has different ideas,

different vocabulary.

 

 

11.

No, [what I write on the web is not different from what I write in my journal]. I
think it is pretty much the same, just faster on the computer. [I publish things
about] fractions, decimals, things like that. One time, I put that it was really
cold in here, but would not write that in the web journal. On the web journal, I
would write something like I was doing fractions and decimals and stuff came
up with this problem, give the answer and how I got it.

 

 

170

 

 

APPENDIX C (cont’d)

 

12. [What I write in my journal and on the web is] basically the same. I write if the
problem was hard or if I have to remember something to do it better later. I
write it down, so I can remember it.

 

13. [What I write in my journal is] a little bit [different than what I write on the
web] because you show what you’ve done, but you do the same thing in your
journals, so it’s not that very much different.

 

14. [What I write on the web is] not really [different from what I write in my
journal] because when if I write something in my journal, I usually write that on
the web.

 

15. I think [what I write in my journal is different from what I write on the web]
because I always make things longer when I type so I can explain it better than

when I do it in my journal because I know what I’m talking about in my journal.

 

 

16. [What I write in my journal is] not really [different from what I write on the
web.] It has the same kind of main idea.

 

 

171

 

m1!“
1
D

APPENDIX C (cont’d)

 

[ Question 3: Would you rather write in your journal or on the Web?

 

1.

[I’d rather write] on the web [than in my journal] because it helps with your
typing skills and also other people get to see it from all over.

 

2.

[I’d rather publish] on the web [than write in my journal] because I don't really
write in my journal. I don’t really know what to put in there, but on the web,
there’s somebody else who can help you word it right or whatever when we are
publishing.

 

[I’d rather publish on the web than write in my journal] because you get to
review it while you are typingLit.

 

 

4. It doesn’t really matter [if I publish or write in my ﬁumal].
5. Yeah, I’d rather write on the web because it isn’t right in front of you and the

web is not like that. I can’t explain why.

 

No, well, I like the web better. You don’t have to have a pencil, so you can just
publish.

 

[ I’d rather publish] on the web [than write in my journal] because it is more fun
typing than writing it down.

 

I’d rather write on the web because it is kind of fast and more fun because you
get to see interesting things and see new stuff. With pen and pencil, you get
used to it and gets boring.

 

9.

Either one ’5 fine. I like to do them both [writing on the web and in my own
journal].

 

10. [I’d rather publish] on the web [than in my journal] because it is fun to type and

to have people see your ideas.

 

11. [I’d rather write] on the webjthan write in my journal]

 

12. [I’d rather publish] on the web [than write in my journal] because I like it better.

I like using computers.

 

13. [I’d rather publish] on the web [than write in my journal] because it helps other

kids learn new things and it’s fun typing.

 

 

14. [Compared to writing on my journal, I’d rather publish] on the web because it

just gives you a chance to show your ideas and thoughts.

 

172

 

 

APPENDIX C (cont’d)

 

 

Question 4: You know how we have a new student in this class or Choose One of
your friends who is not in this class. How would you explain what the web journal
is all about to him or her?

 

1. Every once in awhile someone gets chosen to go and write about what we’re
doing in class, you can choose a subject. Like, how to find the sums of adding
fractions and publish what you wrote.

 

N

It’s fun becausgyou get to type and pick what you want to write about.

 

5”

It’s [Publishing is] like notes. At the end of the day, you just have to write down
what you did.

 

4. You pick something, people and they publish it so other people can look at it....
To help other people on the Internet learn.

 

5. You have to explain what you did in class and a solution for how to do it and
give them a couple of examples.

 

6. [Publishing is] like you explain things, explain problems, how to do special
techniques, and easy ways to do things, and you just write it, just type it on.

 

7. I would tell him that you organize it so that we have different partner and that
when you publish, you just pay attention to what’s going on in class and on the
computer, you can type it down on the computer and it will go through. You
still have to type examples and how you got the answer or like a problem you
are doing or checkingwer.

 

8. When you publish, you write what you learned from one of the lessons and
show how you did it.

 

9. I’d say just write what you did in class on the journal, like a problem we did
today. Solution strategy that he learned from Mr. Watkins.

 

10. I’d tell him that it’s like writing in your own journal. Just think of something
and type it, then ask you [Jones] if it is alright. You would put things we are
doing in class. ‘

 

11. On the web, you have to explain to other people what you are doing and how
the problems work.

 

12. You write what you did in class and how to do it, the instructions to do that
paper or problem and write down the ideas.

 

13. The web journal is a web site on the computer, you log into and have your own
password and you publish things you did in class, things that you talk about in
school.

 

14. First of all, you have to learn the stuff, then type all that you know about it... all
the stuff we’ve been learning in class.

 

 

15. You just write about investigations you do and explain it to people like how
you got your answer and what you are doing in school.

