These programs can perform symbolic and numeric calculations and plots. 10 ...... In K. Hastrup & K. Ramløv (Eds.), Feltarbejde, oplevelse og metode i etnografien. .... Education, http://www.math.uncc.edu/~sae/dg3/steinbring.pdf, Norwich, UK.
Mathematical Writing Morten Misfeldt
Dissertation submitted for review, with the purpose of obtaining the degree of Ph.D.
Danish University of Education, Learning Lab Denmark
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Acknowledgements
The following dissertation is the final product of my enrolment as a Ph.D.-student at Learning Lab Denmark. I would like to thank director Hans Siggaard Jensen and the Science Technology & Learning group, for their never failing support and their belief in my research, and for providing a truly inspiring workplace. I would like to thank my supervisors Carl Winsløw and Kasper Hornbæk for indispensable help, support, review and the countless valuable discussions we have had during my time as a Ph.D.student. I would also like to express my gratitude to Nicolai Paulsen, Robin Engelhardt, Lisser Rye Ejersbo, Mette Andresen, Thorkild Hanghøj, Raymond Duval, Uri Leron, Rudolf Strässer and Sherry Turkle for their valuable comments and criticism. A one-semester long visit to MIT Media Lab in the fall of 2002 taught me a lot about valuable attitudes towards research. I am grateful to the Lifelong kindergarten group for showing me how research can be playful and appreciative, while remaining intellectually challenging and very serious. I would in particular like to thank Professor Mitchel Resnick for his supervision, critical review and help. This dissertation is based on a collection of papers, some published and others not. The papers have been rewritten so that they – hopefully – tell a coherent story (see appendix B for a more detailed description of how each chapter relates to previously published papers). This means that I am in debt to a number of reviewers and conference working groups. Appendix A is a preprint of a book chapter. I bring this chapter unedited, because it was written together with Niels Grønbæk and Carl Winsløw. This text represents an important educational implication and framing of my research among undergraduate students, which is why I wanted to include this chapter in the dissertation. I would like to express my gratitude to the subjects of my empirical investigations, the researchers that were kind enough to describe their working processes to me and to the students who allowed me to observe their collaborations. Finally I want to thank my family, my wife Sille in particular, for endless support, patience, and a lot of hard work during my preparation of this dissertation.
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Resumé
The subject of this dissertation is mathematical writing, and how various technologies support this activity. The body of the dissertation consists of two main parts; an empirical part and a theoretical part. The empirical part is presented first and consists of two investigations, one about mathematicians’ use of different media for mathematical writing and another about students’ collaborative writing. Following the introduction, the first chapter is used to argue that a closer look at mathematical writing has a potential for improving the computer support for this activity, and to introduce the theoretical perspectives that I have used. Chapter two comprises the first investigation. This investigation is based on interviews with professional researchers of mathematics and it consists of eleven semi-structured interviews. During the interviews, the researchers explained what purposes their writing served, both in their personal working process, and in connection with collaborative work. The investigation shows that writing is very important to the researchers in almost all phases of their work. When they are clarifying the initial ideas about a problem, pen and paper is used in a way that supports thinking, and writing and calculations are crucial when an idea has to be developed further. Moreover, writing is central in collaboration and for storing ideas. Researchers tend to use pen and paper for calculations and other types of writing that support thinking, and many of them only use computers late in the writing process. Chapter three reports on a field study of undergraduate students’ collaborative work in connection with a writing assignment. This investigation is based on several data sources. Over a period of two month, I followed two groups of students working with mathematics projects, by having a frequent contact (approximately once every week) with the groups, and by video-observing seven of their group meetings. Furthermore, each of the students kept a detailed diary of one week’s work with their mathematics projects. The survey shows that collaborative writing in mathematics can be challenged for example by a need to do individual work in the middle of a meeting, for instance to do a calculation. The survey also shows that face to face communication on mathematics is heavily dependent on ostensive use of paper based representations. Chapter four relates the two empirical investigations to the literature concerning (alphabetic)
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writing processes. The question of mathematical discovery through writing is addressed, and the role of planning and rhetorical goal setting as a driving in this process is discussed. In chapter five I present a semiotic framework to and use it to discuss the role of external representations such as formulas and diagrams in mathematical thinking. A special type of external representations called commutative diagrams found in the empirical investigation is used as a case. It is argued that the ways mathematical symbols are organized in two dimensions are crucial to mathematical thinking. Chapter six is concerned with face-to-face meetings in mathematics. Taking outset in the processional mathematicians’ reflections on the need for being face-to-face I consider aspects of common ground and coordination. The use of various media during face-to-face interactions on mathematics is discussed. In chapter seven I adopt a semiotic interaction model based on turn-takings in order to correlate the three aspects with the interaction with electronic (computer) media. This model allows me to describe the structure of interactions with electronic media and with pen and paper, and to describe how interactions with other people are mediated. Chapter eight concludes the dissertation. The main contributions are identified as an empirically grounded description of the mathematical writing process, and knowledge about how different media supports this process, especially the persistent dependence on handwriting in mathematical work, both in personal work and during meetings. On the theoretical level, the contribution is a result of considering mathematical writing as both writing, and mathematical problem solving, leading to an attempt to combine these two theoretical perspectives into one semiotic model.
