UFDC Home  myUFDC Home  Help 



Full Text  
PAGE 1 IN SEARCH OF A MATHEMATICS DI SCOURSE MODEL: CONSTRUCTING MATHEMATICS KNOWLEDGE THROUG H ONLINE DISCUSSION FORUMS By MADELINE ORTIZRODRGUEZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 1 PAGE 2 2008 Madeline OrtizRodrguez 2 PAGE 3 To my husband, Arturo BirdCarmona, and our children, Maniel BirdO rtiz and Nianti BirdOrtiz. Their constant support and validation have been invaluable. 3 PAGE 4 ACKNOWLEDGMENTS My dissertation journey started much before the beginning of my coursework or the preparation of the proposal. My immediate fam ily always supported and encouraged me to continue studying. My colleagues at the Inter American University of Puerto Rico also pushed me to continue my postgraduate studies and to fulfill my dream of completing a doctoral degree. I especially want to thank Prof. Wanda Ortiz Carrin, Dr. Julie Bruch, and Prof. Edna Muoz. I also want to thank my committee chair, Dr. Kara Dawson, and committee methodologist, Dr. Mirka KoroLjungberg, as well as to my co mmittee members, Dr. Richard Ferdig and Dr. Ricky Telg, for their time and support. Thei r thoughtful recommendati ons, suggestions, and encouragement are embedded in this dissertation. Special thanks goes to Whitney Waechter for her enormous editorial assistance, making this project read in E nglish instead of in Spanishlike English. I am also grateful to Angie Coln, MLS, the Center of Information Access Director, at the Inter American Univ ersity of Puerto Rico in Fajardo, and to Dr. Iris M. Gmez for proofreading mathematics technica l descriptions included in this dissertation. We all worked together and collaborated to complete this proj ect. Their help has made possible the completion of this work. Finally, I want to acknowledge the economic support of the Inter American University of Puerto Rico that made possible my stay at Gainesville, Florida during the first four years of this journey. There is always a group of people that help you fulfill your goals. Sometimes they are behind the scenes and invisible to most observers. However, they are around you, with you, in your thoughts, and in your heart. They support you in so many ways beyond your studies. They become your support group and help you keep going forward. I thank Winnie Cook, for believing in me when I started at UF, allowing me to tutor math. I thank my friend Dina Mayne who became my lunch buddy and someone with whom to talk. I thank her for hearing me and 4 PAGE 5 sharing the little free time she had. I thank my sister, Lourdes OrtizHouman, who supported and encouraged me since the first day I got to Florid a, allowing me to feel at home even though I was so far away from Puerto Rico. And I thank my husband and my kids, who have always believed in me. Their love is in my heart and keeps me warm wherever I go. And I thank everyone else who in one way or another helped me maintain focus on my studies. The distance between our dreams and our goals is as big or as small as the time and commitment we put into reaching them. We are part of a whole where ever we go: at home, work, or school. We become part of a commun ity and serve that community in one way or another. Without it, well be lost. Thats why I want to thank all my professors and classmates, as well as my supervisors and coworkers. Together, with my immediate family, they were part of my community during my time in Gainesville allowing me to fulfill this goal. Finishing my dissertation is our accomplishment. 5 PAGE 6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4LIST OF TABLES .........................................................................................................................10LIST OF FIGURES .......................................................................................................................11ABSTRACT ...................................................................................................................... .............12CHAPTER 1 INTRODUCTION ................................................................................................................ ..13Purpose ....................................................................................................................... ............16Significance of the Study ........................................................................................................17Research Question ..................................................................................................................19Definition of Terms ................................................................................................................19Delimitations ................................................................................................................. ..........202 REVIEW OF LITERATURE .................................................................................................21Using Technology in Mathematics Education ........................................................................23Writing in Ma thematics ........................................................................................................ ..28Communication in Mathematics Learning .............................................................................37Using Discussion Forums ................................................................................................37Discussion Forums Research in Mathematics Education ................................................39Communities of Practice .........................................................................................................44Interaction and Cooperation ............................................................................................45Interaction, Negotiation, and Collaboration ....................................................................46Communities of Practice in Mathematics ........................................................................47Summary ....................................................................................................................... ..........493 QUALITATIVE RESEARCH FOUNDATIONS ..................................................................52Qualitative Research Fundamentals .......................................................................................54Qualitative Research Approach ..............................................................................................55Epistemology: Constructionism ..............................................................................................56Theoretical Perspective: Social Constructionism ...................................................................57Social Constructionism in Edu cation: Alternative Pedagogies .......................................58Reflexive deliberation ..............................................................................................59Polyvocal pedagogy .................................................................................................59Collaboration ............................................................................................................60Summary ..................................................................................................................61 6 PAGE 7 Social Constructionism Assumpti ons and Mathematics Knowledge ..............................61Knowledge is socially constituted ............................................................................62Linguistic signals, expe rience, and culture ..............................................................63Verification of theory through re search is rendered suspect ....................................64Subjectivity Statement ............................................................................................................65Research Setting .............................................................................................................. .......67The Math Forum @ Drexel Web Site .............................................................................68History of the Math Forum @ Drexel .............................................................................68The Math Forum @ Drexel Services ...............................................................................69alt.math.undergrad Discussion Group .............................................................................71Reflections about the Pilot Study ...........................................................................................72Lessons Learned through the Pilot Study ........................................................................74Criticisms .................................................................................................................... .....754 METHODOLOGY AND METHODS ...................................................................................82Discourse Analysis Methodology ...........................................................................................82Gees Discourse Analysis Method ..........................................................................................87Working with Transcriptions ...........................................................................................88Building Tasks .................................................................................................................89Inquiry Tools ...................................................................................................................90Discourse Models ............................................................................................................92Reviewing the Preliminary Discourse Model ..................................................................92To Summarize .................................................................................................................9 3Data Collection Procedures ....................................................................................................93Data Analysis Process in this Dissertation Project .................................................................95Questioning the Data .......................................................................................................97Application of Gees Disc ourse Analysis Method ..........................................................97Validity in Qualitative Research .............................................................................................9 9Limitations ................................................................................................................... .........1045 CONSTRUCTING MATHEMATICS KNOWLEDGE THROUGH ONLINE COLLABORATION ............................................................................................................110Preamble ...................................................................................................................... .........110Introduction to Augusts Data ..............................................................................................111Part 1: Analysis of Ac tivities and Connections ....................................................................113Problems, Questions, and Inquiries Introduction ..........................................................113Setting: Use of time, space, and characters ............................................................113Catalyst: Analysis of problems and questions ........................................................114Problem Evaluation and Solution Generation ...............................................................118Augusts Coda(s): Additional Information ....................................................................129Part II: Analysis of Sign System s, Knowledge, and Identities .............................................130Symbols of Common Language and Mathematics Notation .........................................130Identities of the Fo rum Participants ..............................................................................133Moving toward a Discourse Model: Summ ary of Augusts Da ta Analysis .........................134 7 PAGE 8 6 NEGOTIATING MATHEMATICS MEANING AND UNDERSTANDING ....................141Introduction to Septembers and Octobers Data .................................................................141Part I: Analysis of Ac tivities and Connections .....................................................................142Problems, Questions, and Inquiries Introduction ..........................................................142Setting: Use of time, space, and character..............................................................143Catalyst: Analysis of problems and questions ........................................................143Problem Evaluation and Solution Generation ...............................................................147Septembers and Octobers Coda(s): Additional Information .......................................153Part II: Analysis of Identities ................................................................................................155Reviewing the Preliminary Mathematics Discourse Model .................................................1587 GENERATING NEW MEANING AND UNDERSTANDINGTHROUGH ONLINE POLYVOCAL COLLABORATION ...................................................................................168Introduction to Novembers and Decembers Data ..............................................................168Part I: Analysis of Ac tivities and Connections .....................................................................169Problems, Questions, and Inquiries Introduction ..........................................................169Setting: Use of time, space, and character..............................................................169Catalyst: Analysis of problems and questions ........................................................172Problem Evaluation and Solution Generation ...............................................................176Novembers and Decembers Coda(s): Additional Information ...................................184Part II: Analysis of Identities and Relationships ..................................................................187Refining the Preliminary Math ematics Discourse Model ....................................................1918 CONCLUSIONS ................................................................................................................. .197Summary of Major Findings .................................................................................................198Identity of Participants ..................................................................................................198Sign Systems or Alternative Ways to Communicate ....................................................199Activities Related to Mathematics Cognitive Development .........................................200Activities Related to Affective and Emotional Support ................................................202Connections Within, Between, and Among the alt.math.undergrad Forum Threads ...204Connections to Other Resources outside the Math Forum @ Drexel ...........................205In Summary ...................................................................................................................205Implications and Recommendations .....................................................................................207Recommendations for Practice ......................................................................................208Recommendations for Research ....................................................................................209Conclusions ...........................................................................................................................210APPENDIX A PILOT STUDY ................................................................................................................. ....212Using Public Discussion Forums to Construct Mathematics Knowledge ............................212Review of Literature .............................................................................................................213 8 PAGE 9 Methodology ................................................................................................................... ......216Participants .................................................................................................................. ..217Researcher .................................................................................................................... .218Procedure ..................................................................................................................... ..219Results ...................................................................................................................................219Case of the Integral ........................................................................................................220Trigonometric Identity Case ..........................................................................................220Discussion .................................................................................................................... .........221References .................................................................................................................... .........222LIST OF REFERENCES .............................................................................................................224BIOGRAPHICAL SKETCH .......................................................................................................241 9 PAGE 10 LIST OF TABLES Table page 21. Domains of discourse in mathematics ....................................................................................51 31. Math Forum @ Drexel web site resources by section. ..........................................................78 32. Data summary for October 2004: Threads with 10 to 25 postings .........................................79 33. Parts of a story with higherorder structure ............................................................................81 51: General description of th readed discussions in August ........................................................136 61. General description of threaded di scussions in September and October ..............................161 62. Web resources referenced to in the threaded discussions of September and October data ..162 63. Closing message topics of September and October data ......................................................163 71. General description of threaded di scussions in November and December ..........................193 72. Connecting the discussion forum with ot her math resources (from Novembers and Decembers Data) ............................................................................................................194 73. Use of direct and indirect questions in the opening posts of the threads..............................195 10 PAGE 11 LIST OF FIGURES Figure page 21. Software types used to construct knowledge in mathematics ................................................50 41. Components of an ideal discourse analysis ..........................................................................108 42. Inquiry tools used in Gees discourse analysis .....................................................................109 51. Radius of an arc tree. Each color represents a differe nt participant. ................................137 52. Integrate!!! tree. ..................................................................................................... ...........138 53. Graph of 3^x=y .....................................................................................................................139 54. Writing the discourse model .............................................................................................. ...140 61. Graph planarity tree.. ................................................................................................. .......164 62. Finding derivative tree. .....................................................................................................165 63. Consecutive terms tree.. ............................................................................................... .....166 64. Logs tree.. ............................................................................................................ ..............167 11 PAGE 12 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy IN SEARCH OF A MATHEMATICS DI SCOURSE MODEL: CONSTRUCTING MATHEMATICS KNOWLEDGE THROUGH ONLINE DISCUSSION FORUMS By Madeline OrtizRodrguez December 2008 Chair: Kara Dawson Cochair: Mirka KoroLjungberg Major: Curriculum and Instruction The purpose of our study was to examine how participants of a public online mathematics discussion forum collaborated, negotiated, a nd generated new meaning and understanding through dialogue, intertextuality and polyvocality while construc ting undergraduate mathematics knowledge. Our study was conducted under Kenneth Gergens theoretical pers pective of social constructionism and the methodology and methods proposed by J. P. Gee. Analysis of data included five months of threaded discussions divided into three periods of analysis which gave the researcher the oppo rtunity to develop, review, and refine a preliminary mathematics discourse model. Preliminary mathematics discourse models show ed how participants engaged in dialogues that included specific activities and connections Data also showed how participants used common language and mathematical symbols to communicate, the state of mind of the forum participants (social identity), the relationships they engaged in, and how they resolved their mathematical questions, problems, and inquiries. 12 PAGE 13 CHAPTER 1 INTRODUCTION Only dialogue, which requires critical thinking, is also cap able of generating critical thinking. Without dialogue there is no communication, and without communication there can be no true education. Paulo Freire, Pedagogy of the Oppressed (1970, p. 81) Teaching and learning is going through a continuous process of change that since the twentieth century can be thought of as a digital revolution (Robl yer, 2003). This revolution has influenced schools and colleges as well, including the teaching and learning of mathematics at all levels. However, this digital revo lution is not a panacea in education. The impact on education of the use of computers, networks, and the Intern et has been widely criticized (Oppenheimer, 1997, 2003; Bains, 1997; Postman, 1992, 1995; and Stoll, 2000) Nevertheless, it is research in theory and practice which can document the integrati on of new technologies into the educational environment, while helping to answer many of the questions posed by critics and teachers. Changes in education are not new; they ha ve occurred throughout hi story. In the 1800s, for example, teaching and learning incorporated the use of correspondence courses in addition to the existing traditional facetoface e ducation. Later, in the twentieth century, digital alternatives became available to society and rapidly ma de their way into the educational setting. The World Wide Learn (2005) indicated that digital environments in education continue to increase in every subject matter The use of computers, networks, and the Internet has changed schools and colleges environment as well, allowing for mo re studentcentered ac tivities (Knowlton, 2000), different types of interaction (Moore, 1989; Gunawardena, Lowe, & Anderson, 1997), and different types of communication (Morrison & Guenther, 2000; Berge, 2000; Weiss, 2000; and Hacker & Niederhauser, 2000). Research by the Alliance for Higher Education Competitiveness has indicated that quality in online learning experiences is about teachi ng strategies and not technology (Abel, 2005). That is, software and hardware acquisition is not enough; inservice 13 PAGE 14 computer training that focuses on classroom integration technology strategies (Dawson, 1998, p.2), as well as research about formal and informal communities of learners, can make a difference in education. New technological resources in communication and networ king provide students and teachers with the opportunity to study, explore, collaborate, and investigate while joining together in virtual communities of learners. St udents and teachers can communicate regularly via synchronous and asynchronous electronic tools that enable sending and re ceiving feedback in a timely fashion (Berge, 2000; Wheeler, 2002). Togeth er, they become part of a community of learners (Lock, 2002) in which the distance from the educational institution to the students location is no longer a limitation to acqui ring and constructing new knowledge (Simonson, Smaldino, Albright, & Zvacek, 2003). Virtual communities include peopl e from different places and countries where spatial and temporal boundaries are entirely symboli c (Shumar & Renninger, 2002, p. 6). These communities are created and maintained thr ough the use of synchronous and asynchronous communication tools (Lock, 2002) that help devel op environments sustained by the interaction of their members. Synchronous communication tools are used at the same time from different places; examples of such include the use of chat rooms and videoconferences (Simonson, Smaldino, Albright, & Zvacek, 2003). Asynchronous communication occurs at different times and different places and includes tools such as email, discussion forums, and listserves (Simonson, Smaldino, Albright, & Zvacek, 2003). Gilbert and Moore (1998) and Wagner (1994, 1997) defined interact ion as an interplay and exchange of ideas in which individuals and groups collaborate with each other (as cited in Roblyer & Ekhaml, 2000). Different types of in teractions include lear nercontent, learner14 PAGE 15 instructor, learnerlearner, (Moore, 1989), and learnerinterfa ce (Hillman, Willis, & Gunawardena, 1994). Researchers suggested that through interaction, it is possible to promote critical thinking, higherorder thinking, distributed thinking, a nd constructive thinking (Berge & Muilenburg, 2001) (as cited in Tu and Corry, 2003, p. 309) to provide an emotional, supportive space in which to share and find information (Ste in & Glazer, 2003), to de velop a sense of being selfdirected while taking res ponsibility for learning (Lee & Gibs on, 2003), and to interact with others while constructing knowledge be yond independent means (Vygotsky, 1978). An asynchronous online communication tool, th e discussion forum is a tool where people interested in the same topic or subject matter can interact with each other by asking questions, giving answers, and clarifying ideas. It is through this interaction that users develop a reciprocal communication system, one that emphasize[s] le arner developments in cognition, motivation, and social advancement (Chou, 2004, p. 11). This process allows discussion forum participants to develop new skills and construct new knowle dge (Hacker & Niederhauser, 2000). Gilbert and Moore (1998) also stated that social rapport and collaboration c ould lead to grea ter levels of interaction (as stated in Roblye r, 2000). In this way, a cyclical relationship takes place between collaboration and increased interaction and viceversa. Discussion forums can be classified as privat e or public digital spac es. The latter are not password protected or restrict ed for members of a specific organization or in stitution. Anyone with access to an Internet connect ion can log into a public discussion forum to post a message at any time and from any place. It is this flexibility that allows participants to contribute to and to be part of a larger community of learners. An example of such a public environment is The Math Forum @ Drexel This electronic space began as a project to produce compute rgenerated videotapes (Dawning, 1997, first 15 PAGE 16 16 paragraph) and later developed into the Ge ometry Forum (in 1992). This dynamic community continued to grow under the leadership of Sw arthmore College in Pennsylvania (Grandgenett, 2001). Students, teachers, research es, parents, and math enthusia sts can benefit of the tools provided by this environment. It promotes interaction between th ese groups through the availability of chat rooms and discussion forums. It also houses bibliographies related to math education for teachers and researchers, as well as journal abstracts, articles, and other teaching and learning resources. Research about the use of discussion forums in mathematics education is scarce and contradictory. Thus this study tr ies to clarify how knowledge construction takes place in such environments. To do so it examines writing in online asynchronous learning in combination with informal learning environments. That is, comm unities of practice where more knowledgeable others engage in collaborative pr actices with mathematics learners. Purpose Our study connected research from four general areas of study: technology use in mathematics, writing in mathematics, comm unication in mathematics learning, and online communities of practice. Specifically, written communications in an online public discussion forum were examined to describe and analy ze how mathematics knowledge was constructed. The purpose of this study was to examine th e types of online dialogues and discursive collaboration that took place in an online public di scussion forum that faci litated the construction of mathematics knowledge, meaning making, a nd understanding. Construction of knowledge was defined from a social constructionist perspe ctive (Chapter 3). Data analysis was based on Gees (1999, 2005) discourse analysis methods (Chapter 4). PAGE 17 Significance of the Study The use of discussion forums has been research ed in terms of their relationship to distance learning, online courses, and ble nded courses (a combination of facetoface learning and online courses) (Wegerif, 1998; PrezPrado and Thir unarayanan, 2002; Kanuka, Collet, & Caswell, 2002; Da Silva, 2003; Lee & Gibson, 2003; Woods & Ebersole, 2003; Knowlton, 2003; Schallert, et al., 20032004; Fe rdig and Roehler, 20032004; Je tton, 20032004; Im & Lee, 20032004). However, most research studies focus on s ubject matter that is traditionally considered languagebased. Only a few studies have looked at the use of discussion forums in mathematics education. For example, Sener (1997) examined the use of asynchronous environments in a distance education science engineering degree that included mathematics courses. Sliva (2002) looked at an online discussion forum in a hybrid math ematics elementary methods course. Smith, Ferguson, and Caris (2003) studied mathematics in structors experiences at the undergraduate college level in an online discussionbased e nvironment. Lotze (2002) analyzed tutoring effectiveness. Bolin (2003) studied differe nt communication tools and their impact in mathematics learning. Quinn (2005) looked at mathem atics identity. Nevertheless, none of these studies examined the types of dialogue and discursive collaboration occu rring in an online public discussion forum. Nor had they examined how pa rticipants coconstructed knowledge about high school mathematics, and first and second year un dergraduate mathematics, or at how informal mathematics learning could contribute to knowle dge construction. These st udies did not analyze empirical accounts of discursive collaboration, where relati onships between people assuming different roles are described (Satwicz & Stevens, 2008). Sharma and Hannifin (2004) have also identi fied the need for further research about mathematics and technology in informal environments. They stated, 17 PAGE 18 Exploring the impact of differe nt technologies and tools used by learners to develop their learning in informal environments can a llow for more systematic and controlled introduction of informal technologies into fo rmal educational settings. Equally important, such exploration may allow identification of appropriate combinations of structured scaffolding and less structured learning activit ies to support different types of learning. (p. 204) In this way, this study extended the research base beyond languageb ased courses, beyond the point of view of the instructors, beyond formal learning, and beyond prescriptive research. Finally, this research could have implications on mathematics distance education, not only for undergraduate mathematics, but also for math ematics courses offered through virtual schools environments. As Sakshaug (2000) concluded, Distance education is a growing area of instructional delivery. More schools are offering distance education course s than ever before. Mathematics educators must explor e this venue of teaching in order to have input about what and how mathematics is taught (p. 122). Nevertheless, few researches have been conducted in this area. One way to contribute to these needs is to research a public digital space such as The Math Forum @ Drexel where a community of learners al ready exists and where mathematics knowledge is already being constructed through discussion forums (Renninger & Shumar, 2002). Such a study can inform communication and commun ity building as well as informal learning in mathematics. In the Math Forum @ Drexel, participants use text to pose questions and answers, give feedback, recommend resources, and find solu tions. As its participants interact with each other in a textbased environment, they become active learners, reflect about the problems posed, take time to reply, and collaborate with each other to develop new knowledge. In the Math Forum @ Drexel, users appear to be able to communicate effectively and to help each other develop new meanings and understanding. This study will examine how this happens. 