 

 

173

 

APPENDIX C (cont’d)

 

[Question 5: What is the purpose?

 

[The purpose is] to share what you’re learning with people who are reading it.

 

[The purpose is] so other people know how to do it or know what you are talking
about.

 

The purpose is to explain to other people what you are doing. Like, people can
click on Otto and see what we’re doing.

 

If you are stuck on something, you can go and look back. If you forget how to do
something ydu can look back on your notes. It could tell the teacher if you are
paying attention in class or not.

 

The purpose is so that other people in other schools can see what you are doing.

 

[The purpose is] to remember things and if someone else published and you didn’t
understand something, you can go to their publication and read it and understand it.

 

[The purpose is] to help other people and let kids know what you are doing at your
schooL

 

[The purpose is] to help other kids, like maybe in the X school district or all over,
to help them if they don’t know or if they are doing the same thing or if they want
to see how other people are doing it.

 

[The purpose is] so that other people can see what you learned and what you do in
math class.

 

[The purpose is] so that other people, in smaller grades can see what we’re doing if
they haven’t learned it yet, or older people to see what they are doing.

 

[The purpose is] so that other people can see what we are doing and they can learn
some stuff too.

 

[The purpose is] so that people can see what you did in class.

 

[The purpose] I think is to share your ideas. People respond to it. It’s not really bad
if they respond to it because you know that they are reading your stuff and listening
to what you have to say.

 

The purpose is to let other people know what we are doing in class.

 

The purpose is so that people can see what you are learning and so other people can
learn.

 

 

 

174

 

APPENDIX C (cont’d)

 

 

Question 6: What are important things to remember while publishing? Things that
you shouldn’t do?

 

1.

Shouldn’t tell them the wrong way to do a problem. Remember that everybody
is going to read it. Make sure you have the right answer before typing it up.
You shouldn’t publish the wrong answer or the wrong way to write it.

 

[In publishing, you need] to do it step by step. Like if you can say that these all
go together, you have to say 2 and 2/5 plus this is what. Instead of just writing
down what everything is altogether, go step by step and tell what you are doing.
you should go step by step and add it together. Make sure you are spelling it
right and double check the answer to make sure you know it is right.

 

You could do whatever homework you did and explain how you did it. Like
your homework, you can scan it through and write what goes with it and you
can cut out stuff that doesn’t. You shouldn’t mess around or scan stuff that
shouldn’t be scanned.

 

z“

If you find something interestinﬂou can publish it.

 

Don’t write stupid stuff or things you don’t know what you are talking about.
You should write about stuff you know about.

 

To stay on task, to listen to Mr. Watkins too while publishing. You shouldn’t
talk about other things beside our assignment, not listen to Mr. Watkins.

 

[You need to remember] how to do it. Before you publish, you have to know
what you are explaining and how to do it. You know what to do when you get
on there and how to explain your answers. You shouldn’t play around on it.

 

Make sure other people can understand it, make sure it makes sense. You
shouldn’t do bad things, like English when it is about math and something that
doesn’t make sense.

 

Remember your problem, and how you got it, what strategies did you use. You
shouldn’t put the whole problem, but just part of the problem, and the examples.
You could put down your answer and how you got it.

 

10. Don’t put your name. Make sure the sentences are right and not irregular

sentences and make sure what you wrote is right and like how most people do it
and not wrong.

 

 

11. Everybody can see it, your audience you are writing to. Use proper spelling,

punctuation, all the editing stuff and make sure your information is correct. If it
isn’t, everybody is going to get messed up.

 

 

175

.....-“ 7?.

 

APPENDIX C (cont’d)

 

12. Shouldn’t type something off your brain, something you think you learned at
home from your book, something you haven’t talked about yet, something you
don’t know totally about yet.

 

13. To not forget it, have it there and other people can read it. Be serious about it.
Pay attention in class before you publish, so you know what we are talking
about that day. You shouldn’t mess around with the computer and try to focus
on class.

 

14. Just typing and trying to pay attention to Mr. Watkins at the same time, so we
can still get a good participation grade and still have some fun on the comppter.

 

15. You have to explain it really well like how you do the problem, how you get the
answer, what to do. You shouldn’t write something you don’t explain at all and
you should try it on somebody else to see if they understand it, so then other
people will probably understand it too.

 

16. Remember to title your paper and tell what you are doing, to show it and don’t
just put some answers down. Make sure you show how to do it....Don’t just put
the answers, but make sure you tell what the questions are and what you did.

 

17. Write in what you talked about in class. You should reread it to make sure you
put what you wanted to say. Just ﬁll out all the information on the screen.
Things you talk about in school or you can have your paper scanned. Don’t use
foul language and don’t put your full name.

