PRESERVICE TEACHERS’ USES OF THE INTERNET TO SUPPORT THEIR
LEARNING OF MATHEMATICS: THE CASE OF THE PYTHAGOREAN
THEOREM
By
Aaron Brakoniecki

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics Education – Doctor of Philosophy
2014

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ABSTRACT
PRESERVICE TEACHERS’ USES OF THE INTERNET TO SUPPORT THEIR
LEARNING OF MATHEMATICS: THE CASE OF THE PYTHAGOREAN
THEOREM
By
Aaron Brakoniecki
The Internet is a go-to resource for many students, both for academic and recreational
purposes. This dissertation explores how preservice elementary teachers use the Internet
to support their learning of mathematics. Set in the context of needing to get students to
explain a proof of the Pythagorean Theorem and its converse, this study asks preservice
teachers to use the web in order to further develop their own understanding of this
mathematics. This study seeks to investigate the kinds and quality of the mathematical
connections that preservice teachers make after investigating the Pythagorean Theorem
online. It also examines the information seeking strategies employed by the preservice
teachers while completing their investigations. Lastly this study explores possible
connections between the kinds and quality of the connections exhibited by the preservice
teachers and the information-seeking strategies employed by the preservice teachers
during their investigation.

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Copyright by
AARON BRAKONIECKI
2014

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To my family for their infinite love and support,
my teachers for their infinite wisdom and patience,
and my friends for their infinite kindness and loyalty;
I am thankful these groups are not mutually exclusive.

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ACKNOWLEDGMENTS

There are many people who have had profound influence on my thinking and
writing leading up to, and during this dissertation. It is impossible to name them all, but a
few deserve special recognition for their work in helping me grow as a researchers and a
scholar.
My dissertation co-director and guidance committee chair Dr. Sandra Crespo is
responsible for taking this enthusiastic and naïve young graduate student and helping me
become a thoughtful and critical researcher and mathematics educator. Saying “I couldn’t
have done this without her” is perhaps the biggest understatement I could ever make. I
am forever indebted to her for her wisdom, patience, guidance, and love of coffee.
My dissertation was also co-directed by Dr. Ralph Putnam. From my first year in
the mathematics education pro-seminar to these final years working on this dissertation
(with an independent study in between), my thinking pertaining to the use of technology
in mathematics education has become more layered and nuanced because of what I have
learned from his expertise. My research has been forever improved because of his work
with me.
The other members of my dissertation committee also had their work cut out for
them when they agreed to work with me, but they were incredibly successful in
improving my work. Dr. Brin Keller is responsible for helping me think through the
variety of complex interaction between learners, mathematics, and technology. Dr.
Patrick Dickson provided guidance and experience as I situated this study in the context
of learning online and outside of the classroom setting. All four of these individuals

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working together not only made my dissertation better, but also made me a better
researcher, and I am forever grateful.
I have had the privilege of being a part of several writing groups during my time
at Michigan State University. These groups have forced me to reflect on my own writing,
and think critically about what I write and the way I write it. I am grateful for the time,
edits, and comments given by Leslie Dietiker, Aladar Horvath, Kate Johnson, Sarah
Kasten, Ann Lawrence, Mari Levin, Lorraine Males, Aaron Mosier, Sam Otten, and
Justin Thorpe.
I also must acknowledge the faculty, staff, and graduate students at Michigan
State University. I have learned incredible amounts from them, both inside and outside
the classroom. I came into this program having a background in mathematics and very
little understanding of education. Their work has been incalculable in helping me become
a more thoughtful and skilled mathematics educator.
Lastly, I need to acknowledge my parents (Al and Kris) and brother (Adam) for
their support during graduate school. While everyone else listed in these
acknowledgements is mostly responsible for supporting my scholarship and my
researcher, these three people are responsible for supporting me. I am especially grateful
that they didn’t ask too many questions annually when I kept hinting I would graduate,
ultimately to return for “one more” year. This time, I promise, it’s for real.

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TABLE OF CONTENTS

LIST"OF"TABLES ........................................................................................................................................ix"
LIST"OF"FIGURES.......................................................................................................................................xi"
CHAPTER"1 ...................................................................................................................................................1"
Purpose.......................................................................................................................................................................1"
Opportunities and Challenges Present with the Internet............................................................................3"
Goals of This Dissertation ...................................................................................................................................5"
CHAPTER"2 ...................................................................................................................................................7"
Information Seeking Online................................................................................................................................7"
Frameworks of Knowledge .................................................................................................................................9"
Concept Mapping ................................................................................................................................................ 10"
CCSSM and The Pythagorean Theorem...................................................................................................... 11"
Purpose Revisited ................................................................................................................................................ 13"
Research Questions ............................................................................................................................................. 14"
CHAPTER"3 ................................................................................................................................................ 15"
Participants............................................................................................................................................................. 16"
Tasks ........................................................................................................................................................................ 17"
Web Task I (Open Searching).................................................................................................................... 17"
Web Task II (Closed Searching) ............................................................................................................... 18"
Investigating the Effect of Tasks.................................................................................................................... 24"
Concept Mapping (Pre, Middle, and Post)............................................................................................. 24"
Pre Interview.................................................................................................................................................... 26"
Post Interview .................................................................................................................................................. 27"
Vignette of Data Collection Experience ...................................................................................................... 28"
Summary................................................................................................................................................................. 32"
CHAPTER"4 ................................................................................................................................................ 33"
Chapter Structure ................................................................................................................................................. 34"
Frameworks of Information-Seeking ............................................................................................................ 35"
Task I – Open Searching ................................................................................................................................... 39"
Timid Navigator.............................................................................................................................................. 40"
Laborious Navigator...................................................................................................................................... 42"
Divergent Navigator ...................................................................................................................................... 43"
Focus on Detail ............................................................................................................................................... 45"
Focus on Static Content and Focus on Dynamic Content................................................................. 46"
Patterns Across Task 1....................................................................................................................................... 48"
Task 2 – Closed Searching ............................................................................................................................... 50"
Site 1– PBS (Nova)........................................................................................................................................ 50"
Site 2 – GMAT Prep Review ..................................................................................................................... 52"
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Site 3 – RonBlond .......................................................................................................................................... 53"
Site 4 – YouTube Video Khan Academy Proof of PT....................................................................... 55"
Site 5 – YouTube Video Proof of Converse.......................................................................................... 56"
Patterns Across Task 2....................................................................................................................................... 58"
Summary................................................................................................................................................................. 59"

CHAPTER"5 ................................................................................................................................................ 62"
Chapter Structure ................................................................................................................................................. 63"
Mathematical Content of Created Concept Maps..................................................................................... 64"
Unpacking the Counts of Content............................................................................................................. 68"
Mathematical Links Among Content............................................................................................................ 69"
Framework for Forms and Quality of Links ......................................................................................... 70"
Definitions and Examples of Forms and Qualities.............................................................................. 71"
Illustration of Mathematical Link Coding.............................................................................................. 74"
Summary of Form and Quality of Mathematic Links in Concept Maps........................................... 77"
Structures and Purpose of Created Concept Maps ................................................................................... 81"
Structure ............................................................................................................................................................ 82"
Purpose............................................................................................................................................................... 83"
Summary of Results............................................................................................................................................ 84"
CHAPTER"6 ................................................................................................................................................ 87"
Review of Search Strategy Findings ............................................................................................................. 87"
Review of Math Content and Connections Findings ............................................................................... 89"
Relationships Between Search Strategies and Content........................................................................... 91"
Classroom Goals of the Participants ............................................................................................................. 94"
Connecting to the Literature ............................................................................................................................ 98"
Summary.............................................................................................................................................................. 100"
CHAPTER"7 ..............................................................................................................................................102"
Implications ........................................................................................................................................................ 103"
Limitations .......................................................................................................................................................... 106"
Future Directions .............................................................................................................................................. 108"
Conclusion .......................................................................................................................................................... 109"
APPENDIX ................................................................................................................................................111"
REFERENCES ..........................................................................................................................................120"

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LIST OF TABLES

Table"1"–"Description"of"Sites"Used"in"Web"Task"II.................................................................. 23"
Table"2"–"Summary"of"InformationUSeeking"Coding"of"Participants ................................ 39"
Table"3"–"Time"(and"Percentage)"Spent"with"the"Format"of"Content"Presentation... 47"
Table"4"–"Time"Spent"on"Static"and"Dynamic"Content"in"Site"1 .......................................... 51"
Table"5"–"Total"Time"Spent"on"Site"2 .............................................................................................. 53"
Table"6"–"Time"Spent"on"Static"and"Dynamic"Content"in"Site"3 .......................................... 54"
Table"7"–"Time"Spent"Watching"Video"in"Site"4 ......................................................................... 55"
Table"8"–"Time"Spent"Watching"Video"in"Site"5 ......................................................................... 57"
Table"9"–"Content"Included"in"Concept"Mappings .................................................................... 66"
Table"10"–"Summary"Counts"of"Content"in"Each"Concept"Map ........................................... 67"
Table"11"–"Forms"and"Quality"of"Mathematical"Knowledge................................................. 70"
Table"12"–"Examples"of"Forms"and"Qualities"of"Mathematical"Links............................... 73"
Table"13"–"Form"of"Mathematical"Link"Contained"In"Concept"Map.................................. 77"
Table"14"–"Quality"of"Mathematical"Links"Contained"In"Concept"Map ............................ 79"
Table"15"–"Categories"of"Behavior"of"Participants ................................................................... 81"
Table"16"–"Structure"in"Each"Concept"Map.................................................................................. 82"
Table"17"–"Table"of"Categories"from"Findings"(Empty) ......................................................... 87"

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Table"18"–"Table"of"Categories"from"Findings"(Searching)................................................... 89"
Table"19"–"Table"of"Categories"from"Findings"(Searching"&"Math"Connections)........ 91"
Table"20"–"Descriptions"and"Examples"of"Classroom"Goals ................................................. 95"
Table"21–"Table"of"Categories"from"Findings"(Filled) ............................................................ 97"

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LIST OF FIGURES

Figure"1"–"Representation"of"the"Design"of"the"Interview .................................................... 15"
Figure"2"–"Screenshot"of"PBS"Website ........................................................................................... 19"
Figure"3"–"Screenshot"of"GMAT"Review"Site ............................................................................... 20"
Figure"4"–"Screenshot"of"ronblond.com"Site................................................................................ 21"
Figure"5"–"Screenshot"of"Khan"Academy"Video ......................................................................... 22"
Figure"6"–"Screenshot"of"TutorVista"Site ...................................................................................... 23"
Figure"7"–"Example"Augmented"Concept"Map............................................................................ 26"
Figure"8"–"Diagram"of"Common"Aspects"of"the"Internet"Searching"Process ................. 40"
Figure"9"–"Pre"Concept"Map"from"Participant"5 ........................................................................ 62"
Figure"10"–"Post"Concept"Map"from"Participant"5.................................................................... 63"
Figure"11"–"Counts"of"Content"in"Each"Concept"Map............................................................... 68"
Figure"12"–"Augmented"and"Chunked"Concept"Map"from"Participant"5"(Pre) ............ 75"
Figure"13"–"Augmented"and"Chunked"Concept"Map"from"Participant"5"(Post) .......... 75"
Figure"14"–"Forms"of"Mathematical"Links"Contained"in"Concept"Maps .......................... 78"
Figure"15"–"Quality"of"Mathematical"Links"Contained"in"Concept"Map .......................... 80"

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CHAPTER 1
Recent research on the preparation of future teachers of mathematics has focused
on issues related to the experiences preservice teachers have as part of their teacher
preparation programs both in their content and methods courses, and their school field
experiences (Cochran-Smith, Feiman-Nemser, McIntyre, & Demers, 2008; Sowder,
2007). Although this has generated important research, the emphasis of learning inside
the program has overshadowed opportunities for examining the learning of mathematics
that occurs outside of the programs. The learning of mathematics by preservice teachers
outside of coursework is an important issue to consider given the limited time teacher
preparation programs have with their preservice teachers. Already, the Internet is a
resource used by many college students for both academic and non-academic purposes
(Wilber, 2008). To this end, the Internet is an emerging resource to support the teaching
and learning of mathematics for preservice teachers both inside and outside the
mathematics classroom. The use of this resource to learn mathematics is the focus of this
dissertation.
Purpose
Teacher preparation programs consist of both coursework and field experiences.
They include both content courses and methods courses for the preservice teachers
enrolled in the program. The time spent on content and methods in these classes is
limited, and cannot prepare teachers for every piece of content, or mathematical scenario
they will encounter or be responsible for teaching during their time in schools. In
elementary teacher preparation courses, the focus on content is especially limited, as
preservice teachers must focus on multiple content areas. It should thus be expected that

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elementary mathematics teachers may often encounter situations in their classrooms
where they are not familiar with the mathematics content that occurs as part of a
changing curriculum or that appears based on student inquiries. This poses questions
about the approaches used by teachers when they encounter such scenarios. What are the
possible strategies employed by preservice teachers when they need to better understand
mathematics outside of their content classes?
A possible answer to this question is that they might use the Internet as a learning
resource. Most American households now have Internet access (U.S. Census Bureau,
2013) and use the Internet in many difference aspects of their life every day. Users can go
online to interact with businesses, government services, communicate in different
formats, and receive multiple forms of entertainment. People go to the Internet for its
information and services. The characteristic of the Internet as reference is also being used
more frequently by students as a tool to help with their studies (Wilber, 2008). As the
Internet becomes a more prominent tool in education, it is important to understand the
ways in which it is used to support academic learning.
The Internet is filled with diverse content from a variety of sources in a number of
different forms. There are numerous ways that mathematical information can be
presented to users via the Internet. Oftentimes, a traditional, static presentation (similar to
the way it is found in textbooks) has been used to present mathematics content. There are
also new, dynamic presentations available (e.g., through animations and web applets) that
allow users to see and interact with the content. The communicative features of the
Internet can also be utilized for presenting mathematics, which can be seen in YouTube
videos, discussion forums, and web blogs.

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One of the difficulties that this new resource presents to its users is that it is not
always related to or compatible with the curriculum of the classroom. Oftentimes, users
have to search in various locations to try to find the information relevant to the
mathematical situation they are trying to explore (which is often in a form different from
the more structured presentation of relevant information provided by teachers or texts).
This ill-defined structure (Spiro & Jehng, 1990) has unique challenges that require
information-seeking skills beyond what is required with more traditional informational
media used by students (e.g., documents, texts).
Opportunities and Challenges Present with the Internet
The sophistication and omnipresence of the Internet suggest that this venue is an
important tool to investigate how it may support the mathematical learning of preservice
and in-service teachers. When placed in scenarios where learners are solely responsible
for their own understanding of mathematics content, the Internet becomes a valuable
resource for solo learners. Not only is there a variety of mathematical content contained
within the vast span of the Internet that can cover most any inquiry posed by learners, but
that content is also presented in numerous different formats, allowing diverse learners, all
with unique learning styles, to not only locate relevant information, but to locate it in a
format they prefer (Kuiper & Volman, 2008). Additionally, the ability of the Internet to
allow users to communicate and receive feedback with people, groups, and organizations
beyond the user’s immediate geographic area can also add an interactive aspect to this
“solo” learning environment. Physical location is no longer a constraint to collaborative
learning.

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Yet given these opportunities offered by the Internet, there are also challenges
present within this environment. Unique to the Internet is that the burden of determining
the quality of a source of information is often left to the learner, whereas in more
traditional environments, often the quality of a resource is determined by an instructor
before being utilized within a classroom. Users are not often able, or even aware, of a
need to check the quality and reliability of the information located (Kafai & Bates, 1997;
Schacter, Chung, & Dorr, 1998). Additionally, the location of information online often
requires the use of search engines, which necessitate using particular techniques and
syntax to use effectively. Users have different skills and techniques for utilizing search
engines, producing different kinds of resources within their searching (Fidel et al., 1999;
Large & Beheshti, 2000). Also, given the open nature of the Internet, users continuously
reflect on their own understanding, controlling their own pace through new content, and
also influencing what they know to investigate further (Fidel et al., 1999; Hirsh, 1999;
Lawless & Schrader, 2008). In more traditional environments, an instructor would
determine the pace of content and help students reflect on their understanding. These are
just a few of the demands placed on users of the Internet not usually associated with more
traditional resources.
The opportunities and challenges present when one uses the Internet as a resource
(especially the opportunities present beyond those of more traditional resources) raises
questions about the use of the Internet to support preservice and in-service teachers. We
might ask about the contexts under which preservice and in-service teachers go to the
Internet for support with content. We might try to better understand the kinds of
information they are looking for when they go online. We might investigate how they

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navigate this new environment for the purposes of their mathematical understanding.
Additionally, we may ask about how these preservice teachers learn mathematics from
their experiences online, then take that new knowledge and understanding back into other
contexts where they can then utilize what they learned.
Goals of This Dissertation
This dissertation will begin to address a small fraction of this large collection of
issues. The Internet has the potential to serve as a multipurpose resource to support the
lifelong learning of teachers. However, it is not yet clear how preservice teachers might
be using this widely available tool as a resource to support their learning of unfamiliar
mathematics and mathematics for teaching. It is important to study the ways the Internet
is currently being used to learn mathematics as it may suggest strategies that can be
leveraged to support learners both inside and outside of the classroom. Additionally, it
may also be possible to design experiences that could make using the resources of the
Internet more effective for the learning of mathematics. This is what this dissertation
begins to investigate.
This research is important as it seeks to better understand the relationships
between an evolving educational resource and preservice teachers who frequently use the
resource for both academic and non-academic purposes. In particular, it is important to
better understand the academic ways this common tool is being used to support subject
specific learning (in this case, mathematics). While many people become skilled at
exploring the Internet outside of academic purposes, it is not clear how applicable those
skills are for academic purposes, nor if users are able to adeptly transfer the skills to new
environment for new purposes.

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The characteristics unique to the Internet, including its hyperlinked environment,
its dynamic and interactive features, and the ability to communicate among diverse
communities are new features beyond those that exist within traditional mathematics
classrooms and learning environments. Because of these new factors, there could be
differences in what is learned from a newer environment like the Internet, and a
traditional environment like a classroom or a textbook. It is not clear how some of the big
ideas of the mathematics classroom (e.g., reasoning and proving, problem solving) might
be supported on in these digital environments. This dissertation seeks to understand the
ways in which mathematics is understood when preservice teachers use the Internet to
increase their understanding of a mathematical topic.
The Internet is a resource that mathematics educators have at their disposal to
support the teaching and learning of mathematics. The availability and presence of the
Internet outside of the classroom make it a particularly important and potentially
powerful resource that could be harnessed for supporting mathematics learning.
Understanding the mathematics that learners are able to understand from engaging with
these environments, and the ways they went about exploring these environments to
produce that learning would provide insights for teacher educators into how this tool
might be leveraged for further support among learners and teachers of mathematics.