18 PAGE 19 Research Question This research will study the following questio n: How does online dialogue and discursive collaboration facilitate the coc onstruction of knowledge in high school and fi rst and second year undergraduate mathematics via a discussion forum named alt.math.undergrad in the Math Forum @ Drexels web site? This study analyzed digital archived data, a collection of primary sources organized as threaded discussion, available through the Web (Bolick, 2006). It examined how participants in the discussion forum constructed mathematics knowledge during the academic semester of August through December of 2004. Definition of Terms Discussion forums: asynchronous online communicati on tools where a group of people interested in the same topic or subject matte r interact with each other by posting questions, answers, or both (Simonson, Sm aldino, Albright, & Zvacek, 2003). Public discussion forum: discussion forums that are accessed freely through the Internet. Community of learners: group of people who share a common goal or objective and help each other to fulfill this goal (L ave & Wenger, 1991; Rogoff, 1994). High school and first and second year undergraduate mathematics courses: mathematics courses geared toward freshm an and sophomore students, including General Math, College Algebra, Trigonometry, Elementary Statistics, PreCalcul us, Calculus I, and Discrete Mathematics. Voluntary participation: selfinitiated posting by help seeking individuals or more knowledgeable others of a writte n contribution to the discussi on forum that can include but is not limited to asking or answering a que stion, giving a hint, or adding comments. Discursive collaboration: when a group of people create meaning together through language to develop shared meaning (Satwi cz & Stevens, 2008). According to Maye (2003), collaboration may take place in voluntary study groups and in consultation with professors, teaching assistants, and tutors. In this study, collaborati on takes place in an online public discussion forum be tween voluntary participants. Social constructionism of knowledge: discourse that leads to meaningmaking or generation of new understandings (Gergen, 1999). This research analyzes written discourse. 19 PAGE 20 Informal learning environments: learning spaces outside th e classroom with no grades attached. According to Offer (2007), informal sources include peers, friends, and family members. However, in this research, sources may include more knowledgeable others that may or may not be known by helpseeking individuals. Delimitations The following boundaries were established in order to conduct this research: 1. This study will analyze the postings to the public discussion forum alt.math.undergrad located in The Math Forum @ Drexel (http://www.mathforum.org). 2. alt.math.undergrad includes questions and answers from different mathematical topics, but only those threads related to high school and first and second year undergraduate mathematics as defined above will be examined. 3. The number of postings in each threaded discus sion varies in terms of participation, from fewer than five to more than 100. In this research, threads with 10 to 25 posting were selected for analysis. Each thread is a discussi on about a specific topic or question that can be subdivided into several conversations or storie s. These conversations can have two or more postings each. A complete story can include six sections; these are setting, catalyst, crisis, evaluation, resolution, and coda (Gee, 2005). 4. The time period selected for study is one academ ic semester which took place from August to December of 2004. 5. Transcripts were developed by the resear cher and reviewed by committee members. 20 PAGE 21 CHAPTER 2 REVIEW OF LITERATURE Those who have studied the use of technology in mathematics teaching and learning have noted that technology mediates learning. That is, learning is different in presence of technology. K. Heid, TechnologySupported Mathematics Learning Environments (2005, 67th NCTM Yearbook, p. 348) Technology in education opened a new field in educational research studies which included analyzing and exploring the impact of audiovisual tools, instructional systems, vocational techniques, and computers on teach ing and learning processes (Roblyer, 2003). However, research in educational technology is no t about the tools or gadgets themselves (Clark, 1983; Kozma, 1991). Instead, it is about how both the old and the new can be used to develop meaning and understanding; it is about how these new tools and techniques can help students learn in a society that is always changing. Learning environments are organized in different ways. One such distribution is presented by Rogoff (1994), where she summar ized instructional approaches around three models: adultrun, childrun, and communities of learners. The first two organized learning activities from toptobottom and from bottomup, respectively. In adultrun instruction, the teacher makes all the decisions, and the purpose of education is the transmission of knowledge. This type of education is base d on behaviorist principles, one of its major proponents being B. F. Skinner (LeverDuffy, McDonald, & Mizell, 2 003). According to Rogoff (1994), the childrun model has children constructing knowledge on their ow n; adults only contribu te to their learning by setting up learning environments. One of the proponents of this model was A. S. Neill, an English teacher who founded Summerh ill, a school where children gr ew in liberty and learned as much as they wanted (Neill, 1963; Hemmings, 1975). 21 PAGE 22 The third instructiona l model proposed and supported by Rogoff (1994), the communities of learners model, is based on participatory theories, where learners collaborate with each other and where learning is the resu lt of a process of interaction conducive to transformation. As Rogoff stated, this model is not a balance between adultrun and childrun instruction; it is a different type of mode l. In informal learning environments, more knowledgeable others support those in need of sp ecific help. Roles are not static, however, and they will change at different times. According to Rogoff, Lave, and Wenger, however, informal learning environments can also promote the development of authentic communities of practice, where people with similar goals and interests interact and construct new knowledge (Rogoff & Lave, 1984; Lave, 1991, 1996; Lave & Wenger, 1991; Wenger, 2001; Rogoff, 1994). Even more, Resnick (1987) suggested there is a need to br ing together formal lear ning (schooling based on individual performance) and informal le arning (based on soci al interactions). This dissertation investigated a community of learners ma de possible through the use of asynchronous communication, over th e Internet, where informal interaction took place through written communication. In this envi ronment, participants were located in different places and contributed at different times. The use of an asynchronous environment in mathematics learning was explored, specifically an online public di scussion forum, where informal mathematics learning took place. This dissertation examined a textbased website, which allowed its users to construct mathematics knowledge through disc ourse while interacting, collaborating, and negotiating outside of the school setting. It focused on the genera tion of transformative written dialogue whose end result was the development of new mathematical understandings and knowledge. More specifically, this dissertation explored the use of an online public discussion 22 PAGE 23 forum as it related to high school and college first and second year undergraduate mathematics topics and the construction of knowledge developed through informal dialogues. The next sections will present a review of related topics. These include (1) using technology in mathematics education, (2) wri ting in mathematics, (3) communication in mathematics learning, and (4) onl ine communities of practice. Using Technology in Mathematics Education The use of technological devices, gadgets, or machines in mathematics is as old as mathematics itself. From the use of pebbles to count to the use of digital technologies to visualize concepts and communicat e ideas, technology has always played an important role in mathematics education (Dilson, 1968). The use of computers in mathematics dates from the second half of the twentieth century, becoming popular in sc hools in the 1970s and 1980s with mainframe computer systems and standalone personal computers used mostly for computerassisted instruction (Roblyer, 2006). The advent of the Web at the end of the 1980s allowed its users in the following decade to share inform ation and to communicate in synchronous and asynchronous environments in an easier way. These new Internet tools also permitted building virtual communities of learners throughout the world. The following pages analyze research about th e use of computers in mathematics teaching and learning. First, a series of articl es by Crowe and Zand (2000a, 2000b, and 2001) and a master thesis by Morley (2007) researching the use of instructiona l technologies in mathematics are reviewed. As it will be shown in the follo wing paragraphs, Crowe and Zand studied the use of computers from a comprehensive perspective and Morley developed a mathematics computer vision. Next, research by Samantha, Peressini, an d Meymaris (2004) exploring effective learning environments in mathematics supported by t echnology are assessed. Lastly, Galindos (2005) 23 PAGE 24 and Roses (2001) studies showing how the Intern et was used as a source of information and communication will be reviewed. Crowe and Zand (2000a, 2000b, and 2001) researched and summarized computer use in mathematics by collecting data from British, American, and Australian universities. They developed a taxonomy of computer uses in mathem atics that included three general categories: (1) the use of productivity tools (such as sp readsheets), (2) the us e of general purpose mathematics software (such as Mathlab, Maple and Mathematica ), and (3) the use of information tools. The taxonomys second category was subdivided into two general subcategories: first, the use of didactic software packages, and second doing mathematics software. Under didactic software packages, Crowe and Zand included the use of multimedia and assessment. These were tools containing computerassisted instruction so ftware used to complement instruction. The second subcategory, doing mathematics software included the use of programming languages, often associated with developing problem solvi ng skills in mathematics (Papert, 1980). Doing mathematics software included the study of skills that allowed students to choose a particular procedure to solve a problem (Schoenfeld, 1989, as cited in Shield and Galbraith, 1998). This category also included geometric visualizati on and algebraic manipul ation software, both generally used to help students concep tualize abstract ideas (Figure 21). When analyzing information tools, Crowe a nd Zand (2000b) found that participants used three types of Internet resources: (1) speci fic, including textual ma terials (books, digital libraries, tutorials, web pages) java modules (interactive de monstrations, investigative dynamicenvironments), and other media; (2) general, consisting of reference resources, journals, indices, and math web sites; and (3) dialogic communication to ols, related to online 24 PAGE 25 support and online courses. These indicated that mathematics teaching and learning was possible through verbal communication (encouraging words and discourses), visual communication (graphics use and geometric representations), and the use of symbols (algebraic notation). This classification is similar to the one presented by Stonewater (2002), which he called the Rule of Three for its graphic, numeri c, and algebraic components. Computerwritten communication was reported as having more advantages than audio communication (Crowe & Zand, 2000b). According to these authors, the major advantage to computerwritten communication is that it lets users have a permanent record of a dialogue, thus allowing them to revisit the same argument fo r different purposes as many times as needed. Written communication, they stated, could be save d in a file, located, accessed, and edited in the same way as any other file (p. 140). They also found that computerwritten communication allowed for increased speed of response b ecause students could receive prompt feedback. However, Crowe and Zands investigation did not include the use of communication tools (synchronous or asynchronous) available through the web. Crowe and Zand (2000b) concluded that techno logy has a fundamental role to play, not only in teaching the present curriculum but also in shaping the curriculum of the future (p. 146). This view was supported by Morleys (2007) ma thematics technology vision. In her research, she concluded that Using IT [instructional tech nology] in mathematics in struction can nurture positive attitudes toward mathematics while creati ng critical thinkers and lifelong learners (p. 30). Morley contended that through IT, mathematic al concepts could be studied through multiple representations and that IT could reduce the gap between mathematical concepts and real world data. As new technologies devel op and old ones improve Crowe and Zands conclusions seem to be disputed, still Morleys vision seems to stay the same. 25 PAGE 26 Three examples of mathematics education re search that concentrated on the use of technology to help students develop meaning in mathematics are those of Samantha Peressini, and Meymaris (2004), Galindo (2005), and Rose ( 2001). The first studied th e use of calculators and computers in a learnercentered environment; th e second two explored the use of the Internet as a source of informa tion and collaboration. According to Shamatha, Peressini, and Me ymaris (2004), technology can be used to transform mathematics teaching and learning th rough the development of effective learning environments. These, they stated, included co mmunity, learner, knowledge, and assessmentcentered environments. They also agreed with Crowe and Zands position regarding the use of technology in mathematics education when they concluded that technologysupported activities can work meaningfully in a learning environmen t that research proves effective (p. 378). By taking a different angle, Galindo (2005) also explored the use of computers in mathematics education. He looked at how the In ternet is supporting and helping enhance the learning and teaching of mathematics (p. 241). He listed different types of resources available on the Internet for teachers and students, including realtime data projects, existing data sets from different organizations, and collaborative projects These resources, he stated, can facilitate meaningful collaboration among individuals or groups to support mathematics learning (Galindo, 2005, p. 251). He also added the possibilities of consulting with experts by submitting questions and engaging in conversations and co llaborative interactions. Some collaborative efforts listed by Galindo incl uded web sites such as Conectando las Matemticas a la Vida: Proyecto Internacional [Connecting Mathematic s to Our Lives: International Project] ( http://www.orillas.org/math ), Class2Class ( http://www.mathforum.org/class2class ), and Global Schoolnets Global Schoolhouse Organization ( http://www.globalschoolnet.org/GSH ). 26 PAGE 27 Rose (2001) went a step further, using the Inte rnet as an information gathering resource in a college calculus course. Her goa l was to increase classroom d ialogue, discourse, interaction, reflection, and writing about mathematics (pp. 1011) while students gathered information about how calculus was used in their particular majors. Students reported they were able to improve their skills in locating information not only about calculus but also about other mathematics topics. Rose also re ported students were able to sp eak of mathematics in a more positive nature and were able to find for themselves its connections to realworld situations to their areas of study and/or pers onal lives (Rose, 2001, p. ix). Rose (2001), Galindo (2005), and Crowe and Za nd (2000b), looked at th e Internet as a source of information and collaborative effo rt. Galindo (2005) and Crowe and Zand (2000b) referenced The Math Forums Ask Dr. Math ( http://mathforum.org/dr.math ), a component of The Math Forum web site that concentrat es on K12 level problems. Dr. Math is a questionandanswer service for mathematics students and their teachers (Galindo, 2005, p. 254). None of these authors, however, looked at the Discussion Forums section of The Math Forum @ Drexel website, a place to Read, post, browse, search, and subscribe to dozens of discussions, ranging in focus from AP courses and Investigations curricula to history, from policy and news to professional teaching associations, from Sp anish puzzles to software, and more ( The Math Forum @ Drexel 2006). However, the Internet is also a place for formal mathematics learning. The increased number of courses and programs offered online has a direct impact on mathematics education. National reports have pointed to the increas e of distance learning courses and programs, including those that include the teaching and learning of mathematics (Conference Board of Mathematical Sciences (Lutzer, 2000; The Worl d Wide Learn, 2005). In its 2000 report, the 27 PAGE 28 CBMS, an umbrella organization of professional mathematics organizations established in the 1960s, added a new section on distance learning and mathematics for the first time in its history. In addition, the World Wide Learn web site, a directory of higher education online programs, continuously adds new programs and offerings to its list of programs. In 2005, this web site listed more than 25 online programs related to mathematics offered in ten states around the United States. Associations such as the Amer ican Mathematical Asso ciation of TwoYear Colleges (AMATYC) and the Distance Learni ng Committee emphasized the importance of collaboration and communication in online education (AMATYC, 2005). In summary, these authors examined the use of computers in mathematics education in different ways. They also looked at the Internet including its tools and information repository as it related to mathematics learning. They set th e groundwork for the integration of technology in mathematics education. The next section will examine research about writing in mathematics. The introduction will examine the position taken by the National Council of Teachers of Mathematics. Then, different communication domains in mathematics and mathem atics research about cognitive development and writing are examined. To end this section, research studies about writing in mathematics using digital and nondigital onetoone a nd onetomany environments are reviewed. Writing in Mathematics Writing in mathematics is an outcome of the 1960s W riting Across the Curriculum movement (Clarke & Waywood, 1993, p. 235). As a result of it, the National Council of Teachers of Mathematics (NCTM), after a major revision of the mathematics curriculum during the 1980s, added communication sta ndards to all grade levels that included talking, reading, and writing components. In their document, Curriculum and Evaluation Standards for School Mathematics (1989), they stated the importance of writing. 28 PAGE 29 For grades K4: Writing about mathematics, such as describing how a problem was solved, also helps students clarify their thinking and de velop deeper understanding (NCTM, 1989, p. 26). For grades 58: Opportunities to explain, conjecture, and defend ones ideas orally and in writing can stimulate deeper understandings of concepts and principles (NCTM, 1989, p. 78). For grades 912: All students need exte nsive experience listen ing to, reading about, writing about, speaking about, reflecting on, and demonstra ting mathematical ideas (NCTM, 1989, p. 140). This was ratified by the NCTM in a following edition titled Professional Standards for School Mathematics (NCTM, 1991) It was the NCTMs contention that writing in mathematics was connected to deeper understanding. However, as important as it may be, according to Quinn and Wilson (1997), writing activities are not used consistently in school mathematics (as cited by McIntosh & Draper, 2001). This is also the case in collegelevel mathem atics. Burton and Morgan (2000) reported that there has been an increase in the recognition of the importance of communication skills in mathematics by professional organizations a nd researchers but that the training of mathematicians does not appear to include any systematic attention to the development of writing skills (p. 448). They critiqued the teachi ng and learning of mathematics, as if its only purpose was filling students head with facts and skills (p. 450) and not initiating students into mathematical communities. In terms of lifelong learning, Gross (1992) indi cated that people needed to develop new kinds of learning, such as the ability to communicate with colleagues around the world via computer bulletin boards (p. 136), to learn with others, and to learn by teaching. He suggested using the Invisible University when referring to the World Wide Web. Gross argued that what 29 PAGE 30 we learn today will be obsolete in five years an d that learning to communicate through different kinds of media is a necessity for the twentyfirst century. Most writing in mathematics, according to a study by Pearce and Davison (1988), is incidental and algorithmic. It mimics the teachers presentati ons and includes direct copying and notetaking (sic) (p. 13). In the classroom culture, writing is used for knowledge telling, which is, according to Bruer (1993), related to recitation or routine and mechanical activities. He affirmed, however, that the goal related to writin g activities must be geared toward knowledge transformation, which consists of authentic writing tasks and serving larger communicative purposes. Writing with a purpose is located in the latter type of writing, knowledge transformation, which is a type of writing that includes going back and forth between planning and text (Bruer, 1993, p. 246). These authors agre ed that writing must be more than just repeating what was said in the classroom. The purpose of a written piece implies a cult ural and methodological background that will shape its components (Richards, 1991). In this sense, distinct domains of discourse can be identified in mathematics. According to Richar ds, there are at least f our types of domains: research, inquiry, journal, and school mathematic s (Table 21). Richards looked at mathematics as a content area, and his taxonomy is general in scope. However, he sustained that two domains pertain to the teaching and learning of mathematics: the inquiry domain and school mathematics domain. As was stated before by Pearce and Davison (1988), students tend to mimic the teachers presentations, learning by repetition. Richards (1991) called th is writing type the school mathematics discourse and described it as a sequen ce that included three steps: initiation, reply, and evaluation. The problem, he contended, was that it left very little or even no space for 30 PAGE 31 inquiry. Another way of looking at school math ematics was presented by Romberg (1992). He stated that most students are enga ged in traditional settings of instruction instead of authentic instruction; they are led to store up information, knowing what instead of knowing how (p. 52). Inquiry mathematics, where participants engage in dynamic discussions by asking questions, presenting conjectures, listening, re ading, and problem solving, was favored by Richards (1991). He sustained th at it is through reflexivity that participan ts begin to communicate with each other and through collabora tion and negotiation that learners develop meaning together. These activities would help the learners constr uct their own knowledge (Richards, 1991); it is by being active in le arning that students can construct their own knowledge (Romberg, 1992). Developing meaning together by communica ting, collaborating, and negotiating meaning is to Richards (1991) what Vygotsky (1978) referred to as the interpersonal and intrapersonal processes in learning. In order to develop meaning, there is a need to develop inquiry discourse, which includes authentic instruction, engaging in knowing how, interacting, collaborating and negotiating, all of which will be studied in this research. Advocates of writing in mathematics argue th at writing helps the l earner develop deeper understanding (NCTM, 1989, 1991). Activities that e ngage the learner in writing include keeping a math journal during a specific period of time, expository writing, and using writing prompts. The objective of writing, according to Miller an d England (1989), is to focus the students thinking toward a better understa nding of the subject matter (p. 299). It should not be to demonstrate writing ability; inst ead, students should be encouraged to think, reflect, and record (Tichenor & Jewell, 2001). 31 PAGE 32 The benefits of writing were stated by Cook (1995). He indicated that writing will (1) provide the opportunity to restru cture new knowledge, (2) allow the student to review, reiterate, and deepen understanding, (3) enco urage the student to clarify and consolidate new information, and (4) help the student to put in order his/ her thoughts. Dusterhoff (1995) also added that writing helps the student explore, clarify, confirm, and extend hi s/her thinking and understanding (p. 48). Cognitive skills, conducive to understanding, used when writing in mathematics include comparisons, analysis, and synthesis (Miller & E ngland, 1989). By taking these skills one by one and comparing them to the cognitive domain of Blooms Taxonomy, it can be argued that most cognitive levels are covered. In order to make comparisons, basic knowledge needs to be included. To analyze a concept, th e learner needs to break down a whole into its components and find relationships between them. To synthesize, the student will put together separate ideas, producing something new (Bloom, Engelhar t, Furst, Hill, & Krathwohl, 1956). Only two components of Blooms taxonomy ar e missing here: application and evaluation. However, Nahrgang and Peterson (1 986) sustained that the use of journals in mathematics can help students develop intellectual skills such as synthesis, interpretati on, translation, analysis, and evaluation. These skills are us ed in authentic problem solvi ng and writing. Therefore, when learners write in authentic settings, they mostly use higherlevel cognitive skills. The following paragraphs present a series of research projects in which writing in mathematics is explored from different domain perspectives, as formulated by Richards (1991) above. These domains include expository writing or school mathematics domain, journal writing, inquiry and research about mathematics. 32 PAGE 33 Writing with the intention to describe or e xplain a mathematical idea was categorized as expository writing by Shield and Galbraith (1998) In a study where expository writing was analyzed during a three month peri od of time, eighth graders were asked to write letters to an imaginary friend (Shield & Galbraith, 1998). Task s included explaining all about a chosen procedure and explaining a mathematical idea to someone who had trouble with it (p. 37). In this study, Shield and Galbraith (1998) made no aim to influence the development of student expository writing in mathematics [although the teachers tried] to stimulate further elaboration through discussion (p. 44). Accordin g to the authors, stude nts wrote their letters using an algorithmic style. That is, a stepbyst ep process, similar to textbook presentations and teaching practices, or what Richards (1991) cal led school mathematics. Shield and Galbraith (1998) concluded that the argument which stated that writing in mathematics promoted deeper understanding was unsupported. They sustained that increasing students meaningfulness in mathematics writing would require the students to show higher levels of thinking about the ideas they present in writing. What Shield and Galbraith (1998) did not cons ider in their conclusi on, however, was that students were writing letters to imaginary fr iends who never answered back. There was no interaction, negotiation, or colla boration between the students a nd their imaginary friends and therefore no need to go beyond the minimum re quirements of the task. Students had no reciprocity and letters were i ndependent from one another. There was no implicit or explicit dialogue in writing letters that would not be answered. The task students were engaged in was not authentic, nor was it inquiry based. Therefore, in this particular re search, the question still remained the same: can writing in mathematics pr omote the development of higher order skills? 33 PAGE 34 In another research by Stone water (2002), students were give n an essay question for their second exam in a college calculus cl ass. They were told to visit with the professor if they had questions about the exercise. This writing exercise was not new to the students, as they had had several writing assignments before and an essay question on their first ex am. The purpose of this research was to identify criteria that discri minated successful from unsuccessful writers in mathematics. As a result, Stonewater (2002) created The Mathema tics Writers Checklist. The essay question analyzed in this research wa s very specific. It asked for definitions and explanations, as well as for the relationship be tween various concepts. In class discussions, topics were addressed fro m a conceptual rather than a procedural approach and problems were represented in three formats: algebraically, numerically, and graphically. According to Stonewater (2002), successful wr iters developed, elaborated, or clarified mathematical descriptions by using examples and mathematical notations and by being careful to address all the components of the exam question. In Richards (1991) domains, this ex ercise will also be classified as school mathematics. Writing in mathematics can help students orga nize, clarify, and reflect on their own ideas (Burns, 2004). Using journal wri ting as a tool in mathematics, Nahrgang and Peterson (1986) and Shield and Galbraith (1998) invited students to reflect on their own learning. Journal writing was studied in secondary math ematics (grades 712) (Clarke & Waywood, 1993) and in college mathematics (Loud, 1999; Goss, 1998; DiBartolo, 2000) Researchers purposes were to help students see themselves as active learners while constructing mathematics knowledge through internal dialogue. Clarke and Waywood (1993) believed that th rough journal writing, students were able to engage in cons tructive personal dialogue. Loud (1999) found that students did better when incorporating structured complex writing assignments into course requirements (p. 95). 34 PAGE 35 Goss (1998) found that by writing, students engaged in the construction of mathematics concepts and were able to organize and explain their ideas in a more precise and coherent way. Three types of journal entries were identifie d in the data set collected by Clarke and Waywood (1993): recounting entries, summarizing entrie s, and dialogue entries. This last type of entry was more reflective, including (1) explan ations about how students solved a problem and how new topics were related to old ones, (2) the identification and analysis of difficulties, and (3) questioning themselves and asking for help. In th is research, interacti on between teacher and students was limited. Still, Clar ke and Waywood (1993) conclude d that students actively constructed mathematics through writing. The categor ies they chose to divide students entries presented a continuum of students understanding of mathematics. Recounting was classified at the lowest level; that was where students described mathematics. Summarizing implied the capacity of grouping together and integrating mathematical con cepts, and dialogue was at the highest level, where learners created and shaped mathematical knowledge. Shield and Galbraiths (1998) research about expository writing had students write without interacting with others, but Cl arke and Waywoods (1993) resear ch about journal writing had students not only recounting and su mmarizing, but also engaging in personal dialogues. In the latter research, students had a chance to go back and reread what th ey had written before in their journals. This allowed students to move in a learning continuum that started with recounting and summarizing and ended with the engagement of pe rsonal dialogues; they we re able to reflect on and transform their own knowledge. Through jour nal writing, Clarke and Waywoods students engaged in what Bruer (1993) called an authentic writing task. Clarke and Waywoods (1993) research also supported Pearce a nd Davisons (1988) argument indi cating that writing can lead to a deeper understanding and improve d mastery of a topic (p. 6). 