 

18. Just type all that you know about the subject. You shouldn’t do anything else
you weren’t told to do, just do what you were supposed to do and not anything
else. Like don’t mess with things, just type your publication.

 

 

19. Make sure that people can understand what you write and choose your words
carefully. Don’t go to other web pages while publishing.

 

176

 

 

APPENDIX C (cont’d)

 

 

Question 7: Now I am going to ask you about how you like publishing or how you
ﬁnd it interesting, whether you like publishing or find it interesting. So what do you
think about publishing?

 

1. I think publishing is neat because we get to share ideas with others outside this
school.

 

2. It’s fun, but sometimes it takes away from class. It’s hard to type and pay
attention at the same time.

 

3. It’s fun. I think that you should have it next year to help out the kids because
this has helped me a lot in understanding the computer and math because I was
a bad student. I’m surprised I got into enriched math. When I explain it on the
computer, I know exactly what I am saying. That helps me out with my math.
When we have partners, like smart people helping us, you get it because once
you read it, you say, oh, now I get it.

 

 

4. If you don’t have a pencil, you don’t have to write everything down. I like
looking at other people’s journals, what other people have been putting on there
and see if I want to put a response.

 

5. I think it’s a good way to learn how to do different things and learn how to, for
example, if you didn’t know how to do something, you can go back and learn
all different kinds of things. If you didn’t know how to do something and
somebody published it and you were looking around on the computer and found
something you didn’t know how to do and you read it and they explained it
really carefully, you can learn how to do it.

 

6. Publishing makes sense for people to do it, not like all those things that the
teacher makes you have to do. It’s something people want to do... because it is
fun and if you are going to help someone else and if other people are going to
help you. If there is someone else in another city trying to do a problem, I can
write this down and try to help them if they have a problem with it.

 

7. Publishing is fun because in your journal, when you are by yourself, you might
not be able to think of what to publish, but when you have a partner, you can
ask them how to word it right or help you.

 

8. It’s interesting to see how other people learn math instead of only the way you
learn.

 

 

9. I think publishing is interesting. Sometimes, I get on the computer at my
grandmother’s house and get on to see what other people are doing.

 

 

177

APPENDIX C (cont’d)

 

10. I think it’s interesting because it’s a new place to explore. On the web, you can
go back and say that’s really cool, and that’s really cool and I’ve never seen that
before.

 

11. [Publishing is] fun because you get to show other people what you are doing,
and so they can understand it too.

 

12. I think it’s [publishing] fun because you get to write on the computer, put your
ideas, and what you did in class. It’s nice to let people know what you did in
class.

 

13. Yeah, [publishing is interesting] When you do it, its a group thing, compromise
and think of something to put on. It’s just teamwork when you put it on there.

 

14. I think publishing is interesting because this is the only class you get to do this
kind of stuff, like typing your journals, type what you know about what you did
in a certain area.

 

 

15. At ﬁrst, it’s kind of interesting, but it gets boring after a little bit.

 

 

 

178

 

APPENDIX C (cont’d)

 

[ Question 8: How do you like publishing compared to other aCtivities?

 

1. I would rather publish than take a test or would publish and do the worksheets
at home.

 

2. Yeah, [I’d rather publish than do other math activities] because I don’t like
doinghe other things in math class. It’s fun.

 

3. I like publishing. It’s fun. It’s more fun than other activities. When you
publish on the computer, it’s just more fun than sitting and doing your
assignment.

 

4. Yeah, it’s fun because it’s like telling people how to do other things they don’t
know how to do.

 

5. Yeah [I’d rather publish than do other activities] Both of them help you
understand how to do it, but publishing is funner.

 

6. Yeah I like publishing, sometimes in class, it is boring, but when you get to
publish, you can write down what you think about it, how you got the problem.

 

7. Yeah, I’d rather publish than other math activities] because others are boring.
You get to work with the computers in publishing.

 

8. Yeah, [publishing is interesting] because you get to work with other students
that you know. You get to type and work together and to publish something
that everyone can see, that is bigger than just in the classroom, so everybody
that has a computer can see it too.

 

9. Yeah, [publishing is interesting] because you can work others and talk about
what you think is correct and what you should write on there, whatever you
think everybody will understand.

 

10. I think it’s funner than just sitting in class and just payingattention to Mr. W.

 

11. It doesn’t matter [whether I publish or do other math activities] because I
usually publish what I do in math class.

 

12. Yeah, [I’d rather publish than do other math activities] because you have to do
other activities , but you get to sit there, learn how to use the computer, write
about stuff you did in class, instead of just write down things on paper.

 

13. If it’s just dittos, I don’t mind, but like tests, I’d rather publish. It’s just fun. I
don’t really know why, but I just like doing it.

 

14. I’d rather publish than any other subject in school except for gym and in math.
It’s fun typing in publishing. On the other hand, you don’t get to do the other
stuff. You don’t get to learn the other stuff when you are typing.