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Table of contents ACKNOWLEDGEMENTS ............................................................................................................................. I RESUMÉ ........................................................................................................................................................III TABLE OF CONTENTS ................................................................................................................................V INTRODUCTION ........................................................................................................................................... 1 1
ASPECTS OF MATHEMATICAL WRITING ..................................................................................... 5
1.1
RELEVANT PRACTICES ........................................................................................................................... 6
1.2
COLLABORATION IN MATHEMATICS ..................................................................................................... 8
1.3
PSYCHOLOGY OF WRITING PROCESSES ............................................................................................... 11
1.4
SEMIOTIC REPRESENTATIONS IN MATHEMATICAL WRITING ............................................................ 12
1.5
TECHNOLOGIES FOR MATHEMATICAL WRITING ................................................................................ 14
1.6
COORDINATION AND COMMON GROUND ............................................................................................. 17
1.7
HOW TO STUDY MATHEMATICAL WRITING ........................................................................................ 17
2
MATHEMATICIANS WRITING: AN INTERVIEW STUDY AMONG RESEARCHERS.......... 19
2.2
INTERVIEW STUDY AMONG MATHEMATICIANS .................................................................................. 20
2.3
DATA ...................................................................................................................................................... 22
2.4
FIVE FUNCTIONS OF WRITING MATHEMATICALLY............................................................................. 26
2.5
DISCUSSION ........................................................................................................................................... 35
2.6
PARTIAL CONCLUSION.......................................................................................................................... 36
3
STUDENTS’ COLLABORATIVE WRITING .................................................................................... 39
3.2
QUESTION .............................................................................................................................................. 40
3.3
METHODOLOGY .................................................................................................................................... 40
3.4
THE STUDENTS’ WORKING PROCESS ................................................................................................... 43
3.5
A CONVERSATIONAL EPISODE ............................................................................................................. 45
3.6
FLOW OF CONVERSATION AND CONVERSATIONAL BREAKDOWN ...................................................... 49
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3.7
FUNCTIONS OF WRITING IN COLLABORATIVE SETTINGS ................................................................... 55
3.8
FACTORS INFLUENCING CONVERSATIONAL BREAKDOWNS ............................................................... 57
3.9
PARTIAL CONCLUSION.......................................................................................................................... 59
4
MATHEMATICAL WORK AS A WRITING PROCESS ................................................................. 61
4.1
TWO CONCEPTUALIZATIONS OF THE WRITING PROCESS ................................................................... 61
4.2
DISCOVERY WRITING............................................................................................................................ 66
4.3
WRITING AND MATHEMATICS .............................................................................................................. 68
4.4
PRIVATE WRITING IN MATHEMATICS .................................................................................................. 70
4.5
PARTIAL CONCLUSION ......................................................................................................................... 73
5
SEMIOTIC REPRESENTATIONS AND MATHEMATICAL THINKING: THE CASE OF
COMMUTATIVE DIAGRAMS .................................................................................................................. 75 5.1
INTRODUCTION ..................................................................................................................................... 75
5.2
COMMUTATIVE DIAGRAMS .................................................................................................................. 76
5.3
SEMIOTIC REPRESENTATIONS IN MATHEMATICS ............................................................................... 76
5.4
MATHEMATICAL THINKING AND COMMUTATIVE DIAGRAMS............................................................ 82
5.5
PARTIAL CONCLUSION ......................................................................................................................... 85
6
COMMON GROUND AND PRIVATE SPACE; COORDINATING MATHEMATICAL
COOPERATION........................................................................................................................................... 87 6.1
COORDINATION AND COMMON GROUND ............................................................................................. 87
6.2
PRIVATE SPACE AND CONVERSATIONAL BREAKDOWNS .................................................................... 89
6.3
FACE TO FACE INTERACTIONS AND COORDINATION IN PROFESSIONAL MATHEMATICAL
COLLABORATION............................................................................................................................................ 89
6.4
THE IMPORTANCE OF BEING FACE TO FACE - OR NOT ....................................................................... 90
6.5
COLLABORATION, HEURISTIC AND CONTROL TREATMENT ............................................................... 95
6.6
COLLABORATIVE ASPECTS OF DIFFERENT MEDIA............................................................................ 100
6.7
PARTIAL CONCLUSION........................................................................................................................ 102
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COMPUTERS AS MEDIA FOR MATHEMATICAL WRITING: A MODEL FOR SEMIOTIC
ANALYSIS................................................................................................................................................... 105
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7.1
INTRODUCTION ................................................................................................................................... 105
7.2
DYNAMIC SEMIOTICS AND A MODEL BASED ON TURNTAKINGS ....................................................... 106
7.3
MATHEMATICAL TYPESETTING ......................................................................................................... 110
7.4
STUDENT COLLABORATION AND CONVERSATIONAL BREAKDOWNS ............................................... 116
7.5
PARTIAL CONCLUSION........................................................................................................................ 118