35 PAGE 36 A study that engaged second graders and elem entary education majors (undergraduate students) in a pen pal exercise during a 13 w eeks period studied collaborative writing in mathematics (Tichenor & Jewell, 2001). College students emailed letters to the students through the college professor and classroom teacher. Lett ers included openended questions and sentence prompts for the kids. In return, second graders me t individually with their teacher to discuss what they would write about and used the computer to answer questions, complete sentences, and ask math questions of their pen pals. According to Tichenor and Jewell (2001) writing also helped preservice teachers [gain] a better understanding of how children learn, think, feel, and write (p. 304). The authors indi cated that second graders fe lt their math performance was better because of the keypal experience (p. 305) and that teachers developed a deeper understanding of teaching and learning (p. 306). The Tichenor and Jewell (2001) study is an example of how writing in school mathematics can e ngage students in discourse, even if they are from different levels and from remote locations. It also exemplifies the use of technology as an aid in developing communication skills and a community of practice. In Louds (1999) research, students attitudes toward mathem atics positively changed after having written experiences in a college calculus course. DiBartolos (2000) research also found similar outcomes when studying formal and inform al writing in a college mathematics course and when evaluating students written responses to test questions in Set Theory, Combinatory, Probability, and Statistics ( p. 101). Furthermore, by writing a bout the importance of calculus and mathematics in their majors, college stude nts were able to thi nk, reflect, understand, find relevancy, and learn new mathematics (Rose, 2001). In summary, according to the research cite d above, writing had a positive impact on students performance and attitudes toward math ematics. Writing in mathematics was researched 36 PAGE 37 using individual activities such as expository writing, essay writing, journal writing, research paper writing, and pen pal activities. In most cases, writing allowed stude nts to reflect on their work and engage in the development of higher order skills. Communication in Mathematics Learning Traditionally, communication in mathematics was initiated by the teacher with little input from the students. Nevertheless, changes in teaching and learning paradigms opened new windows of possibilities to teac hing and learning mathematics. The use of computers and Internet communication tools has revolutioni zed the way people communicate and the way courses are imparted (from facetoface to hybr id or full online courses). Synchronous and asynchronous communication tools ar e available in many schools, public libraries, and homes. The first happens at the same time, although not always from the same place. The second takes place at different times and usua lly from different locations. Common types of digital asynchronous tools are email, discussion forums, blogs with comments, and wikis. Emails are onetoone or onetomany electroni c communications that allow for personal communications. Discussion forums are mainly onetomany communications that allow users to engage in highlevel discussion by frami ng and presenting ideas, formatting challenging questions for peers, and responding to those questi ons to clarify misconceptions (Hacker & Niederhauser, 2000, p. 55). Using Discussion Forums Regarding the use of discussion forum, most re searchers point to the need to monitor and scaffold students participat ion (Kanuka, Collect, & Caswell, 2002; Tu & Corry, 2003; Wegerig, 1998); to give specific instructions and examples before the discus sions start, including what to do and when to do it (discussion cycles, discussion duration, frequency of participation, depth of discussion) (Knowlton, 2003; Tu & Corry, 2003); to ge nerate an evaluation rubric with specific 37 PAGE 38 criteria together with the stude nts, as to increase collaborati on and selfinclusiveness (Knowlton, 2003); to divide the class into small groups so th at discussions can have more depth (10 to 15 students per group) (Tu & Corry, 2003); and to develop more structured activities at the beginning of its use, moving toward more abst ract activities at the end (Wegerig, 1998). Research continued to develop connecting the use of discussion forums to blended learning environments. A sample of this was pr esented in the Special Issue: ComputerMediated Communications, published in the Journal of Research on Technology in Education (Winter 20032004) edited by Suzanne Wade. Schallert, R eed, et al. (20032004) st arted this issue by considering the benefits of com puter mediated discussions in st udents learning. This was a topic addressed from different perspectiv es by all the authors in this issue. Other topics included advantages, disadvantages, demographics, instructors roles, and pedagogical implications when using discussion forums. Shallert, Reed, et al. (20032004) stated that discussion forums offer the learner the experience of thinking about an issue and commenting on it while reading others comments a valuable learning experience (p. 111). Ferdig and Roehler (20032004) addressed a similar point when they stated, multimedia environments generate higher level questions than students in classes without multimedia (p. 119). Je tton (20032004) added that computermediated discussions facilitated learning by helping students: (1) make connections, (2) gain multiple perspectives, (3) develop problem solving skills (4) add depth to their ideas, and (5) elicit instructors input. Im and L ee (20032004) looked at studenttostudent communication as a major tool to in developing a learning community. Ferdig and Roehler (20032004) also presented five main advantages of using computermediated discussions. These are interactivity, ac tive learning, teacher/stude nt relationships, an 38 PAGE 39 increase in higher order thinking skills, and flexib ility. The idea of interactivity was associated with collaboration, feedback, guidance, teamwor k, and giving students a vo ice. Active learning was related to reflecting and making connections ; promoting teacher/students relationships; and increasing flexibility, not only by allowing participation any time and from any place, but also by giving learners the time to think and struct ure a response, promoting reflexivity. Fauske and Wade (20032004) and Im and Lee (20032004) evaluated the instructors role in computermediated environment. Fa uske and Wade (20032004) identified the following instructors roles: monitori ng discussion, to develop netiquette collaboratively with students (p. 147), determining the appropriate level of structure needed and dire ction (p. 147), modeling responses, and using alternative mo des of communication when needed. Nevertheless, none of these studies is direc tly related to mathematics education. They examine the use of discussion forums from diffe rent pedagogical perspectives unrelated to subject matter or content. This dissertation stud ied the development of th readed discussions in a specific discussion forum, a section of a commun ity of practice, as it related to informal mathematics learning and inquiry learning. Altho ugh these discussions were initiated by a single person, all participants of the forum were able to read, reflect, and reply to the opening post (message). Still, some participants chose to stay in the background rather than actively participate. Discussion Forums Research in Mathematics Education Only a few studies were identified in which discussion forums were used in mathematics education. In the following paragraphs, resear ches concerning lear ning about teaching mathematics are divided into two general categor ies. The first category involves research in which discussion forums were used by preservice and inservice teachers (Sliva, 2002; Smith, Ferguson, & Caris, 2003). The second category in cludes research in which discussion forums 39 PAGE 40 were used to learn how to do mathematics (Lotze, 2002; Bolin, 2003; and, Quinn, 2005). Only Lotzes (2002) research examined a semiinfor mal mathematics learning environment; that is, tutoring sessions that took place outside of the classroom and to which no grades were attached. Slivas (2002) research inve stigated how discussion forums were used to learn about mathematics. This study examined an onlin e discussion forum in a hybrid mathematics elementary methods course with 20 preservice elementary teacher candidates (students). Small group discussions explored five different topics related to ma thematics education. These were NCTM Standards, equity, technology in the ma thematics classroom, brain research, and the Massachusetts Comprehensive Assessment System (p. 84). In this research, students were using the discussion forum to learn about teaching mathematics and not to learn how to do mathematics. By using the discussion forum, students bega n to develop community ties; they started thinking as researchers, became more reflec tive, and communicated openly in what they felt was a nonthreatening atmosphere (Sliva, 2002). This experience also allowed students to become more comfortable when using technology (Sliv a, 2002). According to the author, the main implication of this study was th e possibility of developing a comm unity of learners that would provide support to future teachers, thus decrea sing the sense of isolati on many new teachers feel. Nevertheless, a limitation of this research was th at it was conducted using a passwordprotected asynchronous webbased discussion forum (p. 81 ), and once students finished the semester, they would not have access to the system wh ere previous discussions were located, thus minimizing the opportunities to continue collabor ating and supporting one a nother. This pointed to the need of using public (or open) environm ents with free access over the Web where students could continue developing a mathematical community. 40 PAGE 41 Secondly, research by Smith, Ferguson, and Ca ris (2003) studied how discussion forums were used to learn how to do mathematics from the instructors perspective. The experiences online instructors had in an online, textbased environment were an alyzed. Still, these authors did not analyze students practice. Their goal was to i nvestigate teaching and social issues, as well as differences between facetoface and online teaching. They interv iewed instructors from a wide spectrum of courses and then focused on mathema tics instructors, following up with an extended sample of mathematics teachers. Mathematics instructors in Smith, Ferguson, and Caris (2003) research showed frustration when using textbased tools that limited their trad itional teaching strategies especially when they needed to communicate with formulas, symbols, and diagrams. They complained about the need for greater precision in th e use of [mathematics] language (p. 41). It seems that instructors wanted to teach the way they were accustomed to on a chalkboard and expressed concern about the discussion forums textbased format. This led Smith, Ferguson, and Caris (2003) to state, The consensus is that current Webbased distance learning environm ents do not adequately support mathematics (p. 49). This study did not report about the teacher s technology skills and experience, which could have an impact on how they used technology to write mathematics in alternative ways. The authors, Smith, Ferguson, and Caris (2003), suggested that there was a need to use new tools that would allow instru ctors to insert formulas and diagrams in an easier way. In addition, they stated, there was a need to consid er the social impact that teaching online had on many of these instructors. New technologies no w include the use of wh iteboards (Glover, D., Miller, D., Averis, D., & Door, V, 2007), which can address some of the limitations presented by Smith, Ferguson and Caris (2003). In fact, Lo tzes (2002) research participants used 41 PAGE 42 whiteboards, video, and audio to communicate. Comparisons are difficult to make, though, because Lotzes participants were students and tutors, and he compared online versus facetoface college tutoring in mathematic s and statistics in an effort to help determine the merits and drawbacks of new technologies. In Lotzes (2002) research, stude nts and tutors were paired and met a total of six times (three facetoface sessions alternated with three online sessions). He reported that students with high levels of mathematics and technology anxiety where those with greater difficulties. This led Lotze (2002) to report that some students felt frustration, dissatisfaction, and technical problems during the online tutoring sessions. He also report ed that some students needed more training to learn how to use and manipulate writing implemen ts used with the whiteboard. Still, Lotze (2002) concluded (1) that the medium was conduc ive to learning (p. 125), even though it was not perfect, (2) that online tutoring took longe r in time than faceto face tutoring, (3) that interaction and communication was meaningful, and (4) that lear ning (knowledge construction) could take place in the online tutoring environm ent. Lotze (2002) ended his research hoping for a time when students could communicate with cyb ertutors at any time and from anywhere. Bolin (2003) planned to use di scussion forums in a hybrid college mathematics course so that all students could communicate and negotiate mathematical meanings with their classmates. Interview data, however, showed the researcher that the students lacked the confidence to participate in this environment, had no time or desi re to participate, or did not know how to write the mathematical symbols that they thought we re essential in writing mathematics. Instead, students preferred to use email to communicate w ith the professor and to keep an ejournal about their meaning making processes, both more personal and individuali zed activities. Bolin (2003) recommended new research to examine the conversations that take place between 42 PAGE 43 professors and students and the extensive, backandforth negotia tion of meaning and understanding as they evolve over an extended period of time (p. 108). In Bolins (2003) research, students lack of confidence s eemed to prevent them from becoming part of a community of learners thro ugh the discussion forum because it allowed their peers to see what their mathematical limitations (questions or misconceptions) were. Still, students did ask questions (email), wrote about how to do mathematics (email), and reflected about meaning making in a more personal environment (ejournal). In another study, Quinn (2005) started to addr ess some of Bolins (2003) questions when he studied online experiences and mathematical identity with undergradu ate mathematics online students who volunteered to partic ipate in his study. Participa tion in online communities was examined in relation to selfconfidence, mathem atics anxiety, selfconcept, and gender. Quinn (2005) analyzed data from synchronous and asynchronous online communication tools and concluded that participating in online communication increased se lfconfidence, reduced anxiety, strengthened mathematics selfconcept, and wa s not associated with gender differences. In terms of mathematics identity, Quinn (2005) stated that vol unteers had a relative anonymity when participating in the discussion foru ms that caused a relative comfort. He then recommended studying an environment where student s could use avatars so that they would not have to expose themselves and thei r limitations to their classmates. This supports Palloff and Pratts (1999) idea about the changes that teaching and learning on line have over its participants where teachers or students are no longer at the center of the learning process. Online learning can promote the development of communities of learners. As According to Palloff and Pratt (199 9) stated In the online classr oom, it is the relationships and interactions among people thr ough which knowledge is primarily generated (p. 15). Moore 43 PAGE 44 (1989) added that there is a need for continuous interaction between in structor and student, student and content, and stude nt and student in distance le arning environments. Increased interaction will minimize the sens e of isolation students can feel when studying from remote locations (Moore, 1989). Increased interactions allow students to constitute a community of learners, one that allows participants to colla borate, negotiate, construc t knowledge, and develop understanding (Palloff & Pratt, 1999; Li, 2004; Trentin, 2001; Dunlap, 2004). The instructors resistance in Smith, Fergus on, and Cariss (2003) re search about using discussion forums reflected the changes new te chnologies brought to the way they taught. No longer was the teacher at the center of the lecture; their role had changed to one of a facilitator. A similar case can be seen in Bolin s (2003) study, when students seem to reject discussion forums because of lack of mathematics confidence. Howe ver, Quinns (2005) research points in another direction, one that looks at the positive outcomes related to self confidence, reduced anxiety, and strengthened mathematics selfconcept when using communication tools while learning mathematics. In the next section, communities of practice are furthered examined. Research by Lave (1991, 1996), Wenger (2001), Lave and Wenger (1991) Rogoff and Lave (1984), and others are discussed and related to this research project. Communities of Practice Groups of people with common goals who are sharing ideas, making decisions together, and helping one another constitute a community of learners (Palloff & Pratt, 1999; Stepich & Ertmer, 2003; Trentin, 2001; Fielding, 1996). Together they interact, negotiate, and generate new meanings, taking responsibility for determining what they need to know, and [directing] their activities to effectively research, synthesize, and pres ent their findings (Dunlap, 2004, p. 41). In these environments, members may have di fferent interests, make diverse contributions, 44 PAGE 45 and hold varied viewpoints (Lave & Wenger, 1991, p. 97). In a community of practice (CoP), there is a sense of reciprocal caring that goes beyond individualism (Fielding, 1996). Furthermore, communities of practice are characte rized by social issues of trust, reputation, space, and time that help maintain k nowledge ties (Nichini & Hung, 2002, p. 52). New technologies have enabled the devel opment of communities in which members no longer see each other facetof ace. As technology becomes transparent, unproblematic, and integrated into the communitys activities (Lave & Wenger, 1991), different ways of interaction become possible (Moore, 1989; Hillman, Willis, & Gunawardena, 1994). Computer mediatedcommunication tools are used to communicat e with others. These are synchronous and asynchronous communication tools that help establish ties between group members through active participation, even when members of the community are located at remote locations. Recently called social software, they promot e the development of supportive environments, scaffolding learning at different levels and allowing learners to try out ideas and challenge each other (Ferdig, 2007). Also, Web 2.0 tools such as podcasts, blogs, and wikis foster collaboration and sharing among its user s (Boyd & Danielson, 2007). Researchers relate communities of practice to informal learni ng, saying that they are rooted in everyday activities, and that they take place through demonstration, observation, and mimesis (Lave, 1996). Learning in communities of practice is mutual and reciprocal as opposed to formal learning or training that is directive (Trentin, 2001). The development of communities of practice is based on interaction, collabora tion, scaffolding, and negotiati on. The following subsections further explain these concepts and present an example of a community of practice. Interaction and Cooperation Interaction is defined as in terplay, an exchange of ideas, and reciprocity of events among individuals (Gilbert & Moor e, 1998; Wagner, 1994, 1997; Roblyer & Ekhaml, 2000). When 45 PAGE 46 interacting with others in facetoface or online activities, lear ners might assume different roles. In the case of cooperative groups, learners are usually assigned to a sp ecific role, each being responsible for a specific task or part of a problem (Hathorn & Ingram, 2002). Responsibility is divided, and collaboration might be implicit, but not a requirement. In this cooperative group type of setting, interaction can be minimal, occurring at the beginning when tasks are divided among the memb ers of a group. Once everyone knows what to do, communication among members of a group can beco me minimal. Learning, in this setting, is divided into chunks, and there is no need to share what has been learned. The emphasis is placed on the solution or outcome. Interaction, Negotiation, and Collaboration Nevertheless, in collaborative groups, interaction happens over time; it is loose and voluntary (Hathorn & Ingram, 2002, p. 37). Colla boration implies that learning is a shared responsibility where members take responsibil ity for one another (Hathorn & Ingram, 2002, p. 36). It is by articulating and el aborating their understanding a nd by sharing ideas and possible solutions to a problem that learners generate new knowledge (Dunlap, 2004). In collaborative groups, negotiation is the source of learning (Sorensen & Munch, 2004). By maximizing negotiation, in teraction is enabled, and the generation of new learning is facilitated (Sorensen & Munch, 2004). Collabo rative groups act as zones of proximal development (Vygotsky, 1978) where any of its members can perform as a tutor or more knowledgeable other at different times. In th is way, empowerment and ownership of meaning is encouraged (Sorensen & Munch, 2004). To negotiate is to interact with another (one or more people) and to reach an agreement. It implies continuous interaction, going back and forth, until an agreement is reached (Wenger, 2001). It is a process of interpretation and act ion, of making, remaking, thinking and rethinking, 46 PAGE 47 and of participation and comprehension (W enger, 2001). It includes offering access to information, hearing or reading another persons perspectives, explaining why, inviting others to contribute, making others follow the rules, opening spaces for discussion, presenting a new argument or adding to an old one, sharing respons ibilities, confronting others positions and limits, and more (Wenger, 2001). Negotiation faci litates reflection and learning (Wenger, 2001). This is why learning is not the outcome of th e individual mind but that of a participatory framework resulting from social practice (Lav e & Wenger, 1991), that is, from interaction, negotiation, and collaboration. Communities of Practice in Mathematics There are many different kinds of communities of learners in mathematics. Some meet facetoface every year, and some meet periodically over the Internet or the Web. Three examples of such communities are (1) the Natio nal Council of Mathematic s that groups together mathematics teachers from K12, mathematics te acher educators, and researchers, (2) the Mathematics American Association that represents mathematics professors, scientists, and investigators, and (3) the American Mathema tical Association of Tw oYear Colleges which embodies mathematics teachers and professors in small colleges and universities. These associations have the similar characteristic of meeting facet oface every year in an annual conference and sharing their wo rk in periodic publications and professional journals. Other communities of mathematics learners m eet online, over the Internet or the Web, or through news groups and email. They might not k now each other personally, but they share ideas and collaborate with each other through electronic means. Two examples of such efforts include the MathViaDistance from Erie Community College in New York and the Math Forum @ Drexel in Pennsylvania. The former is an email li st that communicates to its members by email only, and the latter is a more complex we b site with several online components. 47 PAGE 48 The Math Forum @ Drexel is selfdefined as an online community of mathematicians, mathematics teachers and professors, mathematics learners, and enthusiasts. Participants use textbased asynchronous communication tools to pose questions, answer them, give feedback, recommend resources, and find solutions (S humar & Renninger, 2002; Renninger & Shumar, 2002; Galindo, 2005; Crowe & Zand, 2000b). Mathematic s learners interact with each other, become active learners, reflect about the problems posed by others, take time to reply, and contribute to knowledge building in community. On occasions they also watch without participating, following the turns of a discussi on. Although there are limitations present in this environment, participants overcome them and con tinue working together toward the solution of problems. The Math Forum @ Drexel environment is a public web site that can inspire communication and community building in mathem atics. Members are voluntary participants working together in the generation of new know ledge, usually using pseudonyms. The research project presented here will examine one of its many discussion forums, over a period of five months, and how instances of dialogue (inter action, collaboration, and ne gotiation) contributed to knowledge construction in mathematics. More details about this community are included in the next chapter. According to Shamatha, Peressini, and Meymaris (2004) learning through communitycentered activities included students [that] are encouraged and able to articulate their own ideas, challenge those of others, and negotiate deeper meaning along with other learners (p. 263). They proposed that it is through learnercentered activit ies that Students build new knowledge and understanding on what they already know and be lieve (p. 364). These authors indicated that knowledgecentered activities will help students organize what they know, making connections 48 PAGE 49 that will later support planning and strategic thin king. It can be argued th at by being part of a community of learners, students can partake in learne r and knowledge cent ered activities. Summary This chapter reviewed literature related to the use of technology in mathematics education, writing in mathematics, communication in mathematics, and sp ecifically, the use of discussion forums in mathematics education, as well as the development of communities of practice. It examined the concepts related to communities of practice, such as interaction, collaboration, and negotiation. It also connected these topics a nd concepts with the research project presented in this dissertation. The following two chapters look at qualitative research foundations (Chapter 3) and the methodology and methods (Chapter 4) used to develop this study. In the third chapter, you will also find the subjectivity statement, the research setting, and reflections on the Pilot Study. The methods used in this research are based on Gees (1999, 2005) discourse analysis. These two chapters are then followed by the data analysis (Chapter 5, 6 & 7) a nd conclusion chapters. 49 PAGE 50 Doing Math Didactic Packages Doing Math Software Assessment / Testing Topic or content specific Figure 21. Software types used to c onstruct knowledge in mathematics Multimedia packages Audiocassettes and SW Computer graphics Animations, sounds, and pictures Programming Languages Topic or content specific Multimedia packages Geometric visualization Documentary (monologic) Interactive (dialogic) Interactive (dialogic) 50 PAGE 51 Table 21. Domains of discourse in mathematics Domain of discourse Communities using this type of discourse Research math Scientists and professional mathematicians Inquiry math Literate adults includes mathematical discussions. Includes engaging in dynamic discussions by asking questions, presenting conjectures, listening, reading, a nd problem solving. Pertains to teaching and learning of mathematics. Journal math Used in publications and papers School math Used by teachers and students. Includes initiation, reply, and evaluation. Pertains to teachi ng and learning of mathematics 51 PAGE 52 CHAPTER 3 QUALITATIVE RESEARCH FOUNDATIONS social analysis should help to generate vocabularies of understand ing that can help us to create our future together. For the construc tionist, the point of soci al analysis is not, then, to get it right about what is happening to us. Rather, such analysis should enable us to reflect and to create. Kenneth J. Gergen, An Invitation to Social Construction (1999, p. 195) For centuries, natural sciences were consider ed objective, a natural place for positivistic research. However, according to Burbules (2000 ), having established a stronger set of common standards of practice, common vocabularies, and common techni ques of inquiry (p. 323) does not mean that physical and natu ral sciences knowledge does not get negotiated and constructed. Constructing knowledge in any discipline is a social process (Restivo, 1983). Accordingly, the study of the social construction of mathematic s knowledge goes beyond a positivistic research approach. The evolution of mathematics education studies was analyzed by Schoenfeld (1994) in the Journal for Research in Mathematics Education In his article A discourse on methods, Schoenfeld showed how the focus of methodology in mathematics educational research changed from a positivist perspective sta tistical in nature (mostly usin g hypothesis testing and regression analysis designs) to a nonst atistical, narrative, processorie nted methodology (p. 697). This dissertation continues this tr end; it studied knowledge constr uction as it took place through transformative dialogue among groups of pe ople in an online discussion forum using a qualitative approach. The main purpose of this study was to exam ine the use of transformative dialogue to construct mathematics knowledge in a public online discussion forum. Transformative dialogue seeks a means to sustain the process of comm unication, acknowledges self expression, affirms the Other, coordinates actions that generate meaning together, affirms polyvocality, and 52 PAGE 53 promotes selfreflexivity the questioning of ones own position. This research investigated written interactions during a period of one academic semester, archived at the Math Forum @ Drexel web site ( http://www.mathforum.org ) discussion groups se ction, specifically the discussion group identified as alt.math.undergrad The method used to analyze the data set was discourse analysis, as presented by Gee (1999, 2005). This method allowed the researcher to identify activities and connections within, be tween, and among the data set and to further develop a representation of how mathematics was constructe d in an online asynchronous discussion forum, called a discourse model by Gee (1999, 2005). In gene ral, this research analyzed a socially constructed, complex, and ever changing (Glesne, 1999, p. 5) electronic mathematics public environment. A second purpose of this research was to contribute to the sociology of mathematics a recent trend in mathematics education researc h. This trend claims that the reality of mathematics lies in discourse, so mathematics is as real and only as real as ordinary social life (Restivo & Bauchspies, 2006, p. 198). Wittgenstein (1967, as cited in Restivo, 1983) emphasized the art of questioning in mathematics to establish a context from which a proposition could be true. Lakatos (as cited in Restivo, 1983) looked at mathematics as a fallible discipline, one that is constructed through the development of conjectures, criticisms, and corrections and that is in continuous search for proofs and counterexam ples. The sociology of mathematics moves away from the idea that math ematics is a static a nd rigid discipline and moves toward the idea of mathematics as a cultural expression. As such, mathematics is constituted of mentalandphysical activities culture, and history (Restivo, 1983, p. 239). Based on the purposes of this study, the resear ch question for this study is as follows: 53 PAGE 54 How does transformative dialogue and negotiati on facilitate the social construction of mathematics knowledge in high school and first and second year undergraduate mathematics via an online discussion forum named alt.math.undergrad in the Math Forum @ Drexel web site? This research studied an online asynchronous community and analyzed the corresponding digitally archived data during a specific period of time. It examined how participants in the discussion forum interacted and negotiated with each other to construc t mathematics knowledge through the use of transformative dialogue To accomplish these purposes, social constructionism was selected as the research s theoretical perspective, followed by a methodology and method that allowed the researcher to study language in context. This chapter will focus on the foundations of qua litative research and the analys is of the pilot study. It is divided into the next sections: Description of qualitative research fundament als: research approach, epistemology, and theoretical perspective Subjectivity statement Description of the research setting Narrative with reflections about the pilot study Qualitative Research Fundamentals The study presented here on transformative dial ogue generated and negotiated in an online discussion forum searched for breadth and dept h. It looked to unde rstand how mathematics knowledge was constructed and how understand ing of mathematics was developed through written interactions. It also acknowledged that there are multiple representations and multiple ways of learning mathematics. For this reason, a qualitative approach was selected to examine the data. 54 PAGE 55 Qualitative Research Approach Qualitative research, a movement that bega n in the 1970s (Schwandt, 2000), searches for thick descriptions, those that cannot be found th rough statistical methods that are numerical in nature (Geertz, 1973). According to Patton (2002), Qualitative methods facilitate [the] study of issues in depth and detail (p. 14). These methods are associated with data that comes from words obtained from interviews, observations, and documents (Kvale, 1996; Jorgensen, 1989; Hill, 1993). Patton (2002) sustained that qua litative inquiry produces a great amount of information on a small number of persons or cases, thus reducing generalizations. It is through methodology rigor that the research er analyzes data in order to represent findings (Anfara, Brown, & Mangione, 2002). St. Pierre (2000) affirmed that qualitative researchers have th e responsibility to keep educational research in play, increasingly unintel ligible to itself, in order to produce different knowledge and produce knowledge differently as we work for social justice in the human sciences (p. 27). Moreover, Erickson and Gutierrez (2002) stated that qualitative inquiry can make valuable contributions to educational research, and that evidencecareful descriptive research falls within the range of met hods in education that can be called scientific (p. 21, emphasis in original). The cont roversies concerning research a pproach are many, especially now that the No Child Left Behind Act promotes quantitative research. However, as St. Pierre (2000) and Erickson and Gutierrez (2002) argued, qualita tive research can also be scientific. Crotty (1998) identified four el ements of qualitative inquiry to justify a research project. These are epistemology, theoretic al perspective, methodology, and method. In the next sections, the epistemology and theoretical perspective used in this research are discussed. Methodology and method will be elaborated in the next chapter. 