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CHAPTER 2
This dissertation seeks to investigate the ways in which Internet searching may be
a productive means for learning about mathematics. This chapter looks at what is already
known about these individual areas, and other related areas related to this study. As one
of the key parts of this study involves understanding the ways in which preservice
teachers search for information online, the first literature reviewed relates to Internet
searching and information seeking strategies of Internet searchers. This study also
attempts to better understand how preservice teachers may be learning mathematics via
Internet searching. Therefore, the next section of literature reviews frameworks for
understanding and researching mathematics content knowledge, and how that knowledge
can be represented via concept maps. The last main portion of literature centers on the
content that participants will be investigating, the Pythagorean Theorem and its
associated proofs. The chapter concludes with a presentation of the three research
questions this dissertation addresses.

Information Seeking Online
With the emergence of the Internet as a common phenomenon in peoples’
everyday lives, research on how people use the Internet to find information is becoming
more prominent. In general, when users go to the Internet to find information, their
searches have been described along two different lines: finding or fact-based tasks, and
searching or research tasks (Bilal, 2000; Schacter et al., 1998). Finding tasks tend to be
situations where a user is looking for a specific piece of information, such as identifying
show times for a movie at a local theatre, determining the location of a restaurant, or

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locating capitals of different countries. Searching tasks, in contrast, are situations where a
user needs to explore multiple aspects of a topic in order to identify useful information
for their own purposes, such as identifying a mysterious noise your car is making, or
looking for information about a new congressional bill. Research reports mixed results on
students’ success with these types of searches (Bilal, 2001; Schacter et al., 1998).
Research on online learning is clear that users have many challenges with Internet
searching tasks. There is agreement that successful searchers tend to have some prior
knowledge about the topic that they are investigating (Fidel et al., 1999; Hirsh, 1999;
Lawless & Schrader, 2008), making it easier to formulate relevant searches and assess the
information that emerges. Students often have difficulties, however, locating relevant
information on the web, in assessing the relevance of information, and in exploring the
resources of the web beyond the narrow scope of their investigation task (Kuiper,
Volman, & Terwel, 2005). One of the most consistent findings reported of Internet
searchers is their failure to critically asses the quality of the information presented, and
the source of that information (Kafai & Bates, 1997; Large & Beheshti, 2000; Lorenzen,
2001).
There are several different approaches used to quantify and describe the ways that
Internet searchers look for information. Some have focused on the initial investigative
approaches employed by the searchers (Schacter et al., 1998); others have expanded upon
this to look at the patterns of behavior over the course of the investigation (Bilal, 2000,
2001, 2002). Two frameworks in particular have isolated different aspects of the
searching process. Juvina and Oostendorp’s (2004) framework focuses on the navigation
between sources of information from either web searching or web exploration. A second

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framework (Lawless & Schrader, 2008) narrows in on the skill of locating information
within individual resources and how users investigate within a site. These two
approaches—locating sites and investigating a site—provided this study a useful way to
unpack the complex process of information seeking online. I come back to these two
frames in Chapter 4 where I describe how I adapted them for this study.

Frameworks of Knowledge
Mathematics education has a long history of describing ways that learners come
to know mathematics. Many of these frameworks attempt to describe deeper and more
richly connected understandings of mathematics (Brownell, 1938, 1947; Hiebert &
Lefevre, 1986; Skemp, 1976). This deeper understanding of mathematics extends beyond
students and has also been found in teachers (Ma, 1999). All these frameworks attempt to
describe ways of knowing mathematics beyond just the utility of what to do in
mathematical scenarios—to understanding why particular mathematics makes sense in
mathematical problems.
There has also been much work in mathematics education to describe not only the
mathematical knowledge possessed by students, but also the mathematical knowledge
possessed by teachers and how it interacts with teachers’ pedagogical knowledge (Ball,
Hill, & Bass, 2005; Ball & Hill, 2009; Ball, 2000; Hill, Ball, & Schilling, 2008; Shulman,
1986, 1987). This intersection between content and pedagogy has been expanded to
include the knowledge of teaching with technology (Koehler & Mishra, 2009; Mishra &
Koehler, 2006). What all these frames, whether pertaining to student or teacher,

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emphasize is that to effectively do something like mathematics or the teaching of content,
one needs knowledge that is richly understood in different ways.
Star (2005) unpacked an underlying issue that existed in many frameworks of
mathematical knowledge at the time. Star argued that instead of using the terms
procedural and conceptual to describe two aspects of a single phenomenon, the research
literature had instead been overlapping two different phenomena into these labels. Star
unpacked the ideas of the kind of knowledge, commonly referred to in the literature as
procedural or conceptual, from the quality of that knowledge, which he described as rich
versus superficial (Star, 2005). Star argued that in the literature, conceptual knowledge
was often used to mean richly connected knowledge, and procedural knowledge was
often used to indicate superficial knowledge. Instead he argued, conceptual knowledge
(knowledge of a concept) could be richly connected or superficial. Similarly, knowledge
of a procedure could be richly connected or superficial. This frame is useful for
unpacking both the kind and quality of mathematics understanding possessed by
preservice teachers. This framework and how it is used in this study is further discussed
in Chapter 5.

Concept Mapping
Concept mapping is a tool used to express connections between concepts via
diagram. In a diagram, concepts are connected by words or phrases describing why a
connection exists between these two concepts. There can be multiple links to a concept
from other concepts, and multiple links emerging from a single concept to other concepts.
These concept mappings can vary in size, depth, and detail. The construction of a concept

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map gives its creator an opportunity to express how they are making connections and
interpreting the concepts of a topic. Concept maps have been shown to be an effective
way to capture student understanding of content in mathematics (Baroody & Bartels,
2000). Concept maps have also been used to track changes in understanding of
mathematics teachers (Hough, O’Rode, Terman, & Weissglass, 2007). Much of the
research on concept maps looks at the structures that are used in the concept maps and
propose different ways of quantifying these concept maps (Ruiz-Primo, Schultz, Li, &
Shavelson, 2001; Wallace & Mintzes, 1990).
Ruiz-Primo, Schultz, Li, and Shavelson (2001) looked at several different kinds of
concept maps and how students constructed them. Concept maps varied from having all
nodes filled out, leaving students to fill out the connections: having all the connections
filled out, leaving students to fill out the nodes; or providing a word bank and letting
students construct concept maps from scratch. Providing a word bank resulted in the best
representation of students’ knowledge structures as compared to responses to multiplechoice assessments given to the same students. This approach for creating concept maps
was adapted for this study, as described in Chapter 5.

CCSSM and The Pythagorean Theorem
The Common Core State Standards for Mathematics are the new National Standards
that most states will soon begin following. These standards have put forth a list of topics
by grade level that will guide what mathematics teachers teach and when (National
Governors Association Center for Best Practices & Council of Chief State School
Officers, 2010). In the eighth grade level of mathematics, the Pythagorean Theorem is

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discussed under the content of geometry. Here the authors of the standards focus on two
aspects of the Pythagorean Theorem: (a) the proof (and explanation) of the Pythagorean
theorem, and (b) applications of the Pythagorean Theorem. The three standards as they
appear in the Common Core State Standards are:
•

Understand and apply the Pythagorean Theorem.
o 8.G.6. Explain a proof of the Pythagorean Theorem and its converse.
o 8.G.7. Apply the Pythagorean Theorem to determine unknown side
lengths in right triangles in real-world and mathematical problems in two
and three dimensions.
o 8.G.8. Apply the Pythagorean Theorem to find the distance between two
points in a coordinate system.

In a brief description of these standards in the introduction section, the authors of the
CCSSM the authors write that in 8th grade:
Students understand the statement of the Pythagorean Theorem and its converse,
and can explain why the Pythagorean Theorem holds, for example, by
decomposing a square in two different ways. They apply the Pythagorean
Theorem to find distances between points on the coordinate plane, to find lengths,
and to analyze polygons. (National Governors Association Center for Best
Practices & Council of Chief State School Officers, 2010, p. 52)
Of the three standards, this dissertation focuses only on the first—the meaning of
the theorem and explanation of why it works. Preservice teachers will have encountered
the Pythagorean Theorem several times previously as students and as teacher preparation
students within the mathematics content courses for elementary teachers. Such familiarity
of content has been shown to play a role in the success of information queries in online
searching (Fidel et al., 1999; Hirsh, 1999). Additionally, this standard contains both

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content that preservice teachers are likely to be familiar with (the Pythagorean Theorem)
as well as content that is likely to be unfamiliar to them (the converse to the theorem),
making it well suited for further investigation.
Two textbooks were analyzed for their content related to Pythagorean Theorem:
the text used in the content course taken by elementary mathematics majors (Beckmann,
2011) and a common middle-school mathematics curriculum (Lappan, Fey, Fitzgerald,
Friel, & Phillips, 2009). This analysis provided a scope of the mathematics content
around the Pythagorean Theorem, as well as comparing the content within the context of
what teachers might be prepared to teach via the content course text to what they might
be required to teach via the middle school text. This analysis revealed little overlap
between the content of these two texts. Much of the mathematics content around the
Pythagorean Theorem and its proof contained in the middle-school curriculum was not
contained in the text used by preservice teachers. This suggests that teachers will need to
find a way to be familiar with this content beyond their preparation programs and beyond
the curriculum materials they use in and for teaching.

Purpose Revisited
As described above, teacher preparation programs cannot prepare future
mathematics teachers for every piece of content they will encounter over their career
given the limited nature of the program. It is thus important to study how preservice
teachers might be learning mathematics outside their coursework, especially considering
that teachers may be asked to teach mathematics that they are unfamiliar with. Because of
the Internet’s presence as a resource for both academic and non-academic purposes, and

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its commonplace presence in everyday life, it is important to understand how preservice
teachers are using this resource to support their learning of mathematics.

Research Questions
1. What are information-seeking strategies that preservice teachers use when
locating mathematical information online?
2. What form and quality of mathematical connections do preservice teachers make
around the Pythagorean Theorem after locating information online?
3. What relationships emerge between the information-seeking strategies employed
during the research task, and the form and quality of the mathematical
connections made by preservice teachers?

The following chapter describes the design and rational of this study including its
participants, activities and tasks. Subsequent chapters describe the methods for analysis,
findings, results, and what was learned in relation to these research questions.

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CHAPTER 3
This dissertation focuses on three aspects of learning mathematics online: (a) the
strategies used when searching online, (b) the kind and quality of mathematical
connections made when seeking this information, and (c) connections between strategies
used and the connections that may be made. For the first of these goals, I asked
participants to go online in different tasks with the goal of improving their understanding
of a particular piece of mathematical content (in this case, the Pythagorean Theorem). To
better understand the mathematical connections that may emerge with these tasks,
participants were asked to create concept maps before, during, and after each of the Web
tasks. After analyzing the first two research questions individually, the third research
question explores relationships between online seeking strategies and the mathematical
connections documented. Additionally, data collected during interviews with the
participants was used to help provide context for patterns that emerged through the
analysis of the concept maps data.
The basic structure of the sequence of activities can be visualized in Figure 1.

Figure 1 – Representation of the Design of the Interview
The majority of this chapter fleshes out the brief overview I presented here. First, a brief
description of the participants is provided. Next, the individual tasks of the study are
described, which include the motivation behind many of the tasks as well as how the raw

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data was cleaned for analysis. Lastly, I end this chapter with a vignette on participant,
describing from her point of view, her experience participating in this study.
Participants
For this study I recruited seven participants from a large Mid-Western University.
All were enrolled in their teacher preparation programs for elementary school. All of the
participants, in addition to becoming certified to teach elementary school, were also
enrolled in a math minor program, in which they take additional math courses. These
participants were in the senior year of their program, and all happened to be female. They
were recruited in one of the math content courses required for their elementary teaching
major.
This particular population was selected because, as a course instructor, I was
familiar with the mathematics content included in that course, which included the content
around the Pythagorean Theorem. This included understanding a statement of the
Pythagorean Theorem, using the theorem in mathematical scenarios, and understanding a
proof of the theorem. However, in their course work, the theorem was presented as a
single statement, not a bi-directional statement with a forward direction and a converse
direction. Additionally, the proof covered in this course was only a proof of the
“forward” direction of the theorem, not the converse. Thus, the study participants had
some working background knowledge of Pythagorean Theorem during their mathematics
coursework, but the content of their coursework did not present the material in a way
similar to the way that content is discussed in the Common Core standards. This
familiarity with the content that they would be investigating has been suggested by

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research on online information seeking as a crucial criteria for success (Fidel et al., 1999;
Hirsh, 1999; Schacter et al., 1998).

Tasks
The preservice teachers engaged in three kinds of activities during the study. The
first of these were two tasks that required searching for mathematical information online
(an open searching task and a closed searching task). This section focuses on describing
these tasks, including what they are, why they were structured in those ways, and how the
data collected was cleaned for later analysis. After this section, a description of the other
two activities that investigated the effects of these tasks (concept maps and interviews)
will be presented.

Web Task I (Open Searching)
The research on information-seeking online has yet to reach a consensus as to
whether participants are more successful with open searching tasks or closed searching
tasks (Bilal, 2001; Jones, 2002; Schacter et al., 1998). As these two different kinds of
searching environments were shown to be successful with students, I designed this study
to include opportunities for both kinds of explorations. Task I was an open searching task
where participants had free reign to explore the Internet and Task II was a website
focused task where participants were directed to explore a limited number of sites.
The preservice teachers in this study first had their attention drawn to standard
8.G.6 of the Common Core State Standards for Mathematics (Explain a proof of the
Pythagorean Theorem and its converse). They were informed that their investigations

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online during these tasks would be focused on their own understanding of the content of
this standard. The first task (Task I) asked students to go online and try to locate
resources that would help them better understand the content of this standard. As the
preservice teachers identified sites, they were asked to describe the specific aspects of the
sites that they found beneficial. During this time, the preservice teachers had the
opportunity to explore any particular pieces of content they might be unsure of.
In order to make sense of each preservice teachers’ searching during these tasks, a
database of their web searching was created. The database listed each website that was
visited by a participant chronologically. Each site was individually numbered, and the
time spent at each site was noted. This database also included descriptions of the content
of each of the sites and the participant behavior when exploring each site (gathered from
video of the explorations). Additionally, participants’ written and verbal comments about
these sites was included in the databases. Lastly, any comments and reflections I had in
watching the participants explore these resources was also recorded in this database. This
database construction was performed for both web searching tasks (open and closed) and
served as the main data source for analysis when attempting to uncover the searching
strategies used by the preservice teachers.
Web Task II (Closed Searching)
In the second task, the preservice teachers were directed to a series of five sites
that each contain different resources related to the Pythagorean Theorem. As participants
explored the sites, they were prompted to study the content and describe how the content
aligned with their thinking around the Pythagorean Theorem and its converse, and how it
pushed their thinking.

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The first site, of which a screenshot is presented in figure 2, is a part of the PBS
television program titled NOVA. A page on this site displays a 3-4-5 right triangle with
squares built off of the legs of sides of length 3 and 4. The applet instructs users to drag
one of these squares to the outline of a square off of the side of the triangle with length 5,
at which point the square snaps into place. When the remaining square tile is dragged to
also fit into the square of side length 5, it is instead broken apart into unit squares, and
applet users can fit the individual squares around the existing square. This displays how
in a 3-4-5 right triangle, the sum of the areas of the squares off of the legs of this specific
right triangle, must be equal in area to the square off the length of the hypotenuse. This
applet has users work with one specific triangle, and they drag discrete blocks between
the different areas, not following a specific pattern just filling in blank spaces.

Figure 2 – Screenshot of PBS Website
The second website is a GMAT review site and a screenshot of this can be viewed
in Figure 3. The GMAT is the Graduate Management Admissions Test and is a test that is
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often used in determining admissions to management or business schools. The site does
not contain and references to the proof of the Pythagorean Theorem. However, it does
feature connections to a wide variety of related topics all in one location. The site features
pictorial, algebraic, and textual descriptions of what the Pythagorean Theorem actually is.
It also lists a couple examples of applying the Pythagorean Theorem. Additionally, the
converse to the Pythagorean Theorem is briefly described and an example is presented.
There are also connections to Pythagorean Triples on this site as well as special right
triangles (45-45-90 and 30-60-90). This site is included as it has a breadth of content and
users of this site have a potential to see many connections they might not have made
otherwise.

Figure 3 – Screenshot of GMAT Review Site

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The third website is a collection of several different aspects related to proving and
illustrating the Pythagorean Theorem, and a screenshot of the site can be viewed in
Figure 4. The site first defines the Pythagorean Theorem. Then it lists an example of the
Pythagorean Theorem with a 3-4-5 right triangle illustrated in a diagram. It presents three
scenarios where one of these sides might be unknown, and it illustrates how to find the
length of the unknown side in each of these instances. Lastly, the final resource on this
page is an applet that demonstrates a proof of the Pythagorean Theorem. The applet is
both an animation and text presented simultaneously. This proof calculates the area of a
single square in two different ways, sets them equal to each other and determines the
Pythagorean Theorem. This applet lets users change the lengths of the starting triangles,
and also redefine the labels used in the proof. This page is included as it positions the 34-5 triangle as an illustration of the Pythagorean Theorem, while also providing a proof
of the Pythagorean Theorem later on the same page.

Figure 4 – Screenshot of ronblond.com Site

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The fourth website is a YouTube video from the popular Khan Academy series of
videos, and a screenshot of this video page can be seen in Figure 5. This video presents a
proof of the Pythagorean Theorem very similar to the proof that was presenting as part of
the math content course took as part of their teacher preparation program. This video is
presented as it features an individual describing the proof to an audience in real time, as
opposed to the previous websites where the information was visually on the page for the
preservice teachers to put together on their own. This video is presented near the end of
all of the websites so there can be a separation between what preservice teachers
encounter from the static websites prior to this video. Additionally, the Khan Academy is
a popular trend in online education research. The inclusion of it in this study is a small
way of beginning to explore the ways that these videos might potentially play a role in
preservice teacher preparation.

Figure 5 – Screenshot of Khan Academy Video
The last website is a second YouTube video produced by the website TutorVista,
and a screenshot of this video can be seen in Figure 6. However, this video (as opposed to

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the other sites presented in this section) is about the converse to the Pythagorean
Theorem, as opposed to the Pythagorean Theorem itself. This video presents a step-bystep proof of the converse to the Pythagorean Theorem. The steps taken are both written
on the screen and a voice narrates the steps of the proof simultaneously.

Figure 6 – Screenshot of TutorVista Site
The following table shows a summary of the five different sites used in Task 2
and a very brief description of their characteristics.
Site

Description
Series of pages with static and dynamic content with an
illustrated example of the theorem and applications
GMAT Review
Single page of static content covering the theorem and
multiple applications, and special cases of the theorem
Ronblond.com
Single page with statement, application, and a dynamic
applet illustrating proof of theorem
YouTube – Khan Academy Video with narration of proof of Pythagorean Theorem
YouTube - TutorVista
Video with narration of proof of converse to the Theorem
Table 1 – Description of Sites Used in Web Task II
PBS

These sites were chosen to incorporate a variety of both dynamic and static presentation
styles. They cover sites produced by tutoring companies, educational programming, and

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independent websites. Some sites were chosen as the content was spread across a series
of pages and some were contained within a single page. As with Web Task I (Open
Searching), the ways that participants interacted with these five different sites was also
described in the same web-searching database.