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CONCLUSION...................................................................................................................................... 121
BIBLIOGRAPHY........................................................................................................................................ 127 1
APPENDIX A: ASSESSMENT AND CONTRACT-LIKE RELATIONSHIPS IN
UNDERGRADUATE MATHEMATICS EDUCATION............................................................................. 1 1.1
INTRODUCTION. ...................................................................................................................................... 1
1.2
DIDACTICAL CONTRACTS IN UNIVERSITY MATHEMATICS ................................................................... 3
1.3
A DIDACTICAL ENGINEERING PROJECT: THEMATIC PROJECTS IN REAL ANALYSIS ........................... 9
1.4
STUDENTS AT WORK: ADIDACTICAL SITUATIONS OR CONTRACT FOLLOWING ................................ 13
1.5
EVIDENCE OF CONTRACTUAL UNDERSTANDINGS IN STUDENTS’ WRITTEN REPORTS ...................... 18
1.6
CONCLUSIONS ....................................................................................................................................... 21
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APPENDIX B: PUBLICATIONS.......................................................................................................... 25
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Introduction
How can technology support mathematical writing, similar to the way prose writing is supported by word processors? This question made me begin the project ‘mathematical writing processes’. Computers are becoming the most widely used medium for written communication; everything ranging from personal communications and school reports to research articles are developed and presented with a digital writing tool. The increasing use of digital technology is a development with many advantages. As an example, the Internet supports development of online learning communities, and improves the infrastructure for distributed collaboration. These advantages, however, depend nearly exclusively on written communication. Another example of these advantages is that word processors allows for other kinds of writing processes than previous technologies did. It is very easy, for example, to obtain a printed version of the text one can restructure and revise the text, with minimal effort. My motivation for doing this project is the observation that several attempts to activate the potential of digital writing in connection with mathematical activities, particularly in the educational system, seems to come out unsuccessful. The problem of how technology can support mathematical writing, similar to the way prose writing is supported by word processors can be considered a challenging design problem. The objective of this dissertation is not the development of a mathematical writing tool. Rather, I have chosen to try to understand what people do when they write mathematics using digital and analogue technologies. In order to explore the mathematical writing process, I have employed different ethnographical approaches. In workplace studies and computer supported collaboration (CSCW), a pragmatic version of ethnography has proven an effective approach for generating new designs and understanding problems with existing ones. The underlying idea behind using ethnography is that the complexity of many working situation makes the study of people in the context of their workplace a valuable source of information. This dissertation employs two different empirical settings; professional researchers working in the field of mathematics and undergraduate students working on collaborative projects in a real analysis course.
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I have limited my interest of what people do when they write mathematics to the following research questions: 1. How can mathematical work be considered as a writing process, and be understood from existing theoretical frameworks about writing processes? The psychology of writing has been considered from at least two starting points; as a rhetorically driven problem solving process and as a process of discovery. Both of these aspects are meaningful to parts of the data about mathematical writing. 2. What is the influence of external representations on mathematical thinking? One point of view is that semiotic representations are the only empirical access people have to mathematical concepts. The impact of semiotic representations is therefore crucial to mathematical thinking. Another point of view is that mathematical representations are merely re-presentations of mathematical concepts, that neither creates or empowers any insights by themselves. 3. How does the social setting of, say group work affect mathematical writing? Mathematics has traditionally been considered a solitary pursuit, but collaborative work is becoming more and more widespread in connection to mathematics. The goal of this dissertation is: To explore mathematical writing through empirical investigations and theoretical discussions, aiming at a conceptualization of the mathematical writing process that takes psychological, semiotic and social factors into account while maintaining a focus on what media and technologies that can be used to support mathematical writing. I want to emphasize two aspects of the above sentence. Firstly, the goal is not a purely empirical investigation, but has the empirical investigation and theoretical discussion as two competing goals, and secondly I investigate the activity mathematical writing in order to generate knowledge that can be used for developing new mathematical writing tools. I decided to focus on both empirical investigation and theoretical discussion. On the practical level, this dual focus means that the dissertation falls naturally into two parts, an empirical and a theoretical. In the first part of the dissertation, I present two empirical investigations, and in the second part I discuss the results from the investigations from several perspectives. This organisation
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reflects my intention to generate empirically grounded, theoretical knowledge about mathematical writing. In this dissertation, no new technology is developed or tested. Mathematical writing is looked at as an activity, and I try to pay attention to how technology does, and potentially could, support this activity. One should not understand this goal as merely focusing on new digital technology. Whatever people use to write while they work with mathematics, how this works for them, and why, is the scope of this dissertation. The methodological approach is an open ethnographical investigation inspired by Strauss and Corbin (1998), in the sense that I begin by investigating mathematical writing in two different settings. The purpose of these two investigations is to obtain an understanding of the functions of mathematical writing. After I have developed the empirical categories, the second part of the dissertation concerns theoretical explanations and challenges to the empirical results. This discussion is organised around three theoretical perspectives on mathematical writing. First, the mathematical writing process is examined by relating the results from the two investigations to theories of writing process. Secondly, mathematical writing is examined as a problem-solving process where external (written) representations are used to support thinking. And finally, mathematical writing is discussed from a collaborative perspective. After the theoretical discussion, the different perspectives are combined in a semiotic analysis.