55 PAGE 56 Epistemology: Constructionism Epistemology is the theory of knowledge, of meaning making; it is that component of philosophy that studies how we learn. Accord ing to Crotty (1998), epistemology can be organized into three main divisions: objectivism, c onstructionism, and subjectivism. As in a continuum, at one end, objectivists believe t hat meaning, and therefor e meaningful reality, exists as such apart from the operation of any consciousness (Crotty, 1998, p. 8); that knowledge can be objective, complete and unchang ing (Burbules, 2000). At the other extreme, the subjectivist considers that meaning is imposed on the object by the subject [that] it is created out of nothing (Crotty, 1998, p. 9). However, constructionism moves toward the center of this continuum. For Crotty (1998), all knowledge, and therefore all meaningful reality as such, is contingent upon human practices, being c onstructed in and out of interaction between human beings and their world, and developed and tr ansmitted within an essentially social context (p. 42, emphasis in original). It is his co ntention that qualitative researchers tend to invoke a constructionist epistemology where meani ng is created in relation ship. This dissertation looks at knowledge from the constructionist standpoint, in which knowledge is constructed through interaction and not in the individual mind. According to Crotty (1998), meaning making is the result of human beings being consciously engaged with the world. The process of constructing the world takes curiosity, imagination, and creativity (Crotty, 1998). Still, this does not mean falling to subjective, unfounded interpretations. Instead, [a] dialogue with the materials helps the researcher pay attention to the object of rese arch (Crotty, 1998, p. 51, emphasis in original) while following a specific methodology or method rigor. The idea of language and dialogue is at the center of the constructionism epistemology. Berger and Luckmann (1966), forerunners of social constructionism, stated that sociology of 56 PAGE 57 knowledge presupposes sociology of language (p. 185). Sampson (1993 as cited in Gergen, 2000) also indicated that meaning is rooted in social process sustained by conversations occurring between people (p. 149) Therefore, it is through langua ge and dialogue that people generate meaning together (Gergen & Gergen, 2004). Theoretical Perspective: Social Constructionism Following the constructionism epistemology, this research is based on the social construction of knowledge theoretica l perspective. Social constructionists are interested in the collective generation of meaning (Crotty, 1998) as it derives from collaboration and interaction and from reflexive questioning, dial ogue, and negotiation (Gergen, 1999). Gergen (2000) affirmed that meaning is neither the result of the individual mind (cognitive constructivism) nor that of the group (social constructivism) but that it is a byproduct of language use within relationship (p. 150). [For a discussion about the history, principles, similarities, and differences on cognitive construc tivism and social constructivism, see Duffy and Cunningham (1996).] According to Gergen (2000), knowledge is not developed in a single mind or from a single individual, but rather from language use within relationship (p. 150). It is culture that molds knowledge, he says. That is w hy, for the social constructionist, knowledge is a communal creation (Gergen, 1994, p. 207). Social constructionists are aware of th e multiple causes that can produce specific outcomes. Thus, they cannot agree with the idea of causality pr oposed by positivistic researchers, as if the conditions of a specific experiment c ould be controlled to produce a specific human outcome. That is why for social constructionists, numb ers and statistical an alysis are not enough. They sustain numbers and statistics eliminate th e voices of research pa rticipants, especially silencing those without sophisticated knowledge, therefore sile ncing those without power. 57 PAGE 58 Furthermore, social constructionists argue in favor of a plurality of voices and disagree with the idea that there is one true answer to any question (Gergen, 1999, p. 92). As Gergen (1999) stated, each construction has both potential s and limits, both scientif ically and in terms of societal values (p. 93). He al so sustained that there is no need to abandon all voices to save only one. Thus, to say that there is one truth th at can be generalized to the whole population is for the social constructionist a form of cultural imperialism (Gergen, 1999, p. 93). Imposing one view over another as if things were black or white would be imposing the voice of those in power. For this reason, social construc tionists step away from dichotomies and welcome plurality. They search for new ways of looking at things, wa ys that will contribute to the development of a generative theory They also search for new po ssibilities that will include accounts of our world that challenge the takenforgranted conventions of understanding, and simultaneously invite us into new worlds of meaning and action (Gergen, 1999, p. 116, emphasis in original). The social constructionism theoretical perspect ive developed in differe nt disciplines, such as therapy, organizational change, education, an d scholarly expression (Gergen, 1999). However, since the focus of this research project is on the construction of ma thematics knowledge in education, the following two sections will only pres ent (1) how social constructionism is applied to education, including pedagogica l alternatives, and (2) the social constructi onist assumptions as they relate to this research project. Social Constructionism in Educ ation: Alternative Pedagogies As previously stated, social constructionism is an outgrowth of communal relations (Gergen, 1999). Therefore, in education, soci al constructionists fa vor three pedagogical alternatives: reflexive deliberation, polyvocal pedagogy, and collaborative classrooms. In this research project, reflexive deliber ation was used to generate the data set without the intervention 58 PAGE 59 of the researcher. Participants of the discussi on forum were able to reflect before writing a message (post). Different solutions were intr oduced by the participants presenting different views and therefore resulting in collaborative polyvocality. While working together, the discussion forum participants engaged in negotia tion and collaboration pract ices to generate new meanings. These alternative pedagogies are explained below. Reflexive deliberation Reflexive deliberation is the outcome of comm unities of practice. Together, participants learn about the ways a community exists and how it thinks about knowledge, language, discourse, and their relationships and dist ribution in society (Gee, 2000, p. 522). In a community of practice, learners work together and interact with each other (Rogoff &Lave, 1984). They set goals together, negotiate appropri ate means to reach them, and help each other throughout the process. In this research project, the contributions th at mathematics learners made to an online discussion forum were examined. Participants were part of a community where reflection was possible, even encouraged. The di scussion forum allowed learners to present a problem in a post and to read postings by others. It s quality of asynchrony allowed pa rticipants to reflect before replying to others. Together, par ticipants interacted and negotiated one or more answers to the posted questions. Polyvocal pedagogy Gergen (1999) stated that th e Internet is an example of how people can generate new potential. This is possible when body and t echnology merge togeth er, facilitating the development of polyvocality. According to Gergen (1999), the Internet can help students develop multiple voices, [different] forms of expr ession, or [different] ways of putting things (p. 59 PAGE 60 183). McCarty and Schwandt ( 2000) added that through polyvo cal pedagogy, students could participate in a wide range of c onversations and acquire different rh etorical skills. These in turn allow students to take persuasi ve positions by actively participa ting in different conversations. Hatch (2002) also presented polyvocal methods as a means to find multiple perspectives. He stated that polyvocal analysis is interested in the existence of multiple truths; that is, in the multiple voices that are telling a story. In this research, this will translate into the multiple discourses written by the forums participants. Through polyvocality, the researcher tries to identify the components of a story, which, according to Hatch (2002), is always partial, local, and historical. Therefore, the final purpose of polyvocal analysis is to capture the multiplicity of voices present in the scene and to tell as many stories as they generate. Gergen (2003) would agree with Hatchs (2002) vi ew of polyvocal analysis. In this research, polyvocality took place in different ways. After writing an original message (post) in a discussion forum, the participants of this community were able to present different answers to the same question or pr oblem, developing a threaded discussion that branched in different directions. At the same time, they developed different ways of representing an answer (narrative, algebraic, graphical, or geom etric), including different levels of abstraction. This, in turn, allowed the part icipants to look at mathematics from different perspectives. Collaboration Collaborative practices favor dialogue, consen sus groups, and the generation of new ideas and opinions. It also opens the classroom to ne w experiences, permitting students to work within the community, establish Intern et communications around the worl d, and even develop authentic projects (Gergen, 1999). The student is seen as an active participan t in a community of learners (Lave & Wenger, 1991; Lave, 1991; Wenger, 2001). 60 PAGE 61 This research analyzed the conversations in a mathematics online discussion forum available to those with access to a computer with an Internet connection. These conversations, generated asynchronously, made possible the disc ursive collaboration among its participants. Together, participants formed a community of mathematics learners. Learners from different backgrounds and nationalities worked together to find one or more solutions to a problem or question. Summary Reflexive deliberation, polyvocality, and colla boration are pedagogical alternatives in the social constructionism theoretical perspective, available in an online asynchronous discussion forum to those with access to a computer and an Internet connection. With this, the physical boundaries of a classroom disappear and its walls are expanded to the worlds cybernauts, where it is possible to become part of a larger community of learners. In this study, the researcher examined how th e Internet allowed high school and first and second year undergraduate mathematics learners to interact and co llaborate, engaging in transformative dialogue. This research analyzed how participants in an online discussion forum constructed mathematics knowledge through reflex ive deliberation, polyvo cality, and discursive collaboration. The social constructionism theoretical persp ective is based on a set of assumptions. The next section will list these assumptions, as stated by Gergen (1994). Then an explanation of those that apply to this research follows. Social Constructionism Assumptions and Mathematics Knowledge Kenneth J. Gergen (1994) pres ented five major assumptions within the domain of social constructionism that broke away from the empirici st tradition of positivistic research. These are: (1) knowledge is socially constituted; (2) knowledge is embedded in historical developments; (3) 61 PAGE 62 linguistic signals are related to experience and influenced by cu lture; (4) knowledge is influenced by personal values, ideologies, and visions; and (5) verifi cation of theory is rendered suspect. In the following paragraphs, the first, third, and fifth assumptions are discussed as they relate to this research project. Knowledge is socially constituted Gergen (1994) sustained that social circumstances affect knowledge construction. He also maintained the importance of language in people s relationships. As stated before, knowledge is constructed in relationship with others in the collectivity, not in the individual mind but through social processes of communication. The idea of socially constructi ng knowledge, that is, of soci ally constructing meaning in mathematics, is the main focus of this study. As Restivo and Bauchspi es (2006) explained in The will to mathematics: Minds, morals, and numbers, Mathematical objects are things produced by, manufactured by, social beings through social means in social settings. There is no reason why an object such as a theorem should be treated any differently in this sense th an a sculpture, a t eapot, a painting, or a skyscraper. Mathematicians work with notations, symbols, and rules; they have a general reservoir of resources, a toolkit, socially construc ted around social interests and oriented to social goals. The objects they cons truct take their meaning from the history of their construction and usage, the ways they are used in the present, the consequences of their usage inside and outside of mathematics, and the network of id eas they are part of within math worlds and within larger societal worlds. (p. 210) G. H. Hardy (1992), an English mathematician, also supported this idea when he wrote that the function of the mathematician is to do something, to prove new theorems, to add to mathematics (p. 61). Hardy talked about the importance of individually generating mathematical ideas and not about constructing mathematics in relationship. However, as documented in the foreword (by C. P. Snow) of Hardys book and in Beckmanns (1971, p. 138) book, Hardy worked closely with Ramanujan, an Indi an mathematician with little early formal 62 PAGE 63 education. Together, they produced five pape rs of the highest class (Hardy, 1992, p. 36). Together, Hardy and Ramanujan were able to construct new mathematics. This research analyzed mathematical intera ctions and negotiation that took place in an online discussion forum. It studied how learners worked together, searching for one or more solutions to a problem, socially construc ting mathematics meaning. It examined how transformative dialogue was used to construct mathematics knowledge. Linguistic signals, experience, and culture Berger and Luckmann (1966) stated that language has the capacity to transcend the here and now [that it] bridges different zones w ithin the reality of ever yday life and integrates them into a meaningful whole (p. 39). Be it face toface (first degree in Berger and Luckmanns view) or written (second degree, according to Berger and Luckma nn), language is capable of becoming the objective repository of vast accumulations of meaning and experience, which it can then preserve in time and transmit to fo llowing generations (B erger & Luckmann, 1966, p. 37). According to Phillips (2000), language is a hu man construct, and different individuals may construct slightly different things with it, even when they use the same words (p. 4); therefore words can be interpreted in different ways. Berger and Luckmann (1966) also shed light on how the social dist ribution of knowledge works. They sustained that knowledge [is encounte red] in everyday life as socially distributed, that is, as possessed differently by different individuals and types of individuals (p. 46). They sustained that no one knows exactly the same as another. Moreover, Phillips (2000) cited Lorraine Code in saying that knowledge is gr ounded in experiences and practices, in the efficacy of dialogic negotiation and of action (p. 30). In mathematics, meaning and experience ar e mainly represented through algebra and geometry. For many, algebra is a special language. According to Restivo and Bauchspies (2006), 63 PAGE 64 knowledge is constructed through the use of nota tions, rules, and theorems. As time goes on, more mathematical representations and applicati ons are developed. In th is research, discussion forum participants presented solutions in narr ative, algebraic, and geometric forms through words, symbols, and diagrams. Participants inte racted, negotiated, and collaborated with each other to find one or more answers to th e posted questions, problems, and conundrums. Verification of theory throug h research is rendered suspect Gergen (1994) sustained that by testing a hypothesis, the resear cher is already seeking for data that best serve their intere sts. He also affirmed that any intelligible h ypothesis can be verified or falsified (p. 206) From the interpretive point of view, the investigator attempts to document the rules of meaning within a specific context; the documentation serves not as a validating device but as [a] rhetor ical support (Gergen, 1994, p. 206). The investigator, following social constructionist principles, is not in search of specific data or results, nor of specific truths to accept or reject a h ypothesis; instead, the researcher looks for meaning as it comes from the data itself, looking at the di scourse of those that negotiate and collaborate while construc ting meaning. In the social constructionism theoretical perspective, meaning is generated bottomup inst ead of topdown. Gergen (1999) indicated that in topdown analysis, it is t hose in authority who set the rule s. Indeed, transformative dialogue seeks for means of sustaining the process of communication, acknowle dging self expression, affirming the Other, coordinating actions that generate meaning togeth er, affirming polyvocality, and promoting selfreflexivity, the questioni ng of ones own position. It is the idea of togetherness that makes the difference; that is, working together, reflecting together, and coordinating actions together to generate new meanings. In the research presented here, a discourse model was developed from within the data itself, analyzing the transformative dialogues o ccurred in the discussion forum. The model was 64 PAGE 65 then recursively verified twice throughout a period of five months. This research examined how participants generated meaning together and how interactive di alogue, negotiation, and discursive collaboration took pl ace to construct mathematics meaning. To accomplish this, methodology rigor was performed through discourse analysis implemented with Gees (1999, 2005) discourse analysis methods. The next chapter will examine this methodology from a theoretical stance as well as its application in this research project. However, before presenting the methods used in this research and the appl ication of Gees discourse analysis, three more sections are presented below. Fi rst, the researcher is introduced in the subjectivity statement; second, the research setting is described; and lastly, reflections on the pilot study are narrated. Subjectivity Statement1 Social constructionists cannot remain dispas sionate about their work. They acknowledge that there is a reason that guides their work, motiv ating them to invest time in such a project. This researcher has taught junior high, hi gh school, and undergraduate mathematics and is interested in researching the use of online communication tools in learning mathematics. By investigating the use of discussi on forums in mathematics, the researcher was able to explore how students write mathematical ideas and sear ch for meaning while interacting, negotiating, and collaborating with each other. As a graduate student, this researcher studied the teaching of mathematics with technology and the importance of communication in elearni ng environments, as well as the concept of quality in distance education from the students perspective (OrtizRodrguez, Telg, Irani, Roberts, & Rhoades, 2005). Her interests also in clude researching online learning environments 1 The subjectivity statement is written in third person, instead of first person, mainly because of cultural reasons. In the Puerto Rican culture we are taught not to talk about ourselves as we believe that this shows the character flow of lack of humility. The following paragraphs present a picture of who I am in third person. 65 PAGE 66 and how they can help students develop commun ities of practice that promote meaning making and understanding. Mathematics has always been this researche rs favorite subject. In grade school, she was not a high achiever, although she strived to be one. She did not understood why the group of students, who were all supposedly at the same le vel of understanding, completed different sets of problems. By the time she reached the seventh grade, she had caught up with her peer group. By ninth grade, she started helping her classmates with their math homework. At that time, she admired her math teacher and her pedagogy. The teachers daily routine was simple and straightforward, following essentialist and behavi orist practices. Homework was checked first, then a new topic was presented, and finally, th e teacher gave the students some practice and homework problems. The teachers elemen tary algebra class seemed so easy! That time was when this researcher first became a tutor and felt empowered while helping her classmates understand and complete their ho mework. However, when she got to college, she was once again an underachiever in math, lacking the background knowledge needed to excel. Nevertheless, she worked hard to keep learni ng math and managed to complete a degree in Secondary Math Education. She then became a math teacher. From 1961 to 1973, while she attended elementary and secondary school, there were no personal computers. Mainframe systems were a reality in a few school systems throughout the United States; but the public school system in Puer to Rico did not use com puters, at least to her knowledge. The first personal computer widely av ailable in K12 settings was the Apple IIe, commercially marketed after 1977. During the 1970 s, some students used calculators, but she never had one due to a lack of financial resources. With a calculator, students were able to check 66 PAGE 67 the solutions to their homework problems, allowing them to look at patterns and to better develop concept knowledge. The researcher taught math in high school ( 19771978) and in junior high school (19781979, 1991), as well as first and second year under graduate college cour ses (19912002). What she loves the most is helping students feel that th ey can learn how to do mathematics, that they can be successful in mathematics. Finding out there are no secrets behind the numbers can help students understand the concepts at hand. This sense of empowerment in math is hard to accomplish because many students past experiences are grounded in behaviorist practices in which memorization had a major role. The daily r outine that her old nint h grade teacher followed did not work for most students. Thats why she believes that learning mathematics has much to do with constructivist and constr uctionist practices, in which students can do mathematics in collaboration, developing concepts and understanding and applying them in everyday life (NCTM, 1989). Ideally, constructionism should be reached so that learning can focus on the collective generation of meaning (Crotty, 1998, p.58). Communication a nd collaboration are the building blocks for the development of commu nities of learners (Rogoff & Lave, 1984; Lave, 1991; Lave & Wenger, 1991; La ve, 1996; Wenger, 2001; Shamat ha, Peressini, & Meymaris, 2004) and math communities are no exception. Research Setting Our study examined interactions that took place in an online discussion forum located at a public web site and available to those interested in mathematics teaching and learning with access to the Internet. In the next sections, the we b site is presented to the reader, including its history and services. 67 PAGE 68 The Math Forum @ Drexel Web Site Selfdefined as an online math education community center, the Math Forum s mission is to provide resources, materi als, activities, persontoperso n interactions, and educational products and services that enrich and suppor t teaching and learning in an increasingly technological world (Drexel University, 19942006, paragraph 1). To fulfill their mission, the Math Forum has stated five main objectives. Th ese are (1) to encourage communication throughout the mathematical community, (2) to o ffer model interactive pr ojects, (3) to make mathrelated web resources more accessible, (4 ) to provide highquality math and math education content, and (5) to spread news about new resources in the Internet. The Math Forum is a very dynamic community of mathematicians that continually adds more resources and services to its web site. History of the Math Forum @ Drexel In 1992, the Geometry Forum was founded by Eugene Klots, a mathematics professor from Swarthmore College and the developer of the software program Geometers Sketchpad (Kane, 2000). It was not until 1996 that the forums name changed to the Math Forum During this year, Swarthmore College received a three million dollar grant from the National Science Foundation to further expand the s ite. A year later, in 1997, the Math Forum web site was helping close to 100,000 users a month (Downi ng, 1997). For many years, the media portrayed the Math Forum as a homework helper web site and a mathematics online community especially renowned for its Problem of the We ek and Ask Dr. Math sections. On April 6, 2000, The Associated Press reported that the Math Forum received almost 1 [one] million visitors and 12 million hits per month (Popular math Web site is sold to WebCT, 2000). It was in that sa me month, April of 2000, that the Philadelphia Inquirer reported that the Math Forum was sold to WebCT, the online software company (Woodall, 68 PAGE 69 2000).This was mainly a financial arrangement, as its offices stayed at Swarthmore College, and the Math Forum continued to be managed by Eugene Klots (Research and Development) and Steve Weimar (Technology and Education). A year later, in 2001, Drexel University acquired the Math Forum In a special issue of the Math Forum Internet News (volume 6, number 36a), the President of Drexel University, Dr. Constantine Papadaki s, informed the members of the mathematics community that the Forum had a new location and a new name, The Math Forum @ Drexel. He also pointed out that the web s ite would continue to offer the same services while introducing some new cutting edge features ( Math Forum Internet News 2001, paragraph 3). Since then, the web site changed its presence in th e Internet and continued to expand. The Math Forum @ Drexel Services Services provided by the Math Forums web site are divided into four main sections: (1) Main Areas, (2) Projects, (3) Features, and (4) Archives ( The Math Forum @ Drexel 19942004). Main Areas is subdivided into seve ral subsections specially dedicated to students, teachers, parents and citizens, and researchers. The Projects section is geared toward the development of math skills, including a Probl em of the Week area for each secondary math subject: Math Fundamentals, PreAlge bra, Geometry, and Algebra. The Math Forum Newsletter (MFIN) is located under the Features section, as are the Mathematics Discussion Groups. Finally, the fourth section in cludes links to mailing lists, wo rkshops, software, articles, book reviews, and more. (Table 31 for a complete list of the resources provided in the Math Forum web site). The first issue of the Math Forum Internet News was published in October 7, 1996 ( http://www.mathforum.org /electronic.newsletter/ ); since then, weekly issues were published. Eleven volumes, with added issues for special purposes, were published up through 2007. During 69 PAGE 70 the last 20 years, the newsle tter kept the math community updated with new ideas, new developments, and the latest news in mathematics. The Math Forum also works in collaboration with professional organizations such as the National Council of Te achers of Mathematics (NCTM) and the Mathematical American Association (MAA). The publication of the first Math Forum Newsletter issue in 1996 was an extension of a conversation already started, where references to math resources were made and geometry problems were posted through a mailing list. Email correspondence was accepted from community members and answered individuall y. Once the online newsletter was available through the site, it included a sec tion called Check out our Web Sit e at the very end. In the first issue, only three resources were listed; the first was a link to The Math Forums home page, the second a link to the Problem of the Week, and the last to Internet Resources (Steves Dump). It was not until September of 1998 that a link to the Discussion Groups was added in the forums newsletter. This was a special issue (3.39A) dedicated to describing interesting conversations take[n] place during September of 1998 on Internet math discussion groups ( The Math Forum News, September 1998, first paragraph). This issue was also the beginning of a monthly series dedicated entirely to describe differe nt discussion forums and conversations. Two references are made in the Discussion Groups series to the alt.math.undergrad the first on July 28, 1999, and second on June 8, 2000. Both references include the description of the group, a link to its home page, and an example of the inte ractions that took place on that particular day. This information is similar to that given about any other discussion group. The Discussion Groups series ended on August 2000. 70 PAGE 71 alt.math.undergrad Discussion Group Data for this research was located in the Math Forum @ Drexels web site ( http://www.mathforum.org ) discussion section. The alt.math.undergrad discussion forum was chosen for analysis because it included high school and undergraduate first and second year mathematics discussions, courses the resear cher has taught and tutored before. The Math Forum @ Drexel was selected since it is a wellestablished community of mathematics learners with more than two decades of existence and experience. By November 8, 2006, there were a total of 68 active Discussion Groups and 13 inactive groups in this section. The active discussion groups were divided into ten categories. These are Courses, Curricula, Education, History, Math Topics, Online Projects, Policy and News, Professional Associations, sci.math, Software and Inactive. The discussion group titled alt.math.undergrad belongs to the Math Topics category, and it is described as an unmoderated newsgroup focused on undergraduate mathematics (Math Forum @ Drexel 2004). Other discussion groups in the Math Topics category include the following: alt.algebra.help.independent, al t.math.recreational.independent, discretemath, geometry.college, geometry.precollege, geomet ry.puzzles, geometry.research, and Snark Community members freely choose the group in which to participate, posting their messages from anywhere and anytime. Archives in the alt.math.undergrad discussion forum include me ssages posted from July 7, 1996 to the present time. This is an active comm unity that continues to receive messages up to the current date, expanding its archives day by day. As of November 8, 2006, the alt.math.undergrad included a total of 38,797 messages (posts), divided into 8,516 topics (threads). This research project will analy ze the messages posted duri ng the first academic 71 PAGE 72 semester of 2004 (August to December). More details about the data set as well as the selection criteria will be describe d in the methods chapter. Reflections about the Pilot Study A preliminary study was conducted to examine how mathematics know ledge was socially constructed at the undergraduate level in the online public mathematics discussion forum called alt.math.undergrad. This initial phase allowed the researcher to investigate how data was organized, what subject matters were studied, and how discussion was generated. It also allowed the researcher to explore the activities and rela tionships taking place throughout the discussions. The alt.math.undergrad discussion forum was chosen as the research focus because it includes high school and first and second year undergraduate mathem atics topics. This discussion forum has free access through the Math Forum @ Drexels web site (http://www.mathforum.org). The pilot study inves tigated threaded discussions (topics) and their corresponding postings (messages) occurring during a onemonth pe riod. Data was selected from October, 2004, because the pilot study started on that same month. At first, a general analysis of all October, 2004, discussions showed that there were 167 threaded discussions (topics). From these, only three had more than 25 postings (messages), and twelve had between ten and twentyfive postings. A content analysis followed to identify the specific mathematics topics discussed in the latter group. In addition, the nu mber of participants, frequency of participation, and time span of discussion was recorded. Further evaluation allowed the researcher to take general notes about the discussions, draw tree diagrams to identify the flow of interaction, and determine the number of storie s in each threaded discussion. A selection from those threaded discussions cove ring topics offered in high school and first and second year undergraduate mathematics followed. This led to a to tal of five threaded discussions that met the data selection criteria (Table 32). 