Investigating the Effect of Tasks
In order to investigate the effects of participating in the above Internet searching
tasks, the preservice teachers in this study participated in two other kinds of activities that
tried to uncover changes in their thinking. The first of these activities was the creation of
concept maps where the preservice teachers were asked to represent how they were
thinking about the Pythagorean Theorem. The second kind of activity was responding to
interview questions, about the Pythagorean Theorem standard they were investigating,
about their participation in this series of tasks, and about their history of informationseeking online. This section tasks each kind of these activities and describes them in
detail.

Concept Mapping (Pre, Middle, and Post)
To better understand the connections that the preservice teachers were making
among the content of the Pythagorean Theorem, participants were asked to create concept
maps representing how they were thinking about that content. Participants were shown
example concept maps of different styles around both mathematical and scientific content
so that they may be familiar with common features of concept maps. The creation of the
concept maps was done before the participants engaged in either searching task, between

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the two tasks, and again after completing both tasks. The task each time utilized the same
prompt and participants were encouraged to reference the example concept maps, or their
prior concept maps if they wanted to.
Preservice teachers were asked to create a concept map about the Pythagorean
Theorem, using a word bank as a starting guide. As previously discussed, the creation of
a concept map from a word bank captures knowledge structures in a similar way to a
multiple choice test (Ruiz-Primo et al., 2001). As the preservice teachers created each
concept map, they were asked to talk aloud about how they were thinking about the map,
why they were making particular connections, and what they were still unsure about. All
concept maps are created and recorded with a LiveScribe™ pen, which records, in real
time, the creation of the concept map and records the audio that occurs during that time
period. The creation of the concept map can then be recreated digitally, in real time with
the corresponding audio played simultaneously.
Concept maps contain both nodes and links between the nodes. The nodes
describe pieces of mathematical content, and the links describe how those pieces of
mathematical content are connected. However, based on piloting attempts with other
participants, it was found that the link descriptions between the nodes of content were
often left off of concept maps by novice concept map creators. In order to have the most
description of how participants were connecting pieces of content in their concept maps,
the raw concept maps that were created by the participants were augmented with their
descriptions of their concept maps. Using the audio recorded by the Livescribe pen, the
text spoken by participants when they created their concept maps was able to be inserted
in their concept maps. This allowed both the oral and written descriptions for the links

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between nodes of content to be analyzed for connections. The below figure (Figure 7)
illustrates what one of these concept maps looks like when augmented with the preservice
teacher’s spoken text.

Figure 7 – Example Augmented Concept Map
Pre Interview
Before they engaged in either searching task, and after engaging in both tasks, the
preservice teachers were asked about standard 8.G.6 - Explain a proof of the Pythagorean
Theorem and its converse. In each iteration participants were asked what they thought
this standard meant, what they didn’t understand about this standard, why they thought
this standard is included in what students need to know about mathematics, and what
their goals, as teachers of this standard would be for their own classrooms.
This section focuses the preservice teachers towards a specific part of the
Pythagorean Theorem, the proof of the theorem and its converse. This is where some of

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the content the participants encountered as part of their math content course provides a
background for their future explorations. This section also begins to ask the participants
to begin to imagine this content in the context of teaching to others. This is important as it
leads to the next grouping of tasks where the preservice teachers are asked to explore
content they might be unsure of as well as locate some resources that might help them
teach that content.
Post Interview
The post interview had three themes of questions running throughout the
interview: questions asking them about the specific standard they investigated during
these Internet searching tasks (the same questions as the pre interview), questions asking
the preservice teachers to reflect on the task, and questions asking the preservice teachers
about their habits online (see Appendix for detailed protocol). The questions about the
activities asked the preservice teachers to reflect on how they viewed their knowledge
having changed (if at all) as part of these activities, and reflect on what was reinforced
that they already were familiar with. There were also questions asking them to reflect on
using the Internet as a tool for exploring mathematics content. The questions asked the
participants to discuss what parts of the online resources were the most helpful to them,
and what parts of the online resources were the least helpful.
The second theme of questions in this section focused on obtaining information
about the preservice teachers relationships with and habits of using the Internet.
Participants were asked to describe their time online, what they do task-wise online, and
to self rate themselves as Internet users. In asking these questions, a picture began to
emerge of how the participants position themselves as Internet searchers. It was also

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possible to get a sense for the types of activities they typically do, whether they are for
academic or recreational purposes. Their skills as an Internet user was important to learn
about as it provided context to help think about their searching strategies as a whole.
This section has described the activities and interviews that were conducted
during the data collection of this dissertation. Each item was described in detail and the
physical prompts can be found in the Appendix of this document. The next section briefly
describes how these items were administered, as well as the role of the interviewer during
these sessions.
Vignette of Data Collection Experience
So far in this section, I have described the participants, the tasks, and activities
that the preservice teachers were asked to complete as part of this study. The seven
participants had similar experiences in these series of tasks, so now I present a composite
vignette of these experiences. This provides a perspective for what the overall experience
was like for the participants as they went through these activities and completed the
online searching tasks.
In a content course for elementary mathematics majors, I recruited participants to
participate in my dissertation study about learning mathematics online. The interested
preservice teachers signed up on a sign-up sheet and eventually received an email if they
were selected to participate in the study. Each participant was asked to come to an office
in the university mathematics building at a specific time and asked to allot up to two
hours for this study, of which they would be compensated with a gift card.
Upon arrival to the office where the data collection would take place, I greeted the
preservice teacher and engaged in some brief small talk about their day and their

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semester. The session began with going over the consent form with the participant. It was
at this point that the participant also received the compensation for their participation.
Then, the recording devices were turned on and the various activities of the session
began.
I told the preservice teacher that the purpose of the study was to better understand
how preservice teachers learn math online, and that the activities of the day would be
focused on the Pythagorean Theorem. At this point I asked the participant to create a
concept map for how they were thinking about content related to the Pythagorean
Theorem. The preservice teacher was shown examples of concept maps, and given a task
sheet (see Appendix) that listed terms that they may wish to include in their map, but not
required to. They were encouraged to talk aloud as they were creating their maps, and
also to describe their map after they were done. This activity was done with a
LiveScribe™ pen in a special notebook.
After completing the concept mapping, the participant was presented with the
three standards from the Common Core State Standards around the Pythagorean Theorem
in the eighth grade. Their attention was focused on the first standard, 8.G.6 – Explain a
Proof of the Pythagorean Theorem and its converse. The participant was asked a few
questions about their understanding of this standard, questions they had about it, why it
may be important, and what their goals would be for their own classroom students.
Once this pre-interview was done, the participant was introduced to their first
task, the open searching task. The preservice teacher was told that they were to think
about their own understanding of the standard, and they were instructed to try to locate
resources on the web that would help them better understand the content of that standard.

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The participant was provided with a sheet to note any resources that appeared especially
beneficial where they could also list reasons why it seemed helpful. I turned on a screen
and video recorder and gave the preservice teacher the laptop, showing that they would
be using the web browser Firefox, and briefly showing how to use the URL and search
bars. As the preservice teacher searched online, I sat off to the side, watching what they
were doing. The preservice teacher was given no more than 20 minutes for this activity,
though some took less time than that.
After completing the first web task, the participant was again asked to create a
concept map reflecting how they were thinking about the Pythagorean Theorem and its
related content. The participant was allowed and encouraged to look at their previous
concept map, and any notes that they took during the open searching task. Again, the
preservice teacher was encouraged to think aloud during this activity and to describe the
design of their map after completing it. This was again done with the LiveScribe™
recording pen.
The next activity for the participant was the second Internet task, the closed
searching task. The participant was told that the researcher had picked five websites that
he wanted them to explore. They were encouraged to explore these sites in whatever way
they normally would. The preservice teacher was told to focus on the content of the sites,
to see what aligned with what they were thinking and what pushed on their thinking
about explaining a proof of the Pythagorean Theorem and its converse. Again, they were
given a blank sheet so that they could describe the content they were exploring. I again
turned on a screen and video recorder, and directed the preservice teacher to a document
on the computer that had the five links for them to explore. The participant clicked on the

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links, one at a time, and explored each of the sites. After they were done exploring each
site, I asked the preservice teacher what they noticed about the site they just explored.
Following the completion of the second task, the participant was again directed to
construct a concept map from scratch. Again, like the second time the preservice teacher
constructed a concept map, they were encouraged to look at their prior iterations, and any
notes they may have taken. Constructed with the LiveScribe™ pen, they again were
encouraged to narrate their thinking, and describe their construction.
The preservice teacher then was told that the following two activities would just
be interview questions and that they were done with the Internet and constructing concept
maps. The first set of questions that were asked of the preservice teacher was the same set
of questions that were asked before the first task, focusing on the standard 8.G.6, and
how the preservice teacher was thinking about that standard. Following that series of
questions, the participant was asked by the interviewer about the activities they just
completed. They were asked about the ways that they thought these tasks impacted their
thinking, and the ways in which the resources they located also had an effect on their
thinking. Lastly, the participant was asked a few questions about their history of using the
web (and search engines) for both academic and personal purposes. The participant lastly
had the opportunity to ask the researcher any questions that they might have. After this,
the recording devices were turned off, and I thanked the participants for their time and
their participation in this study.

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Summary
This chapter described the participants of this study, the activities that were used
with the participants of the study, how I cleaned the raw collected data for later analysis,
and a composite vignette of what the entire data collection experience was like for each
participant, so as to provide a coherent image of what all of these experiences looked like
together. The next three chapters each focus on the three research questions of this study.
Each chapter will discuss the frames used to analyze the data, along with examples from
the data. These chapters will also present summaries of the analysis performed on the
data with the frameworks. The results of these analyses are connected back to the
research questions, examining what answers to the questions emerge from the data and
what still remains unanswered.

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CHAPTER 4
The recruited preservice teachers completed two Internet-searching tasks as part
of this study. Before, during, and after these tasks, participants also completed activities
that helped illuminate how their understanding of mathematical connections may have
been changing. This chapter focuses on the main activities in these sessions, the Internetsearching activities, while the next chapter unpacks what impact these activities had.
These Internet-searching activities attempted to capture two different methods used by
these participants in order to locate information online. These methods are similar to
methods used to locate information in the real world. Consider the following analogy:
When going to the neighborhood library to find information, two different
processes occur before obtaining that information. Upon entering the library, the person
is confronted with a myriad of books and other resources. The person will need to find a
way to narrow in on a specific book or resource that they will want to look at more
closely for the desired information. Secondly, once a resource has been selected, the
person will then need to examine the resource and attempt to locate the relevant and
desired information within the resource they chose.
This process is similar to what users of the Internet experience when they try to
locate information online. First, upon opening the web browser (entering the library), the
user has access to numerous different resources, although many of these resources may
be unfamiliar to the user. The user is then confronted with how to select one of the
resources. If the user has some experience, they might know some resources they have
used in the past and go directly to these previously used sites. Otherwise, they need a way
to identify the specific resources that are located within the web. In the physical analogy

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of the library, this was traditionally done with a card catalogue. In the digital realm of the
Internet, unfamiliar resources are typically located via a search engine.
After locating a specific resource, there are many ways that people can find
information within that resource. Users of traditional books might look to a table of
contents or glossary to narrow in on information, or they might skim through the pages to
try and quickly identify the structure and scope of the resource. This is similar to how
users of online resources narrow in on information. Some look for organizing information
near the start of web resources, some read lines of text, and some skim over sites to get an
overall sense to the information. The ways that we search for and read for information in
the digital realm is not separate from the ways we have carried out these similar tasks
with more traditional resources, though there are important differences.
This section explores the range of different information seeking strategies that
were used by this sample of preservice teachers as they tried to learn more information
about one standard from the Common Core State Standards for Mathematics. By looking
at the approaches used, both in attempting to locate resources, and in locating information
on those resources, we can begin to identify more and less powerful strategies of
searching. Ultimately, the goal of this chapter is to help us answer the question, what are
information-seeking strategies that preservice teachers use when locating mathematical
information online?

Chapter Structure
The Internet activities that the preservice teachers performed were divided into
two parts. Following this form, this chapter will also discuss the two activities in turn.

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Before presenting the activities and their results, the first part of this chapter will discuss
the frames of Internet searching that were used in the analysis of these two activities. In
the next part of the chapter, the first task will be discussed where participants were free to
use the Internet in whatever way they wanted to locate information they found beneficial.
The third part of this chapter will focus on the second task given to the preservice
teachers, in which all participants explored the same five resources (three sites and two
videos). The final part of this chapter pulls together what was found across the different
activities and the information seeking strategies used by the preservice teachers.

Frameworks of Information-Seeking
Within the research on Internet navigation strategies, two different kinds of
classifications exist within the literature. One kind focuses on the navigational patterns of
Internet searchers. That is, they attempt to describe the ways in which a user goes from
page to page, as they look to obtain information. A second kind of classification attempts
to describe the intent or motivation or focus of the Internet searchers. These focuses on
describing the kinds of sites that users spend time investigating, and looks to classify
their navigation by the pattern of behavior while on the site. The first kind of
classification scheme of navigational patterns of navigating between sites is more directly
observable than the second as it is easier to see changes in navigation between different
pages, as compared to potentially just eye movement on a single page. However, both are
observable to an extent and both provide ways of answering questions about the
information seeking strategies employed by Internet searchers.

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To help classify the navigation patterns of the preservice teachers between
different sites , this study draws upon a classification system that emerged after
researchers analyzed Internet search history of users in a study and categories of behavior
of the participants were identified. These categories included the timid navigators, the
laborious navigators, and the divergent navigators (Juvina & Oostendorp, 2004). In each
category below, I summarize the descriptions of these categories from the research,
describe how I adapted their descriptions for the analysis of the navigation patterns in my
study, and provide an example of what each behavior looks like within the searching
data.

•

Timid navigator
o This group in the original study is described as visiting very few pages and
not venturing far from the homepage.
o For my study, as there was no homepage, I looked at how far my
participants went from their search results page, and if they often only
clicked one link to take them off of the search results page and then
returning back to the results page without exploring their site further.

•

Laborious navigators
o Participants in the original study were said to belong to this group if they
had a high amount of uses of the back button and page revisits.
o For my study, I counted only attempts at revisiting and returning to sites
that contained information they were studying. I did not count returning to
a search result page as this was not revisiting subject matter information

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they had encountered before, but just accessing the results of a previous
search. There actually turned out to be no participants in my study that
exhibited this particular type of navigation.
•

Divergent navigators
o Participants in the original study were said to belong to this group if they
had among the highest number of visits to unique pages and had little or
no revisitations to pages already explored.
o For my study, a participant was said to visit a high number of unique
pages if they visited more than 7 and if almost all of their revisitation was
contained to search result pages.

A different framework was employed to study the information-seeking strategies
employed by the participants while they were searching for information within a single
site. In their handbook chapter of research on new literacies, Lawless and Schrader
(2008) review and summarize research on navigating complex digital environments. In
reviewing research on patterns of navigation behavior, they found several studies
(Horney & Anderson-Inman, 1994; Lawless & Kulikowich, 1996; MacGregor, 1999)
that, although disconnected from each other, came to similar conclusions about general
searching patterns. They found that all the students had the same basic three groups of
navigators, though each had their own names for these groups. These were

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•

Focus on Comprehension - those that had a focus on comprehending the
information they were encountering (called “studier profile”, “knowledge
seekers” and “concept connectors”, respectively),

•

Focus on Static Content - to force a linear structure onto a non-linear environment
(called “book-lover profile”, “apathetic hypertext user” and “sequential studier”,
respectively).

•

Focus on Dynamic Content - those that played with special features embedded in
the resources they were using (called “resource junkies”, “feature explorers” and
“video viewers”, respectively),

There is one final note that cuts across both of the categorizations presented here.
The studies included do not attempt to label any individual as existing wholly within a
single category of any categorization. They note that individuals would often use
different approaches to searching at different times during their navigation. Because of
this, these studies try to describe all the different kinds of navigational patterns that exist
within any particular search, and indicate all the different patterns exhibited by an
individual throughout their navigational session.
With this in mind, I analyzed the navigation of all seven of my participants and
noted which of these strategies participants were using at some point during their
completion of the first activity. Again, participants could be using different strategies at
different points in time. After focusing on the distinction described by this study, I then
turned to the categorization found across all the studies described originally

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(comprehending information, playing with special features, and forcing a linear structure
onto a non-linear environment). The results of this analysis are presented below.

Timid
Navigator
Laborious
Navigator
Divergent
Navigator
Focus on
Details
Focus on
Static
Content
Focus on
Dynamic
Content

Part. 1
X

Part. 2
X

Part. 3
X

X

Part. 4
X

Part. 5
X

Part. 6
X

Part. 7
X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X
X

X

X

X

Table 2 – Summary of Information-Seeking Coding of Participants
Task I – Open Searching
The section below will look at each of the categories described above in the
context of the first Internet searching task. Each section will first examine what a
prototypical example looks like for each category, using examples from the participants
wherever possible. Then the category will be examined across all participants to identify
what, if any, patterns existed among the participants within each individual category. The
following section will then zoom out to look at patterns across all the categories.
The first three groupings of categories focus on how participants move from site
to site, looking at the transitions among web searches, the results of the search, and the
resulting pages that were clicked on for examination. The different patterns of movement
between these various types of pages are difficult to describe via a presentation of the

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actual data. However, these navigational patterns can be described looking the diagram
below, adapted from (Hölscher & Strube, 2000, p. 340). In Figure 8, the common
practices of performing and using an Internet search engine are illustrated, including
launching a search engine, formulating a query, submitting a query and obtaining results,
examining the page of results, selecting and examining a document from the results, and
browsing among other pages. There exist numerous arrows (of which only a few are
listed) that illustrate the flow of a user as they go from using a search engine to
examining results, and back and forth. A few of these included arrows are labeled and the
below sections describe what patterns of navigation utilize these arrows for each of the
categories.

Select / Launch Search Engine
Generate + Select Search Terms
Formulate Query
Submit Query / Get Results
Examine Page of Results
(1)
Select Document from Results

(3)
(2)
Browsing

Examine Document
(4)

(5)

Figure 8 – Diagram of Common Aspects of the Internet Searching Process
Timid Navigator
A timid navigator was one who did not stray far from their search results page.
Here they would perform a search, click on a result to examine a single page, and then
return to the search results page to either click on an additional result or to perform a new
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search. In the figure presented above, a timid search would have the majority of their
navigation described by arrows one (selecting a document from a page of results), two
(going back to the search results page after examining a single document), and three
(getting results from a newly submitted query after examining a document).
One excerpt of this kind of searching behavior could be located within the open
searching of Participant 7. At the start of her searching, she obtains the results from a
page, clicks on one result to examine the site (arrow one), returns to the same search
result to locate additional results (arrow two), before clicking on another result to
examine (arrow one), and finally returning to submit a new query (arrow three). This
same pattern occurs at several other points in her data as well, even when she transitions
to looking at video resources.
Across all participants, this category was the most common, with all the
participants exhibiting a timid navigator pattern at some point during their entire open
searching task. Despite each search result page having at least ten results listed, the most
common activity after exploring a result was for participants to go back and perform a
new query to obtain results (arrow three), than is was to click an additional result from
the same listing of results (arrow two). There were only rare instances that participants
used the results that they clicked on as launching points to then browse further links off
of those pages they already explored. Most stayed very close to their search results page
and the most common behavior across all the participants was to only explore the direct
result of a search.