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1 Aspects of mathematical writing
Formulas, natural language, diagrams, drawings and other forms of representations are used extensively in mathematics, both to support thinking and to communicate ideas. Such mathematical representations have traditionally been generated using pen and paper, but in the last two decades the computer has become a powerful alternative to handwriting. The advantages of using a computer for mathematical writing would appear to be many. Computational technology makes it easy to review the writing and gradually change text without rewriting it all. Computers make it easy to create a publishable document, and we know from other domains that computers can successfully facilitate collaborative writing among students and professionals (Guzdial, Rick, & Kerimbaev, 2000; Posner & Baeker, 1992; Scardamalia & Bereiter, 1993). Despite the advantages of using computers to write mathematics, two observations suggest that mathematical writing might be difficult to support: First pen and paper still plays an important role in mathematics, and second there seems to be empirical indications that computer systems to support collaboration among university students, by having them communicate in written form, are particularly difficult to apply in mathematics (Guzdial, Lodovice, Realf, Morley, & Carroll, 2002). The problem with using computers for mathematical writing could be that mathematical notation is difficult or impossible to write in many computer applications. This problem was approached by Guzdial et al. (2002) by constructing a formula editor for one successful Computer Supported Collaborative Learning (CSCL) system (the CoWeb or Swiki, for a description of the CoWeb see Guzdial et al. (2000)). The students of mathematics however, did not use this improved system, which suggests that either the students did not want to collaborate or that the support for writing formulas was somehow insufficient to facilitate collaboration through such a system. The problems with introducing collaborative technology in mathematics education contrasts somewhat with the results from a large interview-based study by Leone Burton (1999) on how mathematicians come to know mathematics, revealing that mathematical research is moving towards a more collaborative nature (Burton, 1999). Computers are becoming the primary medium for written communication, and since writing in connection to mathematical activities seems to be difficult with a computer, it makes sense to study
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mathematical writing. In the following section I clarify what activities we could study in order to learn about mathematical writing.
1.1 Relevant practices One can study different practices in order to approach potential problems and possibilities in using computers for mathematical writing. I have found at least three relevant practices that might be looked at; mathematics education, education in another topic and mathematical research. At most levels in mathematics education computers are only rarely used for writing: Computers are most often used as advanced calculators (Dreyfus, 1994; Guin & Trouche, 1999; Lagrange, Artigue, Laborde, & Trouche, 2001). Hence, a study of the mathematical writing process, mediated by computers, is not easily conducted in an educational setting. What one can do, however, is to introduce technology for writing in an educational setting as a development project, but I believe that such an introduction could benefit from some knowledge on how computers are used elsewhere for mathematical writing. In other fields, where writing is important, word processors have had a big influence on the writing process (Sharples, 1992) and on the tendency among students to collaborate and share their work (Guzdial et al., 2000). Therefore, it might be relevant to study the potential of word processors in, for example, language education, and that is indeed done by Sharples & Evans (1992). There are a lot of questions regarding mathematics that remain unanswered by such studies, though, because the way external representations are used in mathematics can be highly specialised and differ greatly from other domains (Duval, 2000b), and because of the initial observation that mathematics might present some special problem. In this dissertation, I have chosen to look at two empirical domains; professional mathematicians and undergraduate students working with mathematics. I investigate mathematicians’ use of computers and other media to support their writing. Among the students, I investigate the nature of their collaboration. I look in detail at how they interact and discuss mathematical issues when they meet to work on their projects, and I observe how they organise their collaboration.
1.1.1 The role of research practice for teaching and learning It is evident that the work of mathematicians and the work of students of mathematics differ: The criteria for success differ somewhat (having papers accepted as opposed to passing examinations), the mathematical content domain can differ greatly (even though this can be the case among
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researchers and among students as well) and the work setting, both physically (office or classroom) and process-wise, are very different (e.g., time spent on a research paper vs time spent on weekly homework). Despite these differences, mathematical research activity is often seen as a benchmark for mathematics teaching in several ways: •
The work of mathematicians is used as a metaphor while introducing a more investigative type of activity (The role of research practice for teaching and learning Fuys & Huinker, 2000).
•
The use of the historical development of a mathematical content domain (as it has been developed by researchers) as a model for planning a guided (re-)investigation of the content domain by students (Freudenthal, 1973, p. 109).
•
As a case for discussing a framework for understanding learning of mathematics (Sfard, 1991, p. 12).
Brousseau (1997) claims that: “the intellectual work of the student must at times be similar to this scientific activity” (p. 22), but that this ‘adidactical situation’ cannot be the only type of activity when learning mathematics. The adidactical situation should be accompanied by didactical situations such as instruction by a teacher. The adidactical situation, which is essential to ensure that the students learn, clearly uses the work of mathematicians as important benchmarks, whereas the didactical situation, that is equally important, does not. Burton (1999) has, in a large interview-based study on how research mathematicians come to know mathematics, revealed that the discipline is moving towards a more collaborative nature (p. 137). She describes this as being in contrast to the practices in mathematics teaching and learning. By studying mathematical researchers, Burton tries to provide inspiration for, and reveal potential misconceptions about, the ways in which mathematics is typically taught (ibid., p. 121). Likewise, Burton and Morgan (2000) reveal profound differences in the use of natural language in research papers, and conclude that this diversity is not reflected in the way writing is introduced in mathematics classes. In Burton (2001), it is argued that a model describing professional mathematicians’ way of learning should be applicable to any learner of mathematics. Her argument is based on the notion of communities of practice (Wenger, 1998), and highlights the epistemological practice over the actual knowledge objects.