72 PAGE 73 Next, mathematical stories were constructed using tree diagrams to show the flow of interactions present in each th readed discussion. The branches of the tree diagrams represented the way discussions were conducted, showing who replied to whom, which messages received one or more answers, and those that did not receive any reply. Each tree branch was identified as a story where participants interacted with each other by evaluating a problem, negotiating an answer, and generating new mathematical know ledge. Tree branches were collections of continuous postings that exemplified how inter action, negotiation, and disc ursive collaboration took place. Once the stories were constructed, a d ecision was made to eliminate redundant intertextuality and personal identifications. An alysis focused on how participants developed mathematical knowledge; by eliminating redundant intertextuality, an understanding of thought processes in each story was facilitated. Since the focus of the study was on knowledge construction, the participants identity was not re levant. This alone could be the object of future research. At that point in the study, a question a bout methodology arose. Was content analysis enough? Was open coding and grounded theory th e path to follow? Was phenomenology the correct methodology? What had to be the main focus when identifying how participants developed knowledge? Discourse an alysis as stated by Gee (1999) was then chosen to further analyze the data set. Gee provi ded guide questions to analyze activities, which allowed the researcher to go beyond the mathematical conten t of each threaded discussion in search of negotiation and collaborative strategies In Gees discourse analysis, the Activity building looked at how participants use cues or clues to assemb le situated meanings about what activity or 73 PAGE 74 activities are going on (Gee, 1999, p. 86). Gee focu sed on the specific actions that take place throughout discourse. To questi on the data, Gee (1999) formul ated the following questions: What is the larger or main activity (or set of activities) going on in the situation? What subactivities compose this activ ity (or these activities)? What actions (down to the level of things like requests for reason) compose these subactivities and activities? (p. 93) Questioning the mathematical stories allowed the researcher to identify activities and specific actions conducted by the participants. Howe ver, at that point, ea ch story was analyzed independently. Connections withi n, between, and among the data set were not considered in the pilot study. As an experiment, the researcher then deci ded to go a step further into Gees methodology and tried to identify the parts of a story pres ent in each mathematical story (or tree branch). The researcher found that tree branches with five or more messages included all or most of the elements (body parts) of a stor y identified in Gees discourse analysis methods. These include setting, catalyst, crisis, evaluation, resolution, and coda (Table 33). Dividing the mathematical storie s in this way that is, us ing the body parts allowed the researcher to identify types of activities that were conducted by the par ticipants throughout the discussions. These activities were based on previ ous research about discussion forums. A report was then written based on these results (See Appendix A for the complete Pilot Study report). Lessons Learned through the Pilot Study Conducting the Pilot Study allo wed the researcher to iden tify a methodology that fit the theoretical perspective as well as the data set. It helped explain how par ticipants of a discussion forum constructed knowledge. The use of tree di agrams made it possibl e to represent the threaded discussions in a graphical format, to id entify the flow of convers ations, to identify the stories present in each threaded discussion, to breakdown the stories in to body parts, and to complete discourse analysis questioning th e data and searching for breadth and depth. 74 PAGE 75 The pilot study showed that users willingly posted questions answers, suggestions, and references to Internet resources. Participants in the discussion forum worked together to find new meanings, presenting different ideas, some in algebraic form and others in graphical form through words. Polyvocality was present when part icipants presented similar ideas in different ways. As a community of practice, participants helped each other in different ways, sometimes identifying new resources and other times giving support to each other. Their voice helped the researcher build knowledge from the ground up thr ough the analysis of transformative dialogue. Knowledge in the discussion forum was c onstructed through ac tive participation. Criticisms As stated, the pilot study allowed the researcher to conduct a preliminary analysis; however, underanalysis occurred at different le vels (Antaki, Billig, Edwards, & Potter, 2002; Burman, 2003). First, a summarylike report wa s presented, losing details and subtleties incorporated in the data. This was the result of spending more time organizing the data instead of analyzing utterances. Secondly, only isolated quo tations were present in the report; even though a complete story was included as an example, very little analysis followed that is, questioning the data was minimal. Mental constructs combined with previous research findings on discussion forums were used to find out these same constructs in the data, not allowing the data to speak for itself. The researcher moved toward general assumptions instead of going back and forth between the general and the sp ecific (Antaki et al., 2002, section Underanalysis through Spotting, paragraph 4). Finally, at the tim e the pilot study was conducted, sociology of mathematics was not yet identified in the research literature, limiting the analysis, findings, and conclusions. However, after reviewing discourse an alysis literature beyond Gees methodology (Austin, 1962; Foucault, 1972; Parker, 2001; Hepburn & Potter, 2003; Van Dijk, 2003; Potter, 75 PAGE 76 2003a, 2003b, 2004; Ainsworth & Hardy, 2004; Fairclough, 2004; McKenna, 2004; Rogers, 2004; Stevenson, 2004; Clarke, 2005), the researcher had a more globa l overview of this type of research methodology and methods, which, in turn, allowed the researcher to develop a more representative discourse model(s) of the data. Nevertheless, as Gergen (1999) stated, this model will not be final. Even Gee (2005) affirmed that Discourse models, though they are theories (explanations), need not be complete, fully forme d, or consistent (p. 85). Moreover, as Foucault (1972) indicated in his book The Archaeology of Knowledge Discourse is the path from one contradiction to another: if it give s rise to those that can be seen, it is because it obeys that which it hides (p. 151). The analysis of discourse is, for Foucault, a way to hide and reveal contradictions. However, contradic tions were not identified in th e pilot study, or at least they were not reported. The pilot study was just the firs t step of analysis, and a path had yet to be found. In summary, in spite of the many limitations previously identified, it was through discourse analysis that the researcher was able to explore the transformative dialogue occurring in the alt.math.undergrad discussion forum. The researcher star ted to explain why and how things happen as they do (Gee, 2000, p. 196); that is, how mathematics knowledge was constructed in this digital environment. (A copy of the pilot study report is located in Appendix A: Pilot Study. Also see Appendix B: UF Institutional Review Board Letter, and A ppendix C: Request for Copyright Permission.) The following chapter presents discourse analysis as a methodology and a method. It begins with a revision of discour se analysis literature, and ends with Gees discourse analysis methodology. This is then followed by the process of applying Gees discourse analysis to study how discussion forums are used to construct ma thematics knowledge in the particular discussion 76 PAGE 77 forum under study. Information about validity in qualitative research and Gees discourse analysis, and the limitations of the study will be analyzed at the end of the chapter. 77 PAGE 78 Table 31. Math Forum @ Drexel web site resources by section. Section Resources Main Areas Search for Math Resources Student Center Teachers Place Parents & Citizens Research Division Math Resources by Subject Math Education Key Issues in Math Projects Ask Dr. Math Teacher2Teacher Math Fundamentals: Problem of the Week PreAlgebra: Problem of the Week Geometry: Problem of the Week Algebra: Problem of the Week Active Problem Library Math Tools Features Dynamic Geometry Software Teacher Exchange Internet: Mathematics Library Math Forum Newsletter Mathematics Discussion Groups Math Awareness Month Math Forum Showcase Whats New? Archives Articles & Book Reviews Geometry Newsgroups / Topics Internet Software: Mac & PC Learning & Math Discussions Mailing Lists & Newsgroups Math Software Mathematics Teacher Bibliographies Math Forum Workshops The Math Forum Quick Reference (2006). Retrieved November 8, 2006, from http://www.mathforum.org/special.html 78 PAGE 79 Table 32. Data summary for October 2004: Threads with 10 to 25 postings Thread Title & General Description Evaluation 1 Units, and algebraic integers Total Postings: 10 Frequency of participation: 1 post 6 people 2 posts 2 people Time span: 2 days Topic: Advanced Algebra Storylines: 8 Notes: All storylines have two postings. The initiator of this thread presents an argument that is refuted by all others. They consider it flawed. 2 A Question about Math Curriculum (Math Instructors and Professors Please Respond) Total Postings: 25 Frequency of participation: 1 post 2 people 2, 3, 4, 6 posts 1 person 8 posts 1 person Time span: 11 days Topic: Undergraduate Math Curriculum Story lines: 8 Notes: A student asks for feedback from math specialists to evaluate the curriculum changes in his program of study. 3 Need help with some integral! Total Postings: 12 Frequency of participation: 1 post 3 people 2 posts 3 people 3 posts 1 person Time span: 3 days Topic: Advanced Calculus Storylines: 4 Notes: Topic not taught in first and second year undergraduate mathematics. 4 2 sinA versus sin2A Total postings: 14 Frequency of participation: 1 post 4 people 2 posts 3 people 3 posts 2 person Time span: 4 days Topic: Trigonometry Storylines: 7 Notes: Topic is taught in first and second year undergraduate mathematics. 5 Extrema / Diff Total postings: 14 Frequency of participation: 1 post 5 people 2 posts 2 people 4 & 8 posts 1 person Time span: 4 days Topic: Calculus I Storylines: 8 Notes: Topic is taught in first and second year undergraduate mathematics. 79 PAGE 80 Table 32. Continued Thread Title & General Description Evaluation 6 Tan to Slope Total postings: 11 Frequency of participation: 1 & 2 posts 2 people 5 posts 1 people Time span: 4 days Topic: Trigonometry Storylines: 3 Notes: Topic is taught in first and second year undergraduate mathematics. 7 Norms!!!! Topic: Vectors Notes: Topic not taught in first and second year undergraduate mathematics. 8 Truth Tables Help Topic: Logic Notes: Topic not included in study. 9 Uniform Convergence Topic: Advanced Mathematics Notes: Topic not taught in first and second year undergraduate mathematics. 10 Statistics Total postings: 10 Frequency of participation: 1 & 3 posts 2 people 3 posts 1 people Time span: 8 days Topic: Central Tendency Storylines: 4 Notes: Topic is taught in first and second year undergraduate mathematics. 11 Need this explained Total postings: 10 Frequency of participation: 1 post 2 people 2 posts 4 people Time span: 7 days Topic: Logarithm Storylines: 3 Notes: Topic is taught in first and second year undergraduate mathematics. 12 Latest fuss, my apologies Total postings: 19 Frequency of participation: 1 post 7 people 2 & 3 posts 2 people Time span: 4 days Topic: Apologetic Argument Notes: Not relevant for this study. Decision: Will not be used in pilot study. 80 PAGE 81 Table 33. Parts of a story with higherorder structure Body parts Description Setting Sets the scene in terms of time, space, and characters Catalyst Sets a problem Crisis Builds the problem to the point of requiring a resolution Evaluation Material that makes clear why the story is interesting and tellable Resolution Solves the problem Coda Closes the story Gee, J. P. (2005). Discourse Analysis: Theory and method (2n d ed., p. 131). New York: Routledge. 81 PAGE 82 CHAPTER 4 METHODOLOGY AND METHODS To teach someone the meaning of [a] sentence is to embed them in the conversational sea in which [the] sentence swims. James P. Gee, An Introduction to Discourse Analysis: Theory and Method (2005, 2nd ed., p. 46) In this chapter, the methodology and met hods used to study the construction of mathematics knowledge through the use of a particular discussion forum is presented to the reader. Based on Crotty (1998), methodology includes the strategy, plan of action, process or design lying behind the choice and us e of particular methods (p. 3) He stated that the methods are the techniques or procedures used to gather and analyse (sic) data re lated to some research question (Crotty, 1998, p. 3). In general, me thods are more specific than methodology. However, in this research, the methodology and methods are intertwined in the same concept. They are both classifi ed as discourse analysis: methodol ogy in general terms, and more specifically, Gees (1999, 2005) di scourse analysis processes as a method. The next sections explore discourse analysis as a methodology, Gee s perspective of discourse analysis as a method, the application of Gees methods to this study, the validity, and the limitations. Discourse Analysis Methodology Setting the basis for discourse analysis, Au stin (1962), a forerunner, looked at the components of language. His analysis was that of u tterances, that is, the an alysis of sentences. Others looked at discourse analysis from different perspectives. For example, McKenna (2004) discussed the analysis of the r elationship between language and society (p. 10), Joworski and Coupland (1999) talked about the analysis of language in use (p. 1, as cited in Clarke, 2005, p. 148), and still others used discour se analysis as a research to ol (Fairclough, Graham, Lemke & Wodak, 2004). In general, research ers looked at discourse analysis as a way to analyze language 82 PAGE 83 form and function (Fairclough, Graham, Lemke, & Wodak, 2004; Gee, 2004, p. 19, emphasis in original; Austin, 1962). However, most authors looked only at discourse as spoken language. That was the case of Austin (1962), who stated that to say somethi ng is to do something (p. 108). Years later, we found Foucault (1972) paraphrasing Austin when he wrote to speak is to do something (p. 209). Finally, McCarty and Schwandt (2000) sustained that in atte mpting to talk, people enter the world of discourse (p. 5556). These authors mostly looked at discourse as spoken language, but discourse is also found in text (W odak, 1996; Mishler, 1990; Vygotsky, 1962). Discourse analysis is applied across the soci al sciences, including education and learning processes (Potter, 2003a; Potter, 2003b; Rogers MalancharuvilBerkes, Moley, Hui, & Joseph, 2005; Rogers, 2004). It is also considered by ma ny as a social action (Van Dijk, 2003; Wodak, 1996; McKenna, 2004; Potter, 2004). That is because, as Fairclough, Graham, Lemke, and Wodaks (2004) indicated, different theoretica l, academic and cultural traditions push discourse in different directions (p. 4). Two current classifications of discourse an alysis, one by Joworski and Coupland (1999) and another by Fairclough, Graham, Lemke, a nd Wodaks (2004), show the focus discourse analysis takes in research. Th ere are several similarities betw een both frameworks, which make possible their combination. In general, they can be listed as (1) the sp eech and conversational analysis or the study of indivi dual text and talk, (2) the nego tiation discourse in social relationships or the social agen ts and social change, and (3) the power/knowledge, ideology, and control discourse, or the anal ysis of social agents and social change, respectively. Representing the negotiation discourse in social relationships category, we have Faircloughs (2004) and Gees (1999, 2005) discourse analysis methodology. Conversely, two 83 PAGE 84 authors who focused on power/knowledge discou rse are Foucault (1972) and Van Dijk (2003); the former presented his views as the archaeolo gy of knowledge and the latter as Critical Discourse Analysis. In 1998, Gee and Green wrote, Discourse anal ysis approaches have been developed to examine ways in which knowledge is socially co nstructed in classrooms and other educational settings (p. 119). To study disc ourse, that is to study language inuse, is, according to Gee (2004), inherently political and has implications on status, so lidarity, distribution of social goods, and power. (p. 33). Rogers and others (2005) also stated that Gees theory is inherently critical in the sense of asserting that all discours es are social and thus ideological (p. 370). In this way, Gees discourse analysis has similaritie s to that of Van Dijks (2003). As such, Gees discourse analysis also has a critical stance, is embedded in social constructionism practices, and maintains humanistic principles. The critical component of disc ourse analysis cannot be considered neutral, because it is caught up in political, social, racial, economic, religious, and cultural formations (Rogers, MalancharuvilBerkes, Moley, Hui, & Joseph, 2005, p. 369). It is in this sense that discourse analysis is critical. Following th is view of discourse analysis, Gee (2004) differentiated between discourse analysis using lower case and uppercase letter s; that is, he differentiated between critical discourse analysis (cda with lowercase) and Critical Discourse Analysis (CDA with uppercase). One way to look at his representa tion is through a continuum. At one end, Gee (2004) presented critical discourse analysis as an ecdotal reflections on writt en or oral texts (p. 20), and at the opposite end, he pr esented Critical Discourse Anal ysis as related to political proselytism (p. 20). 84 PAGE 85 According to Gee (2004), the distinction be tween critical and noncritical discourse analysis is related to how social practices are st udied; if social practices are treated solely in terms of patterns of social interaction (p. 32), then the study would be noncritical (cda). However, critical approaches go further and tr eat social practices not just in terms of social relationships [but also] in terms of implica tions for things like stat us, solidarity, distribution of social goods, and power (p. 33) (CDA). As an example, Gee (2005) presented how language is used as a gatekeeper in a job interview. Th e same happens with mathematics serving as the gatekeeper for whitecollar professions in area s related to engineering and medicine (Restivo, 1983; Moses & Cobb, 2001). Gee (2004) also wrote about how discourse analysis is applie d to education. He suggested that it needs to show how a distinctive commun ity of practice is constituted out of specific social practices (across time and space) and how patterns of participation systematically change across time, both for individuals and the community of practice as a whole (p. 39). He added that learning is a type of so cial interaction in which knowledg e is distributed across people and their tools and technologies (Gee, 2004, p. 19) By moving across time and space and adding more data to the analysis, discourse analysis allows the researcher to identify different stories in a data set (polyvocality). However, data might not at all times be consistent, and finding discontinuities is always a possibility (Hatch, 2002; Foucault, 1972). New data helps the researcher refine the previously iden tified discourse models (Gee, 1999, 2000). That is why Gees (2005) discourse an alysis allows a researcher to think more deeply about the meanings that people give to words so as to make ourselves better, more human people and the world a better, more human place (p. xii). Gee believes language has meaning only in and through social practices practices which often l eave us morally complicit 85 PAGE 86 with harm and injustice unless we attempt to transform them (p. 8, emphasis in original). Therefore, discourse analysis is an important human task; it is a way to better understand those around us. In general, Gees discourse analysis fo cuses on sociocultural practices. Through his method, researchers can decompose text while sear ching for breath and dept h, for the details that tell a story. Discourse analysis helps the rese archer understand the sociocultural practices happening in a specific community of practice. For this reason, Gees (1999, 2005) method was chosen to analyze the community of mathema ticians that constructed knowledge in the discussion forum alt.math.undergrad located at the Math Forum @ Drexel s web site. In this study, discourse analysis is defined as a search for meaning situated in specific sociocultural practices and experiences (Gee, 2000, p. 195). This search for meaning came from the analysis of data in an asynchronous comm unication system, that is, the analysis of text generated without the interventi on of the researcher. It was through the analysis of threaded discussions that the multiple discourses of thos e collectively constructi ng high school and first and second year undergraduate mathematics know ledge in an online discussion forum were identified. The analysis included threaded discu ssions from five months. The first month allowed the researcher to construct a preliminary discou rse model. Data from subsequent months were used to revise and refine the previous discourse model, genera ting a new discourse model of how people constructed mathematics knowledge through the use of asynchronous threaded discussions after each period of analysis. It was discourse analysis as stated by Gee ( 2005) that allowed the researcher to construct a set of models coming from the data itself, let ting the online participants speakout and introduce themselves to the researcher as they cons tructed mathematics knowledge and generated new 86 PAGE 87 meanings. This type of mode l was identified by Gee as a cultural model in 1999 and 2000 and later as a discourse model in 2005. The next section will examine details about G ees discourse analysis method, that is, Gees discourse analysis processes. Gees Discourse Analysis Method Gee (2005) stated that All life for all of us is just a patchw ork of thoughts, words, objects, events, actions, and interactions in Discourses (p.7). When presenting his method, Gee started by making a distinction between lit tle d and big D in their re lationship to discourse. Little d is about how language is used on site to enact activities a nd identities [that is, about] languageinuse (Gee, 2005, p. 7). It can be asso ciated to the study of form and function in language. Yet, big D has to do with what accompanies language, that is ones body, clothes, gestures, actions, interactions, symbols, tools, technologies (be they guns or graphs), values, attitudes, beliefs, and emotions ., and all at the right places and times (Gee, 2005, p. 7). In this research, little d is used to analyze the form and function of language which makes possible the construction of high school and first and second year undergraduate mathematics, that is, the types of questions and inquiries posted and the re plies that promoted or limited interaction. big D is used when anal yzing specific actions and interactions made possible through the use of technology, specifi cally an online async hronous communication tool, the public discussion forum alt.math.undergrad located at the Math Forum @ Drexel ( http://www.mathforum.org ). Gee (2005) identified two types of discourse analysis: one that studied general correlations between form (struc ture) and function (meaning) in language (such as Austins, 1962), and another that studied spe cific interactions between language and context (p. 54). It is this last type that Gee adopt ed to study discourse. Gee (1999, 2005) proposed a method that 87 PAGE 88 included the following general steps: (1) analyze raw transcriptions and prepare transformed and theorized transcriptions; (2) use building tasks (and the co rresponding questions) and inquiry tools to identify themes; (3) write a preliminary Di scourse model; (4) test with new data, as in a recursive analysis; and fi nally, (5) validate the Discourse model. This method is explored in the following paragraphs. And after, details of how Gees method was applied in this research are explained. Working with Transcriptions The first step in Gees methodology is to analyze raw transcriptions and prepare transformed and theorized transcriptions. Gee (2 005) suggested looking f or patterns and links within and across utterances ( p. 188). In this way, the resear cher starts making conjectures about the meaning of text from the start. When working with text, the re searchers have to say the sentences of the text in their minds. To do this, they must choose how to break them down into lines .Such choices are part of imposing a meaning (interpretation) on a text and different choices lead to different interpretations (Gee, 2005, p. 126). These lines will reflect the information st ructure of a text (Gee, 2005, p. 127). In turn, Gee called these sets of lines stanzas, which are devoted to a single topic, event, image, perspective, or theme (Gee, 2005, p. 127). Th ese are also identif ied by Gee (2005) as microstructures (p. 127). Subsequently, as stanzas accumulate into larger pieces of information, macrostructures are created (G ee, 2005, p. 128). Gee compared macrostructures to stories, and as any othe r story in literature, they can be subdivided into different components. Gee (1999, 2005) suggested the followi ng six body parts to a st ory: setting, catalyst, crisis, evaluation, resoluti on, and coda (Table 33). Gee (1999, 2005) also noted that or ganizing text in this fashi on allowed the researcher to check for patterns in peoples speech or text and find basic themes. During the phase of analysis, 88 PAGE 89 the researcher would shuttle (sic) back and forth between th e actual lines [raw transcript/transformed transcript] and the idealized lines [theorized tr anscript] (Gee, 2005, p. 129). Gee stated that a line and stanza representation of a text [can] simultaneously serve two functions. First, it represents the patterns in terms of which the speaker has shaped her meanings [and] second, it represents a picture of [the] analysis, that is, of the meanings [the researcher is] attri buting to the text (p. 136). Groups of stanzas can be organized into stor ies and then divided into story body parts, connected blocks of information that can be used to discover structure in information, and which, in turn, can help the researcher to look more deeply into the text and make new guesses about themes and meaning (p. 136). Working with transcriptions is the first st ep in Gees analysis. Building Tasks The analysis will continue by questioning th e data already converted into theorized transcriptions with stories divided into body parts. For this ph ase, Gee (1999, 2005) developed a set of building tasks, divided into seven areas of reality. These are significance, activities, identities, relationships, politics, connections and sign systems and knowledge (Figure 41). Each one of the building task s generates a small part of the full picture (Gee, 2005, p. 110) because they are deeply interrelated ( p. 104), and together [they] constitute a system (p. 102, emphasis in original). Analysis is then produ ced by answering a set of questions related to each one of the building tasks. The reflection that accompanies this stage helps the researcher to give meaning to language (p. 110) and to theorize with the data, creating a theorized transcript. However, according to Gee (2005), Actual analys es usually develop in detail only a small part of the full picture (p. 110); that is, not all building tasks are used in every analysis. 89 PAGE 90 Two building tasks were selected to conduct this research; these are the activity and connections buildings. More details about this deci sion are included in the next section. Inquiry Tools In addition to the building tasks, Gee (1999, 2005) uses a set of inquiry tools to help the researcher develop a Discourse model (Figure 42). The inquiry tools include the analysis of social languages, intertextualit y, Conversations, Discourses, s ituated meaning, and Discourse models. These are described by Gee (1999, 2005) in the following way: Social languages are varieties of languages that, in general, have two purposes: first, to express different socially significant identiti es and second, to enact different socially meaningful activities (Gee, 2005, p. 35); they are a representation of what we learn and what we speak (Gee, 2005, p. 37), the result of our cultural experiences and environment. Intertextuality happens when spoken or written te xt alludes to, quotes, or otherwise relates to, another [text] it relates to words that other pe ople have said or written (Gee. 2005, 21). Voithofer (2006) explained that texts do not exist in a discursive vacuum, that intertextuality interrelates cultural, literary a nd historical factors that come together in a moment within a text (p. 204). However, as wi ll be shown below, Voit hofers conception of intertextuality is cl oser to Gees notio n of Conversations. Expanding the notion of intertextuality, Conversations are related to themes, debates, or motifs that have been the focus of much talk a nd writing in some social group with which we are familiar (Gee, 2005, p. 21). When conversations in clude debates, people usually take sides. Conversations as an analytical tool will study wha t sides there are and what sorts of people tend to be on each side (Gee, 2005, p. 35). As Voithofer (2006) indicat ed, Conversations are related to cultural, literary, and historical factors. 90 PAGE 91 Discourses study the who and the what: who ar e those who speak or write their identities (their socially situated identity ), and what are they doing (the activities they are completing; that is, the socially situated activity ). Through discourses, the researcher can identify multiple entities (Gergen, 1999). Gee (20 05) sustained that Different social identities (different whos (sic) ) may seriously conflict wi th one another (p. 25, emphasis in original). Foucault (1972) identified this type of conflict as contradictions, irregul arities in the use of words, incompatible propositions (p. 149). In studying the who and the what, G ee (2005) listed the following Discourse characteristics: Discourses are always embedded in social institutions, have no clear/discrete boundaries, can be split into two or more, or can meld together. Discourses can change over time, emerge as new ones or die; be limitless. In addition, Discourses are defined in relationship with others; they are social practices and mental entities. The next two inquiry tools, situated m eaning and Discourse models, are closely interrelated, as were intertextuality an d Conversations. According to Gee (2005), situated meanings are images and patterns, initially develope d from a word and turned into a theory. Situated meanings are shared within the community in which people live; th ey are rooted in the practices of the sociocultural group to which th e learner belongs (p. 60). To find meaning in context is to find the material se tting where people are present; it is to find what they know and believe, their social relationships ethnic, gendered, and se xual identities, as well as cultural, historical, and institutional factors (p. Gee, 2005, 57). Gee (2005) agrees with Barsalou (1992), in considering situated meanings, midlevel patterns or generalizations (p. 66, emphasis in original), betw een two extremes, the general and the specific. However, situated meanings, like Discourses, are neither static nor definitions. 