 

15 . I probably like the other things [math activities] better. I don’t know why, I just
do.

 

 

16. I think kids would rather publish than do other things in class. It’s different
because it let’s you explain things instead of letting the teacher tell you how to
do it. You can put how you think is easy to do it your own way.

 

 

179

 

APPENDIX C (cont’d)

 

Question 9: Do you think publishing is different from other activities that we do in
class (makes it fun)? How?

(Probe) Next year, we are thinking about having publishing but trying to
understand whether we should have it. Do you think we should have it? Why
should I have publishing? or what is it about publishing that you like?

(Probe) How is it fun? What makes it fun?

 

 

1.

[Publishing is different from other activities] because you are using a computer
and sharing it with everybody else, whoever looks at the Internet, share what
you did in class.

 

[Publishing is different from other activities,] like with worksheets, you have to
do that alone, but with publishing you get to go up and work with a partner.

 

[Publishing is different from other math activities] because you get to use the
computer, you get to put what you say, so everyone in the entire world can read
it. You can put it in your own words, but you have to make sure other people
can understand it.

 

Yeah, it’s funner because of typing and looking through the stuff like what you
wrote. Like when you go through the stuff and see what goes with the stuff. It’s
fun going through and seeing what other people wrote. And you have a
password so you can put in your own thing.

 

Publishing is different because with a worksheet, you just put down the answer,
but with the publishing you have to explain your answer, how you got it. It’s
better to explain your answer than just right down the answer because if you get
the answer from somebody, you can’t explain it. But if you publish, you have to
explain it.

 

It’s different [from other math activities] because, in our classroom, the only
people who get to see what you are doing is in your class, but when you

publish, other kids get to see it too.

 

Publishing is different because you use computers and in other activities we
don’t use the computer as much.

 

Publishing is different because you use computers, you write solution strategies,
not just talk about them, so you can remember them. It’s not just problems, but
solution strategies, ways to solve stuff and you can make up your own
problems, too.

 

9.

Yeah publishing is different. If you are just sitting there listening to Mr.
Watkins, it gets boring, but publishing is fun.

 

 

10. Yeah, [other math activities are different] like with the overhead, we draw

graphs and shades things in, and we don’t do that in publishing.

 

180

 

 

 

APPENDIX C (cont’d)

 

11. [Other math activities are] sort of [the same as publishing] because you write
down what you did down in class, so it is sort of the same, but you don’t just do
it, you write about what you did.

 

12. [Publishing is] not really [different from what you do in other math activities]
You just talk about what you are doing. When you publish, you talk about the
ditto sheet or what you did in class today.

 

13. I think publishing is different from other activities. Usually in other activities,
you don’t get to type, you just write problems. Here you can explain why you
put something down, why or how you came up with the answer.

 

14. Publishing helps other people and gives you a chance to do other things you
don’t know how to do. If you publish and let everybody see it, you feel happy
because you taught somebody how to do something that they didn’t know how
to do. If you didn’t have publishing, most people couldn’t learn as much stuff
because they couldn’t look through and see if anybody has put anything they
did not know.

 

15 . When you publish, you are not by yourself, and sometimes kids don’t do the
work because they have to sit by themselves or do it by themselves and don’t
understand it. But when you are publishing and you put down a problem, you
have a friend there who can help you solve the problem or a friend who can
help you think of a problem to put down. It is also fun because it lets you show
other kids, nogjust those you are working with, it shows other kids.

 

16. Yeah, because other people can learn how you learn and it’s fun.

 

17. Yeah [I think you should have publishing next year.] I think it is interesting
because it gives kids a break from Mr. W. talking and it is a new thing to learn.

 

18. With other stuff, all you got to do is work. This is fun, though. ...On the web,
you don’t have to do tests or other assignments. You just got to write down one
example problem and tell what it is about.

 

19. When you are publishing on the computer, you give other people chances to
learn, other people on the Internet.

 

20. I don’t know [why publishing is fun]. I think it is that you just basically get to
work with your friends.

 

 

21. You miss out on the other stuff sometimes. It’s just fun doing things on the
computer.

 

 

181

 

APPENDIX C (cont’d)

 

 

Question 10: Do you think that publishing on the Web leads you to do math or
makes you want to do math more? Why?

 

 

1.

It makes learning math more interesting because you get to type on the
computer and write about what you are doing in class.

 

2.

Once you do your math more, you get to publish more. If you don’t do your
math, you can’t publish because you don’t have anything.

 

If you are publishing and others are doing the Classwork, the publishers usually
have to do it at home and turn it in. I don’t mind because you have all night to
do. I’d rather publish and do that at home. Except when I don’t know what we
are doigcovering in class.

 

When you are publishing, you are talking about math problems, and it just
makes you want to do math more.