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Laborious Navigator
The laborious navigators were those who revisited sites they had already
explored, possibly as reminders of content, or to contrast with new content that was
encountered. Again, as a reminder, this category includes only revisits to sites or results
clicked off of a results page. This does not include revisiting the search result page itself.
Distinguishing whether they return to a page of search results or a page of content
emphasizes what makes a laborious navigator different from a timid navigator. In the
figure presented above, this would be characterized by repeated or frequent navigation
along arrow four, where after content in a document was examined, it was then
reexamined, at some point in time, after its original examination.
Again, this category of navigation did not appear among the navigational patterns
of any participant at any point within their open-searching activity. There are several
contributing factors that may have influenced this pattern of behavior. It may that after
clicking on a link from the search result page, the color of the link changed, signaling to
the participant that they already explored a given resource. Additionally, it may be that
participants knew they had a finite amount of time to complete the task and would rather
explore new content with that time than revisit content they already encountered.
Additionally, it could also be that participants felt they completely understood and
remembered all of the content of a site and felt it unnecessary and redundant to revisit
any site they already explored. All of these reasons, and others, may have contributed to
why there was a lack of the laborious navigator strategy exhibited by the participants in
this study.

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Divergent Navigator
The pattern of divergent navigator consisted of moments where a participant used
the links within one of the pages they had clicked on from the search results, to then go
and further explore new sites that may or may not have been included in the search
results page. These additional sites are explored after having clicked on a link contained
within a page, and not having been clicked from a listing of results from a search. In the
diagram described above, this pattern of action would be described by arrow five, where
after exploring the results of a page, the participant then transitions to a pattern of
browsing, exploring additional sites that are linked off of that document. This is different
as compared to those who go back to the search results page for additional sites to
explore or a new query (timid navigator), or those who revisit the same page for
additional exposure to the same content (laborious navigator).
Within the data collected as part of this study, an instance of the divergent
navigation pattern can be identified with participant 1. At least twice during her open
searching task, she exhibits the behavior of a divergent navigator. The first instance of
this occurs when she clicks on a PBS site from a search results page, which then causes
her to explore additional links off of that original PBS site, clicking on two additional
pages, before backing back out to the results of the search. Later, in a different search,
participant 1 clicked on a Common Core test prep site and browsed on several different
subpages off of the original page numbering a total of seven pages in all within that
Common Core test prep site. This are both illustrations of arrow five, where after
examining a site, a participant then goes to browse further information based on links off
of that site, and not based off of the search results.

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This pattern of navigation was not commonly observed among the patterns of
navigation exhibited by the participants. Only two out of the seven participants exhibited
the divergent navigator pattern at some point in time during their open searching activity.
Sometimes the additional pages explored were for pages located within the original site
clicked (as was the case for participant 1), and sometimes they were for pages outside of
the site that was clicked (as was the case for participant 4). It may be the case that these
sites just did not have a lot of outside links contained within each page, or at least not a
lot of links that the participants identified as having relevant, useful information given
their task at hand. It could be that the participants felt that the page that they were
directed to in a site from the search results would be the one page in the site that
contained the most relevant information. Therefore, other pages within that site would be
unlikely to contain more relevant information. This may explain why the majority of the
strategies employed by the participants were the timid navigator strategy where
additional, pertinent information is located by going back to explore other pages among
their results or by performing a new query search, rather than continuing to explore
within the site, or off of the site, as with the divergent navigator.
The next section moves away from looking at how participants were navigating
from page to page, and site to site in utilizing the results they were obtaining from their
search results. Instead, the focus is on the describing how the participants were
interacting with the content and information on the individual sites that they were
visiting.

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Focus on Detail
The focus on detail categorization describes the attention that was being paid to
the content by the participants when they were searching for information on the different
pages that they visited. Here, it is examined whether, at any point during their
explorations, participants spent sustained, focused attention to the content available on
the page they were examining. This is, admittedly, a bit more inferential than some of the
categories. Participants’ eye and head movements were examined to determine whether
they were reading text, examining a picture, or intently watching a video.
Within the data of this study, an instance of the focus on detail can be found with
participant 4. In her searching, she ended up clicking on the Wikipedia.com site entry that
discussed the Pythagorean Theorem, after scrolling and doing some brief exploration of
the page, she came to the section focusing on algebraic proofs of the Pythagorean
Theorem. It was here that the participant spent an extended amount of time, reading (lineby-line) the algebraic proof and comparing the written proof with the included
corresponding diagram to the right of the text. In this section, you can see her eyes,
slowly scrolling across the part of the screen where the text is located, they jump of her
eyes from the text to the diagram and back, and also the movement of the mouse along
the words of the proof that are included on the site. This held her focused attention for
over one minute and twenty-one seconds.
Every participant as some point in their searching exhibited this focus on detail.
For many of the participants, it occurred at several points during their explorations of the
different pages and sites with content. It should also be noted that all of the participants
also had moments where they were obviously not focusing on details. In these moments,

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participants skimmed quickly over portions of pages, or entire pages as well. Here the
participants seemed to be attempting to get a broad understanding of the kinds of
information located within a part of individual pages. Based on these skimming moments,
participants would then later go on to either read more closely, or select different sections
or pages to explore.

Focus on Static Content and Focus on Dynamic Content
The final categories of this analysis are described together here as they refer to
very similar phenomena. The sites that the participants explored contained two formats of
content presentation, static and dynamic. The static content does not change in time while
a page is being viewed. This would include things like text and images. Dynamic content
does change in time. This would include things like interactive web applets, embedded
videos, and animated .gif images. The total amount of time that was spent exploring the
content sites was calculated for each participant, noting how much time was spent with
static content and how much time was spent on dynamic content. If the amount of time of
time spent with static content was over 30% of their total time spent with all content, they
were said to have exhibited some focus on static content. Similarly, they were said to
have exhibited a focus on dynamic content if they spent over 30% of their total time with
dynamic content. The results of this analysis are summarized below (Table 3).
For the focus on static content, this is perhaps best illustrated by participant 5,
who spent all of their time exploring content solely within sites that utilized a static
presentation of the content. This participant only focused on content that was textual, or
that also included diagrams of the content as well. This is contrasted with someone like

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participant 3 who was categorized as focusing on dynamic content. For this participant,
88.4% of the time she was looking at content, it was dynamic content. For participant 3,
the majority of this was online videos about the Pythagorean Theorem, it’s proofs,
examples, and applications.
Static Time
Dynamic Time
Total Time
Participant 1
08:04 (89.5%)
00:57 (10.5%)
09:01
Participant 2
08:08 (45.4%)
09:48 (54.6%)
17:56
Participant 3
02:30 (11.6%)
19:08 (88.4%)
21:38
Participant 4
11:39 (55.1%)
09:29 (44.9%)
21:08
Participant 5
04:34 (100%)
00:00 (00.0%)
04:34
Participant 6
05:36 (45.5%)
06:43 (54.5%)
12:19
Participant 7
04:01 (39.1%)
06:15 (60.1%)
10:16
Table 3 – Time (and Percentage) Spent with the Format of Content Presentation
Overall, there tended to be two types of groups within the data, those who spent
an almost split amount of time exploring both static content and dynamic content (and
hence, those categorized as both “focus on static content” and “focus on dynamic
content”) and those who spent the majority of their time with one format of the content as
opposed to the other (and hence, only was categorized as either “focus on static content”
or “focus on dynamic content). Participants 2, 4, and 6 were the most evenly split with
the formats of the content that they focused on (around a 45%-55% split for one or the
other format). Participants 1, 3, and 5 were heavily focused on either the static content
(89.5% & 100% for participants 1 and 5 respectively) or the dynamic content (88.4% for
participant 3). Participant 7 was somewhere in between these two categories, showing
more of a focus toward the dynamic presentation of content (60.1% of her time), but still
spending enough of her time with static content (39.1% of her time), to still be
categorized as both. It was interesting that in the interviews with students after the
completion of all tasks, most all professed to having strong opinions about whether or not

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they liked to learn mathematics from web videos. Some of those preferences can be seen
in the summary presented here.
This section looked at the different ways of describing how participants were
searching for information, whether it be in using web searches to locate resources (such
as how one would locate books within a library), or in describing the ways in which they
searched for information within individual sites (such as how one would locate
information within an individual book). The next section looks across these individual
codes to look for patterns across the codes.

Patterns Across Task 1
All of the participants exhibited behavior that would label them as “timid
navigators”, that is, they mostly only clicked on one link off of a search result page,
explored that single page without clicking any additional links, and then returned to the
search result page. Only two participants exhibited behavior at any time that would be
considered “divergent navigator”. Looking at how these participants searched for
information within individual sites, it’s noted that every participant at some point had a
“focus on content”, devoting an extended period of time to particular content on a
specific page.
However, there was a notable distinction that emerged when looking at the kinds
of resources the participants utilized during their searching. For some of the participants
(Participant 1, 3, and 5), tended to focus only on one type of content, either static, or
dynamic, but rarely spent time looking at both. For others participants (Participants 2, 4,
and 6), they exhibited both a focus on static content and focus on dynamic content,

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meaning that the participants spent an almost equal amount of time looking at both static
and dynamic resources. Only one participant (Participant 7) had a somewhat split focus,
though looking at her times, it is clear she still spent more significant time with one type
over the other.
Across all of the participants, two commonalities occurred among all the
participants. In general, they were all categorized as focusing on the content they were
encountering. As all the participants were focusing on completing a specific task
designed to push their understanding of familiar content, this finding is not particularly
surprising. Also across the participants, I notice that all were categorized as being
categorized at some point as “timid navigators”. This indicates that one of the most
common traits of participants, no matter the group, was to click on a search result,
explore the single page, and then return to the search results to either click a new result or
perform another search. There are numerous different reasons that this pattern may be
prevalent, from the structure of the websites (mostly self-contained) to the design of the
task (clearly identified information was focused upon) to the searching preferences of the
participants. This uniformity across participants here does not appear to explain the
differences in groupings that appeared in the categorization of the math content.
This section focused on the information seeking behaviors of the participants as
they completed activity 1, the open searching task. The next section looks at the behavior
of the participants in activity 2, the closed searching task.

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Task 2 – Closed Searching
After their open explorations, and the creation of their second concept maps,
participants were all asked to explore the same five sites, again with the focus of helping
themselves better understand standard 8.G.6., Explain a proof of the Pythagorean
Theorem and its converse. This section separates each of the five sites and looks at how
participants engaged with the sites separately. At the end of the section, an overall
summary of this part of the activity is presented. Throughout this section, the coding
scheme described above of how users focused their attention on individual pages has
been used.
The nature of this second task directed students to examine specific sites. Only
two of the sites (Site 1 and Site 3) contained opportunities to engage with both static
content and dynamic content within the same site. Therefore, these are the only two sites
that are analyzed with respect to the dichotomy of static vs. dynamic. Also, because this
task directed students to examine these sites, all the participants were coded as “focus on
details” for the first three sites. A different proxy for “focus on details” was created for
the two videos included in this activity, and that scheme is discussed in the description of
those sections.
Site 1– PBS (Nova)
This resource is one of two resources is this section whose authors are somewhat
known in education. In this case, this site is hosted by the Public Broadcasting Service
(PBS) and specifically, their NOVA television program. This site presents a brief,
elementary explanation of what the Pythagorean Theorem is, using general language and
phrases such as “this side”. Links off of this main page allows users to find more

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information about who Pythagoras was, see either an interactive, or static visual example
of the Pythagorean Theorem relationships holding, and they can also look at several
mathematical scenarios that require the Pythagorean Relationships to solve them. This
site does not contain information on a general proof of the Pythagorean Theorem, nor
does it contain information about the converse to the Theorem.
The below chart shows how much time the participants focused on the static
content, and the dynamic content within this overall site.
Static Time
Dynamic Time
Total Time
Participant 1*
0:42+0:39 (100%)
00:00 (0.00%)
01:21
Participant 2
00:49 (94.2%)
00:03 (05.8%)
00:52
Participant 3
01:29 (69.0%)
00:40 (31.0%)
02:09
Participant 4
01:23 (100%)
00:00 (0.00%)
01:23
Participant 5
00:48 (100%)
00:00 (0.00%)
00:48
Participant 6
02:03 (65.4%)
01:05 (34.6%)
03:08
Participant 7
01:44 (100%)
00:00 (0.00%)
01:44
* This participant originally came across this site in their open exploration. Their time spent in the original
exploration has been added to the time spent in the secondary exploration.

Table 4 – Time Spent on Static and Dynamic Content in Site 1
Here we can use the coding scheme from the above section looking at how participants
read for content in individual sites. Every participant would be coded as “focus on static
content” as every participant had at least 30% (actually at least 65.4%) of her time
focusing on the static content contained throughout the PBS site. Only two participants
engaged for any significant amount of time with the dynamic content (Participant 3 and
participant 6) of this site and thus these two would be coded as “focus on dynamic
content” as well. It should be noted that the interactive content was only available on one
of the pages within this site. Some participants never clicked on the particular page and,
thus never had the opportunity to engage with the dynamic content. Others made it to the
web applet, but never clicked on the applet to interact with it.

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This site has at least thirteen unique pages directly related to the content of the
Pythagorean Theorem immediately linked among each other, and has access to numerous
other pages that get further from the mathematical content, and closer to other pages of
the television program, and network. However, the majority of the participants only
explored less four or fewer of these sites. The only exceptions were participant 6 (who
explored eight of the sites) and participant 7 (who explored ten of the sites). Additionally,
there were two comments that stood out from those that explored this section. The first
was from participant 1 who commented on the authority of the site, recognizing it (PBS)
and exclaiming that she thought that a site would have more information and links on it,
but that she knew it was a “trusted site”. She was the only participant to comment on the
reliability of this site.
Site 2 – GMAT Prep Review
The company “Platinum GMAT”, which specializes in preparation for the GMAT
test, compiles this resource. All the information for the Pythagorean Theorem is
contained on this single page, and organized in a table of contents at the beginning of the
page. The site begins with a statement of the theorem, and examples of the theorem
working. Then a statement to the converse of the Pythagorean Theorem is also presented
with an example. The site continues to include sections on Pythagorean Triples, Special
Right Triangles (30º-60º-90º, and 45º-45º-90º), and then a final sample problem from the
GMAT that requires the Pythagorean Theorem to solve.
The chart below shows just the total time the participants focused on the content
of this site (which was all of a static format).

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Time On Site
Participant 1
00:43
Participant 2
03:44
Participant 3
01:37
Participant 4
04:07
Participant 5
02:07
Participant 6
02:56
Participant 7
02:04
Table 5 – Total Time Spent on Site 2
Using the above rubric for coding how participants were engaging with content would
result in every participant being coded as “focus on details” (as all participants were
instructed to examine the site) and “focus on static content” (as there was no dynamic
content located on this site. The interesting pattern to note focuses on the total time spent
on this site. Participants 1, 3, 5, and 7 spent around two minutes or less looking at this
site. Participants 2, 4, and 6 spent around three minutes or more looking at this site. This
distinction falls along very similar lines to the mathematical grouping distinction
presented in the previous chapter, and the search strategy grouping distinction presented
earlier in this chapter.
Site 3 – RonBlond
This website begins with a statements of the Pythagorean Theorem and utilizes a
color-coded diagram illustrating the different sides and their relationships within the
formula. After, a diagram of a 3-4-5 right triangle is presented with squares drawn off of
each side, continuing the color-coded theme. Below this example are three instances of
the formula being used to find the dimension of each of the three sides, if the lengths of
the other two sides are given. This site also includes a web applet that contains
instructions, a visual display area, and a bottom section containing descriptions of the
visual display area. In this applet, users can pick a side of a right triangle to make the

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“unknown” side, rename sides and angles, show areas of squares off the lengths of the
sides, and view a dynamic illustration of a proof that will appear step by step. Directly
before this applet is a link to a page with static images that presents a sequential
presentation of this same proof.
The below table shows how much time the participants focused on the static
content, and the dynamic content within this overall site.
Participant 1
Participant 2
Participant 3
Participant 4
Participant 5
Participant 6
Participant 7

Static Time
Dynamic Time
Total Time
00:25 (39.7%)
00:38 (60.3%)
01:03
00:33 (34.3%)
01:03 (65.6%)
01:36
00:56 (56.0%)
00:44 (44.0%)
01:40
00:46 (21.3%)
02:50 (78.7%)
03:36
01:28 (55.7%)
01:10 (44.3%)
02:38
00:55 (45.8%)
01:05 (54.2%)
02:00
00:29 (19.7%)
01:58 (80.3%)
02:27
Table 6 – Time Spent on Static and Dynamic Content in Site 3

As opposed to the first website explored by students which had the dynamic content
embedded in one of its many pages, this website had almost its entire main content on the
main page, including the web applet at the bottom. While most of the static information
at the top of the page was basic, the applet at the bottom was more involved and was
obvious as the focal point of the page. Coupled with the fact that the applet contained
several different tools for the participants to explore, all the participants spent a
significant portion of their time with the web applet, and thus all the participants would
be coded as “focus on dynamic content” for this site. What was also unique was that
many participants divided their time with the static part of this content as well. Only two
participants did not also receive a code of “focus on static content”, participant 4 and
participant 7.

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Site 4 – YouTube Video Khan Academy Proof of PT
The first of two YouTube videos in this collection, this YouTube video is
authored as part of the series of math videos put out by Khan Academy. This video runs a
total of 8 minutes and 49 seconds. The beginning of the video presents a review of
terminology relevant to right triangles. Then a proof of the Pythagorean Theorem is
presented, similar to proofs presented in other resources. This video also contains a
description/explanation of why a particular shape in the construction must be a square
(have right angles), which is often omitted in other proofs. The video is narrated in real
time as the drawings and constructions are produced with different color descriptions.
The table below lists the total time spent watching the 8:49 second video and also
displays what percentage of the total video was watched.
Participant 1
Participant 2
Participant 3*
Participant 4
Participant 5
Participant 6
Participant 7

Time Watching Video
01:36 (18.1%)
05:02 (57.1%)
07:37 (86.4%)
05:41 (64.5%)
03:52 (43.9%)
05:05 (57.7%)
03:52 (43.9%)

*This participant originally came across this video in their open exploration. Their time spent in the
original exploration has been listed here to capture the time spent in every participants’ initial viewing
Table 7 – Time Spent Watching Video in Site 4
As noted for the first three sites that students were directed to as part of this activity,
using the coding of “focus on details” presents a challenge as all students paid some
attention to parts of these sites and would have all earned that code. However, for the two
YouTube videos contained in this study, the only mathematical content in the site is
contained within the videos and so we can look at how much focus was paid to the videos
by looking at the total time students spent watching the videos.