91 PAGE 92 Instead, situated meanings are f lexible transformable patterns th at come out of experience and, in turn, construct experience as meaningful in certain ways and not others (Gee, 2005, p. 67). As an inquiry tool, situated meaning is a thinking device (Gee, 2005, 70). The inquiry tools used in this research include soci al languages, intertextuality, discourses in terms of what people are doing and how they are interacting, and situated meanings how they generate meaning together. These should help the researcher deve lop a discourse model explaining how mathematics knowledge was constr ucted through the use of the discussion forum alt.math.undergrad at The Math Forum @ Drexel Discourse Models Initially called cultural m odels (Gee, 1999, 2000), discourse models (Gee, 2005) start by making assumptions and end with preliminary explanations of the world we live in. They help us make sense of things, understanding texts and the world; they help us pr epare for action in the world (Gee, 2005, p. 75). Still, discourse models need not be complete, fully formed, or consistent (Gee, 2005, p. 85). Moreover, discourse models are shared by peopl e belonging to specific social or cultural groups (Gee, 2005, p. 95); distributed across the di fferent sorts of expertise and viewpoints (Gee, 2005, p. 95). They are what Foucault (1972 ) called discourse fo rmations; that is, positivitys. Gee (2005) also sustained that Dis course models link to each other in complex ways to create bigger and bigger storylines ( p. 96), which in turn would approximate to Foucaults (1972) conception of the archaeology of knowledge. Th e development and refinement of a discourse model will help validate the research findings. Reviewing the Preliminary Discourse Model Once the researcher is ready to construct a Discourse model, s/he has reached the third level of Gees discours e analysis. The initial Discourse model statement is then tested with more 92 PAGE 93 data in a recursive analysis (stage 4). As stated before, discourse models can change in different ways by adding, changing, refining, and deleting co mponents until an explanation closer to the data set can be reached. In this research, the preliminary discourse mode l is the result of Augus ts data analysis and interpretations. This model will then be re vised and refined as more data is analyzed. To Summarize Gees method offers the researcher the methodol ogy rigor necessary to pay attention to the object of research. The process includes theorizing with data, questioning the data from different standpoints (building tasks), usi ng inquiry tools to further anal yze the data set, developing a discourse model (an explanation), reviewing, refining, and validating it with additional data. This process will help the researcher examine the transformative di alogue generated in an online asynchronous discussion forum, in which participants constructed new mathematical understandings about high school and first a nd second year undergraduate mathematics. Data Collection Procedures As previously stated, this research analyzed asynchronous data archived in digital format from an online public discussion forum, a leading center for mathematics and mathematics education on the Internet ( The Math Forum @ Drexel 2004). Digital archived data are free and accessible to all Internet us ers (Bolick, 2006, p. 122) and take advantage of computers characteristics. In this research, data was the product of voluntary participation in the alt.math.undergrad discussion forum. Data can also be clas sified as primary sources, since it was recorded while participants of the discussion forum interacted with each other without the intervention of others. Data collection was limited by the time period under analysis, which included one academic semester, starting in August of 2004 and ending in December of 2004. During this 93 PAGE 94 period of time, the alt.math.undergrad discussion forum listed 761 threads that generated more than 3,800 postings. The selection of threads (topic s) was narrowed down to select those with a specific number of postings and content areas of fered in high school and first and second year undergraduate mathematics. The number of postings (messages) in each thre aded discussion (topic) varied in terms of participation, from less than five to more than 100. However, only threads composed of 10 to 25 messages each were chosen for analysis. The number of postings (messages) was important because the analysis incorporated identifying si x body parts to a single mathematics story. This decision was based on the results of the pilot stu dy that showed stories could be subdivided in their body parts for indepth analysis. Threaded di scussions are developed in a hierarchical style and include one or more stories, like branches of a tree. Th erefore, having fewer than ten messages would make it difficult to find all of the components of a story (Table 33). A total of 37 threads met these criteria. In addition, permission from the University of Floridas Institutional Review Board02 was solicited, but reviewers indicated that no consent form was require d from the participants since the archived data was previously collected (See Appendix B). Besides, no personal contacts with the users of the Math Forum @ Drexel discussion group were established. Only the postings with the indicated characteristic s were chosen for analysis. In summary, threaded discussions (topics) selected for analys is consisted of those with 10 to 25 postings (messages). They included topics related to high school and first and second year undergraduate mathematics. A time limit was established for analysis, and only those postings written from August to December of 2004 in the alt.math.undergrad discussion group of the Math Forum @ Drexel web site, located at a university in the east of the USA, were analyzed. 94 PAGE 95 Participants who generated the data for this st udy were voluntary users of the discussion forum. Data was asynchronously generated, digitally archived, and accessible through the Internet. Although no consent forms were ne cessary (See Appendix C), the Math Forum @ Drexel was contacted and permission was granted to conduct this research. Data Analysis Process in this Dissertation Project Gees (1999, 2005) discourse analysis was chos en to analyze the different storylines presented in the threaded data located at the alt.math.undergrad discussion forum. This type of analysis involved asking questions about how la nguage, at a given time a nd place, is used to construe the aspects of the situ ation network as realized at that time and place and how the aspects of the situation network simultaneously give meaning to that language (remember reflexivity) (Gee, 2005, p. 110). Following up on Gee s statement, in this research, the time of analysis was one academic semester (August to December, 2004), the place was the alt.math.undergrad at the Math Forum @ Drexel web site, the setting was the online public discussion forum, the language was mathematics, and the simultaneous network was related to the different types of interacti on, negotiation, and discursive collaboration practices taking place in each threaded discussion. Threaded discussions are bounded by a beginning and an ending. As a result, the activity building task was selected as the main source of analysis. The connections building task helped establish relevancy within, betwee n, and among threads. The inquiry tools used in this research (social languages, intertextualit y, discourses, and situated meani ngs) supported the analysis of the data. This analysis permitted the identification of instances where coordinated actions conducted in collaboration with others generated meaning (Gergen, 1999). This type of analysis also allowed the researcher to iden tify the sources of polyvocality present in the data. According to Gee (2005), human language supports the performance of 95 PAGE 96 social activities and social identities [as well as] human affiliation within cultures, social groups, and institutions (p. 1). Data in the alt.math.undergrad discussion foru m was organized by topics (threaded discussions). The threads, composed of ten to twentyfive messages/postings, were further analyzed. Each topic (threaded disc ussion) was represented in a tree diagram to identify the flow of conversati on and the interactions that took place between its participants. Each branch of the tree diagram represented a storyline. This also provided a means to identify the different parts of a story and to compare th em with one another. In order to inform the research question, the content of each storyline and its corresponding body parts were then analyzed using the activity and c onnections building task s questions designed for this research (for details go to the next section). The use of inquiry tools and building tasks as presented by Gee (1999, 2005) allowed the researcher to develop a model of how mathema tics knowledge was constructed and negotiated in an online discussion forum. This type of model was categorized by Gee in 2005 as a discourse model and previously, in 1999 and 2000, as a cultural model Although both categorizations relate to the same concept, G ee, in 2005, decided to emphasize its discursive component. Still, the researcher prefers cultural model over discourse model because it not only takes into concern that knowledge is developed through the analysis of discourse, but it also emphasizes the context in which it was developed. The cultural compone nt that evidences the idea of a mathematics community is emphasized in this research in accordance with Restivos (1983) conception of sociology of mathematics. 96 PAGE 97 Questioning the Data The specific questions used in this research were as follows: A. From the activity building task: 1. What was the main activity going on in each threaded discussion? [Look at the main question in the first post/message, a nd the setting and catalyst body parts of the threaded discussion.] 2. What subactivities compos e this activity? [Look at th e crisis, evaluation, and resolution body parts.] 3. What actions took place that composed th e subactivities of the activity? [Look at the crisis, evaluation, an d resolution body parts.] 4. What types of solutions were presented to the participants of the threaded discussion? [Look at the reso lution and coda body parts.] B. From the connections building task: [Comparisons within the stories of a threaded discussion, and between and among threaded discussions] 1. What types of connections are made with in a storyline, between storylines, and within a threaded discussion? 2. What types of connections are made with other threaded discussions and are there different threaded discussions studying the same question? 3. How is intertextuality used to creat e connections withi n, between, and among threaded discussions? 4. How do connections help to constitute coherence in the discussion forum? [validity] Application of Gees Discourse Analysis Method The procedure or method used in this resear ch included the followi ng steps; one through five identify how data was organized, and st eps six through fifteen states the methods of analysis. These are as follows: 1. Identify the threads by mont h (August to December, 2004). 2. Choose the threads related to high school and first and second year undergraduate mathematics in each month with 10 to 25 postings. 97 PAGE 98 3. Make a summary table with descriptive statisti cs of the threaded discussions related to this research by period of analysis, including content area, number of threads, number of postings, number of participants, and time span of discussion (from fi rst to last posting). 4. Take each topic (threaded discussion) and draw a tree diagram showing the flow of conversation. 5. Convert each tree branch of each tree diagram into a mathematics story. In terms of Gees discourse analysis, these are called macrostructures. 6. Analyze each story (macrostructure) to identif y the story components; these are the body parts of a story, as stated in Gee (Table 33). 7. Divide mathematical stories into its component s: (1) setting, (2) cata lyst, (3) crisis, (4) evaluation, (5) resolution, and (6) coda. Th ese components were used to answer the analysis building task questions. 8. Analyze the story lines (microstructure) to identify activities (or sequence of actions) taken place (building activities ta sk from Gees discourse analysis) use of the questions listed above. 9. Look across the threaded discussion stories fo r themes; that is, types of activities the participants engaged in. These helped find patterns of communicati on, similarities, and differences and in tur n, a discourse model. 10. Analyze the storylines to identify connections within and between stories in a thread, and between and among different threads. 11. Look across the threaded discussion stories for connecting themes taken place between the stories in each thread, and between and among the threads. These helped find patterns, similarities, and differences and in turn, a discourse model. 12. Develop a preliminary Discourse model/Cultural model. 13. Repeat steps six through twelve twice (one for data from September and October and another for data from November and December) and modify the Discourse model/Cultural model as needed. 14. Develop a cultural model, a theory that represented the data set. 15. Establish validity. (For more details on va lidity, see section below.). The following sources were used to review and refine the Discourse model: a. Reflexive notes written throughout the pr ocesses of organization and analysis b. Summaries developed throughout the or ganization and analysis of data c. Expert audits dissertation committee 98 PAGE 99 Validity in Qualitative Research Finding validity in qualitative research is a bout building credibility and trustworthiness (Mishler, 1990; Kvale, 1995; Lincoln & Denzin 2000; Patton, 2002). For this reason, qualitative researchers open their work to the reader to be evaluated by them. They ove rtly present the trail that led them to the research interpretations. St ill, from a postmodern view, no interpretation can be considered a final statement (Lincoln & Denz in, 2000) or a generalizat ion or universal truth (Kvale, 1995). To explore validity in qualitative researc h, arguments by Mishler (1990), Kvale (1995), and Patton (2002) are discussed below, including notes of how their id eas of validity were established in this research study. Then, a discussion will follow about how validity was explicitly accomplished in this research project. Mishler (1990) presented valid ity as a process through which a community of researchers evaluates the trustworthiness of a particular study (p. 415). As such, validity is established in different ways, depending upon the type of study conducted by the researcher. In his article, Mishler presented three exemplary studies and sh owed how validity was es tablished in each one. The first study was about Life History Narratives and Identity Formation by Mishler himself. In this study he demonstrated validity by maki ng full transcripts and ta pes available to other researchers. Methods were also used to link data, findings, an d interpretation. The second study, Narrativation in the Oral St yle, illustrated validity by showing full texts and its representations, by theorizing with the transcri pts, and by illustrating how interpretation was reached. In a third study, in which a narrativ e strategy was used, the author followed a sequence of steps, a structural model for the analysis of [a] passage (p. 433), validating his work by making the process visible. 99 PAGE 100 According to Mishler (1990), v alidation of findings is embe dded in cultural and linguistic practices (p. 435); theref ore, validation is provisional. Qual itative researchers need to show consistency in their work and thought pr ocesses so that their conclusions prove trustworthiness. In this research, data was availa ble freely from the Internet, and a trail can be followed from the theorized transcripts to the data itself. Another author who studied va lidity as it relates to soci al constructionism was Kvale (1995). He defined validity as a form to dete rmine if a specified method can be used to investigate what it is intended to investigate (Kvale, 1995, On va lidity and truth section, first paragraph). However, for Kvale, that is not enough. He suggested the need to look at the researchers ethical integrity as well. For Kvale, that is a criti cal component used to establish the quality of scientific knowledge (Kvale, 1995, Validity as quality of craftmanship section, second paragraph). Kvales conception of validity also include d the process of re cursively questioning, checking, and interpreting th e research findings. Some of the stra tegies used in this process are checking meaning of outliers, us ing extreme cases, following up surprises, looking for negative evidence, making ifthen tests, ruling out spur ious relations, replicat ing a finding, checking out rival explanations, and getting feedback from informants (Kvale, 1995, Validity as quality of craftsmanship section, forth paragraph). He also stated that, in general, to validate is to question (Validity of the validity question se ction, second paragraph), to question everything. In this research, recursive questioning and r eevaluation of a discourse model was accomplished throughout the process of analysis and interpretation across time. Lastly, Pattons (2002) concepti on of validity is related to three areas: (1) the use of rigorous methodologies; (2) the es tablishment of the researche rs credibility; and (3) the 100 PAGE 101 presentation of philosophical beliefs. To de termine the use of rigorous methodologies, Patton included Kvale (1995) questioning as well as tr iangulation, design, and a pplicability. According to Patton, triangulation can be accomplished in at least eight different ways. These are (1) using different data collection methods (2) using different data sources, (3) having multiple analysts, (4) using multiple theories or perspectives to interpret data, (5) including participants reviews of data and interpretations, (6) including the audien ce feedback, (7) adding personal reflections, and (8) using expert audits, such as doctoral committe es and peer reviewers. A selection of these strategies is made to establish findings credibility. In general, this selection is restricted by theoretical perspectives and me thodologies. In this research, triangulation was possible through the use of reflexive deliberation made throughout the process of organization and analysis of data. It was also possible due to the suppor t and guidance of the dissertation committee. Methodology rigor was accomplished by selectin g epistemology, theoretical perspective, methodology, and methods that matched each other. To determine the researchers credibility, Pa tton (1995) stated that a qualitative report should include some information about the rese archer (p. 566). This information would be related, in one way or another, to the research project. Additionally, background characteristics of the researcher and any personal information that might influence interpretations must be included. Peshkin (1988) called this type of pe rsonal report a subjec tivity statement. The researchers credibility is al so accomplished through intellectua l rigor. Reviewing the data and rechecking interpretations over and over again to make sure they make sense increases the quality of analysis (Patton, 2002, p. 570). In this study, the research er established credibility by exposing her background and person al characteristics to the reader in the subjectivity section. Intellectual rigor was possibl e by closely following Gees discourse analysis methods. 101 PAGE 102 Finally, validity is traditionally classified as in ternal or external. Internal validity refers to how findings correspond to reality; the closer the findings are to reality, the better. However, from a postmodern perspective, all findings and interpretations are medi ated, viewed through the researchers eyes, through her/ his beliefs, and through her/his background. Therefore, research findings and interpretations are limited by the implementation of methodol ogy and the researcher her/himself. This is closely re lated to what Patton calls intell ectual rigor, discussed above. In qualitative research, external validity that is, the genera lization of interpretations or implications of a study is limited to the cont ext of the research. Th erefore, many qualitative researchers will not talk about generalizations; instead, they will present preliminary findings based on the research data and its context. In Gees (1999, 2005) discours e analysis methods, the researcher presents a preliminary explanation in the form of a discourse model or cultural model. This research will present a discourse model of how mathematics know ledge was constructed through transformative dialogue in an online discussion forum. (chapters 5, 6, and 7). Validation is a process that happens thr oughout the process of analysis. Gee (1999) identified four sources of valid ity in discourse analysis. These are convergence, agreement, coverage, and linguistic detail. Convergence is reached when analysis offers compatible and convincing answers to the research questi ons (p. 95, emphasis in original). Agreement is attained when other sorts of research tend to s upport our conclusions (p. 95). Coverage is related to the applications that can be extrapolat ed from the data, or being able to predict the sorts of things that might happen in related sorts of situations (p. 95). Linguistic details are linked to the grammatical devices used in language Still, validation in qualitative research is much more. 102 PAGE 103 In this research, linguistic details validity was present when analyzing the transcripts, converting them into transformed transcripts (u ntil storylines and body pa rts were identified), and then creating theorized transcripts. Examples of such are included in the data analysis chapters, evidencing case studies with raw data, corresponding transformed transcripts and theorized transcript (Mishler, 1990) This process included questioni ng the data in different ways and following Gees methodology rigorously, which ev idenced the researchers ethical integrity and philosophical beliefs (Kvale, 1995; Patton, 2 002). To triangulate data, the researcher used summaries, personal reflections, a nd expert audits (Patton, 2002). Convergence validity was then achieved when using theorized transcripts to answ er the research question. Therefore, internal validity was accomplished through the use of linguistic details and convergence (Gee, 1999, 2005). The research presented here analyzed a fi vemonth period of interactions in the alt.math.undergrad discussion forum located at the Math Forum @ Drexel web site. By using Gees (1999, 2005) methods and consistently following the set of steps presented above (intellectual rigor), a Discourse /Cultural Model was constructed. This model was first based on the findings of the first analyzed month and was then reviewed and refined with additional data of the following four months. Diffe rent storylines were analyzed to identify consistencies and contradictions, incorporating them to the initi al Discourse/Cultural Model. The final Cultural Model allowed the researcher to find a prelimin ary answer to the research question: How does transformative dialogue and ne gotiation facilitate the social construction of mathematics knowledge in first and second year undergraduat e mathematics via an online discussion forum named alt.math.undergrad in the Math Forum @ Drexel web site? Gee (1999, 2005) called this process coverage validity. Other researchers call it external validity. 103 PAGE 104 Limitations Several limitations were present in this research project, some of which are identified in the following lines, although not in any particular or der. First, this research used archived data, available through a public discus sion forum on the Internet. Sin ce the data was archived, no interviews were made, nor were in teractions with the participants of the forum possible. Still, data included discursive collabo rations, embedded in threaded discussions where mathematics knowledge was constructed. For this reason, the main limitation present in this research was not being able to clarify with the participants th e messages (postings) they made in the threaded discussions. The use of triangul ation by including participant s reviews of transformed and theorized transcripts and interpretations was not po ssible. However, to keep the interpretations as close as possible to the data, the researcher kept a journal with notes about the data for every month studied, wrote personal refl ections throughout the processe s of organizing and analyzing the postings/messages in each threaded discussion, and worked closely and in collaboration with expert auditors the dissertation committee. A second limitation related to th e lack of interviews or personal communications with the participants is that of mainly focusing on the cognitive domain (how people develop mathematics knowledge) instead of the affective do main (what are their values, ideologies, or visions). This had a direct impact on the methodology, limiting the building tasks and inquiry tools used to analyze the data. Other limitations were related to the theoretical perspective chosen for this research. By moving away from traditional ideas, the social constructionist theoretical perspective has generated many critiques mainly because for the so cial constructionist knowledge is constructed in relationship, in collaboration, in the communit y, and not in the individual mind. This has set 104 PAGE 105 the stage for criticisms about realism, experien ce, and other mental stat es such as relativism (Gergen, 1994). One major concern of those who criticize social constructionism is their view of what is real, of what is true, what really happened, what must be the case (Gergen, 1994, p. 223). However, social constructionists value multiple views, multiple voices or discourses, and multiple perspectives. Social constructionism opens the door to transformative dialogue, negotiation, and increased possibili ties of new understandings. For the social constructionist, social analys is should help to gene rate vocabularies of understanding that can help us create our fu ture together (Gergen, 1999, p. 195). Moreover, meaning is contextual, implying that mean ing, understanding, and therefore knowledge are related to the context in which it is de veloped, to a particular time and space. The example of the Earth being flat or round has been used to sustain one side of this controversy or the other (Phill ips, 1997, section Kenneth Gergen, paragraph 5; Gergen, 1994, p. 223). At one time in history, the world was thought to be flat, and that wa s true at that time and space, but later the study of the stars and ne w technological developments made it possible for Pythagoras (500 BC) and Aristotle (350 BC) to find new ways to describe the Earths shape. A new conception of the Earths shape was devel oped, and today it is belie ved that the Earth is round. This example presents the importance of time and space and of context when making interpretations. A second limitation to the theoretical persp ective selected is re lated to the idea of experience and mental states. Thes e are contested by social construc tionists, mainly because they lend themselves to an ideology of individualism with all it s invitations to alienation, narcissism, and exploitation, closing the door to explore possibili ties of alternative 105 PAGE 106 constructions (Gergen, 1999, p. 227). To know what is re al or objective, or even what is true or false, is not as important as th e consequences they might have a nd the practical implications that can be made under a specific contention of truth (Gergen, 1999). However, for the constructionist, being settled in a position is never the end of th e story there can always be other ways of looking at things. In this sense, mathematics is a great example. The mathematical competencies that promoted new developments in the seventeenth century and the work of mathematicians to find alternative constructions for writing mathematics (notation development) would have not been possible if ma thematicians had been content with previous findings or work. New possibili ties and new ways of looking at mathematics generated new understandings and facilitated the construction of new mathematics. As was noted before, mathematics is a cultural expression (Restivo, 1983). A third criticism of social constructionism ha s to do with the incohe rence of skepticism (Gergen, 1999, p. 227). Some critics ask, Isnt the social constructionist position itself a social construction? (p. 228). The social constructionist would answer in the affirmative; and they will address the importance of reflexivity and new po ssibilities. (For more on this issue see the section titled Social constr uction in education at th e beginning of Chapter 3.) Again, there is no final word for the constructionist. Because of this, relativism is probably the main argument used against the soci al constructionism theo retical perspective. Critics may ask how moral and poli tical deliberations can be made if there is no final word. Social constructionists would an swer with other questions such as: Whose moral or political stance must be taken? Is ther e something good for everyone, a uni versal truth? Why not accept that there are multiple and compe ting realities, a local morality, or local goods? For the social 106 PAGE 107 constructionist, no single voice could ever be the answered; instead, there should be conversation, dialogue, and negotiation; that is, meani ng making in relationship. In this research, transformative dialogue interaction, discursive collaboration, and negotiation was examined in order to devel op a discourse/cultural model (Gee, 1999, 2005) of how mathematics knowledge was constructed in an online discussion forum. However, the researcher does not intend this model to be the final word about mathematics knowledge construction in online environments. This model can be contested and reviewed by others, including mathematics sociologists, mathematics educators, and mathematics researchers. This model could also be the continua tion of a conversation that addr esses the use of technology in mathematics; it could even be an invitation for re searchers to collaborate in finding new ways to teach and learn mathematics, new ways of meaning making th at adds to the notion of sociology of mathematics The possibilities are endless. 