 

S"

No, [publishing does not make me want to do math more], not that much.

 

Yeah, [publishing does make me want to do math more] because when I publish
I start writing something and think, oh my gosh, it does really work and start
using it.

 

Yeah, [it makes me want to math more] because if other kids like it from
wherever, then your teacher might like it too. If you are with your partner and
doing a problem and they like it too, the teacher might like it too.

 

9°

Not really, [publishing doesn’t make me want to do math more.]

 

Yeah, [it makes me want to do math more] because you want to learn stuff.
When others login, they can see your initials and decide to read yours. They can
learn from them. I think it’s cool. I get to see other solution strategies and I get
more involved in math because I learn more stuff everyday. And I get to write
more stuff. After I type stuff, I want to learn more about that subject and
publish more on that subject.

 

10.

No, I think it is a more interesting way of taking a break. Instead of saying that
you are going to put down your pencil and take a break, you can just go to' the
computer and take a break while you are working.

 

 

11.

Yeah, [publishing makes me want to do math more] because it’s easier to
explain and you’ll be able to explain more stuff.

 

 

182

 

APPENDIX C (cont’d)

 

12. Yeah, [publishing makes me want to do math more] because it is fun. You get
to learn more about what you are doing that way. That makes you understand it
more. Once you think about it more to write it down, it makes you understand
the problem more.

 

13. Yeah, [publishing makes me want to do math more] because if you don’t do the
homework, you won’t be able to do that because you don’t know what you are
doing, so it helps the kids because it is fun to do that, so they get to do that, so
they should do the homework. It makes them think I better do this if I want to
write on the computer tomorrow.

 

14. Yeah, sometimes, if I am in science, I think about things I would put on the
web. Usually the class after because I think about that I could have put that on
before instead of this or something... or what I may want to do.

 

15 . Yeah, [publishing makes you want to do math more because] it makes you want
to get to math class or for first class because you are eager to do it.

 

 

16. No [publishing does not make me want to do math more.]

'95!‘

5" 'i v “ J‘

F.‘ '.V

 

 

 

183

"In 1- “I .

 

APPENDIX D

AUDIENCE

 

. [Publishing is sort of different from writing in the individual journals] because

when you publish in your journal it’s just for you, but when you publish on the
Internet, it’s for everybody else.

 

Yeah [publishing the journal on the web is different from writing in the
individual journals], because everybody else can see it and they don’t have to
ask you. They can just get on and see it.

 

Yeah, [the web journal is different from writing in your own journal] because
other people can read them not just teachers, and they can login from home to
read it.

 

Yeah, [what I write in my journal is different than what I write on the web.]
You go into more detail on the web. People explain it better so other people
can understand it.

 

3"

Because you are using the computer, sharing your ideas.

 

[Publishing], it’s fun because it gives you a chance to explain to other kids, how
to do a problem or how you got the answer or examples of what you did.

 

I like publishing on the web [versus just writing on the computer] better
because other people can see what you wrote. If you just typed it on something,
not everybody would see it.

 

Publishing is interesting because kids that don’t know how to do things get a
chance to learn about computers, not only computers, but the Internet. People
respond to their journals and may get better ideas of what they have been doing.
For example, I put a math problem, an example on the Internet and Shanon
corrected it because I made a couple of mistakes.

 

9.

[I’d rather write] on the web [than in my journal] because it helps with your
typing skills and also other people get to see it from all over.

 

10. On the web, you have to explain to other people what you are doing and how

the problems work.

 

11. The purpose is to explain to other people what you are doing. Like, people can

click on Otto and see what we’re doing.

 

12. The purpose is so that other people in other schools can see what you are doingL

 

 

13. [The purpose is] so that other people can see what you learned and what you do

in math class.

 

 

184

 

[mg-m- 112-1!- "I. - .( 1.

 

APPENDIX D (cont’d)

 

14. [The purpose is] so that people can see what you did in class.

 

15. [The purpose] I think is to share your ideas. People respond to it. It’s not really
bad if they respond to it because you know that they are reading your stuff and
listening to what you have to say. ’

 

16. The purmse is to let other people know what we are doing in class.

 

17. Make sure other people can understand it, make sure it makes sense. You
shouldn’t do bad things, like English when it is about math and something that
doesn’t make sense.

 

18. Everybody can see it, your audience you are writing to. Use proper spelling,
punctuation, all the editing stuff and make sure your information is correct. If it
isn’t, everybody is going to get messed up.

 

19. Make sure that people can understand what you write and choose your words
carefully. Don’t go to other web pages while publishing.

 

20. I think publishing is neat because we get to share ideas with others outside this
school.

 

 

21. [Publishing is] fun because you get to show other people what you are doing,
and so they can understand it too.