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In this section, I have so far described how mathematical research can be considered to be a benchmark for mathematics teaching and it seems to be clear that the practice of researchers can be examined in the planning of educational activities of students from a variety of educational perspectives. Burton, Brousseau and the metaphorical approach of Fuys and Huinker all point out the similarity in practice, with only little reference to the content of mathematical research. On the contrary, the guided reinvention approach focuses on how the historical development of mathematical knowledge can be used to inform the planning of didactical activities. Neither of these approaches claim that research practice should be the only source for inspiration when one is planning teaching.
1.1.2 The mathematical writing process The dissertation has an underlying assumption: that the ‘mathematical writing process’ has some distinctive features that are found both in educational settings, at least at the tertiary level, and among researchers. The assumption that one can speak of the mathematical writing process across several different levels and settings is not unproblematic. Pimm (1987, p. 198) explores the structural metaphor of “mathematics as a language”. If we take the structural aspect of the metaphor seriously, it makes sense to talk about mathematical conversations, mathematical argumentation, mathematical vocabulary and mathematical writing. Furthermore, I have argued that the activities of researchers can (and often does) serve as a guiding principle or inspiration for planning students’ learning activities. This does not mean that insights concerning mathematical writing among researchers are immediately transferable to educational settings, or vice versa. In this dissertation I investigate mathematical writing from a number of perspectives. In the rest of this chapter I will describe mathematical writing with outset in (1) the type of activities that collaboration in mathematics involve, (2) the writing process involved in mathematical work, (3) the representations used in mathematics, and (4) the typical tools that are used for writing and collaboration in mathematics. The dissertation attempts to take all these aspects into account and they are introduced in this chapter.
1.2 Collaboration in mathematics Mathematical research has traditionally been described as mainly individual, for example by Hadamard (1945) and Polya (1971), but recently Leone Burton (1999) has interviewed a large
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number of researchers in mathematics and found that most professional mathematicians collaborate quite often, and that there is a tendency among professional mathematicians to collaborate more now than twenty years ago. There is no obvious explanation for this change, but one reason could be the introduction of electronic communication and another, pointed to by Burton, that the amount of mathematical knowledge needed to do research has become so large that one person does often not hold it. Collaborative writing has been investigated from a number of viewpoints, see (Sharples, 1993) or (Kim & Eklundh, 2001) for an overview. One important approach by Posner and Baeker (1992) was to interview respondents from a large variety of topics (“medicine, computer science, psychology, journalism, and freelance writing” p. 127). They found a great diversity in the respondents’ approaches, but developed a framework for describing collaborative writing as consisting of roles played in the collaboration, activities performed in the writing process, document control methods used, and the employed writing strategies. Kim and Erklundh (2001) have investigated collaborative writing and reviewing activities on researchers in a variety of the natural sciences and engineering including one (of 11) in applied mathematics. They find that, in collaborative writing, there is tendencies to have one person manage the document throughout the entire writing process. Their investigations also show that pen and printouts are preferred over electronic collaborative tools when group participants review documents. In university education the teaching has long reflected the idea that mathematics it is best done alone. This means that university teaching in mathematics often does not address the social aspects of learning. Of course mathematics education has always included social dimensions, but in a very traditional understanding of mathematics education, this is mainly seen as a shared validation and institutionalization (Brousseau, 1997) of mathematical knowledge, as it is expressed in lectures and exercise sessions where ‘perfect’ solutions are shown. While exploration and investigation of mathematics are mainly considered a personal endeavour (this organisation of teaching is elaborated in Appendix A, where it is described as the ‘Lecture Problem Class’ contract). Recently we have seen an increased focus on the social aspects of learning in university-level mathematics, and there have even been attempts to introduce computer supported collaborative activities in mathematics education such as: Programming to learn advanced mathematics (Cottrill
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et al., 1996; Leron & Dubinsky, 1995); the use of computer algebra systems1 as a vehicle for learning mathematics (Winsløw, 2000, 2003); and the use of interactive geometrical software (Shaffer, 2002). A common feature with these attempts is that the collaborative setting is around the computer. Here the computer can be used either in an ostensive way where a student or teacher uses a computer to explain an argument or result, or leads an entire class’ investigation, using a computer (Winsløw, 2003); it can be used as a microworld for exploration of certain mathematical ideas (Laborde & Strässer, 1990; Papert, 1980; Shaffer, 2002) or it may be used as a cognitive tool or instrument (Guin, Ruthven, & Trouche, 2004) augmenting the cognitive abilities of the user by performing specific types of tasks such as calculations or graphing. A much less developed collaborative use of computers in university mathematics education is the use of computers for writing and communicating mathematical ideas. Web based systems that support communication among students as a substitute or supplement to the face to face interactions in class has been widely introduced in university education. A fair amount or research supports the value of such systems (Guzdial et al., 2000; Koschmann, Hall, & Miyake, 2002). With respect to mathematics however, Mark Guizdial (2002) reports on serious problems and active resistance when using the collaborative environment Coweb in undergraduate classes in mathematics and chemical engineering. There are several possible explanations for this resistance. In their paper Guzdial et al. (2002) points to two explanations: Firstly, collaboration was not a well established part of the culture in the mathematics and chemical engineering classes where Coweb was tested, neither among faculty nor among students. Secondly, many of the mathematics exams had a very competitive character that did not endorse healthy collaboration across the class. Nevertheless, the authors do not feel they can fully explain the problems in mathematics. These results reflect an unfortunate state of affairs because the use of computer based communication systems to augment face to face teaching holds a promise of supporting students’ participation in an out of class learning community, and the asynchronous and written nature of the communication have been shown to support personal reflection (Guzdial et al., 2000; Mason, 1994; Salmon, 2000; Scardamalia & Bereiter, 1993). This promise is relevant to mathematics because the dependence on written representations and need for individual reflection are at least as important in
1
Computer algebra systems (CAS) are professional mathematical software packages such as Maple or Mathematica. These programs can perform symbolic and numeric calculations and plots.