107 PAGE 108 Figure 41. Components of an ideal discourse analysis Activity Sequence of actions Significance Sorts of meaning Sign systems Equations, graphs, images, diagrams Politics Distribution of social goods Building Tasks Identities Participant roles Relationships Social interactions Connections Going back and forth, relevant to others 108 PAGE 109 Social languages Situated meanings Discourses Conversations Intertextuality Discourse/Cultural Model Figure 42. Inquiry tools used in Gees discourse analysis 109 PAGE 110 CHAPTER 5 CONSTRUCTING MATHEMATICS K NOWLEDGE THROUGH ONLINE COLLABORATION Preamble This study analyzed five months of data from a discussion forum in the Math Forum @ Drexels website. Discourse analysis, as stated by Gee (1999, 2005), was used to perform data analysis. Specifically, it followed the steps introd uced in Chapter 4. In general, analysis was organized by period, thus provi ding the researcher the opportunity to develop a preliminary mathematics discourse model after the first period of analysis (August Chapter 5) and then to review and refine the model throughout the fo llowing two periods (September and October Chapter 6, and November and December Chapter 7). Each case of study was a threaded discussi on, a set of related messages. These were analyzed to differentiate the stor ies present in each thread. Their structures were then converted into tree diagrams that showed several branch es and subbranches, which in turn identified different dialogues or stories wi thin a discussion. These stories were rewritten into theorized transcripts and were further analyzed. Each thread had a specific mathematics topic introduced as a question or problem (catalyst). These were included in the first message of the discussi on (original post). On occasion, they also included a specific set ting. The messages that followed examined the question or problem (evaluation and crisis co mponents of a story) until a resolution was achieved. Some threads also included a coda addi ng additional information to the threads. This chapter is the first of three data anal ysis chapters and incl udes the analysis of Augusts data. The organization of this chapter is similar to that of the next two data analysis chapters which include the analyses of the following months until December. The first three sections of each chapter explain the analysis of the main components of each story. The first 110 PAGE 111 includes the analysis of the se tting and catalyst; the second co ntains the analysis of the evaluation, crisis and resolution; and the third provides the analysis of the coda. The researcher used the activity and connections building tasks as the main source of analysis, using data questions listed in Chapter 4. However, data also allowed the researcher to explore other building tasks. Further details are included in the introduction of each chapter. Finally, to end each chapter, a preliminary mathematics discourse model is presented to the reader. It shows how mathema tics knowledge was negotiated a nd constructed throughout each period of study in the onli ne discussion forum of the Math Forum @ Drexels website. The first month, August, provided the basis for a general ma thematics discourse model. The patterns that started to emerge at this point were revi ewed and refined throughout the following months. Nevertheless, the final model can only be taken as a tentative way of constructing mathematics knowledge in a specific online discussion forum. Introduction to Augusts Data A total of four threaded discussions met the se lection criteria in the month of August. As stated in Chapter 4, these were threaded discus sions with 10 to 25 messages (posts) that included mathematics topics ranging from General Mathem atics to Calculus. In the first period of analysis, the following mathematics topics were present: PreCalculus (Exponents), Calculus (Radius of an Arc and Integ rate!!), and Discrete Mathema tics (Probabilities). Table 51 shows the number of postings in each threaded discussion. They fluctuated between 10 and 17 messages and produced from one to 10 st ories in a single th readed discussion. Tree diagrams were constructed to follow the co nversations that took place in the threaded discussions (Figures 51 and 52). These were sets of posts (m essages), represented as rectangles, that followed an original communication (origi nal post). Arrows coming out of a rectangle indicated that one or more replies were made to a message. Starting at the original post and 111 PAGE 112 following the arrows downward until there are no more arrows, a branch of the tree diagram was identified. Each tree diagram represented a th readed discussion and included one or more branches. Branches were identified as Stories and were noted where they occurred in the diagrams. Examples of stories with two messages are found in Figure 51 (Stories 1, 9, and 10) and in Figure 52 (Story 3). Some stories included simila r posts, and in the extreme cases only the last post was different. That is, they had more than one ending (differe nt codas). A total of 20 stories were identified among the four threaded discussion s selected for analysis in the August data. Part I of this analysis describes data usi ng the activities and connections building tasks. Both building tasks provided the tools to identify main activities and subactivities related to the construction of mathematics knowledge. These were organized into three main sections: Problems, Questions, and Inquiries Introduction, with information about the setting, about how the authors of a problem or question of an original post guided the participants, and about how the authors positioned th e questions; Problem Evalua tion and Solution Generation, concerning how inductive reasoning, algebra, an d geometry were used throughout the evaluation, crisis, and resolution of the di fferent stories; and August Coda (s): Additional Information, suggesting how to use the forum to promote a better discourse among its participants. Data from the first month also allowed the re searcher to explore tw o more building tasks: sign systems and knowledge, and iden tities (Part II of the analysis). First, the sign systems and knowledge building task was used to analyze data that showed how authors used common language and mathematics notati ons as well as their comments regarding its importance. Second, the identities build ing task helped to examine the F orum participants, thus allowing 112 PAGE 113 the researcher to describe those who inter acted in the forum to construct mathematics knowledge. Part 1: Analysis of Activities and Connections Problems, Questions, and Inquiries Introduction The setting and catalyst of each threaded discussion in the August data were included in the original post (first message of a thread). Ac cording to Gee (1999), the setting sets the scene in terms of time, space, and characters, ( p. 112) and the catalyst sets the problem. Setting: Use of time, space, and characters Using the Math Forums website, the Discussion Forums space was controlled by the software itself. It provided participants two textboxes; the first was used to add a topic (which worked as a title) and the second to add a text message. This second textbox had no limitations on the number of words, symbols, lines of text, or equations used. It was in this second textbox that participants added questions or problems to further analyze them in collaboration with other participants. This site had no constraint of tim e, since it was available 24 hours, seven days a week. Data showed that three out of four original posts did not include a specific problem setting; that is, they did not make reference to time, character, or space related to the math problem itself. The only thread that made reference to time a nd character was the Probability thread. Data follows: I SETTING & CATALYST Stanza 1: Presenting the problem Alfredo 1 Recently 2 a prominent political figure stated 3 that the probability of finding 2 persons 4 with the same Birthday date 5 in a group of 50 6 is almost 99% 113 PAGE 114 7 is it true? 8 how do i calculate that? In terms of time, the author indicated that the problem was recently" stated. Therefore, this problem was probably stated during the pr evious weeks, around the months of July or August. There were, however, no more indications of time. In this same message, a character was identified when indicating who proposed the pr oblem, that is, a prominent political figure. However, no other characteristics of this person were added. Finally, there was no indication of where the author heard the problem. The fact that only one out of the four thread s selected for analysis included a setting and that it did not include many details pointed to the idea that stating a setting in a math problem in this environment was not as important for the participants as stating the problem itself. This was also a reflection of how mathematics is taught at most secondary schools and college. There is an emphasis on abstract ideas instead of concrete or real life prob lems. The following chapters will continue to examine how the setting was used to present mathematics problems and questions to confirm or reject this preliminary outcome. Catalyst: Analysis of problems and questions Most authors started the threaded discussions with the presentation of a question based on a specific problem. These were introduced in diffe rent ways. Participants included explanations and examples and paraphrased questions or problem s. For example, in the thread Radius of an Arc, the question was written at the begi nning of the message and was followed by an explanation that included a para phrased question. The author th en ended the message with a comparison to another problem. The following portion of data shows this interaction. I CATALYST Stanza 1: The Problem Sebastian 1 Is it possible to find 114 PAGE 115 115 2 the radius of an arc 3 given just two points 3a (coordinates) 4 of the arc? Stanza 2: Explanation 5 In other words, 6 every arc 6a is part of a circle 6b with a certain radius. 7 If two coordinates 7a (x1, y1) and (x2, y2) 7b of the arc are given, 8 can the radius 9 be worked out 10 using those two coordinates? Stanza 3: Comparison 11 I have been able to do this 12 given three points of the arc 13 but the application 14 I am working with 15 has only 16 two known points 16a of the arc. In the first Stanza, Sebastian presented the main question of the problem to the forum participants and included specific conditions and an alternative way to identify the points in a graph (the coordinates). The s econd Stanza started with the conn ection clause In other words and was used to elaborate on the ques tion presented in the first stanza. As shown in the data sample above, Sebastian wanted to make sure that the question he posed was understood. For this reason, he decided to use different methods to state his problem. First, he presented the question (L ines 14), and then he included an explanation (Lines 57a). In this way, he stated the conditions under whic h the problem had to be evaluated. He also paraphrased the initial ques tion (Lines 810) in an attempt to make clear what he initially meant. To end this original post, Sebastian added a comparison to a similar problem (Stanza 3), an PAGE 116 explanation of when he was able to find the radius of a circle. He guided future participants in a specific direction so that the evaluation of his problem was as accurate as possible. Thus, the author tried to be very clear and specific about his question. A counterexample was found in the Integrate thread, where the author asked for help by stating a direct question. I CATALYST Stanza 1: Presenting the problem Mara 1 I need some help 2 to find step by st ep integral!!!! 3 sqrt(1+x^2)dx =?????? In this case, no details are offered to the pa rticipants; there is just a question with no examples or paraphrasing. No pattern seems to be used to present the mathematics question to the participants of the forum. The only commonality between this post and others was that like most of the original pos ts, there was a question. During the first month of analysis, most partic ipants (three out of four) used direct or indirect questions to introduce their problems or questions. For example, participants stated, I need some help to find ., Is it possible to fi nd .?, Can the be worked out?, Is it true? How can I calculate .? Through these indirect and dire ct questions, original posts authors took the voice of those who seek fo r support from more knowledgeable others. Some included a tone of skepticism (Is it true?); some asked for assuran ce (I have been able to. .); some overtly sought step by step procedures; and still others asked for analysis (Is it possible to .). In most cases, partic ipants entered the Math Forum to find help, and they asked specific questions to locate more knowledgeable ot hers who could provide that help. Having the question at the begi nning or end of a message di d not significantly influence the number of replies in these threaded discussions. Original posts with questions at the 116 PAGE 117 beginning of the messages generated between eleven (11) and sixteen (16) replies. Those with questions at the end of the messa ges generated nine to fourteen (14) replies. However, not all original posts included a speci fic question. In the thread Expone nts, the original post included a problem with a possible solution and an example. I CATALYST Stanza 1: Presenting the problem Frank 1 Raising 3^x = 0 to 1/x 2 We get 3 = 0^(1/x) Stanza 2: Presenting a possibl e solution of the problem 3 Using 3^x = y 4 taking the natural log 5 we have x ln 3 = ln y 6 or x = (ln y)/(ln 3), 7 a simple log function 8 not defined at y = 0 As shown above, there was no specific questi on in this thread; only a problem and a possible solution were stated. Franks mathemati cal statement was a statement that supported a specific idea instead of a ques tion that looked for more inform ation, help, or corroboration. In this post, there was no overt guidance of what the author wanted or needed. Instead, Frank showed how he started solving the problem by explaining how he had analyzed the problem. This post presented the opposite situ ation of that of Sebastian (above ). Nevertheless, this did not discourage participants of the forum, and they engaged in an interchange of ideas on the topic that led to a discussion between two partic ipants, a controversy interrupted by a more knowledgeable third party who ended the thread. In summary, two major ideas can be gathered from these examples: first, most threads included a question in the original post, and second, the location of the question did not affect the number of replies. Additionally, in most threads (three out of four) the first message did not 117 PAGE 118 include a specific setting to the problem. Instead, authors chose to present the problems (catalysts) using different methods such as questions, paraphrased questions, examples, and other comments that guided the participants of the fo rum to answer the question presented to them. Only one thread included a partial setting that ga ve the time and character associated with the problem, but this information was not relevant to answering the question. The questions in the catalyst section were in cluded in different areas of the messages. Whether at the beginning or at the end of the post, the place ment of the questions did not influence the number of replies received. A count erexample was found when the author of one post did not include an overt que stion (no direct or indirect question) but only included a problem with a possible solution. Although this did not discourage the users of the forum from engaging in the evaluation of the problem, only one story with two participants was generated from this original post, and it ended with a message from Tom, a third, more knowledgeable participant who clarifie d and thus ended the discussion. The author of this thread did not participate in the conversation; instead he stayed in the backgr ound until the problem was resolved by Tom. Problem Evaluation and Solution Generation After initiating a new thread by asking a question or presenting a problem, voluntary participants engaged in a dial ogue that promoted the constr uction of mathematics knowledge. Participants worked together to build the problem to the point of requiring a resolution (crisis), including material that [made] clear why the story [was] interesting and tellable (sic) (evaluation) until they found one or more solu tions (resolution) to the problem (Gee, 1999, p. 12). The following pages include the analysis and inte rpretation of the participants interactions that exemplify how they collaborated to construct mathematics knowledge. First, algebra 118 PAGE 119 manipulations were used to solve mathematics problems. However, using algebra as the sole source of information to find the solution to pr oblems was not always enough. The principles and definitions behind a problem pointed to specific algebra uses and manipulations. Participants also used geometric definitions and drawing explorations to develop mathematics knowledge. For example, in the thread Radius of an Arc, participants engaged in a series of posted contributions that included specific and general algebra solutions as well as geometric explorations. An example of algebraic misinter pretation was found in th e thread Exponents. In the thread Radius of an Arc, participants evaluated Se bastians question in different algebraic ways. For example, in Story 2, Ral ph introduced inductive reas oning through algebraic manipulation. He tested different coordinates that sa tisfied the premises stated by another participant of the thread in a previous message. Stanza 15: Building a generic ca se through inductive reasoning: Case 1 Ralph 104 How about: x^2 + (y 4)^2 = 25 105 Let's test it for (3, 0): (3)^2 + (0 4)^2 = 25 9 + 16 = 25 25 = 25 check! 106 [test it for] (3, 0): (3)^2 + (0 4)^2 = 25 9 + 16 = 25 25 = 25 check! 107 So a circle 108 of radius 5 with center (0, 4) 109 passes through (3, 0) and (3, 0). Stanza 16: Case 2 110 Or how about: x^2 + (y 5)^2 = 34 111 [test it for] (3, 0): (3)^2 + (0 5)^2 = 34 9 + 25 = 34 119 PAGE 120 34 = 34 check! 112 [test it for] (3, 0): (3)^2 + (0 5)^2 = 34 9 + 25 = 34 34 = 34 check! 113 So a circle of radius sqrt(34) 114 with center (0, 5) 115 passes through (3, 0) and (3, 0). Stanza 17: A generic case 116 Or how about: 117 x^2 + (y 6)^2 = 45 118 x^2 + (y 7)^2 = 58 119 x^2 + (y 10)^2 = 109 120 and even 121 x^2 + (y k)^2 = k^2 + 9 In Stanzas 15 and 16, Ralph included detailed work, showing all th e steps (algebraic manipulation) he made to test a possible solu tion. In Stanza 17, he presented three more equations (lines 117 to 119), but he did not includ e all the details, leaving it for the reader to complete the same way he previously did in St anzas 15 and 16. In Line 121, Ralph presented a generic equation that could be us ed to find more specific cases. In Story 4 of the same thread, Jack also built on Joes coordinates to find a solution to the problem. He stated a similar argument to that of Ralph but from a more abstract position (See Stanza 15 below). Equations were stated, and pos sible values for the variables were given (Stanza 16 and 17), but deta ils were left out for th e reader to complete. Stanza 15: Equation: Abstract statement Joe 104 x^2 + (yb)^2 = c^2 105 will be a circle 106 through both 107 (3,0) and (3,0) 108 whenever 3^2 + b^2 = c^2, 109 which will be true 110 for infinitely many pairs 120 PAGE 121 111 of values for b and c Stanza 16: Cases of study 112 In particular, 113 for b = 0, c = 3 and 114 for b = 4, c = 5, 115 so right here we have 116 two circles of different radii. Stanza 17: Arriving to a generic solution 117 In fact, for every real number b, 118 let c = sqrt(9 + b^2) and 119 you will get as many circles, 120 x^2 + (yb)_^2 = c^2, 121 as there are real number values for b, 122 all passing through both 123 (3.0) and (3,0). RESOLUTION Stanza 18: Stating a solution 124 So the number of circles 125 is uncountably (sic) infinite, 126 much "larger" than 1 Starting with a general argument (Stanza 15), Ja ck wrote an equation, an abstract statement (line 104) that was explained in the lines that fo llowed. He based his argument on Joes previous example (Stanza 4) of two possible coordinates that belong to a ci rcle. Jack went on to show two particular cases where the equation held (Sta nza 16, Lines 113 and 114), demonstrating that it was possible to find more than one circle passi ng through the same two co ordinates. Then, in Stanza 17, Jack presented a gene ral solution to the problem. Ralph and Jack both used a similar perspec tive when analyzing this problem. They both used their knowledge of algebra to build a soluti on to the problem. Ralph started his presentation by showing the readers all of th e details on how to solve the e quations he posed. Jack, however, left these details to the reader. His presentation was more concise but also more abstract, leaving much of the work to the reader. The Radius of an Arc question was also analyzed from other 121 PAGE 122 mathematical perspectives in this thread. In the section Geometry references and drawing explorations, the geometric counterpart of this analysis will be presented to the reader. First, however, a look at algebraic misinterpretations will follow. When users of math do not take into account the principles and definitions related to a specific problem, they could engage in algebraic misinterpretati ons. This is not only related to computation errors; it also includes the use of sp ecific principles to analyze a problem and find a solution. That was the case presente d in the thread titled Exponents. Technical explanation: In pure mathematics, the equation 3^x = 0 has no solution. This equation reads three to the x equals zero or thr ee risen to the exponent x equals zero. To find a solution to the equation 3^x = 0, you would need a value for x that makes both sides of the equation equivalent, but that value does not exist. By definition 3^0 = 1, and as you substitute x for a positive integer, you will get a value greater than one, that is a positive integer such as 3^1 = 3, 3^2 = 9, and 3^3 = 27. If you use positive fractions to substitute for the exponent x, the equation will still give you a positive value. Fo r example, 3^(1/2) = 1.73 (the square root of three) and 3^(1/3) = 1.44 (the cube root of thre e). Otherwise, if you cons ider negative fractions or negative integers such as 1, 2, 3, the solution to this equa tion will still be positive. Other examples can be 3^(1) = 1/3 = .33 and 3^(1/2) = 0.58. Thus, when evaluating 3^x, only positive solutions are possible; that is, the solutions are gr eater than 0. Therefore, 3^x is never equal to zero, and 3^x = 0 has no solution. Figure 53 shows a graph of 3^x = y (where y is any possible solution). It shows how the curve of 3^x is above the xaxis, producing only positive solutions. A close examination of the graph shows that the cu rve approaches the xaxis in the negative side (to the left hand side on the horizon tal axis), but never touches it. 122 PAGE 123 This problem was the source of controversy de veloped by Mike and Jack, two of the four participants in this thread. Th e author of the thread, Frank, stayed in the background without making any other comments. The forth participant, Tom, also stayed silent until the end, at which time he presented the resolution. Throughout th e evaluation and crisis of this thread, Mike and Jack engaged in a dialogue in which Jack tr ied to convince Mike of the possibilities of finding a solution to the equation 3^x=0. As shown above, in the technical explanation, such a solution was not possible. The controversy was not resolved until a third party interrupted the controversy. Mike and Jack star ted their interaction as follows: Stanza 3: Problem evaluation Mike 9 The exponential function 3^x 10 never takes (sic) the value zero. 11 You are starting from an equation 12 that can never be satisfied. Stanza 4: New possibilities Jack 13 True, 13a if were dealing 14 only with real numbers, 15 but in the extended real numbers 15a [oo, +oo] 16 3^x = 0 16a when x = oo In Stanza 4, Jack tried to introduce a new math ematical interpretation to the problem. He explored the idea of the exponent as a negative infinitive (Line 16a ), but Mike tried to maintain the definition of the equation base d on mathematical principles. In the lines that followed, they pushed each other to try to interpret and reinterpret the equation beyond its mathematical meaning. Jack was trying to generate new m eaning and Mike was keeping his ideas tied to mathematics fundamentals. Their voices took different positions, pres enting their ideas in different ways. 123 PAGE 124 This was an example of how algebra is re stricted by mathematic al definitions and principles. Although a math rule allows one to complete a step in algebra, taking that step does not necessarily make sense. There is more to algebraic manipulation than simply following the rules. In this particular case, the forum served as a means to explore possibilities that might have been dismissed at once in the classroom due to a lack of time for discussion. It also allowed the space for a more knowledgeable other to present th e definition, origin, and use of exponents, not without first questioning the controversy genera ted by Mike and Jack ab out extended numbers. As Tom stated in the following stanza: Stanza 16: Evaluating the controversy Tom 96 What could be the motivation 97 of someone 98 to want to consider 98 Inf and Inf as extended "numbers"? 99 That is not helpful Afterwards, Tom went on to introduce the definition and history of exponents (Stanzas 17 & 18). He also made a reference to a professional article published in The American Mathematical Monthly by Knuth (1999). In this way, th e forum expanded its frontiers to professional literature, which could serve as an invitation to go beyond the discussions and controversies taking pl ace in the forum. In summary, using algebraic manipulations to solve problems was used in the forum in different ways. These included three formats: one, giving all details of the work or stepbystep solutions; two, giving enough information to solve a problem but leaving out the details for the readers to complete; and third, giving abstract solutions to solve a problem to generalize a solution. The problems posted in this month also showed how algebraic manipulation was not always enough to solve a problem. It also showed how the use of mathematical definitions as 124 PAGE 125 well as the use of mathematical properties is important to reach mathematically supported solutions. Otherwise, confusion and misinterpretations can result. In addition to algebra, geometry was also used to explore problems and find solutions. Geometry, the study of lines, surfaces, points, and curves, helped the participants of the forum to evaluate the problem posed by Sebastian. A rich set of possibilities was submitted and elaborated on throughout the messages of this thread by multiple participants. For example, John first used the concepts of perpendicul ar bisector and three noncolli near points, Second, Joe wrote about the endpoints of a diameter ; third, Jack introduced the c oncept of the minimum possible radius, and finally Ben showed a way to expl ore this problem on the Cartesian graph. These ideas were spread throughout different stories of the same threaded discussion, thus connecting different ideas among the stories of this thread. Participants were collab orating with each other and helping Sebastian interpret the problem from different geometric angles. John presented the following resolution and evaluation: Stanza 4: Answering a question John 17 No 18 Its not possible to 19 uniquely determine 20 the center of the circle 21 given only two points 22 on the periphery Stanza 5: Perpendicular bisector 23 Each point 24 on the perpendicular bisector 25 of the side connecting 26 the two points 26a (x1, y1) and (x2, y2) 27 will do. Stanza 6: Three noncollinear point s and three perpendicular bisectors 28 If you work with 29 three noncollinear points 125 PAGE 126 30 the center of the circle 31 is the intersection 32 of the three perpendicular bisectors 33 of the sides connecting the points 33a (x1, y1) and (x2, y2), 33b (x1, y1) and (x3, y3), 33c and (x2, y2) and (x3, y3). In Stanza 5, John gave a solution to the problem without an explanation. The reader needed to have an understanding of th e geometry terms introduced in the message in order to understand it. The same happened in Stanza 6 when John contin ued to debate about other ways to find the solution to the problem posed by Sebastian. This time John introduced two concepts in the same stanza. These were three noncollinear poin ts and three perpendicular bisectors with their corresponding coordinates. To understand what John was saying, the readers could draw their interpretations on a piece of paper, use computer software to sketch out the propositions stated, or they could mentally draw the diagrams posed by John. A second geometric analysis of the problem wa s posed by Joe when he used the concept of endpoints of a diameter to clarify when it was possible to find a singl e circle. This idea was previously posed by other members of the foru m, when presenting the algebraic solution. According to Joe, Stanza 19: About endpoints Joe 116 It's true that 117 if 118 (3,0) and (3.0) 119 are the endpoints of a diameter 120 then 121 there is only one circle. Stanza 20: Infinite circles 125 In fact 126 there are infinitely many circles 126 PAGE 127 127 that contain the points 128 (3,0) and (3,0). Stanza 21: Using an equation back to algebra 128 Any equation of the form 129 x^2 + (yk)^2 = 9 + k^2 130 (for any real k) 131 is the equation of a circle 132 that contains those two points. In Stanza 19, Joe used the logic construction ifthen to introduce the concept of endpoints of a diameter. With this construction, he stated the conditions under which only one circle can pass through the coordinates (3,0) and (3.0). As Joe stated, this can occur when the coordinates are the endpoi nts of a diameter. In Stanza 20, he fu rther explained that there can be an infinite number of circles passing through these coordinates. He did not explain why or how this was possible in this story. Instead, he connected his geometric explanation with the algebraic solutions given before by other members of the forum. Furthermore, Joe explained his last proposition in Story 2 when he stated that any point on the y axis / could be a center of the circle with coordinates (3.0) and (3,0) (Lines 22 and 22a). He implicitly stated that the radius did not need to be three (3) at all times. Jack, in Story 10, also referred to this point when he stated the following: Stanza 4: Minimum possible radius Jack 18 One can find 19 a minimum possible radius 19a half the distance 19b between the points 20 but no maximum The idea that there was no maximum number fo r the radii suggested the infinite number of solutions in this problem. This last point was also explained by J ohn in Story 3 when he suggested looking at the Cartesia n graph and drawing some circles (See lines 145 to 149 below). 127 PAGE 128 Stanza 24: Ends of a diameter John 135 if you have the points 135a (3,0) and (3,0), 136 those do not have to be 137 the ends of the diameter 138 of the circle 139 They could be points 140 near the top of 141 a really big circle 142 They could be 143 a little above or below 144 the diameter of a circle Stanza 25: Cartesian graph 145 Look at a Cartesian graph 146 and draw some circles147 you'll see that you can draw 148 a LOT of circles 149 going through 3,3 on the xaxis. In summary, by contributing to the analysis of the problem and working with different concepts and viewpoints, particip ants of the forum in this particular thread engaged in a distributed discussion giving pi eces of the solution throughout their messages along different paths of the tree diagram. This showed how the messages in the threaded discussion, though divided into 10 different stories (Figure 51), were connected to one another. One reason that made this possible was that five of the seven pa rticipants, including the author of the original post, wrote from two to five messages each. Their dialogue represen ted an instance of intertextuality, where one idea was completed by th e same person at another time, or even when somebody took the idea of another person and comp leted it. The asynchronous characteristic of the medium used in the forum allowed this to happen. In general, participants constructed mathematics knowledge in four ways: (1) using algebraic manipulations, (2) clar ifying algebraic misconceptions, (3) using geometric concepts, and (4) recommending the use of geometric constructions. 128 PAGE 129 Augusts Coda(s): Additional Information Codas were used to show how to address a question or idea to the forum participants. For example, in a followup message, the author of the Radius of an arc thread disagreed with another participant by insisting on the co rrectness of his position. He stated: Stanza 5: Insisting on his point Sebastian 23 Im convinced 24 there is a way 27 You may have overlooked this 38 there is only ONE point 39a on the y axis 40 and NOT infinite points Even though they included several misunde rstandings, these statements were not challenged overtly in any of the immediate messa ges. Instead, participants gave examples and presented different ways to look at the problem from algebraic and geometric points of view; they also clarified misunderstandings. The significance of Sebastians post was purposely overlooked by other participants of the forum who took a more c onstructive position and showed him how to evaluate the problem without ove rtly saying that he had a misunderstanding. Sebastians post generated four messages that led to even more replies. However, it was not until later, in Story 7, that Joe, in Post 13, addressed Seba stians statements and gave him some alternative ways to present his id eas to the participants of the forum. CODA from Story 7 Stanza 19: Recommendations Joe 122 make statements like 123 "I don't see how 124 that can be true" 125 or 126 "But it seems to me 127 that 2 points are enough" 128 or 129 PAGE 130 129 "Can you explain further 130 why there are 131 an infinite numb er of circles? 132 I can imagine 133 only the one with center at (0,0) 134 and radius 3." Stanza 20 : Apology Sebastian 139 My sincere apologies 140 to everyone :( Joes propositions evaluated the manner in wh ich Sebastian presented his ideas to the forum. He showed Sebastian three alternative ways to followup on his initial question (See Lines 123124, lines 126127, and Lines 129134). At that point, Sebastian was only able to apologize for his initial comments (See Stanza 20 ). His message had already been evaluated by other members of the forum who clarified his mi sconceptions with a variety of algebraic and geometric examples and arguments. Nevertheless Joes message was not unimportant; he gave others options and ideas about how to address the forum when asking followup questions in a way that would promote a better discourse between the participants of the forum. Part II: Analysis of Sign Syst ems, Knowledge, and Identities Symbols of Common Language and Mathematics Notation In presenting their mathematics questions or problems, authors mostly used words, but on occasion they also used basic mathematics notati on (symbols) or abbreviations. For example, in the thread Radius of an arc, Sebastian identified two coordinates as (x1, y1) and (x2, y2). In mathematics, coordinates of a point use numbers as subscripts, but the discussion forum system did not provide the opportunity to format text in any way other than simple text (like that of a typewriter). Here Sebastian decide d to write the number after the letter, which did not change its interpretation. This same notation was used by other participants of this thread to identify other geometric constructions. 