 

22. I think it’s [publishing] fun because you get to write on the computer, put your
ideas, and what you did in class. It’s nice to let people know what you did in
class.

 

23. Yeah, [publishing is interesting] because you get to work with other students
that you know. You get to type and work together and to publish something
that everyone can see, that is bigger than just in the classroom, so everybody
that has a computer can see it too.

 

24. Yeah, [publishing is interesting] because you can work others and talk about
what you think is correct and what you should write on there, whatever you
think everybody will understand.

 

25. [Publishing is different from other activities] because you are using a computer
and sharing it with everybody else, whoever looks at the Internet, share what
you did in class.

 

26. It’s different [from other math activities] because, in our classroom, the only
people who get to see what you are doing is in your class, but when you
publish, other kids get to see it too.

 

 

27. Yeah, [it makes me want to math more] because if other kids like it from
wherever, then your teacher might like it too. If you are with your partner and
doing a problem and they like it too, the teacher might like it too.

 

 

185

APPENDIX E

TEACHING AND LEARNING

We can teach others

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1. Yeah, [the web journal and my journal are different] because in the math
journal, the only people who see it are you (Jones) and Mr. Watkins, but on the
web, other kids who are having problems or whatever, they can look on our
sheet

2. No, [it isn’t the same if students just wrote their journal on the computer and
not the web] because on the web, other people can see what you do and it can
help them more. [This is an importantﬂaecause other people need to learn too.

3. [I’d rather publish] on the web [than write in my journal] because it helps other
kids learn new things and it’s fun typing.

4. You pick something, people and they publish it so other people can look at it....
To help other people on the Internet learn.

5. [The purpose is] to share what you’re learning with people who are reading it.

6. [The purpose is] so other people know how to do it or know what you are
talking about.

7. [The purpose is] to help other people and let kids know what you are doing at
your school.

8. [The purpose is] to help other kids, like maybe in the X school district or all
over, to help them if they don’t know or if they are doing the same thing or if
they want to see how other people are doing it.

9. [The purpose is] so that other people, in smaller grades can see what we’re
doing if they haven’t learned it yet, or older people to see what they are doing.

10. [The purpose is] so that other people can see what we are doing and they can
learn some stuff too.

11. The purpose is so that people can see what you are learning and so other people
can learn.

12. Yeah, it’s fun because it’s like telling people how to do other things they don’t
know how to do.

13. Yeah, because other people can learn how you learn and it’s fun.

14. [I’d rather publish than write in my journal because] of typing, telling how to do
it, and publishing.

 

186

 

 

 

APPENDIX E (cont’d)

We can learn

 

1.

[The purpose is] to remember things and if someone else published and you
didn’t understand something, you can go to their publication and read it and
understand it.

 

It’s fun. I think that you should have it next year to help out the kids because
this has helped me a lot in understanding the computer and math because I was
a bad student. I’m surprised I got into enriched math. When I explain it on the
computer, I know exactly what I am saying. That helps me out with my math.
When we have partners, like smart people helping us, you get it because once
you read it, you say, oh, now I get it.

 

Publishing makes sense for people to do it, not like all those things that the
teacher makes you have to do. It’s something people want to do... because it is
fun and if you are going to help someone else and if other people are going to
help you. If there is someone else in another city trying to do a problem, I can
write this down and try to help them if they have a problem with it.

 

Publishing helps other people and gives you a chance to do other things you
don’t know how to do. If you publish and let everybody see it, you feel happy
because you taught somebody how to do something that they didn’t know how
to do. If you didn’t have publishing, most people couldn’t learn as much stuff
because they couldn’t look through and see if anybody has put anything they
did not know.

 

Yeah, [publishing does make me want to do math more] because when I publish
I start writing something and think, oh my gosh, it does really work and start
using it.

 

Yeah, [it makes me want to do math more] because you want to learn stuff.
When others login, they can see your initials and decide to read yours. They can
learn from them. I thinks it’s cool. I get to see other solution strategies and I get
more involved in math because I learn more stuff everyday. And I get to write
more stuff. After I type stuff, I want to learn more about that subject and
publish more on that subject.

 

Yeah, [publishing makes me want to do math more] because it is fun. You get
to learn more about what you are doing that way. That makes you understand it
more. Once you think about it more to write it down, it makes you understand
the problem more.

 

 

It’s interesting to see how other people learn math instead of only the way you
learn.

 

 

187

 

 

 

APPENDIX F

OWNERSHIP

 

1. [If the teacher picked what you had to write about instead of letting you pick
want you wanted to write,] I don’t think it would be done as well. If you publish
about what you want to publish, then you know what you are going to say, and
if you have to publish what you want us to publish, you have to ﬁgure out what
you are going to say and how to figure it out instead of what we already had
planned for it or what was already in our journal.