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In the prompt, students were told to interact with the sites however they normally
would, which included skipping backwards and forwards in videos if that’s what they
would do on their own. Every student took liberties to skip ahead for this site with no one
watching the entire video. Three of the participants ended up watching less than half of
the video (participant 1, 5, and 7). The remaining four participants watched over half of
the video, of which participant 3 watched almost the entire video, while the remaining
participants (participant 2, 4, and 6) all grouped closer together, centering near the 60%
range. It should be noted here that participant 3 originally came across this video during
their exploration during task 1 and so the breakdown of time watching this video was
taking from that initial viewing of the video. As such, watching this video under the
constraints of a different task may have affected how much time she spent watching this
video. It may be the case that she would have spent less time watching this video if her
initial viewing came during the second activity, further into her total interview session.
Though not as dramatic as the other activities, it is possible to note that a similar division
appears between those who spent less time watching the video and those who spent more
time watching the videos (with participant 3 as an outlier to this pattern for this video).
Participants 2, 4, and 6 are once again behaving very similarly.
Site 5 – YouTube Video Proof of Converse
This site is the second of two YouTube videos, this one produced by the company
TutorVista. This video also almost always showed an ad before the video that was able to
be skipped after 5 seconds. This video contains an slideshow like presentation of a
rigorous mathematical proof of the converse to the Pythagorean Theorem. Totaling 2
minutes and 22 seconds, this video is narrated by a woman often using the exact words

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and phrases displayed in the proof. First a statement of the converse is presented, and
then a rigorous proof is presented, with animated diagrams being utilized at each step of
the proof. The proof utilizes the Side-Side-Side triangle congruence test to demonstrate
how a triangle with side lengths that fall into a Pythagorean relationship, must in fact be a
right triangle.
The table below lists the total time spent watching the 2:22 second video and also
displays what percentage of the total video was watched.
Time Watching Video
Participant 1 01:45 (73.9%)
Participant 2 02:22 (100%)
Participant 3 01:49 (76.8%)
Participant 4 04:29 (189%)
Participant 5 01:26 (60.6%)
Participant 6 02:22 (100%)
Participant 7 02:22 (100%)
Table 8 – Time Spent Watching Video in Site 5
As was the case in the last site, we can use the amount of time spent watching this video
as a proxy for how focused the participants were on content. Two differences exist
between this YouTube video and the prior YouTube video. First, this video is much
shorter (almost ¼ the length of time) and so participants did not have as much
opportunities to skip. Secondly, the content of this video was less familiar to the
participants in this study, as the content in the previous video was of a proof, nearly
identical to the proof that was covered as part of their content course.
Within the time spent watching this video, two groups emerge: those that watched
the entire video, and those that skipped parts of the video. Participants 2, 4, 6, and 7
watched the entire video, while participants 1, 3, and 5 watched less than the entire video.
This grouping is again very similar to those that have been noticed at other points in both

57"

the math and the searching strategy analysis. As has happened with other analyses,
participant 7 sometimes aligns more closely with participants 1, 3, and 5, and sometimes
aligns more closely with participants 2 4, and 6. In this case, she happens to align most
closely with the later group.
One unique occurrence happened during the watching of this video that seems
important to note. Participant 4 not only watched the entire video, but came very close to
watching the video twice. After letting the video play through once from start to finish,
this participant skipped backward in the video to rewatch the video almost from the
beginning. While we may be tempted to say that those who watched all of the video had a
greater “focus on details” than those who skipped parts of the video, it should be noted
that the effort put forth by participant 4 to watch the video a second time most certainly
shows a focus to the detail of the content.

Patterns Across Task 2
In this second activity, participants were directed to a series of five sites that
contained both static and dynamic content (though not necessarily both within a single
site). Each individual site was analyzed for how the participants were focusing on the
information contained within the sites. With the sites where both dynamic and static
content was presented, the attention was centered around whether the attention of the
participant was focused on the dynamic content, the static content, or both. For the
videos that were included in the list, the focus shifted to describing the percentage of the
overall video that was viewed by the participants. This served as a proxy for describing
how focused on details the participants were.

58"

Across the sites, participants 2, 4, and 6 often behaved very similar to each other.
Among this behavior includes spending the most time exploring many of the sites. They
spent the most time examining the second site as out of all participants, were in the four
highest percentages of the video watched in fourth site, and also were three of the four
participants who watched all of the video in site five. This is in contrast to participants 1,
3, and 5 who typically spent the least time engaging with any of the five sites. Participant
7 swung back and forth, sometimes behaving in ways similar to the first group and
sometimes behaving in ways similar to the other.

Summary
Overall, this section looked at the searching behavior of the participants in this
study as they searched online for information. The analysis was broken into two parts,
first focusing on the open searching activity, and then the second activity where sites
were provided to the participants. In this analysis, where appropriate, I examined both the
ways in which participants used Internet search engines to find resources (similar to
examining how someone finds books in a library), and also the focus of attention when
searching for information within an individual site (similar to examining how someone
finds information within a book). A couple patterns of behavior were uncovered as part
of this study.
In general, the participants all had very similar information seeking behaviors
when they were searching for information between pages and content sites. There did not
appear to be many noticeable distinctions in participants based on this type of analysis.
When focusing on how participants searched for information within individual pages, it

59"

was found that all participants again spent at least some time focusing intently on the
content of the pages.
A unique distinction appeared when looking at the kinds of content these
participants spent time investigating. One group (made up of participants 2, 4, and 6)
examined a diverse format of content, splitting their time almost equally among static and
dynamic content (identified during their searching within Task 1). This same group was
also found to spend, on average, the most time engaging with the content of a site, and
watching online videos (identified during their searching within Task 2). Contrasting with
this first group was a second group (made up of participants 1, 3, and 5) that showed a
preference for one type of format of content, either static, or dynamic (identified during
their searching within Task 1). This second group also was more likely to spend the least
amount of time exploring sites (identified during their searching within Task 2).
Participant 7 fluctuated in behavior between these two groups.
This chapter investigated the question, “what are information-seeking strategies
that preservice teachers use when locating mathematical information online?” Based on
the performance of these seven preservice teachers, we know that there tended to be a
strong pattern of timid navigation when navigating between search engine results and
individual websites. Additionally, when looking at information in individual sites, there
was a strong pattern among all the participants of focusing on the content within the sites.
However, two different behavior groups emerged when examining the resources
available online. One group tended to spend longer examining individual resources, while
also seeking a balance between static and dynamic content. A different group tended to
spend less time with resources while also preferring a single format of resource, either

60"

static or dynamic. It was particularly interesting that there seemed to be a strong
correlation between the amount of time spent exploring resources, and whether there was
a preference for a single format of resource, or a balance of resources. This work shows
that there appear to be information-seeking strategies common to the preservice teachers
in this study, as well as different patterns of information-seeking strategies.
In the next chapter, the results of the prior chapter (which looked at the
mathematical connections that participants were able to make in their concept maps), and
this chapter are compared and contrasted. Patterns of similarity and divergence are
discussed, as well as what might be underlying influences that lead to these correlations.

61"

CHAPTER 5
To assess the impact of searching for mathematical information online,
participants constructed concept maps before, during, and after their Internet searching
tasks. These concept maps depicted how they were structuring their understanding of the
content around the Pythagorean Theorem. The ultimate goal of this analysis is to better
understand the connections that the preservice teachers were making between pieces of
mathematical content and how these connections changed as participants progressed
through the two searching tasks. To do this, the concept maps (and specifically the links
between the content included in the concept maps) is analyzed in this chapter.
The following figures are two concept maps created by a single participant in this
study. The first was constructed at the start of the interview session before either activity
was administered (Figure 9), and the second was at the end of the session after the
completion of both activities (Figure 10).

Figure 9 – Pre Concept Map from Participant 5

62"

Figure 10 – Post Concept Map from Participant 5
Although these two maps seem very similar with their general structure and form, there
are important differences to note, both with the content of the maps and the links that this
preservice teacher included in each of the maps. This chapter describes the work I did to
unpack these maps, and all of the maps produced in this study. I will return to these two
maps in each section of this study to help illustrate some of the concept that I describe.
Chapter Structure
Although the main goal of this study is to investigate the connections included in
the concept maps, this chapter first begins with an analysis of the mathematical content of
the concept maps. Next the framework used to analyze the links in the concept maps is

63"

presented before continuing to the results of the analysis. Next, a brief look at the overall
structure of the concept maps is presented. Finally, the chapter ends with a summary of
the main findings, as well as how these findings help answer the question of what form
and quality of mathematical connections do preservice teachers make around the
Pythagorean Theorem after locating information online?

Mathematical Content of Created Concept Maps
As described in the methods section, during each construction of the concept map,
preservice teachers were provided with a list of 12 terms that will sometimes be brought
up in conjunction with the Pythagorean Theorem. Participants were instructed that this
was only a partial list and they were free to use some, none, or all of this list when they
were creating their own concept maps that reflected their current thinking around the
Pythagorean Theorem. This section examines the content that participants included in
each of their concept maps including how much of it came from the provided list and
how much was content beyond what was provided.
As stated in the literature review and methods section, this study use the
“construct-a-map” technique where preservice teachers were provided a list of content
that they may wish to include in their maps (Ruiz-Primo et al., 2001). Table 1 below
shows the content that preservice teachers used in each of their mappings. Each grouping
of three columns shows the content in each participant’s Pre (Pr), Middle (Mi), and Post
(Po) mapping. The first twelve rows of the table are the content that was included in the
activity prompt. The subsequent rows are content that the preservice teachers included
without prompting.

64"

We can use the two concept maps (Pre and Post) from Participant 5 presented at
the start of this chapter to help illustrate the idea of content. This preservice teacher
included in her pre map six pieces of content. Five of these pieces were from the original
list provided as part of the prompt (right angle, right triangle, side, hypotenuse,
Pythagorean Theorem) and one was content not included in the original list (the formula).
Looking at the concept map this participant created after completing the activities (her
post map) we see that she included eleven pieces of content. Seven pieces of content were
from the list provided: right angle, right triangle, hypotenuse, and Pythagorean Theorem
from before, and now adding leg, proof and converse. Four pieces of math content were
not included as part of the list provided: the formula from before, and not adding
Pythagorean triples, the definition, and applications of the theorem. A listing of all the
different content included in each of the maps created is contained in Table 1.

65"

Content
Triangle
Angle
Right Angle
Right Triangle
Side
Hypotenuse
Leg
Length
Square
Pythagorean Theorem
Proof
Converse
Formula
Triples
Definition
Perimeter
Area
Application
Pythagoreans
Algebra
Congruency
Coordinate System
Diagonals
Rectangles
Construction
Base
Wheel of Theodorus
SSS
Special Cases

Participant 1
Pr Mi Po
X
X
X
X
X
X
X
X
X
X
X

X
X
X
X
X
X
X
X
X
X

X
X
X
X
X

Participant 2
Pr Mi Po
X
X
X
X
X X
X
X X
X
X
X
X
X
X
X
X

X
X
X
X
X

X
X
X
X
X
X
X

Participant 3
Pr Mi Po
X X
X
X
X

X
X

X
X

X
X
X
X
X
X

X
X
X
X
X
X

X
X
X
X
X
X
X

Participant 4
Pr Mi Po

X
X
X
X
X

X

X
X
X

X

X
X

X

X
X
X
X

X
X

X
X

X
X

X
X

Participant 6
Pr Mi Po
X X
X
X
X
X X
X
X X
X
X X
X

X
X
X
X

X
X
X

X
X
X

Participant 5
Pr Mi Po

X

X

X

X
X

X
X
X
X
X
X

X

X
X

X
X
X
X
X

X
X
X
X

X

X
X

X

X

X

X
X

Participant 7
Pr Mi Po

X
X
X
X
X

X
X
X
X
X

X
X
X
X
X

X
X
X
X

X
X
X
X

X
X
X
X
X

X

X

X
X

X
X
X

X
X

X

X
X
X
X
X
X
X

Table 9 – Content Included in Concept Mappings

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X

The content described in Table 9 is summarized in Table 10 and visualized in Figure 11
below. This new table shows the counts of tallies broken apart by how much content
included in the concept map came from the list provided as part of the task, and how
much was from outside the list provided.

Participant 1

Participant 2

Participant 3

Participant 4

Participant 5

Participant 6

Participant 7

Content
Given
7
7
5
8
7
9
9
9
10
6
4
4
5
6
7
6
10
8
8
8
8

Map – Pre
Map – Middle
Map - Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post

Content
Not
Given
2
4
3
3
7
7
0
1
1
4
1
1
1
1
4
3
3
7
1
1
3

Total
Content
9
11
8
11
14
16
9
10
11
10
5
5
6
7
11
9
13
15
9
9
11

Table 10 – Summary Counts of Content in Each Concept Map

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Figure 11 – Counts of Content in Each Concept Map
Unpacking the Counts of Content
When looking at these two tables and figure, a couple patterns begin to emerge. For most
all of the preservice teachers, they increased the amount of content included in their
concept maps with each successive mapping (5 of the 7 participants). The decrease in the
variety of content for Participant 4 is a result of her increasing focus solely on the proof
of the Pythagorean Theorem. The early peak of content in Participant 1 seems to be due
to her choice of consolidating information in her final map, as opposed to a continued
expansion that was typical of many others.
In general it is important to note that, as a whole, the amount of content increased
with each successive generation of concept maps produced by participants. This suggests
that as the preservice teachers engaged with the two tasks, they chose to represent how
they were thinking about the Pythagorean Theorem with more content, which then allows

68#

for these additional ideas to be incorporated and connected with other content in the later
mappings.
This section has taken a look at the content that was present in the mappings
produced by the participants. The next section will explore the different forms of links
that participants described between the content that they included in their concept maps.
From this we will be able to get a snapshot of the forms and quality of mathematical
connections that the preservice teachers had among the content.

Mathematical Links Among Content
In the prototypical concept map, nodes (content) are connected with words and
phrases that help explain the relationships between two different pieces of content.
Because the preservice teachers in this study were novice concept map makers and might
not connect their nodes with phrases, the concept maps created during the activities were
later augmented with the text spoken by the preservice teachers. This text may have been
spoken as they were creating the concept maps, or later when they were asked to provide
an overview for the concept maps they created. This text was included in an effort to
provide additional information that would help make visible the connections participants
had between the pieces of content they were joining.
The augmented concept maps were broken into smaller chunks, isolating the links
that participants were including. These chunks were coded for the mathematical links that
were contained in each chunk. Each mathematical link was given a code based on two
aspects of that mathematical relationship, the form and the quality of the mathematical
relationship. This section begins by defining the forms and qualities of mathematical

69#

links utilized in the analysis. Later, results from the coding of the concept maps are
presented with an eye towards identifying patterns that emerged out of the coding.

Framework for Forms and Quality of Links
In his 2005 article, Jon Star unpacked an underlying issue that existed in some
frameworks of mathematical knowledge. Instead of using the terms procedural and
conceptual to describe two aspects of a single phenomenon, Star argued that some
research literature had been overlapping two different phenomena into these labels. Star
unpacked the ideas of the forms (kinds) of knowledge, commonly referred to in the
literature as procedural or conceptual, from the quality of that knowledge, which he
described as rich vs. superficial (Star, 2005). Star argued that in the literature, conceptual
knowledge was often times used to mean richly connected knowledge, and procedural
knowledge was often times used to indicated superficial knowledge. Instead, it was
argued that conceptual knowledge (knowledge of a concept) could be richly or
superficially connected. Similarly, knowledge of a procedure could just as well be richly
or superficially connected. This could also be illustrated with a table as in Table 11
below.
Quality of Knowledge
Rich
Forms of

Conceptual

Knowledge

Procedural

Superficial

Table 11 – Forms and Quality of Mathematical Knowledge

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Although this distinction was elaborated by later researchers (Baroody, Feil, & Johnson,
2007), the fundamental distinction of paying attention to the form of mathematical
knowledge separate from the quality of that mathematical knowledge remains a crucial
frame for describing mathematical content. Just as this lens allows us to help understand
the form and quality of the mathematical connections students are making as they engage
with the mathematical content, we can also see the form and quality of the mathematical
connections preservice teachers are making.
The distinction between conceptual knowledge and procedural knowledge seems
at the surface to be a fairly easy distinction to make based upon how it is discussed, either
the piece of knowledge is about an underlying mathematical concept, or it is about a
procedure, an application of that concept. However, when trying to classify pieces of
content in practice, the distinctions between concepts and procedures become much more
nuanced. This is complexified when analyzing concept maps as these representation will
often times be comprised of brief words or phrases, which don’t make clearly visible
some of the meanings involved. In these cases, the augmenting of the concept maps with
spoken text helps contextualize these phrases and provides more information around how
the content is being described.
Definitions and Examples of Forms and Qualities
From these augmented concept maps, the two forms of knowledge that were coded in
each of the concept maps are described according to the following descriptions:
•

Conceptual – Link describes an underlying mathematical phenomena (i.e.
defining it, listing properties of it), or extension of mathematical phenomena to

71#

related contexts (i.e. real world examples of phenomena, special cases of
phenomena)
•

Procedural – Link describes a mathematical process (i.e. describing a step or
series of steps) or represents a phenomenon (i.e. drawing a picture, writing a
formula).

While the process of augmenting the concept maps provided some new insights into the
reasoning preservice teachers were using to structure their concept maps in particular
ways, the text they spoke didn’t always make every decision transparent. In particular, it
was common for participants to explain why they included particular pieces of content,
but it was less common for them to explicitly say what they chose to link two pieces of
content, thus allowing for a clear picture of the math connection they were making.
Sometimes the reasoning for these links were embedded within their descriptions of the
content, which made coding the forms of math connections more challenging.
In addition to the codes around the forms of mathematical links, there were also
codes around the quality of that mathematical links as it was described in the augmented
concept maps. The below distinction was used in categorizing the quality of the included
content contained in each map as either rich or superficial:
•

Rich – The link contained a description of a mathematical phenomena (i.e.
mathematically accurate definition) or described connections between
mathematical phenomena.

•

Superficial – The default quality of mathematical content coded. Described
phenomena was either incomplete or mathematically incorrect (i.e. an incomplete
definition) or possibly contained a mathematical error.

72#

The brief text included in these maps created by these novice concept map makers
presented some difficulties in assigning a quality code to the mathematical content
contained in each map. The default code for quality of these mathematical links was
superficial unless there was present enough description (either in the actual or augmented
map) that allowed for me to see some rich links of mathematical content. By having the
default code of quality be superficial, the results obtained as part of this coding are, by
nature, more conservative than having the default code of rich.
The below table shows the four different combinations of forms and qualities of
mathematical links and provides an illustrative example of each.
Quality
Rich

Superficial

Concept

Form

Procedure

Table 12 – Examples of Forms and Qualities of Mathematical Links
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My understanding of the different aspects of these codes became greater and more
nuanced as I went through more and more coding experiences with the data of this study.
In order to ensure that the coding that occurred during data analysis was as uniform as
possible, the data from the beginning of the analysis process was recoded a second time
after a complete pass through all the data. This was an attempt to ensure that the codes
given to early participants were assigned through the same understanding of the rubric as
those later participants.
Illustration of Mathematical Link Coding
Before beginning this analysis, in an effort to focus on understanding what
mathematical connections were closely related to the Pythagorean Theorem, two texts
that describe the Pythagorean Theorem, one used in the preservice teaching elementary
mathematics content course (Beckmann, 2011), and a textbook commonly used in middle
school classrooms (Lappan et al., 2009) were analyzed. From this analysis, a list was
created of mathematical connections related to the Pythagorean Theorem. This served as
the original starting list of connections for the identification and coding of mathematical
connections in the augmented concept maps.
Below are the same two concept maps created by Participant 5 both before the
beginning of the activities and after completing the activities. However, these two
concept maps, the pre (Figure 3) and post (Figure 4) have also been augmented with the
text spoken by the participant when describing their concept maps. After this
augmentation, the maps were chunked into the different links that the participants were
making to make the entire map easier to analyze.