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mathematics as in any other area. It is therefore interesting to understand the interplay between writing, collaboration and information technology in connection to mathematics. While Kim and Eklundh (2001) are mainly concerned with the reviewing process, the studies that are described in this dissertation examine collaboration from a much earlier stage in the working process. In contrast to the broad scope of Posner and Baeker (1992), this dissertation deals with the writing process of professional mathematicians and university students. And where Burton (1999) is an epistemological study concerned with the learning history of professional mathematicians, this study is more pragmatic in the sense that it describes existing practices of mathematicians and university students and is motivated by the practical problem of why it seemingly is so difficult to use computers for mathematical writing.
1.3 Psychology of writing processes Writing has a long history as an object of study, but in the 1960’s people began to study the ‘writing process’, that is, the human activity that leads to written text. Writing was typically studied from a personal and psychological point of view. In an overview of the cognitive theories of writing processes, Linda Norris (1994) takes a report from the American department of health, education and welfare (Rohman & Wlecke, 1964) as a starting point for theories of writing processes. The report divides the writing process into three phases: Planning, Writing and Re-writing. These three ‘phases’ are central if we want to understand the difference between the various conceptualisations of the writing process. One of the important motives for studying the writing process has been to investigate the claim that new knowledge is generated in the writing process. Looking at the empirical investigations of writing processes, we find at least two different ways to describe this creative potential of writing, namely as a, partly rhetorically driven, problem-solving process and as a process of discovery. When writing is performed as a rhetorical problem-solving process careful planning becomes an important element of the writing activity (Flower & Haynes, 1980). The writing activity is seen as a problem-solving process fuelled by an attempt to explain your message in a way that could be understood by a potential reader. Sometimes this rhetorical process influence the content-related problem-solving because the need to explain a concept can make the writer aware of lacks in his own understanding of this concept.
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Viewing writing as a process of discovery highlights the constant interplay between the text and the writer (Galbraith, 1992). The creative potential is then considered a product of this interaction. Writing is viewed as a process of figuring out what the author means and knows on a topic. The process of externalising a sentence in writing will in itself provoke a contrasting or associated sentence to come to the authors mind. In that sense the writing process can be seen as a personal negotiation between conflicting viewpoints, in order to develop a personal stand on a topic. This conceptualization of the writing process makes private writing a very meaningful activity (Elbow, 1999).
1.4 Semiotic representations in mathematical writing Representations play a central role in mathematics. Often the development of new notation is critical to breakthrough in the history of mathematics (Sfard, 1991), and often mathematical problem-solving draws extensively on different representations. Navigating among different forms of representations of a mathematical object is a central ability in mathematics. Choosing the right representation for the right task is very important, and often difficult (Duval, 2000a).
1.4.1 Sign and medium From a semiotic viewpoint, the basic element in human communication is the sign. A sign consists of material signifiers intended by the producer to stand for something else, the signified (Saussure, 1966, 1916). The material environment for signification is called a ‘medium’. The medium is the channel through which the message is sent. From communication theory we have the following picture of communication (Guiraud, 1975).
Referent
Sender
medium
message
Code
Figure 1.1: The standard model of communication.
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Receiver
Figure 1.1 represents a communicative situation where a message, representing a referent, is sent in a medium relying on a code both in order to be created by the sender and in order to be interpreted by the receiver. Examples of media are paper, speech, gestures. The sender and receiver can be the same person as for instance in personal book keeping (Winsløw 2004).
1.4.2 Signs in mathematical work Mathematical work can be very dependent on sign systems. Mathematical objects are typically not there to be pointed at as anything but a semiotic representation, and when it comes to more abstract mathematics, this access trough semiotic representation can be of a very technical nature. This fact is described by Raymond Duval (2001), who stresses that all mathematical objects has more than one semiotic representation and that it is a crucial and typical error to mistake one of these representations for the mathematical object itself. Looking at the communication model, Duval states that there are no empirical observable referents for mathematical signs, even though two different mathematical signs can easily refer to the same referent. Duval points to many learning difficulties in mathematics that can stem from the complicated nature of mathematical signification. Duval describes two qualitatively different types of transformations of semiotic representations: treatments and conversions. Treatments are transformations inside a semiotic system, such as rephrasing a sentence or isolating x in an equation. Conversions are transformations that change the system while maintaining the same conceptual reference, such as going from an algebraic to a geometric representation of a line in the plane. For Duval, the importance is not the psychological entity of a ‘conceptual reference’ but the fact that ‘something’ (the conceptual reference/conception) is constant under the process of conversion. A semiotic system that allows for treatments inside the system and for conversion of sign to signs in another system referring to the same conceptual object is designated a ‘register’. Duval shows empirically (Duval, 2000a, figs. 4 and 5) that conversions are very difficult for students in some situations. The conversions that seem to be easiest for students are the ones that are congruent, meaning that the representation in the starting register is transparent to the target register (Duval, 2000a, pp. 1-63). One example of congruent conversion is when a sentence in
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natural language can be translated into an algebraic expression keeping the same order of signs and just translating each word to a similar algebraic symbol. The relation between semiotic representations and conceptual understanding can then be described through Figure 1.2.