 

2. Yeah, it would be different [if you told us to type on a speciﬁc topic instead of
letting us choose our own] because the stuff that you want to type, you’ll know
and the stuff you want us to type, we might not know about.

 

3. Yeah, because if you don’t know what you are doing, you’ll just be sitting there
writing about something you don’t know. If you don’t know what you are
going to do right then, you can think of something.

 

 

4. [Compared to writing on my journal, I’d rather publish] on the web because it
just gives you a chance to show your ideas and thoughts.

 

5. It’s fun because you get to type and pick what you want to write about.

 

6. First of all, you have to learn the stuff, then type all that you know about it... all
the stuff we’ve been learning in class.

 

7. Don’t write stupid stuff or things you don’t know what you are talking about.
You should write about stuff you know about.

 

8. [You need to remember] how to do it. Before you publish, you have to know
what you are explaining and how to do it. You know what to do when you get
on there and how to explain your answers. You shouldn’t play around on it.

 

9. Just type all that you know about the subject. You shouldn’t do anything else
you weren’t told to do, just do what you were supposed to do and not anything
else. Like don’t mess with things, just type your publication.

 

10. I think publishing is interesting because this is the only class you get to do this
kind of stuff, like typing your journals, type what you know about what you did
in a certain area.

 

11. Yeah I like publishing, sometimes in class, it is boring, but when you get to
publish, you can write down what you think about it, how you got the problem.

 

12. I think kids would rather publish than do other things in class. It’s different
because it let’s you explain things instead of letting the teacher tell you how to
do it. You can put how you think is easy to do it your own way.

 

 

13. To not forget it, have it there and other people can read it. Be serious about it.
Pay attention in class before you publish, so you know what we are talking
about that day. You shouldn’t mess around with the computer and try to focus
on class.

 

 

188

APPENDIX G

COLLABORATION

 

. Publishing is like doing math, but doing it on the computer and writing it out
instead of just reading it and doing it. You are writing it out. I think it would be
more boring if you just did your own journals because, on the web, you get to
work with a partner. When you look at other people’s work, you can cut out
the stuff that doesn’t go with it. If they put, ‘oh, I forgot to mention I got an A.’
You can cut that out because that has nothing to do with it.

 

. No, it’s not that different [from publishing if you get to work with other people
in another activity.] It don’t make any difference because you still have kids
helping you.

 

. It [publishing] is different from how we did math because we used to just have
the blackboard and no journals. It makes it a little more interesting and different
because you get to work with someone else and both of the people are trying to
ﬁgure it out. I like that better.

 

. [I’d rather publish] on the web [than write in my journal] because I don't really
write in my journal. I don’t really know what to put in there, but on the web,
there’s somebody else who can help you word it right or whatever when we are
publishing.

 

. Publishing is fun because in your journal, when you are by yourself, you might
not be able to think of what to publish, but when you have a partner, you can
ask them how to word it right or help you.

 

. Yeah, [publishing is interesting] When you do it, its a group thing, compromise
and think of something to put on. It’s just teamwork when you put it on there.

 

. When you publish, you are not by yourself, and sometimes kids don’t do the
work because they have to sit by themselves or do it by themselves and don’t
understand it. But when you are publishing and you put down a problem, you
have a friend there who can help you solve the problem or a friend who can
help you think of a problem to put down. It is also fun because it lets you show
other kids, not just those you are working with, it shows other kids.

 

 

. I don’t know [why publishing is fun]. I think it is that you just basically get to
work with your friends.

 

189

 

 

 

 

 

REFERENCES

190

REFERENCES

Ball, D. L. (1992). Halves, pieces, and twoths: Constructing and using representational
contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. A. Romberg
(Eds), Rational numbers: An integration of research, (pp. 157-195). Hillsdale,

NJ :Erlbaum.

Bereiter, C., & Scardamalia, M. (1989). Intentional learning as a goal of instruction. In
L. Resnick (Ed.), Knowing, learning, and instruction, (pg 361-392). Hillsdale, NJ:
Erlbaum.

Borko, H., & Putnam, R. T. (1996). Leaming to teach. In D. C. Berliner & R. C. Calfee
(Eds), Handbook of educational psychology, (pg 673-708). New York:Simon &
Schuster Macmillan.

Brown, A. (1992). Design experiments: Theoretical and methodological challenges in
creating complex interventions in classroom settings. The Journal of the Learning
Sciences, 2, 141-178.

Brown, A. L., Ash, D., Rutherford, M., Nakagawa, K., Gordon, A., & Campione, J. C.
(1993). Distributed expertise in the classroom. In G. Salomon (Ed.), Psychological
and educational considerations, (pp. 188-228). NY:Cambridge University Press.

Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the
representational view of mind in mathematics education. Journal of Research in

Mathematics Education, 23, 2-33.