74#

And so from these two
we can come up with
this formula

I just started off with like the
Pythagorean Theorem itself, cause
that’s what we were focusing on

Chunk 4

Chunk 1
and you have to use
that with a right triangle
we had the three sides
which was one, is the
hypotenuse and the two
remaining sides, and the
hypotenuse in my mind
was c, and the remaining
sides, one was a, and the
other one was b

Chunk 3

Chunk 2
I described what the
right triangle was so
have the right angle

Figure 12 – Augmented and Chunked Concept Map from Participant 5 (Pre)

the hypotenuse and the

other legs which would
Chunk
2
be a-squared plus b-

and then I added in the
Pythagorean Triples,
which is something
important to keep in
mind

squared equals csquared

I put only right triangles
as opposed to, like, just
right triangles leaving
out the only just so
students know, to make
it more clear

Chunk 1

With the Pythagorean
Theorem in the middle
and with the three lines

Chunk 3

Chunk 4

you can prove that with
the converse and just
adding in the definition
of the converse

Chunk 5

then I added
in resources

Chunk 6

Chunk 7

I added them in here, but then I crossed them out because it was kinda confusing and I feel like if I put
resources and then added like the video, here’s the video, the proofs, this is a good site to find the proofs.
Um, just to kinda make it a little more linear. Um, cause over here this was kinda confusing cause there were
two different resources where I can just combine them and make them more specific that way.

Figure 13 – Augmented and Chunked Concept Map from Participant 5 (Post)

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For Participant 5’s pre concept map (Figure 12) we can see the links that the
participant included in their concept map around the vocabulary of and properties of
shapes (Chunk 2 and 3), the stating and meaning of the Pythagorean theorem (Chunk 4)
and the Conditions and Generality of the Pythagorean Theorem (Chunk 1). All of these
links were of the conceptual form, and all were superficial.
When looking at this participant’s post concept map (Figure 13), we see that there
are more links among the mathematical content included, 8 in total. This time the links
still included the vocabulary and properties of shapes (Chunk 2 and 3), the stating and
meaning of the Pythagorean Theorem (Chunk 2) and the conditions and generality of the
Pythagorean Theorem (Chunk 4). However, this post concept map also includes linkss
around the proof of the Pythagorean Theorem (Chunk 6), applications of the Pythagorean
Theorem (Chunk 7), stating and meaning of the converse to the theorem (Chunk 5), and
applications to the converse (Chunk 1). Again, most of the links made by this participant
appear to be of the conceptual form and superficial quality. However, the links made
around the proof and the applications of the Pythagorean Theorem were procedural links,
and the information included when discussing the conditions and generality of the
theorem suggested a rich understanding of the links. From this we can see links in the
post mapping that weren’t included in the pre mapping for Participant 5. Not only were
there more links included in this mapping, but those links were of a greater variety (both
concepts and procedures) and of different qualities (evidence of a rich link). The
following section contains a summary of this analysis for all of the concept maps
produced as part of this study.

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Summary of Form and Quality of Mathematic Links in Concept Maps
The table and figure below (Table 5 and Figure 2) are a representation of the
forms of mathematical knowledge (procedural or conceptual) contained in each of the
concept maps based on the categories of links. In the table, the three concept maps of
each participant are presented sequentially, with first their pre map, middle map, then
post map.

Participant 1

Participant 2

Participant 3

Participant 4

Participant 5

Participant 6

Participant 7

Procedure
1
2
5
3
2
2
1
1
4
4
4
5
0
0
2
5
4
6
1
1
1

Map – Pre
Map – Middle
Map - Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post

Concept
5
5
6
6
6
8
6
5
6
5
4
5
4
5
6
3
4
5
5
4
7

Total
Form
6
7
11
9
8
10
7
6
10
9
8
10
4
5
8
8
8
11
6
5
8

Table 13 – Form of Mathematical Link Contained In Concept Map

77#

Figure 14 – Forms of Mathematical Links Contained in Concept Maps
Noticed that for every participant, the amount of mathematical links included in
the concept mappings is highest in the final, post mapping created during each
administration. The smallest amount of links included in each mapping is contained
either the pre or mid mapping. The implication here is that after doing these tasks of
searching for mathematical information online, the preservice teachers included more
mathematical links in their concept maps than they did before they completed both tasks.
Focusing specifically on the form of mathematical links included within each
concept map (Table 13 and Figure 14), it is noticed that in almost every mapping, the
amount of conceptual links included is greater than the amount of procedural links. This
may be due to the nature of the activity which asked the preservice teachers to create a
“concept map”, which suggests that it may be more likely to include conceptual links
rather than procedural ones. Additionally, when looking at the procedural links included

78#

in concept maps, two different patterns emerged. For some preservice teachers, the
amount of procedural links remained fairly constant during subsequent iterations of their
maps (Participants 2, 4, 6, and 7). Contrasting this consistency were the participants
whose procedural links went up in their post mapping, as compared to their first two pre
and mid mappings (Participants 1, 3, and 5). For all three of the participants in this group,
this was due to the inclusion of links related to the proof of the Pythagorean theorem,
proof of the converse to the Pythagorean theorem, or both.

Participant 1

Participant 2

Participant 3

Participant 4

Participant 5

Participant 6

Participant 7

Superficial
4
6
11
7
7
7
7
6
10
5
6
7
4
5
7
5
3
10
6
5
8

Map – Pre
Map – Middle
Map - Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post
Map – Pre
Map – Middle
Map – Post

Rich
2
1
0
2
1
3
0
0
0
4
2
3
0
0
1
3
5
1
0
0
0

Total
Quality
6
7
11
9
8
10
7
6
10
9
8
10
4
5
8
8
8
11
6
5
8

Table 14 – Quality of Mathematical Links Contained In Concept Map

79#

Figure 15 – Quality of Mathematical Links Contained in Concept Map
When focusing on the quality of the mathematical links included in the concept
maps (Table 14 and Figure 15), one general pattern that emerges across all of the concept
maps is that the amount of mathematical links that are richly described is almost always
less than the amount of links that are superficially described. As stated before, the default
code for this category was “superficial”. The brief phrases that many preservice teachers
utilized during the creation of their concept maps, or in describing their maps, did not
always provide explicit evidence that there were rich links between pieces of content.
This might help explain why the superficial understanding codes seem to outweigh the
rich understanding codes.
It is also noted that for four of the preservice teachers (Participants 1, 3, 5, and 7)
each produced at least one iteration of their concept map that contained no rich
mathematical links (Participants 3 and 7 never included any descriptions in any iteration

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of their maps that were coded as rich). This contrasts with Participants 2, 4, and 6 which
had some rich mathematical links contained within each iteration of their concept maps.
Putting together the patterns noticed in both the forms of mathematical links
coding and the quality of mathematical links coding, I notice a correlation. For
Participants 1, 3, and 5, they included few procedural forms of links in their pre and
middle maps, but noticeable more in their post map. These participants also had at least
one iteration (usually more) of their concept map that contained no richly described
mathematical links. This is in contrast to Participants 2, 4, and 6, where the amount of
procedural links included in all iterations of their map remained about the same while
also producing maps that always contained some rich links. Participant 7, though
displaying behavior that could put her in each of these groups, seems to have produced
maps that are much more similar to Participants 1, 3, and 5.
Characteristics of
Forms of Links

More balance between
conceptual and
•
procedural links
• Consistent amount of
procedural links in each
mapping
Characteristics of
• At least one mapping
• Every map contains at
Qualities of Links
lacking any rich links
least one rich link
Participant
• Participants 1, 3, 5, & 7
• Participants 2, 4, 6
Table 15 – Categories of Behavior of Participants
•

Heavy reliance on
conceptual links
More procedural links
appear in later mappings

•

This section has sought to uncover some patterns that emerged among the
preservice teachers concept maps when looking at the forms and quality of mathematical
links they were including in their maps. The next section looks at the structure of the
concept maps as a whole.

Structures and Purpose of Created Concept Maps

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The previous sections looked at the mathematical content and links among the
content in the pre, mid and post mappings of the participants. These were analyzed for
each individual mapping, and then the results were compared across mappings to see
what changes or alterations occurred. This was done as a change in the iterations of the
concept map might represent that a change of understanding has occurred within the
individual. On a larger scale, all of the content and links in these concept maps are
represented with overall structures. Large changes in understanding may be made visible
through changes to the overall structures of the maps. This section looks at the structures
utilized by the participants when creating their maps, and how these structures may have
changed over the course of the activity.

Structure
There are several different forms that concept maps can take. Originally, maps
were said to always require a hierarchal structure, with concepts and content arranged in
an ascending or descending order. However, others have also accepted radial structure
maps with an idea is located at the center of a map and branches (and sub branches)
radiate outward from the original idea (Williams, 1998). The below table lists the general
structure used in each of the concept maps created by participants.
Participant
1
2
3
4
5
6
7

Pre Map
Mid Map
Tree
Tree
Tree
Tree
Radial
Radial
Radial
Radial
Radial
Radial
Radial
Radial
Radial
Radial
Table 16 – Structure in Each Concept Map

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Post Map
Tree
Tree
Radial
Radial
Radial
Tree
Radial

One observation from this table is noticing that almost every participant chose to
utilize the same structure for their concept map in every iteration of construction. Despite
constructing each new concept map from scratch there was a strong tendency for the
structure a participant used during their first construction to be utilized during each
subsequent construction. This suggests that overall, these tasks did not produce large
changes in how participants were understanding the content of the Pythagorean Theorem,
and therefore, there was no need to significantly alter how they were representing their
understanding.
Purpose
During this activity, I noticed that the participants who utilized a tree hierarchy
for their concept maps seemed to have a different purpose for their concept maps,
compared to the diagrams that utilized a radial hierarchy. Those that made us of the tree
hierarchy seemed to have an instructional or pedagogical purpose for the maps that were
constructed. Here, the preservice teachers commented or wrote that particular pieces of
content, or particular links they were making between pieces of content, were being made
of the benefit of students, or it was something they wanted to teach in a particular way.
The suggestion, either explicit or implicit of an instructional sequence was absent from
those who utilized the radial sequence. This may be because the nature of the tree
structure lends itself well to a sequential presentation of ideas. This is in contrast to the
radial structure which does not lend itself well to an order of first, second, third, etc.
The nature of the task and the background of the participants make it unsurprising
that many of these maps would incorporate not only the content, but also the teaching and

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learning of that content. However, the attention paid to these pedagogical themes
occurred only with the tree hierarchy structure and not the radial hierarchy structure.

Summary of Results
This section focused on the mathematical content included by preservice teachers
in concept maps they created before, during, and after completing Internet searching
tasks. The main analysis of this chapter focused on describing the form and quality of the
links between pieces of content in these maps. Additional analysis also examined the
content itself, and the overall structure of these mappings.
When analyzing the links among the content that was included in the concept
maps, two patterns of behavior appeared. There was one group of participants
(Participants 2, 4, and 6) who included both procedural and conceptual links in their
concept maps, maintaining roughly the same amount of procedural links in all of their
mappings. This group also included rich links among their content in every one of their
mappings that they produced. This pattern of behavior is in contrast to that of a different
group of participants (Participants 1, 3, 5, and 7). This group created concept maps that
had a heavy reliance on conceptual links with almost no procedural links (until their final
mappings, which did include some procedural links). Additionally, this second group
always had at least one of their three mappings (and usually multiple mappings) contain
no rich links.
The analysis of the content in the mappings of the preservice teachers, showed
that the participants tended to include more content after completing each activity, as
compared to the map they previously constructed, suggesting that these tasks may have

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had an impact on the additional content. There was a significant amount of content not
contained in the provided list that which suggests that the preservice teachers were
incorporating content they knew or that they picked up from the explorations into their
representations. In the investigation of the overall structure of the concept maps, I found
that the participants overall did not change the structure of their concept maps each time
they reconstructed it. Additionally, the participants who used a tree hierarchal structure
for their maps also included pedagogical comments or descriptions in their mappings.
There is one note to make about the results of this analysis. This chapter
examined the mappings of the preservice teachers in this study, who created their own
representations of how they were thinking about the content and links among the content.
I do not claim that these concept maps are perfect representation of how these preservice
teachers are thinking about the content. Additionally, I do not claim that the absence of
content, or a superficially described link among content means that the preservice teacher
doesn’t know the content or have a rich understanding of the content itself. All that is
being analyzed is how these preservice teachers are choosing to represent their
understanding of the content via these concept maps.
This chapter investigated the research question of, what form and quality of
mathematical connections do preservice teachers make around the Pythagorean Theorem
after locating information online. In general, it was found that the preservice teachers
tended to represent their connections with conceptual links among content maps that
tended to be superficially described as a whole. However, while this was generally true of
all the participants, two patterns of behavior emerged, one very similar to the overall
pattern, and a different group that had an increase in the amount of procedural links in

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their mappings and also an increase in the amount of richly described links in their
mappings too. Overall, there is evidence that after locating information online,
participants represent their understanding of the content with a greater amount of content
and more links among that content. Additionally, there is also evidence that some
participants have a more varied understanding of the content (both concepts and
procedures), and choose to represent that understanding in visibly rich ways too. This
shows when completing the tasks of this study there was an increase in both the range
(amount) and depth of understanding (quality) represented in the concept maps.
The next chapter looks at the results of the previous searching strategies chapter,
and this math task chapter and looks for patterns in the results. This chapter will connect
the ways in which the searching strategy behavior of participants might also be related to
the forms and quality of mathematical connections the preservice teachers included in
their concept maps.

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CHAPTER 6
The previous two chapters explored two aspects of investigating mathematical
information online: the information-seeking strategies that preservice teachers employed
when completing Internet-searching tasks, and the mathematical content and links that
preservice teachers displayed in concept maps when exploring this environment and
content. This chapter explores what connections exist between these two different aspects
of the investigation. Throughout this chapter, I will be reviewing the findings from the
analyses performed in this dissertation and how they help answer the research questions
of this study: As these results are presented, I fill in the table below (Table 17) to help
organize the information and the patterns that emerge from these results.

RQ1

RQ2

RQ3

Format of
Accessed
• What are information-seeking strategies that preservice
Resource
teachers use when locating mathematical information online?
Time Spent
Exploring
Form of
• What form and quality of mathematical connections do
Links
preservice teachers make around the Pythagorean Theorem after
Quality of
locating information online?
Links
Related
Patterns of • What relationships emerge between the form and quality of
the mathematical connections made by preservice teachers, and
Behavior
Initial Goals the information-seeking strategies employed during the research
task?
at Task
Launch
Table 17 – Table of Categories from Findings (Empty)

Review of Search Strategy Findings
In chapter four, I presented the findings of this dissertation as they related to the
information seeking strategies utilized by the preservice teachers when completing two
tasks. That chapter was broken into two sections, describing the information seeking

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strategies employed during the open-searching task, and those employed during the
closed-searching task. During the open-searching task, I found that all participants used a
timid-navigator approach and only two participants, at any point in time during their
exploration, used a divergent navigator approach. When exploring they ways that
participants explored information within a particular results page, it was found that all
participants, at some point, focused on the content, giving sustained, close attention to
some aspect of content on a page. It was also discovered that participants fell into two
groups with regards to the kind of content that they were focusing on (either static
content or dynamic content). One group of participants clearly favored one format over
the other format while the other group had more balance with the forms they explored.
When analyzing the results of the information seeking that occurred during the
closed searching activity, many similar trends were found. Again, at some point during
this activity, every participant focused on some content, providing a sustained, close read
of the information within the pages. Additionally, those who accessed a balance of
formats in their first task were also the ones who had longer exploration times with sites
during the second task. Those who spent the majority of their time with one format of
information during the first activity were also the participants who had shorter
exploration times during the second task.
The differences in the two different groups found when looking at the information
seeking strategies can be displayed in the updated table below (Table 18). The different
groupings had two characteristics, one pertaining to the formats of the resources they
accessed, and the second with the amount of time they spent exploring the accessed
resources. These groupings were one of the main results found when answering the first

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research question about the searching strategies used by the preservice teachers. Next I
review the results obtained when looking at the second research question.

RQ1

RQ2

RQ3

Format of
• Preference for one format
• Preference for both formats
Accessed
(either static or dynamic) of
(static and dynamic) of content
Resource
content
Time Spent • Shorter exploration times
• Longer exploration times
Exploring
Form of
• What form and quality of mathematical connections do
Links
preservice teachers make around the Pythagorean Theorem after
Quality of
locating information online?
Links
Related
Patterns of • What relationships emerge between the form and quality of
the mathematical connections made by preservice teachers, and
Behavior
Initial Goals the information-seeking strategies employed during the research
task?
at Task
Launch
Table 18 – Table of Categories from Findings (Searching)

Review of Math Content and Connections Findings
In chapter five, I presented the findings of this dissertation as they related to the
mathematical content and connections that preservice teachers included in their concept
maps they constructed during their experiences exploring the Internet for this study. With
regards to the content that they included in the concept maps, the main finding was that
the content increased in frequency as participants created successive iterations of their
maps after exploring more content.
When looking at the connections between the content that was included in the
concept maps, two different aspects were investigated, the form of links that was
included (procedural vs. conceptual) and the quality of those links (rich or superficial). I
found that the majority of the links that were included in the concept maps were of the
conceptual form and the majority of all the connections were of a superficial quality. Like

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the exploration of the content, I also found that the links increased in quantity as the
activity progressed.
From this exploration among the links, two groups emerged in the classifications.
One group of participants included most almost all conceptual links in their concept
mappings. Additionally, these participants had at least one iteration of their mapping (and
frequently multiple iterations of their mapping) that contained no evidence of any rich
links. A second group of participants had more of a balance between the conceptual and
procedural links included in their maps. These participants always included enough detail
in each iteration of their concept maps to provide evidence of at least one rich link.
Lastly, when exploring the overall structure of the concept maps, it was found that
participants utilized two different kinds of concept map structures: tree (hierarchal) and
radial. Despite constructing each iteration of the concept map from scratch, only one
participant changed the overall structure of their concept map, while most everyone
decided to make minor alterations to the content and links of previous concept maps.
Additionally, it was found that those who chose to represent their understanding of their
concept maps with a tree structure also indicated some attention to pedagogical issues
when describing their map.
The differences in the two different groups found when looking at the links
among the content in the concept maps can be displayed in the updated table below
(Table 19). The different groupings had two characteristics, one pertaining to the forms
of the links included in the concept maps, and the second pertaining to the quality of
those links. These groupings were one of the main results found when answering the

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second research question about the form and quality of mathematical connections that
preservice teachers included in their concept mappings.