Concept
Treatment Register A
Register B
conversion
Figure 1.2: from Duval (2000) figure 6.
Conceptual understanding can then be described as a person’s degree of freedom towards various semiotic representations of the same mathematical concept (Winsløw, 2004).
1.5 Technologies for mathematical writing In this section I describe some of the most typical technologies used for mathematical writing, namely LaTeX and Word. These two programs are examples of different approaches to computer support for mathematical writing. LaTeX is a command based system where the equation editor in Word is based on a combination of menus and hotkeys. Furthermore, I describe some of the advantages of paper and handwriting that are difficult to obtain with computer-based technologies.
1.5.1 LaTeX LaTeX is a typesetting system widely used in the mathematical community. The author writes a source code in a text editor, like the source code for a program. This text file is processed by the
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LaTeX program to a print file (see Oetiker, Partl, Hyna, & Schegl, 2002, p 17). An author would write: $\int_a^b f(t)dt=F(b)-F(a)$ in a text editor to typeset: b
∫ f (t )dt = F (b) − F (a) a
in the print file. There is no immediate response by the system, the user have to remember the exact commands for every mathematical symbol and mathematical structures he uses, and will often generate time-consuming errors from the LaTeX, program for instance because of a missing bracket or type error.
Figure 1.3: A text editor and a previewer in LaTeX.
1.5.2 Word The type of system where symbols and structures are selected with the mouse is used in Word’s equation editor, and is the typical way for novices to manipulate mathematical symbols in a computer environment. This type of system is fairly straight forward to use, and sufficient minor documents, but it is tedious and physically challenging to make large equations with this system. It
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is also difficult to arrange mathematical symbols and structures in menus in a way that is reasonably transparent, and it can be difficult to remember in which menu a specific command is found.
Figure 1.4: The equation editor in MS-word.
The type of menus described above is, in Words equation editor, combined with “hotkeys” for the most used functions. The hotkeys can save a lot of time and make the work much more fluent, but they are often difficult to remember because they are not semantic commands. In Word’s equation editor you type: [CTRL]+[SHIFT]+F for fraction, in LaTeX you type \frac. In general LaTeX tends to use the name of the symbol (or an abbreviation of this name) to express the symbol, whereas hotkeys ideally are just the first letter in the symbol name.
1.5.3 Paper Paper is used a lot in intellectual work, but this medium has typically not been the subject of academic investigation before the book “The Myth of the Paperless Office”, in which Sellen and Harper (2002) investigate paper use in various organisations. An important underlying concept in their work is ecological psychologist J.J. Gibson’s notion of ‘affordance’ of different technologies. The term affordance refers to the various functions of use that the physical properties of an object allow (Sellen & Harper, 2002, p. 17). Hence the affordances of a given technology refer to what a person can do with it, rather than what the technology can do for that person. Sellen and Harper describes how the physical properties of paper are rich and unique and hence claims that digital
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technologies is not going to replace paper any time soon, and that it is more likely that digital technology will change the way we use paper.
1.6 Coordination and common ground Pragmatic concepts for describing coordination of collaborative processes are developed, in the literature on computer support for cooperative work (Schmidt & Simone, 1996; Yoneyama, 2004, p. 59). The overall division of labor, i.e. the way that tasks and work are coordinated, is called the ‘cooperative work arrangement’. The subject of work is called ‘common field of work’ and the work performed to establish and maintain the cooperative work arrangement is called ‘articulation work’ (Yoneyama, 2004, p. 60). The concept of common ground denotes the mutual knowledge, mutual beliefs, and mutual assumptions that collaborators may share (Clark & Brennan, 1993, p. 222). In order to communicate effectively individuals that collaborate need to assume a large amount of shared knowledge when they refer to a given concept. Clark and Brennan (1993) describe how different media support grounding, and they identify central ‘constraints’ on grounding that are different for different media, and they describe how the ‘costs’ of communication is different in different media.
1.7 How to study mathematical writing In this chapter we have seen that computers hold a large potential for supporting calculations, and how, in the discipline mathematics education, the existing knowledge about new technology and mathematical activities are almost exclusively concerned with understanding how the computer can be used as a calculator. The effect of the computer as a communicational device is only sparsely investigated in relation to mathematics. Therefore it is useful to study ICT and mathematical communication. ICT is currently revolutionizing written communication in connection to intellectual work. Furthermore writing is one of the, if not the, most important ways of communicating via computer. And it would seem that writing is exactly the mathematical activity that it is most difficult to support with computers.