130 PAGE 131 In the thread about exponents, Frank also used the symbols available on the computer keyboard to present his problem. He stated th e equation ^x = 0, wh ere x was the exponent. The notation he used included the caret (^), which was common in scientific and graphic calculators. These devices show the symbol on their screen but not on the keypad, where exponent keys, such as x2, xy and xn are used. The computer keypad shows the caret symbol (^) on the number 6 key. In the thread Integrate Mara presented a third example of how alternative ways to write mathematics symbols were used. She wrot e the abbreviation of square root as sqrt in line 3, the same way it is used in some spreadsheet programs such as Microsoft Excel She also spelled out the word integral without using abbreviations. There was no symbol available on the keyboard to substitute for the word integral, nor was th ere an abbreviation available in common software. These examples show that transfer from us ing other technologies such as calculators, computer hardware (keypad), and computer software ( Microsoft Excel) occurred when writing about mathematics problems. In an environment th at only allowed for simple text, the writing limitations were not an obstacle for the forums participants; these limitations were overcome by using other techniques borrowed from other technologies. Members of the forum used abbreviations, symbols available on the keypa d, or even spelled out words to clearly communicate the problems and que stions they were posting. Although critics stated before that simple text environments were not conducive to mathematical language in online environm ents (Smith, Ferguson, & Caris, 2003), the participants of these threads found ways to introduce and evalua te mathematical terms. They transferred notation from calcula tors and computer software pr ograms and spelled out complete words when no other symbols were available. In this way, the authors of the threads and those 131 PAGE 132 with whom they interacted were able to express themselves, to present mathematical questions or problems to the forums participants, and to devel op a set of possible solutions that allowed them to reach a resolution. Mathematics notation is an important tool in writing mathematics. According to Tom, in the Exponents thread, the proper notation helps mathematicians in a number of ways. He made the following argument when addressing Mike and Jack and their controve rsy about the use of infinite numbers. 100 notational conventions are put in place 101 for reasons of efficiency 102 of thinking and writing 103 and an aid to instant 104 recognition / apprehension 105 of complicated expressions, 106 to simplify and clean up 107 our writing of math. In lines 101 through 107, Tom explained w hy mathematical notation was important, giving three main reasons: (1) effi ciency of thinking and writing (Lines 101 to 102), (2) aid to instant recognition/apprehension of complicat ed expressions (Lines103 to 105), and (3) simplification of mathem atics writing (Lines 106 to 107). In this way, Tom reinforced the importance of using mathematical notations. In summary, writing in mathematics is not rest ricted to words or text; it also includes the use of symbols. As Tom indicated, symbols ar e used to shorten written explanations or expositions. Symbols are also a part of the mathematics languag e used by all mathematicians. Even though their meanings can cause confusion to new users of math, th e participants of the forum promoted their use and clarified misunders tandings when they arrived. They transferred their knowledge from other technologies such as calculators, computer software, and keyboards to express their mathematical ideas. In writing mathematics, participants of the forum generally 132 PAGE 133 used symbols, abbreviations, or spelled out words in their messages just as do authors of mathematics books. Identities of the Forum Participants Little information is known about the partic ipants of the forum. They were voluntary participants who could be usi ng pseudonyms. In this study, partic ipants of the forum were not contacted at any time, and pseudonyms were used throughout the analysis to protect their online identities. One reference was made about the type of par ticipants of the forum. Tom, in the thread titled Exponents presented his concern about participants becoming confused when arguments were unclear. He identified three ty pes of participants, as follows: Setting 15: About previous discussion 69 novice or 70 casual students or 71 users of math In this way, Tom took the position of a different type of participant, a more knowledgeable other who was concerned about the learner. Par ticipants also included those who provided extra information, thus complementing and even suppl ementing the forum discourses. For example, Tom and Joe gave references to a journal arti cle and a web page, respectively, that further explored the topic under consideration. Therefore, there were at least two general roles engaged in th e discourses of the forum: (1) on one side the novices, casual stude nts, and users of math; and (2) on the other side, the more knowledgeable others, possibly mathematics teachers, instructors, or professors. This concludes the analysis of data for the first period of analysis, the month of August. Putting together the findings st ated above, a preliminary discourse model is presented to the reader in the next section. 133 PAGE 134 Moving toward a Discourse Model: Summary of Augusts Data Analysis To develop the first discourse model the resear cher decided to use a set of questions that could help her organize and summarize the data set. She used six basic questions: what, who, when, where, why, and how. The main focus of this research was answered with the how question which listed the activities and connectio ns the participants engaged in (Figure 54). The discourse model presented in this chapter is just a partial account of what the final proposed model will be. This model is based on da ta from four threads of the discussion forum. Patterns are not clear at this moment; they ar e just beginning to emerge. This preliminary discourse model only indicates how participants could construc t knowledge in such an online environment. It will be revisi ted and reevaluated in the followi ng months. This first model is a tool of inquiry for the following periods of analysis. Novice, casual students, or users of mathematic s, together with more knowledgeable others collaborated in evaluating and findi ng different kinds of solutions to mathematics problems in the areas of precalculus, calculus, and disc rete mathematic, during the month of August. Participants used the following techniques to collaborate while c onstructing mathematics knowledge: Locating questions in different parts of the original message of a thread and accompanying such questions with examples, explan ations, and paraphrased questions Presenting inductive algebraic reasoning and st epbystep procedures, presenting partial procedures so that others could complete the work, presenting genera lizations and abstract statements, and connecting algebraic manipulations to mathematics definitions and principles Presenting different geometric concepts and co nstructions related to a single problem to explore different ways of anal yzing and solving a problem. Connecting algebra and geometry to explore a single problem. Using common language as well as mathematics symbols and abbreviations to communicate. 134 PAGE 135 Complementing and supplementing evaluate d problems by presenting references to documents outside the forum, such as a webpage and a professi onal journal article. Promoting the use of a more open and inviti ng language to ask for followup information by providing specific examples. The next chapter presents the data analysis for the months of September and October. It concludes with a review of this Pr eliminary Mathematics Discourse Model 135 PAGE 136 Table 51: General description of threaded discussions in August Title Content Area Number of Postings Number of Stories Number of Participants Time Span in Days Radius of an arc Calculus 17 10 7 4 Exponents Precalculus 10 1 4 2 Probability Discrete Math 14 4 9 2 Integrate!!! Calculus 12 5 9 3a Note: a) The threaded discussion Integrate!! wa s developed in three diffe rent periods: one day in March, one day in July, and one day in August. 136 PAGE 137 Story 8 Story 6 Story 5 Story 3 Story 2 Story 4 Story 10 Story 9 Story 1 Post 2: Ralph Evaluation, Crisis, and Resolution Post 5: Ralph Algebraic Solution with details Post 8: Jack Crisis and evaluation with examples Resolution Post 6: Sebastian Coda Post 7: John Resolution explanation and reference to the Cartesian plane Post 9: Israel Coda: Asked followup question Post 15: Joe Resolution and coda Post 10: Sebastian Coda: Personal comment Post 11: Jack Coda: Related comment to threads topic Post 12: Israel Coda: Answers personal comment Post 13: Joe Coda: Related comment to threads topic Post 14: Sebastian Coda: Apology Post 4: Sebastian Disagrees with Joe Post 17: Jack Resolution Post 16: John Resolution and Coda Post 3: Joe Geometric and Linear Algebra Solutions Original Post: Sebastian Presents a question, reframes it, and adds work completed. Story 7 Figure 51. Radius of an arc tree. Each color represents a differe nt participant. For example, the author of the original pos t participated five times in this thread (see yellow boxes) and was the one with the most participation. All other participants wrote from one to three messages each. This threaded discussi on generated a total of 17 posts, organized into 10 stories, with 6 participants. 137 PAGE 138 138 Story 4 Story 3 Story 5 Story 2 Story 1 Original Post: Problem Post 1 From Mara Post 7 Santos Corroborates answer Presents solution Post 4 Victor Pr ese nt s so l u ti o n Post 5: Louise R e s tat es so l u ti o n Post 6: Louise Co rr ec t s h e r se lf Post 2 Armando G i ves hin t Post 3 Domingo Followup Post 8: Gregorio G i ves Hin t Post 9: Penelope Pr ese nt s n ew p r ob l em Post 10: Louise Pr ese nt s alt e rnati ve so l u ti o n Post 11: Louise Co rr ec t s h e r se lf Post 12: Armando Presents new hint Figure 52. Integrate!!! tree. This threaded discussion has a total of 12 posts, organized into 5 stories with 8 participants. White color boxe s show participants who only contributed once to this discussion. PAGE 139 Figure 53. Graph of 3^x=y 139 PAGE 140 Why?To find and provide help in learning math What? topics Math Figure 54. Writing the discourse model Mathematics Discourse Model Who? Helpseekers and more knowled g eable others How? Activities and Connections When? August December 2004 Where? The Math Forum @ Drexel Where? Discussion Forum: alt.math.undergrad 140 PAGE 141 CHAPTER 6 NEGOTIATING MATHEMATICS MEANING AND UNDERSTANDING Introduction to Septembers and Octobers Data After stating a Preliminary Mathematics Discourse Model at the end of the last chapter, new data was analyzed to review the model. Following the same steps introduced before (in Chapter 4) and based on Gees discourse model, th is chapter presents the September and October data analysis. It comments, revise s, and adds details to the previ ous analysis. At the end of this chapter, the Preliminary Mathematics Discourse Model is restated accordingly based on the analysis this new data showed the researcher. The next chapter will include the analysis of Novembers and Decembers data. In this way, the Preliminary Mathematics Discourse Model will be reanalyzed and refined to state a workin g Mathematics Discourse Model, one that came from the analysis of five months of threaded di scussions in an online pub lic forum located at the Math Forum @ Drexel. The months of September and October provided th e researcher with a total of thirteen new threaded discussions, seven from September and si x from October (Table 61 for details). They each contained between ten (10) and twentyfiv e (25) messages and examined topics from General Mathematics through Calculus. Specificall y, discussion topics were related to General Mathematics (two), Algebra (one), Statistics (one), Discrete Mathematics (two), Trigonometry (two), PreCalculus (one), and Calculus (fou r). As shown in Table 61, postings in each discussion included from ten (10) to twentythr ee (23) messages, and included the participation of two to nine people. Data did not include a thr eaded discussion with only one story as it did in August. The number of stories in each threaded discus sion ranged from two (Figure 61) to nine (Figure 62). However, this did not mean the stories were all independent. In many occasions, 141 PAGE 142 stories were intertwined with one another as th e following data analysis will show. An extreme case was represented in Figure 63 where a single conversation was divided into eight stories. The next sections include the analysis of da ta using the activities and connections building tasks. Both provided the tools to identify how th e opening post authors and how participants in general constructed and negotia ted mathematics knowledge. As in Chapter 5, activities and subactivities were organized by secti on. In comparison to Chapter 5, th is chapter includes all of the same sections except for the Chapter 5 sect ion titled Common Language and Mathematics Notation. Each section compares its contents with Augusts data and incorporates new details to the previous analysis to include data from September and October. The second and third month of data also allowe d the researcher to co ntinue exploring the identity building task. A set of I statemen ts taken from lines throughout the threads of September and October were used to identify who was seeking for help and what their state of mind was. Using these lines, the socially situat ed identities of the threads opening posts authors were explored and presented in the sectio n titled Identities of the Forum Participants. Part I: Analysis of Activities and Connections Problems, Questions, and Inquiries Introduction The first message (opening post) of each th readed discussion included the setting and catalyst of the stories. It was there where a pr oblem was proposed, where one or more questions were asked, and where the author asked for he lp or corroboration. This post introduced the setting of the thread, which according to Gee (19 99) sets the scene in terms of time, space, and characters (p. 112). Most authors also added a dditional information in the first post, including partial work done. The following subsections will analyze the opening posts and their components. 142 PAGE 143 Setting: Use of time, space, and character The setting during the second period of anal ysis was the same as that stated in the previous chapter. For most par ticipants of the public online disc ussion forum (twelve (12) out of thirteen (13), 92%), the time, sp ace, and character of the problem were not relevant. Instead, they used the opening post to introduce one or more questions or a problem. As in Chapter 5, only one thread included a specific setting. Also, as in the previous chapter, this problem was related to probabilities. In this particul ar case, the author wanted to know how to calculate the probabilities of winni ng a legal case because he was going to court. The fact that the topic of this problem was the same as that of Chapter 5 points to a pattern in statistics, indicating that probabi lities are most often tied to a sp ecific setting. This was the only case out of 13 threaded discussions (approximate ly 8%) where a specific setting was described. Taken together with data from August, there are two out of 17 thread s (11.8%) that present a setting, and both are related to probability problems. Data from next month will clarify this outcome. Catalyst: Analysis of problems and questions Topics in the threads were introduced in various ways. Par ticipants stated a problem by introducing one or more questions ; they demonstrated how they started their analysis by substantiating their questi ons with partial work done before po sting; they looked for clarification, corroboration, and more. Catalysts were also used for different purposes, from trying to find the answer of a specific problem (to complete an assignment or answer a personal question), to trying to conceptualize a concept, or even to attempting to develop new mathematics. As in data from the previous month, mo st threads added one or more questions, sometimes at the beginning of the thread and so metimes included at the end of the problem presentation or in between the problem explan ation(s). If authors di d not include sufficient 143 PAGE 144 details about their question, they were asked to guide the participants of the forum and indicate the specific help they needed. Participants wanted particulars of the work already done to address a specific question. They overtly st ated that they needed more de tails, that is, more information about what the authors misconceptions were so that they could answer the opening post authors questions. In the Basic Measures thread, for example, Leon pr esented a problem without any question. Here the author gave no gui dance to the participants of th e forum as to what he needed or what his misunderstanding was. He then received an immediate reply asking for more information. In this case, Leon was asked to sh ow his work; he received a reply with the following question: What have you done so far? [ Basic Measures : Story 1, Line 38]. Authors of most threads substantiated their questions with partial work already done. On some threads, this meant stating definitions of terms used or including the algebraic proc edures used to start solving a problem. On other threads, it meant asking for corroboration or looking for understanding. This supports the idea that presen ting a problem alone or a simple question at the end of a problem was not enough to guide the part icipants of the forum to negotiate meaning. In the thread Planarity of a Graph, Norm an introduced a problem and then went on to state the definitions related to the specific prob lem. His was a Discrete Mathematics problem and algebra was not needed to solve the problem; he needed to write a proof. Norman did not ask for a solution or a stepbystep process to solve the problem; instead he asked for tips. This thread is unique because it was the author himself who st ated a possible solution at the end of the first story. A total of three participants (including the author of the or iginal post) shared their ideas and collaborated to find a possibl e solution to the initial problem. 144 PAGE 145 Problems related to calculus were supported using algebra. For example, in the thread Trouble Finding Derivative (Figur e 62), the author demonstrat ed stepbystep procedures followed by little or no explanation of why he ch ose a particular method. His was an indirect question: first, he stated the problem, then he argued that Ive done this halfdozen ways, and none gets me the right answer [Lines 45], and then he showed two algebraic attempts at solving the problem [Lines 715 and 16 to 20] There was no specific question in the opening post of this thread, only the step bystep procedure to substantia te his problem. The author was looking for clarification. When solutions seemed inco rrect, authors asked for corrobo ration. Such was the case of Jonathan in the thread Tan to Slope, in wh ich he introduced a problem, showed his algebraic work, and indirectly asked for corroboration. Stanza 3: Looking for Corroboration Jonathan continues 13 And yet, while the slope is correct, 14 the 'b' is incorrect. 15 I don't see where 16 I could be going wrong 17 in such a short, short process. He was confused because although he believed that he was following the right procedure, the answer he got as a result of his work was not the same as the one provided by a solution list. This thread generated 11 messages divided into thr ee stories. In this case, the forum was used to confirm the results of the authors work. In another thread, John ended his opening pos t with the statement, Any help / clearing this up / would be very much appreciated [ Derivative : Lines 2931]. He was looking forclarification and understanding. His opening messa ge presented a problem regarding finding the first and second derivative of a logarithmic function. In his posting, he used algebra to show the 145 PAGE 146 work he had done, as well as written explanations (using words), as if he was thinking out loud. John also used the textbook to find out if his answer was correct but did not understand how the solution was found. His confusion was evident when he asked 23 what I dont understand 24 Is how we got 25 The term 2 + ln t. 26 Where did the + come from? Two more authors who looked fo r clarification were Josefina and Gabriel. Josefina was confused with a trigonometric nota tion. She stated, I assume / 2 sin A is not the same as sin2A / Right? [ 2 sin A vs. sin 2A: Lines 24]. Her question had to do with the concept of multiplication and the commutative law, as if in trigonometry the variable A was a specific number instead of an angle of which you had to find the sin function equivalence. This thread generated 14 messages divided into seven stories. In Gabriels case, he sought an explanati on to further understand a logarithm rule (Figure 64). He wanted to know why this is [ Logs : Line 5]; he wanted to conceptualize a rule that was given to him. This thread generated three stories and included a total of ten messages with a total of six participants. Three of these participants John, Josefina, and Gabriel, used the forum to develop deep understanding of mathematics instead of surface learning. They were not satisfied with memorizing a classroom rule without knowi ng what it meant. For that reason, they looked for more details, information, and examples to clar ify their misunderstandings with help from the Math Forum @ Drexel participants. They negotiated mean ing and understanding with the more knowledgeable others present in the forum. Deep understanding was not only related to homework problems. In Bernices case, the goal was to find help in devel oping new mathematical equations. Hers was the case of a highschool math addict [ Consecutive Terms : Line 75], as she identified herself, who was interested 146 PAGE 147 in finding general algebraic equations to solve consecutive terms that she was developing on her own. In summary, the authors of the threaded discussions in the months of September and October looked for the following types of answers: corroboration, clarification, deep understanding, tips, and specific answers. Some original posts authors were interested in completing homework questions or problems; others were looking for specific answers and understanding. Close to a third (four out of thirteen, 31%) of the au thors in this period, overtly stated that they were trying to construct d eep understanding of math ematical concepts. Problem Evaluation and Solution Generation Besides using algebraic manipulations, geomet ry references, and drawing explorations, as participants did throughout the August data, foru m participants interacted with one another to clarify definitions and notations Intertextuality was used to break down messages, to cite specific portions, to add, comment, and correct mathematics errors, and to answer questions. Most threaded discussions included citations from other posts or ev en from other stories. In this way, participants were able to analyze speci fic ideas presented by others while trying to minimize further misconceptions. In Graph Planarity, Norman ended the presentation of his problem with the question, Any tips? (Story 1, Line 39, Figure 61). To answ er his question, two part icipants, Wilfred (in Story 1) and Sam (in Story 2), re evaluated the original post and presented a theorem that could help Norman clarify the problem. However, the theorem still was not fully understood by Norman because of the notation iff (if and only if ). In mathematics, this notation is used to present a condition that needs to be satisfied. No rman was reminded of this by Wilfred and then again by Sam. 147 PAGE 148 Story 1 Stanza 11: Notation clarification Wilfred continues 86 Common mathematical usage 87 for 'if and only if' is 'iff'. Story 2 Sam Stanza 7: Clarifying use of notat ion iff in Kuratowskis theorem > From Norman >39 Any tips? >> From Wilfred >>63 Kuratowski's theorem: >>64 G planar iff G has >>65 no subgraph isomorphic >>66 to K_5 or K_3,3, >>67 the Kuratowski graphs. > From Norman >>>77 Yes, >>>78 but what about graphs >>>79 that are unplanar and >>>80 do not have subgraphs >>>81 isomorphic to K[5] and K[3,3]. 53 They don't exist. 54 ('iff' = 'if and only if'.) The use of intertextuality in this story allo wed its participants to develop a conversationlike interaction. As shown above, the greater than sign (>) was us ed to identify the contribution of each participant. In Story 1, the notation iff is expanded to its mathematics meaning, if and only if. In Story 2, Sam used three portions of previous messages note the number of greater than signs used to indicate who said what. Some misconceptions in mathematics occu r because mathematics has its own language. In order to fully understand the mathematical th eorems, corollaries, definitions, and propositions, one must understand what the notation and voca bulary means. Norman did not understand what if and only if meant. This limitation hindered him from fully understanding the topic of study. 148 PAGE 149 Even though Wilfred and Sam tried to help Norm an find a way to solve the problem, his limited understanding of the notation obstr ucted his learning. In this th read, no resolution was reached. In the thread Extrema, Jonathan asked for help in differentiating the function f(x) = e^(x)*sin x. Part of the discussion led to the analysis of e^(x) and how to differentiate this factor. Five participants in teracted and negotiated m eaning until a solution to the problem was reached. From a total of eight stor ies in this thread, two (Story 5 and Story 6) evaluated Jonathans work in de tail. Rogelio identified an error in Jonathans work and stated, Thats incorrect [Story 5 and 6, Line 18], and then corrected Jonathans work using intertextuality to explai n why he thought there was an error. Jonathan followed up with another question, and two other particip ants, Morgan and Jake, answered by explaining the difference between positive and negative exponents when diffe rentiating a function. In this thread, the use of intertextuality allowed the pa rticipants to introduce explana tions and to reach a solution. Besides misconception of mathematics principl es and theorems (presented in Chapter 5), computation errors also generated confusion in le arning mathematics. Participants corrected each other using intertextuality, and those who had made the errors accepted being corrected. For example, in Trouble with Derivative (Figure 62) Flores, one of the six participants of this thread, identified a computation error by Domingo (observe intertextuality in Lines 31 to 34) and presented the solution in the following way. Stanza 7: Making a correction Flores > From Domingo >31 So the function is the same as >32 8x ^ 9/2 7x ^ 7/2. 40 OK, so far. >33 Now differentiate and get >34 16/9x ^ 2 + 49x ^ 9/2 149 PAGE 150 35 I get 36x^(7/2) + 49/2 x^(9/2) IV RESOLUTION Stanza 8: Agrees with Flores observation Jonathan 36 So did I, 37 and it was the right answer. Stanza 8: Accepts correction Domingo 41 You're right, I stand corrected Flores used intertextuality to evaluate Domingos work. Flores restated the error (lines 3334) and then added a line (35) with the correct solution. A thir d participant, J onathan, agreed with Flores, confirming the correction, a nd then Domingo accepted the correction. A second thread, Logs (Figure 64), incl uded a correction concerning notation. This time it was an error in an explanation. Once again, intertextuality was used to present the error, and as before, the correction was presented in a single line. Although six participants negotiated meaning in this thread, only two participated in this interchange of ideas. Stanza 5: Correction of previous notation Javier > From Joe >6 a^t = b is the same equation as >7 t = log_b(a), the baseb log of a. 15 ^^^^^^^^ log_a (b) Stanza 6: Accepts correction Joe 16 Omigosh. Thanks for correcting me. When correcting errors, participants also corrected themselves. This was the case of Timothy who noticed an error afte r posting a message. In the thread titled Integral, he added a post to correct himself. He stated 150 PAGE 151 Stanza 6: Correction Timothy continues 50 Whoops 51 book's answer is ok. 52 And I had a sign mistake 53 in my answer: 54 should be + sqrt[x x^2]. 55 Otherwise the two answers, 56 while different in appearance, 57 are reconcilable. In this thread, Timothy presented the correct answer and explaine d why his answer was incorrect. This did not happen in the previous examples where participants only included the lines with error(s) and a line with the correction. Another source of misconceptions in mathem atics comes from the use of tools and the limitations embedded in them. An example of such is presented in the Zero Story. In this thread, Bernice, the author, examined division by zero. She made a reference to a graphing program when stating the following: Bernice : 20 Also 20 my graphing program 21 on my computer agrees: 22 y=2/x 23 if you find where x=0 24 the y will be at infinite. Her argument was disputed by two other particip ants of the thread. First, Pepin conditioned his answer to the type of problem, one that in cluded vertical asymptot es (lines that bound a graph, where a function has no meaning). Pepin continues 52 Graphing programs are 53 not to be trusted if 54 there are vertical asymptotes. 151 PAGE 152 And second, Mike supported Pepi ns points when he stated: Mike 86 Well, no program 87 graphing or otherwise 88 should be trusted "blindly". Mike asked for careful consideration of any pr ogram used to solve a mathematics problem. Tools that help solve mathematics problems are not always one hundred percent accurate and can lead to confusion when their limitations are unkn own. This was the case of Bernice. The forum participants then helped her clarify her miscon ceptions regarding division by zero. They used the formal mathematics definition of division as we ll as other examples and references to web articles. In summary, participants of the forum evaluated problems and negotiated meaning by following up on questions, by adding comments, ne w information, and hints, by clarifying and defining concepts, mathematics terms, and notat ion, by presenting specific (stepbystep) and general examples, by introducing an d explaining math rules and properties, by relating answers to other math content (Table 62), and by co rrecting mistakes. They also supported math teachers arguments, questioned book solu tions, and analyzed graphing programs. Interaction between participants was possibl e through the use of intertextuality, allowing authors of original posts and othe rs to develop conversationlike interactions that promoted the development of new understandings and meanings as well as the clarification of different kinds of misconceptions. Corrections in the threaded di scussions were accomplished in three different ways. First, participants correct ed themselves; second, participants corrected anothers work; and third, participants confirmed other participants corrections. Participants who made mistakes accepted being corrected. 152 PAGE 153 Septembers and Octobers Coda (s): Additional Information Closing a story in a thread was like closi ng a chapter in a book. Participants of the discussion forum added different ty pes of comments that could or could not be related to the topic of the thread itself. In the last chapter, we saw how participan ts used the coda to show other participants how to address a que stion or idea. In this period of analysis, data showed endings with generic statements that included one or more of the following: gratitude (62%), antagonistic or mocking remarks (15%), and different kinds of recommendations (15%). About a third of the endings were specific to the prob lems stated in the threads (31%). For more details see Table 63. The following are examples of extreme codas presented in the forum, those that showed gratitude and those that presented antagonistic remarks. More than half (8 out of 13 threads, or 62%) of the endings included generic statements of gratitude in the form of thank you notes from the opening post authors. The following is from a simple statement in Zero Story: Stanza 14: Gratitude Bernice 172 Thanks Mike 173 this [is] sorta (sic) exactly 174 what I was looking for. Next is a statement that assessed the authors learning in Probability: Stanza 21: Goodbye post Gary 174 To everyone: 175 Thanks for the education. 176 If nothing else, I've learned 177 to "correctly state the problem" The following are statements that acknowle dged new understanding in Trouble finding a Derivative: 153 PAGE 154 V CODA Stanza 15: Finds understanding and gratitude Jonathan 80 That makes it all 81 make /so/ much more sense. 82 I thank you. Most authors of opening posts expressed their gr atitude to the particip ants of the forum. On a more negative side, 23% of the thread s (2 of 13 threads) included antagonistic or mocking remarks. For example, in Extrema Jonath an (the author of the thread) and Jake (one of the additional eight participants) engaged in a controversy because Jonathan did not include details of his misconceptions when he posted the original message. Jake stated the following: Stanza 14: Continues reply to comment about laziness Jake 155 If you do work on the problems 156 as you say, that's comendable (sic). 157 But in that case, then 158 you would gain a >lot 