RQ1

RQ2

Format of
Accessed
Resource
Time Spent
Exploring
Form of
Links

• Preference for one format
(either static or dynamic) of
content
• Shorter exploration times

• Preference for both formats
(static and dynamic) of content

• Heavy reliance on conceptual
connections
• More procedural connections
appear in later mappings

• More balance between
conceptual and procedural
connections
• Consistent amount of
procedural connections in each
mapping
• Every map contains at least
one rich connection

• Longer exploration times

Quality of • At least one mapping lacking
Links
any rich connections
Related
Patterns of • What relationships emerge between the form and quality of
the mathematical connections made by preservice teachers, and
Behavior
RQ3
Initial Goals the information-seeking strategies employed during the research
task?
at Task
Launch
Table 19 – Table of Categories from Findings (Searching & Math Connections)
The next section takes the results of these first two results chapters that focused on

information-seeking strategies and mathematical connections and looks for patterns that
emerge out of the results. This is done with the ultimate goal of answering the third and
final research question, what relationships emerge between the form and quality of the
mathematical connections made by preservice teachers, and the information-seeking
strategies employed during the research task?

Relationships Between Search Strategies and Content
In the searching strategy analysis, the group that tended to prefer a single format
of site, and spent shorter amounts of time with the sites they explored was made up of
Participants 1, 3, 5, and 7. The group that preferred a balance of formats in their

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searching and spent longer amounts of time with the sites they explored was made up of
Participants 2, 4, and 6. In the math content and connections analysis, the group that
included almost exclusively conceptual links, and always had at least one of their
mappings contain no rich links was made up of Participants 1, 3, 5, and 7. The group that
had more of a balance between their conceptual and procedural links and also always
included at least one rich connection in their mappings was made up of Participants 2, 4,
and 6. There was a strong relation between the pattern of behavior identified when
analyzing the searching strategies, and the patterns of behavior identified when looking at
the concept maps. Across both of these analyses, two groups emerged. These two group
are unpacked briefly here.
One of the groups identified (made up of participants 1, 3, 5, and 7) seemed to be
oriented to the big ideas around the Pythagorean Theorem. These participants exhibited
behavior during their searching that suggested a particular preference for one format of
information presentation, and also a more brief engagement with the content presented.
During the first open-searching task, these were the participants who decided to engage
with mostly only static content, or mostly only dynamic content, and very rarely engaged
with content that was not the format of their preference. During the second, closed
searching task, these were the participants who spent the least amount of time engaging
with the static website, and the least amount of time with the videos. When representing
how they were thinking about the Pythagorean Theorem, they primarily only described
concepts, rarely procedures in their maps, and frequently had maps lacking the detail to
see rich connections.

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One possible explanation for this interaction between the searching strategy and
the concept mapping may be that the focus on only one type of presentation format, and
the more limited engagement with the content didn’t allow for a variety of mathematical
connections, nor a rich understanding of those connections, to form. It may be the case
that this single format, brief search approach is beneficial for providing Internet searchers
with broader ideas of the connections and interactions of mathematical phenomena,
allowing Internet searchers to gain a high-level overview of the mathematical terrain. In
this way, this quicker and narrower searching can be beneficial for those who are looking
to get a survey of the content and begin to understand its connections.
The second group identified (made up of participants 2, 4, and 6) seemed to be
oriented to details around the Pythagorean Theorem. These participants exhibited
behavior during their searching that suggested a balanced approach to the format of the
information presented, and also a more extended engagement with the content presented.
During the first open-searching task, these were the participants who split their time
across two different formats of presentation (dynamic and static) spending almost equal
amounts of time with each format. During the second, closed searching task, these were
the participants who spent the most amount of time engaging with the static website, and
the most amount of time with the videos. When representing how they were thinking
about the Pythagorean Theorem, they described (though still heavy on concept) more of a
balance between concepts and procedures, and always produced maps that had enough
detail to provide evidence of a rich understanding of the mathematical connections.
One possible explanation for this interaction between the approach taken during
the information-seeking activities and the concept map production may be that the

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exploration of different formats of content, and the extended amount of time spent with
that content allowed for a more diverse amount of mathematical connections to form, and
allowed for some of that content and those connections to be understood in a rich way. It
may be the case that not only does an extended amount of time with content help produce
richer understandings of diverse kinds of content, but the time exposed with the content
could also be spent looking at a variety of presentation styles of that content. In this way,
those looking to understand specifics of content in rich ways need to not only devote time
to their investigation, but also explore multiple ways of presenting information.

Classroom Goals of the Participants
From the earlier chapters, there appeared to be two different approaches taken by
the participants of this study, a more zoomed-out (big idea) orientation, and a more
zoomed-in (detail) orientation. After discovering the existence of these two groupings, I
wondered if there might be anything that may have been contributing to the distinctions
observed. I wondered if the participants were going into the searching tasks with different
goals and I looked to the pre-interviews to try and identify the goals of these preservice
teachers.
Before the Internet searching tasks began, the participants were asked four
questions about the standard (8.G.6) that they were investigating. These questions
focused on what they understood already about the standard, what they were unsure about
with regards to the standard, why they thought this standard might be important for
students, and what their goals would be for their classroom. When looking at the first
three questions, all the participants gave the same, or nearly identical answers. They all

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thought the standard was centered around understanding and proving the Pythagorean
Theorem and its converse. They all had questions about what exactly was meant by the
converse to the Pythagorean Theorem, or what converse actually meant. They all thought
that this standard was included as it was important for students to understand proofs.
With regard to these three questions, the participants seemed to be entering the Internet
searching tasks with very similar orientations.
The last of the four questions asked the preservice teachers what their own goals
would be as a teacher in their own classrooms as they had students work toward this
standard of “explain a proof of the Pythagorean Theorem and its converse”. As I looked
at the responses that the participants gave to this question, I noticed differences in how
they talked about their own classroom goals. The table below (Table 20) shows the
descriptions two types of responses I saw, as well as examples of those types of
responses.

Descriptions

Examples

General
• Goal of understanding
• Common aspects to all
proofs
• Student work before
teacher
“just, to get 8th graders to
understand why a proof work
and how it works, it would be
cool”

Specific
• Mathematics unique to the
Pythagorean Theorem
• Acknowledgement of multiple
representations/proofs of the
theorem
“although the proofs may be different
they’re, still end up with a-squared
plus b-squared equals c-squared”

“I think that there’s so many visual
“I would want the children to
ways to think about the Pythagorean
work it out first before I show, Theorem"
demonstrate it for them”
Table 20 – Descriptions and Examples of Classroom Goals

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The responses of both of the groups had an overarching goal of students
understanding proof as the goal of the classroom. However, there was a difference in the
focus of the responses that existed in the two groups. In the general orientation, these
participants spoke in general terms of the understanding they wanted students to walk
away with including highlighted “higher thinking” (Participant 1) and “just getting 8th
graders to understand why a proof works and how it works” (not necessarily the
Pythagorean Theorem proof) would be “cool” (Participant 5). While most of these
responses did have a thread of student understanding running through them, there was a
lack of detail included in their responses about what parts of the Pythagorean Theorem
that they wanted students to talk away with. The participants in this group were
Participants 1, 3, 5, and 7.
These responses are contrasted with the responses produced by the specific
orientation. The responses here still carried the thread of student understanding, but also
included either an attention to the mathematics behind the proof or details specific to the
Pythagorean Theorem that they might not have included when discussing other proofs.
Here, participants talked about multiple proofs for the Pythagorean Theorem and wanting
students to see “difference between the proofs and also make connections on how they’re
the same” (Participant 4) or that “there’s so many visual ways to think about the
Pythagorean Theorem” (Participant 6). Additionally these participants sometimes made
comments against the process of just memorizing individual steps in a procedure. The
participants in this group were Participants 2, 4, and 6.
In asking these preservice teachers to consider their own classrooms, and to
interpret this standard (8.G.6) for their own students, two different kinds of

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considerations emerge. Across all the participants, there existed this uniform goal of
understanding the proof of the Pythagorean Theorem. However, there were also those
that listed goals that were unique to the Pythagorean Theorem and paid attention to the
specific mathematics that went into understanding the proof of that theorem. These
participants seemed have a different entry point to the task than those that gave general
descriptions. Those participants in that grouping talked about bigger ideas of
understanding proof in general and understanding why something worked. The
participants in these groupings were divided in exactly the same way that they were in the
groupings of the previous two chapters. The different starting points of these two groups
may have had an impact on the ways in which the preservice teachers performed their
information-seeking tasks and constructing their concept maps. With this discovery, I
now complete the table used in this chapter (Table 21).

RQ1

RQ2

RQ3

Format of
Accessed
Resource
Time Spent
Exploring
Form of
Links

• Preference for one format
(either static or dynamic) of
content
• Shorter exploration times

• Preference for both formats
(static and dynamic) of content

• Heavy reliance on conceptual
connections
• More procedural connections
appear in later mappings

• More balance between
conceptual and procedural
connections
• Consistent amount of
procedural connections in each
mapping
• Every map contains at least
one rich connection
• Detail (Zoomed In)
Orientation

• Longer exploration times

Quality of • At least one mapping lacking
Links
any rich connections
Related
• Big Idea (Zoomed Out)
Patterns of Orientation
Behavior
Initial Goals • Attention to understanding
• Attention to specific
at Task
proof in general
mathematics unique to
Launch
Pythagorean Theorem proof
Table 21– Table of Categories from Findings (Filled)

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It should be noted that the research question of this chapter (what relationships
emerge between the form and quality of the mathematical connections made by
preservice teachers, and the information-seeking strategies employed during the research
tasks) was a question of identifying the existence of a relationship. It only looked to gain
insight if different patterns of behavior happened at the same time. There was no
assumption of causality. However, with the description of the initial goals of the
participants at the launch of the task, I introduce what might be a causal variable. It might
be the case that the different goals of the preservice teachers produce different behaviors
during the tasks and activities in this study. Further study would be needed to confirm
whether this was a causal effect for the other variables, or simply just appeared to be
related to them.

Connecting to the Literature
Beyond the seven participants in this study, there are a numerous ways in which
the findings of this study speak back to the literature and inform mathematics education
more generally. As was stated before, the characteristics of tasks and the personal
intentions of the users performing the tasks affect the ways that participants behave
during the tasks (Lawless & Schrader, 2008). This aligns with the finding presented in the
previous section, namely that the goals that the preservice teachers had for their
classrooms affected the ways that these teachers then performed their searching tasks and
then represented their knowledge.
This is important as it suggests that the knowledge of preservice teachers does not
solely determine the behavior of teachers in the classroom. In other words, though two

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teachers might have the same content knowledge and understanding of mathematics, this
does not mean they will enact lessons in the same way, nor have the same goals for their
classes. The goals that these preservice teachers have for their students seems to be
influential in affecting the ways that they think about content which then influences how
they look for information and then represent their understanding.
This study found that the distinctions in categories of information-seeking
behavior found in previous studies (Horney & Anderson-Inman, 1994; Juvina &
Oostendorp, 2004; Lawless & Kulikowich, 1996; MacGregor, 1999) also seemed to be
productive ways of categorizing the information seeking behavior in this study, with a
different population, and newer technological resources. This study also extends these
findings to a new content area of mathematics.
Of particular interest is the extension of the work around dynamic and static
content, and whether users focused on one, or the other, or both. Mathematics education
research has often described the impact that technology can have on the teaching and
learning of mathematics. The advantages presented by technology often include the
ability to dynamically present material (Kilpatrick, Swafford, Findell, National Research
Council (U. S.), & Mathematics Learning Study Committee, 2001; National Council of
Teachers of Mathematics, 2000; National Governors Association Center for Best
Practices & Council of Chief State School Officers, 2010). Given the ability of the
Internet to present content dynamically, and the advantages of dynamic presentation of
mathematics, it is important to understand the ways that Internet searchers might prefer,
or not prefer, to interact with this dynamic content.

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Lastly, in terms of the mathematical content, one of the most important findings
was that given only a quick exploration of online resources, participants were able to
increase the amount of content they included when describing how they were thinking
about the Pythagorean Theorem, and they were also able to increase the amount of
connections between that content. We know that practicing teachers have limited time to
learn any mathematics they are unfamiliar with and it is unreasonable to expect that
drastic changes in understanding would occur in short periods of time. However, when
participants had a familiarity with the content initially, similar to what was shown in prior
research (Fidel et al., 1999; Hirsh, 1999), they were then able to increase the content and
connections in the concept maps they were able to use to represent their thinking. Again,
this shows that for preservice teachers, given their limited available time, the Internet
may be able to be leveraged as a resource to support their learning.

Summary
This chapter looked at the main results from the first two findings chapters that
examined the searching strategies used when locating mathematics information online,
and the forms and qualities of mathematical connections made when locating information
online. The patterns of behavior that were found in each of these analyses were found to
be highly related. The participants that performed one way in their information seeking,
also behaved in similar ways when they constructed their concept maps. The two
groupings tended to focus on big ideas and have a zoomed-out orientation, or were more
focused on details and had a zoomed-in orientation. This chapter also looked at the ways
that the participants described their goals around this standard of proving the Pythagorean

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Theorem and its converse as they imagined their classrooms. I discovered that the
participants in the detail oriented group seemed to also have classroom goals that were
unique to the proof of the Pythagorean Theorem, while those in the big idea group
seemed to have classroom goals about proof in general.
The next chapter will offer a discussion of the study as a whole, including
limitations of this study, and possible future directions for research based on the findings
of this study.

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CHAPTER 7
In this dissertation, I investigated the information-seeking strategies of preservice
teachers when looking for information online about a proof of the Pythagorean Theorem
and its converse. I also examined how these preservice teachers represented their thinking
around this topic in the form of concept maps. Lastly, I looked for patterns of behavior
between the information seeking strategies used, and the ways that these participants
were representing their understanding of the content. Two patterns of behavior emerged:
one pattern of behavior seemed to have a big idea (zoomed out) orientation, while the
other seemed to have a detail (zoomed in) orientation.
Returning back to the beginning of this dissertation, this study was posed within
the larger context of teachers learning online. The Internet is a potential resource for
teachers to learn mathematics content. However, there are multiple challenges to learning
from and with online resources as opposed to learning with and from more traditional
educational resources. Included in this list of challenges is the Internet’s non-linearity, its
inclusion of multiple formats, and the burden of critically evaluating the reliability of
information being placed on the Internet user. This dissertation explored how preservice
teachers (those about to embark on their careers as teachers) were navigating these
challenges, among others, to make sense of mathematical information they encountered
online.
This chapter contextualizes results from the previous chapters in the broader field
of mathematics education. The following section highlights implications of these findings
in relation to current research in mathematics education and educational technology. I
also discuss the limitations of the study, describing the extent to which findings may be

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generalized. Lastly, this chapter considers how this study might be continued, or
expanded to learn more about how people learn mathematics online.
Implications
This dissertation focused on how preservice teachers may be using the Internet to
learn mathematics content on their own. Previous research had explored patterns of
information-seeking behavior of learners, but it was unknown what behaviors would
appear in college-aged, preservice teachers and if they would be any different from
younger Internet users. This study is an initial exploration into connecting questions
about what preservice teachers are learning about mathematics online, with questions
about how preservice teachers are learning mathematics online. The results of this study
have implications that help mathematics educators better understand what processes
future teachers of mathematics naturally use as they engage with content through new,
digital environments.
One of the main findings was revealing two patterns of behavior among the
participants—a big-idea orientation, and a detail orientation. I do not claim that either
pattern of behavior is better. Both orientations are necessary at different times. While the
detail orientation might be thought to be preferred (as it seems synonymous with a depth
of knowledge), the big idea orientation is also important in mathematics education.
Shulman identified curricular knowledge (Shulman, 1986), Ball and colleagues identified
Horizon Content Knowledge (Ball, Thames, & Phelps, 2008), and Ma identified
Profound Understanding of Fundamental Mathematics (Ma, 1999) as big idea types of
knowledge critical for effective mathematics teaching. This is knowledge of content and
how it fits in with the big ideas found in mathematics, as well as prior and future content.

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Additionally, this study adds to the conversation that teacher knowledge of
mathematics is not sufficient to predict how mathematics will be enacted in the
classroom. During the preinterview, when participants were first asked about the content
of standard 8.G.6, all participants gave nearly identical answers when asked about the
content of the standard. However, when asked to imagine their practice and their
classroom goals for achieving mastery of this standard, participants showed different
ways of how explaining the Pythagorean Theorem and its converse would be enacted in
their classrooms. While teacher content knowledge is considered to be important to the
mathematics that occurs in the classroom, this kind of knowledge does not account for all
the variance that is observed in classroom practice (Hill, Blunk, et al., 2008). Teachers’
pedagogical goals and how they enact this content also shapes practice.
Teachers do not have many opportunities to learn new content within the hours of
the regular school day. Additionally, teachers predominately work alone with little time
to collaborate with their fellow teachers, or learn new content from others with expertise
(Stigler & Hiebert, 1999). This study shows that even in a brief amount of time for
exploration, the tasks used here seemed to lead the preservice teachers to represent their
knowledge with more content and more connections. This suggests that the Internet may
indeed be a productive place to study learning.
Lastly, it is clear that there are possible connections between the goals of the
preservice teachers thinking about this content, the information seeking strategies they
used while searching for mathematical information online, and the ways in which they
represented their knowledge. There are still many questions left unanswered regarding
the relationship among these factors and how they may be interacting or affecting each

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other. A better understanding of these factors could provide mathematics teachers with a
diverse set of tools they can use when reflecting on their understanding of content and the
strategies they could use to strengthen their content knowledge or learn new content.
Mathematics educators (and to an extent, any subject-specific educator) can also
benefit from the findings of this study. Knowing that preservice teachers tend to have two
different orientations to the mathematics content of their classroom is a potential tool that
can be leveraged during teacher preparation programs. Lesson planning in particular is
one moment when preservice teachers are asked to think critically about not just the
content within a particular lesson, but how that content fits in with prior and future
content of the course. Here, mathematics educators can have discussions about detail
orientation and big idea orientations with preservice teachers, as these mentalities are
already present in their thinking about content. Additionally, as mathematics educators
try to provide strategies that will enable preservice teachers to become lifelong learners,
they can also have discussions about different efficient and effective strategies they can
use when using the Internet to learn content. Mathematics educators who study teacher
preparation programs can benefit from the discovery of these two orientations as well. It
is important for researchers who examine the content knowledge of mathematics
teachers, and the content of mathematics classrooms to be aware of mathematics that
connects across a breadth of content, as well as within a depth of content.
This differentiation between big idea orientation and detail orientation also has
implications for those who work with educational technology. Technological Pedagogical
Content Knowledge (TPACK), a popular framework that examines the knowledge
required for teachers’ uses of technology, attempts to identify the interactions between

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content knowledge, pedagogical knowledge, and technology knowledge (Koehler &
Mishra, 2009; Mishra & Koehler, 2006). However, this framework lumps all knowledge
within any category together, no matter the grain size. This research shows that the
different orientations of learners have an impact on what is learned. Within content,
pedagogical, and technological knowledge (and all of their intersections), learners can
focus on better understanding small grain size details, or putting those details together in
larger grain size big ideas, still within a single kind of knowledge. This study has
provided a finer grain lens that can be used to identify the kinds of knowledge important
to the use of technology in teaching.