Cognition and Technology Group at Vanderbilt ( 1991). Technology and the design of
generative learning environments. Educational Technology,31 , 34-40.

Cognition and Technology Group at Vanderbilt (1992). The Jasper Series as an example
of anchored instruction: Theory, program description, and assessment data.
Educational Psychologist, 27, 291-315.

Cohen, M., & Riel, M. (1989). The effect of distant audiences on students' writing.
American Educational Research Journal, 26, 143-159.

Greeno, J ., Collins, A, & Resnick, L. (1996). Cognition and learning. In D. C. Berliner
& R. C. Calfee (Eds), Handbook of educational psychology, (pg 15-46). New
York:Simon & Schuster Macmillan.

Jones, S.C. (1997a). Designing technology to support motivation. Unpublished
manuscript, Michigan State University.

191

 

 

Jones, S.C. (1997b). Mathematical thinking and engagement in a project-based problem
context. Unpublished manuscript, Michigan State University.

Kamii, C., & Livingston, S. J. (1994). Autonomy: The aim of education for Piaget. In C.
Kamii, & S. J. Livingston (Eds), Young Children Continue to Reinvent Arithmetic:
3rd Grade Implications of Piaget's Theory, (pg 59-65). NYzTeachers College Press.

Lampert, M. (1987). Knowing, doing, and teaching multiplication. Cognition and
Instruction, 3, 305-342.

Lappan, G., Fey, J ., Fitzgerald, W., Friel, S., & Phillips, E. (1996a). Connected
Mathematics. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J ., Fitzgerald, W., Friel, S., & Phillips, E. (1996b). Connected
Mathematics: Bits and Pieces II. Palo Alto, CA: Dale Seymour Publications.

Leikin, R., & Zaslavsky, O. (1997). Facilitating student interactions in mathematics in a
cooperative learning setting. Journal for Research in Mathematics Education,
28,331-354.

Millroy, W. L. (1992). An ethnographic study of the mathematical ideas of a group of
carpenters. Journal for Research in Mathematics Education, Monograph Number 5,
ii-ix, 1-51.

National Council of Teachers of Mathematics (1991). Professiorﬁl Standards for
Teaching Mathematics. Reston, VA: Author.

National Research Council (1989). Everybody counts: A report to the nation on the
future of mathemagcs education. Washington, DC: National Academy Press.

Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). What is street mathematics? In
T. Nunes, Schliemann, A. D., & D. W. Carraher (Ed.), Street mathematics and school
mathematics, (pp. 1-25) Cambridge, England: Cambridge University Press.

Pea, R. D. (1993). Practices of distributed intelligence and designs for education. In G.
Salomon, (Ed.). Distributed cognition: Psychological and educational con_siderations.
(pp. 47-87). NY: Cambridge University Press.

Ryan, R., Connell, J. P., & Deci, E. L. (1985). A motivational analysis of self-
deterrnination and self-regulation in education. In C. Ames & R. Ames (Eds),
Research on motivation in education, 2, (pp. 13-51). Orlando: Academic Press.

Scardamalia, M., & Bereiter, C. (1991). Higher levels of agency for children in

knowledge building: A challenge for the design of new knowledge media. The
Journal of the Learning Sciences, 1, 37-68.

192

 

Scardamalia, M., & Bereiter, C. (1994). Computer support for knowledge-building
communities. The Journal of the Learning Sciences, 3, 265-283.

Schoenfeld, A. H. (1989). Leaming to think mathematically: Problem solving,
metacognition, and sense making in mathematics. Learning from Instruction, 334—
370.

Schwartz, J. L. (1993). A personal view of the Supposer: Reﬂections on particularities
and generalities in educational reform. In J. L. Schwartz, M. Yerushalmy, & B.
Wilson, (Eds), The Geometric Supposer: What is it a case of?, (pp. 3-15). Hillsdale,
NJ: Erlbaum.

Songer, N. (1996). Exploring learning opportunities in coordinated network-enhanced
classrooms: A case of kids as global scientists. The Journal of the Learning Sciences,
5, 297- 327.

TELE Team. TELE-Web. [On-line].http://zhao.educ.msu.edu/tele.start.fcgi.1998

Wood, T., & Sellers, P. (1996). Assessment of a problem-centered mathematics program:
third grade. Journal for Research in Mathematics Education, 27, 337-353.

Yackel, E., Cobb, R, Wood, T., Wheatley, G., & Merkel, G. (1990). The importance of
social interaction in children's construction of mathematical knowledge. In T. J.
Cooney, & C. R. Hirsch (Eds), Teaching and learning mathematics in the 1990s, .(pp.
12-21). Reston, VA: National Council of Teachers of Mathematics.

Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interactions as source of learning

opportunities in second-grade mathematics. Journal of Research in Mathematics
Education, 22, 390-408.

193