Limitations
This study contributed to understanding how educators may be learning
mathematics in contexts other than inside the classroom, in particular by researching
content in online educational materials. Yet, despite furthering our understanding in this
area, this research was still a small slice of a much larger pie. This section looks at the
limitations of this research, clarifying the ways in which many important questions have
still yet to be investigated.
By studying preservice teachers, this study looks at a starting point of where
educators begin their careers and the ways in which they use online resources to learn
content. However, this study does not assume (nor would it be expected) that experienced
in-service teachers would use the resource of the Internet in identical ways as preservice
teachers. It is not yet understood how experienced teachers may be using this tool in
productive ways to learn unfamiliar content. Understanding these different approaches is

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crucial for helping preservice teachers transition to possibly more effective information
seeking strategies used by practicing teachers.
In addition to the population limitation of the study described above, there are
questions associated with the gender differences in terms of their learning styles and
interactions with technology. Research has shown already that male and female Internet
users exhibit different information seeking strategies (Hirsh, 1999; Large & Beheshti,
2000; Schacter et al., 1998). The participants in this study all happened to be female,
despite attempts to involve male participants. It is not clear what, if any, differences may
have emerged if males had participated in this research as well.
The content of the tasks utilized in this study also has limitations. The content of
the Pythagorean Theorem was selected intentionally as it draws upon numerous areas of
mathematics including algebra, geometry, and reasoning and proof (among others).
Additionally there are numerous ways of proving this theorem, and many different ways
of illustrating/representing this relationship visually. Because of the broad reach of this
content, I knew that there would be many resources available online that the preservice
teachers in this study could draw upon. However, it is not clear if the tasks used here
would be as successful if alternate content was used that may be harder to find online. It
may be the case that the results seen here are dependent upon the mathematics content
chosen for the task. Different patterns of behavior may have been observed had other
content within the same grade, or even different grades, been used.
One final limitation of this study involves the assessment technique used to
examine the mathematical understanding of the participants. This study had the
preservice teachers construct concept maps to represent their own thinking. With this

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technique, I was only able to get a glimpse of what the participants chose to represent. It
is very likely that participants had much more content and connections than they
represented in their maps. Additionally, this study assumes that they participants are able
to accurately represent their thinking with this technique. The absence of content or
connections in the concept maps produced does not mean that they participants do not
understand that content, just that they didn’t represent it.

Future Directions
Based on the implications and limitations described so far, there are several
opportunities for future study. This study showed a correlation between the orientation of
participants beginning these tasks, the information seeking strategies they used during the
tasks, and how they chose to represent their thinking in the concept maps they produced.
Future studies might attempt to control for the individual factors to examine the extent to
which any factor influences the others. Additionally, future studies might also explore
other potential influences that might also help explain the correlation of these factors.
As described in the limitations section, the focus on preservice teachers in this
study is the first step of a much larger plan. Performing this series of activities with
practicing teachers would illuminate how this, more experienced population is learning
from online resources. Once these patterns of behavior were understood, it would allow
researchers and practitioners to examine any transitions that may exist in patterns of
behavior, as preservice teachers become in-service teachers. This would provide
mathematics educators with opportunities to give preservice strategies that may enhance
how they learn from online resources.

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The mathematical content chosen for this study – The Pythagorean Theorem –
was selected because it is a very well connected piece of mathematics content with a very
rich history. It is not clear if the patterns of behavior found here (both in the information
seeking, and in the concept map construction) would also be visible with the exploration
of other content. Future studies could look at different content within the same level, or at
earlier and later levels, to better understand the impact of specific content on this task.
Lastly, though the participants in this task were asked to locate information that
helped them better understand the content of this standard, it was visible the some
participants extended the task to also attempt to find ways that they could teach this
content. Indeed, teacher understanding of the content is just one step in the process of
teaching the content to students. Future studies might also examine how preservice or inservice teachers are using information seeking strategies to learn not just the content they
are teaching, but the different ways in which this content might be taught.

Conclusion
This study raises new questions about learning mathematics online. The results
when analyzing information seeking patterns of behavior and the content and connections
of concept maps suggest a relationship between how people search and what people
learn. This study serves as a first step to help understand the ways in which teachers may
be taking control of their learning outside of teacher preparation programs. As
mathematics teacher educators, the more we understand about this process, the better
equipped we become to help our beginning teachers become life-long learners after they
leave our institutions.

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This study extends the research on teacher learning beyond the contexts of teacher
preparation programs and their school field experiences. While examination of the
learning in these environments is important, it is also important to recognize that these
are not the only environments where learning takes place. The Internet currently is, and
will continue to be, a source of learning for many preservice and practicing teachers. The
discovery of different orientations towards the exploration of mathematics content
uncovered study serves as a starting point towards understanding both what can be
learned online by mathematics teachers and how it can be learned.

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#

APPENDIX

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#
Thank you for helping out with this study.
We know that there is no way any amount of math classes can prepare teachers for every piece of
mathematics they will encounter over their career. Teachers need to have support that they can
use when they leave their teacher preparation program and encounter unfamiliar mathematics
The purpose of this study is to help us better understand what kind of supports teacher might
benefit from as they expand their math knowledge on their own. To that end, this study is not to
evaluate what you know and don’t know. It’s to help us understand how teachers might be using
the Internet to support their learning of, and knowledge of mathematics for teaching.
This study will have you complete several tasks and use the Internet as a resource to complete
several of the tasks.
Imagine yourself teaching an 8th grade geometry class about the Pythagorean Theorem (this is
usually when the Pythagorean Theorem is taught to students). What do you imagine you would
be responsible for teaching students about the Pythagorean Theorem?

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#
A concept map is a visual representation that connects different concepts together with
connecting phrases around a central theme. Create a concept map around the theme of
“Pythagorean Theorem”. You may wish to use some, all, or none of the terms listed below:
• Triangle

• Leg

• Angle

• Length

• Right Angle

• Square

• Right Triangle

• Pythagorean Theorem

• Side

• Proof

• Hypotenuse

• Converse

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#
Below are the Common Core State Standards related to the Pythagorean Theorem that 8th grade
students are expected to know and do.
8.G.6. Explain a proof of the Pythagorean Theorem and its converse.
8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right
triangles in real-world and mathematical problems in two and three dimensions.
8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a
coordinate system.
Imagine that some day, you are expected to teach your class about the Pythagorean Theorem and
are using these standards as a guide.
Focusing on standard 8.G.6:
• What do you think is meant by this standard? Can you rephrase this standard in your own
words?
• What are some questions you have regarding what is meant by this standard?
• Why do you think this standard might be included in what students need to know about
mathematics?
• What do you think your goals might be for this standard?

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#
With the ultimate goal of having students be able to explain a proof of the Pythagorean
Theorem and its converse (Standard 8.G.6.), use the Internet to help further develop your own
abilities in explaining a proof of the Pythagorean Theorem and its converse. (~About 20
minutes)
For each site you find useful, identify what about the site and its contents you find helpful.
Site:
Description:
Site:
Description:
Site:
Description:
Site:
Description:
Site:
Description:

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#
Explore each of the links below and study the math content on its pages. How does it align with
what you are currently thinking about proving the Pythagorean Theorem and its converse, and
how does it push your thinking so far? (~About 25 minutes)

http://www.pbs.org/wgbh/nova/proof/puzzle/
Describe:

http://www.platinumgmat.com/gmat_study_guide/pythagorean_theorem
Describe:

http://ronblond.com/MathGlossary/Division03/Pythagorean%20Theorem/index.html
Describe:

http://www.youtube.com/watch?v=r382kfkqAF4
Describe:

http://www.youtube.com/watch?v=wxUnodhYGEQ
Describe:

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#
Revisit your concept map from before. Think about what additions/modifications/deletions
would you make to your map after having engaged with this task. Using the theme of
“Pythagorean Theorem”, create a concept map that contains the following terms:
• Triangle

• Leg

• Angle

• Length

• Right Angle

• Square

• Right Triangle

• Pythagorean Theorem

• Side

• Proof

• Hypotenuse

• Converse

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#
Below are the Common Core State Standards related to the Pythagorean Theorem that
8th grade students are expected to know and do.
8.G.6. Explain a proof of the Pythagorean Theorem and its converse.
8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in
right triangles in real-world and mathematical problems in two and three dimensions.
8.G.8. Apply the Pythagorean Theorem to find the distance between two points in
a coordinate system.
Imagine that some day, you are expected to teach your class about the Pythagorean
Theorem and are using these standards as a guide.
Focusing on standard 8.G.6:
• What do you think is meant by this standard? Can you rephrase this standard in your
own words?
• What are some questions you have regarding what is meant by this standard?
• Why do you think this standard might be included in what students need to know about
mathematics?
• What do you think your goals might be for this standard?

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#

Post-Interview Questions

In what ways do you think this task has impacted what you know about the Pythagorean
Theorem?
What are some questions you still have around the Pythagorean Theorem after doing this
task?

In what ways did you find the different Internet resources beneficial?
In what ways could the Internet resources have better helped you?

About how often do you go online for academic use? For personal use?
What kinds of tasks do you do tend to do online?
How would you rate your skills as an Internet user?
How would you rate your skills in using Internet search engines?

What year and major are you?
What math classes have you taken?

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REFERENCES

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#
REFERENCES#

Ball, D. L. (2000). Bridging practices: Intertwining content and pedagogy in teaching and
learning to teach. Journal of Teacher Education, 51(3), 241–247.
Ball, D. L., & Hill, H. C. (2009). The Curious - and Crucial - Case of Mathematical
Knowledge for Teaching. Phi Delta Kappan, 91(2), 68–71.
Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who
knows mathematics well enough to teach third grade, and how can we decide?
American Educator, 29(3), 14–17, 20–22, 43–46.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content Knowledge for Teaching: What
Makes It Special? Journal of Teacher Education, 59(5), 389–407.
Baroody, A. J., & Bartels, B. H. (2000). Using Concept Maps to Link Mathematical
Ideas. Mathematics Teaching in the Middle School, 5(9), 604–609.
doi:10.2307/41181767
Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An Alternative Reconceptualization of
Procedural and Conceptual Knowledge. Journal for Research in Mathematics
Education, 38(2), 115–131. doi:10.2307/30034952
Beckmann, S. (2011). Mathematics for elementary teachers with activity manual. Pearson
Education, Inc.
Bilal, D. (2000). Children’s Use of the Yahooligans! Web Search Engine: I. Cognitive,
Physical, and Affective Behaviors on Fact-Based Search Tasks. Journal of the
American Society for Information Science and Technology, 51(7), 646–665.
Bilal, D. (2001). Children’s Use of the Yahooligans! Web Search Engine: II. Cognitive
and Physical Behaviors on Research Tasks. Journal of the American Society for
Information Science and Technology, 52(2), 118–136.
Bilal, D. (2002). Children’s Use of the Yahooligans! Web Search Engine. III. Cognitive
and Physical Behaviors on Fully Self-Generated Search Tasks. Journal of the
American Society for Information Science and Technology, 53(13), 1170–1183.
Brownell, W. A. (1938). Two kinds of learning in arithmetic. Journal of Educational
Research, 31(9), 656–664.
Brownell, W. A. (1947). The Place of Meaning in the Teaching of Arithmetic. The
Elementary School Journal, 47(5), 256–265.

121#

#
Cochran-Smith, M., Feiman-Nemser, S., McIntyre, D. J., & Demers, K. E. (Eds.). (2008).
Handbook of Research on Teacher Education: Enduring Questions in Changing
Contexts (3 edition.). New York!: Manassas, VA: Routledge.
Fidel, R., Davies, R. K., Douglass, M. H., Holder, J. K., Hopkins, C. J., Kushner, E. J., …
Toney, C. D. (1999). A visit to the information mall: Web searching behavior of
high school students. Journal of the American Society for Information Science,
50(1), 24–37. doi:10.1002/(SICI)1097-4571(1999)50:1<24::AID-ASI5>3.0.CO;2W
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics:
An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural
knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence
Erlbaum Associates.
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking Pedagogical Content
Knowledge: Conceptualizing and Measuring Teachers’ Topic-Specific
Knowledge of Students. Journal for Research in Mathematics Education, 39(4),
372–400.
Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., &
Ball, D. L. (2008). Mathematical Knowledge for Teaching and the Mathematical
Quality of Instruction: An Exploratory Study. Cognition and Instruction, 26(4),
430–511.
Hirsh, S. G. (1999). Children’s relevance criteria and information seeking on electronic
resources. Journal of the American Society for Information Science, 50(14),
1265–1283. doi:10.1002/(SICI)1097-4571(1999)50:14<1265::AIDASI2>3.0.CO;2-E
Horney, M. A., & Anderson-Inman, L. (1994). The ElectroText Project: Hypertext
reading patterns of middle school students. Journal of Educational Multimedia
and Hypermedia, 3(1), 71–91.
Hough, S., O’Rode, N., Terman, N., & Weissglass, J. (2007). Using concept maps to
assess change in teachers’ understandings of algebra: a respectful approach.
Journal of Mathematics Teacher Education, 10(1), 23–41. doi:10.1007/s10857007-9025-0
Jones, B. D. (2002). Recommendations for implementing internet inquiry projects.
Journal of Educational Technology Systems, 30(3), 271–291. doi:10.2190/T1BVXAPG-PR0N-4BV4
Juvina, I., & Oostendorp, H. van. (2004). Extracting semantic and pragmatic information
from web navigation data for user modeling. In Editor (Ed.), Comprehension of
text and graphics: Basic and applied issues (pp. 102–107). Amsterdam: ACM
Digital Library.

122#

#
Kafai, Y., & Bates, M. J. (1997). Interet web-searching instruction in the elementary
classroom: Building a foundation for information literacy. School Library Media
Quarterly, 103–111.
Kilpatrick, J. F., Swafford, J., Findell, B., National Research Council (U. S.), &
Mathematics Learning Study Committee. (2001). Adding it up: Helping children
learn mathematics. Washington, DC: National Academy Press.
Koehler, M. J., & Mishra, P. (2009). What is technological pedagogical content
knowledge? Contemporary Issues in Technology and Teacher Education, 9(1),
60–70.
Kuiper, E., & Volman, M. (2008). The web as a source of information for students in K12 education. In J. Coiro, M. Knobel, C. Lankshear, & D. J. Leu (Eds.),
Handbook of Research on New Literacies (pp. 241–266). Taylor & Francis.
Kuiper, E., Volman, M., & Terwel, J. (2005). The Web as an Information Resource in K12 Education: Strategies for Supporting Students in Searching and Processing
Information. Review of Educational Research, 75(3), 285–328.
Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2009).
Connected Mathematics: Looking for Pythagoreas - The Pythagorean Theorem.
East Lansing, MI: Pearson Education, Inc.
Large, A., & Beheshti, J. (2000). The web as a classroom resource: Reactions from the
users. Journal of the American Society for Information Science, 51(12), 1069–
1080. doi:10.1002/1097-4571(2000)9999:9999<::AID-ASI1017>3.0.CO;2-W
Lawless, K., & Kulikowich, J. (1996). Understanding Hypertext Navigation Through
Cluster Analysis. Journal of Educational Computing Research, 14(4), 385–399.
doi:10.2190/DVAP-DE23-3XMV-9MXH
Lawless, K., & Schrader, P. G. (2008). Where do we go now? Understanding research on
navigation in complex digital environments. In J. Coiro, M. Knobel, C.
Lankshear, & D. J. Leu (Eds.), Handbook of Research on New Literacies (pp.
267–296). Taylor & Francis.
Lorenzen, M. (2001). The land of confusion?: High school students and their use of the
World Wide Web for research. Research Strategies, 18(2), 151–163.
doi:10.1016/S0734-3310(02)00074-5
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding
of fundamental mathematics in china and the united states. Mahwah, N.J.:
Lawrence Erlbaum Associates.
MacGregor, S. (1999). Hypermedia Navigation Profiles: Cognitive Characteristics and
Information Processing Strategies. Journal of Educational Computing Research,
20(2), 189–206. doi:10.2190/1MEC-C0W6-111H-YQ6A
123#

#
Mishra, P., & Koehler, M. J. (2006). Technological Pedagogical Content Knowledge: A
Framework for Teacher Knowledge. Teachers College Record, 108(6), 1017–
1054.
National Council of Teachers of Mathematics. (2000). Principles and Standards for
School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Governors Association Center for Best Practices, & Council of Chief State
School Officers. (2010). Common Core State Standards for Mathematics.
Washington, DC: National Governors Association Center for Best Practices,
Council of Chief State School Officers.
Ruiz-Primo, M. A., Schultz, S. E., Li, M., & Shavelson, R. J. (2001). Comparison of the
reliability and validity of scores from two concept-mapping techniques*. Journal
of Research in Science Teaching, 38(2), 260–278. doi:10.1002/10982736(200102)38:2<260::AID-TEA1005>3.0.CO;2-F
Schacter, J., Chung, G. K. W. K., & Dorr, A. (1998). Children’s internet searching on
complex problems: Performance and process analyses. Journal of the American
Society for Information Science, 49(9), 840–849. doi:10.1002/(SICI)10974571(199807)49:9<840::AID-ASI9>3.0.CO;2-D
Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching.
Educational Researcher, 15(2), 4–14.
Shulman, L. S. (1987). Knowledge and Teaching: Foundations of the New Reform.
Harvard Educational Review, 57(1), 1.
Skemp, R. R. (1976). Relational understanding and instrumental understanding.
Matheamtics Teaching, 77, 20–26.
Sowder, J. T. (2007). The mathematical education and development of teachers. In F. K.
Lester (Ed.), Second handbook of research on mathematics teaching and learning
(pp. 157–224). Charlotte, NC: Information Age Publishing.
Spiro, R. J., & Jehng, J.-C. (1990). Cognitive flexibility and hypertext: Theory and
technology for the nonlinear and multidimensional traversal of complex subject
matter. In Cognition, education, and multimedia: exploring ideas in high
technology (pp. 163–206). Hillsdale, NJ: Lawrence Erlbaum Associates.
Star, J. R. (2005). Reconceptualizing Procedural Knowledge. Journal for Research in
Mathematics Education, 36(5), 404–411.
Stigler, J. W., & Hiebert, J. (1999). The Teaching Gap: Best Ideas from the World’s
Teachers for Improving Education in the Classroom. New York: Free Press.
U.S. Census Bureau. (2013). Computer and Internet Use in the United States (No. P20569). Retrieved from https://www.census.gov/prod/2013pubs/p20-569.pdf
124#

#
Wallace, J. D., & Mintzes, J. J. (1990). The concept map as a research tool: Exploring
conceptual change in biology. Journal of Research in Science Teaching, 27(10),
1033–1052. doi:10.1002/tea.3660271010
Wilber, D. J. (2008). College students and new literacy practices. In J. Coiro, M. Knobel,
C. Lankshear, & D. J. Leu (Eds.), Handbook of Research on New Literacies (pp.
551–581). Taylor & Francis.
Williams, C. G. (1998). Using Concept Maps to Assess Conceptual Knowledge of
Function. Journal for Research in Mathematics Education, 29(4), 414–421.
doi:10.2307/749858

125#