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This dissertation is about mathematical writing. I will not restrict myself to study the way computers are used for writing mathematical formulas and other types of representations. This is not a dissertation that is mainly concerned with usability. I attempt to explore the interplay between the material and semiotic variables (media, tool, register, codes and agency) and the function that writing (using these media tools etc.) serves the author and other people (audience etc.). This goal implies that I will investigate how both new and old media are used for mathematical writing. The term ‘function’ of mathematical writing is to be understood pragmatically as designating why the author involves himself in the writing activity and what he gains from this activity. A central idea in this dissertation is that the semiotic and material variables are not ‘pure’ arbitrarily chosen mediators of an internalized knowledge, but that they affects the author’s ability to express and manipulate the knowledge and hence maybe the knowledge itself. The purpose of the dissertation is to determine the pragmatic functions of writing in mathematics, and to investigate what types of tools, media and representations that people use to support these functions.
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2 Mathematicians writing: an interview study among researchers
In chapter one, I listed number of perspectives on collaborative mathematical writing, namely (1) the type of activities that collaboration on mathematics involve, (2) the writing process involved in mathematical work, (3) the representations used in mathematics, and (4) the tools and media that are used for writing in mathematics. In this chapter, I describe how and for what purposes professional mathematicians write, how their writing depends on the collaborative situation they are in, and how the situation relates to the tools and media that they use. The chapter comprises my first attempt to clarify what functions writing serve in mathematics. I do that by studying the mathematical writing process through interviews with professional mathematicians. The aim in this chapter is to describe the personal writing process of the professional mathematicians, and to investigate if this process is influenced by being part of a collaborative project. I shall describe what purpose writing serves to researchers in the field of mathematics, the interplay between person and media, and how this interplay is influenced by possible collaborative boundary conditions.
2.1.1 Empirical basis This chapter describes an interview study with eleven professional mathematicians. The interviews concerns writing processes and collaborative writing. I distinguish between five functions that writing serves the professional mathematicians. Two of these functions have to do with supporting a personal thinking process, and the rest have to do with information storage, communication and production (e.g. publication). Based on the result from the interviews, the last part of the chapter comprise a discussion of the implications for design of systems to support collaboration through writing among professional mathematicians, students of mathematics and in general, and possibilities and problems with introducing such systems in mathematics education.
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2.2 Interview study among mathematicians In the fall of 2002 I conducted an interview study among eleven professional mathematicians. The interviews were conducted in two rounds approximately two months apart. The first five of the interviews where conducted in Denmark and the latter six, some two month later, in the greater Boston area MA, USA. where I was a visiting scholar at Massachusetts’ Institute for Technology during the fall semester 2002.
2.2.1 Question/purpose The objective of the investigation was to understand the writing process of professional mathematicians involved in collaborative projects, from early idea to finished paper. In particular, I compared the purposes that writing served to the mathematician to the types of representations he/she used and the media (computer or pen and paper) he/she chooses to use. The focus was on the respondents’ personal writing/working process and on how he/she communicated with his/her collaborators throughout this process.
2.2.2 A developing interview guide The interviews were made during the summer and fall of 2002. Between the two interview periods, the first interviews were transcribed and analysed. The results of this preliminary analysis were used in the design of the interview guide for the last six interviews. The five initial interviews were conducted with a very open interview guide focusing on two themes: The use of various media and representations during the personal writing process, and communication throughout collaborative projects. In the last six interviews, the guide had the same overall structure but was narrower because of the insights gained from the first five interviews; the major example of this was an increased focus on developing and verifying a classification of the various functions that writing served the mathematician. The role of the last interviews hence became to enlighten and test the categories and conceptual ordering (Strauss & Corbin, 1998, p. 19) that was developed in the preliminary analysis of the first interviews.
2.2.3 Participants The eleven participants or ‘respondents’ will remain anonymous and called R1-R11. Several of the respondents asked to be quoted only in general terms, and the transcripts will accordingly not be published, and the description of the respondents will be kept at a general level. The respondents are
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employed at different universities in the urban areas of Copenhagen, Denmark and Boston, MA. All the Copenhagen mathematicians (R1-R5) were native speakers of Danish and these interviews were conducted in Danish. The respondents from Boston had various mother tongues but they were all fluent in English, and these interviews were conducted in English. All the respondents were male, their age varied from around 35 to around 60, and all were active researchers. The respondents were chosen randomly. In the Copenhagen area they where chosen from a pool of positive responses to a request to participate sent to everybody employed as researcher of mathematics in the Copenhagen area. In Boston the respondents were chosen randomly from lists of faculty and then asked if they where willing to participate; all but one gave a positive response.
2.2.4 Interview method and coding I met and interviewed all eleven respondents in their offices. The interviews lasted between thirty and ninety minutes. The interviews were all organised as described in (Kvale, 1997); semi-structured conversations taking outset in the interview guides mentioned above. The interviews focused on recent and current work. Where appropriate, we discussed papers on the respondent’s desks or from his archive. If a sheet of paper was particularly interesting, if for example, it contained a form of representation that was new to me, or because we had had an interesting conversation about it, I would ask for a photocopy of that sheet. The interviews were taped, transcribed and analysed, and in the analysis of the interviews I brought in samples of working papers where relevant. I took an open approach (Strauss & Corbin, 1998, pp. 101) to the coding of the first five interviews in particular. This was done to determine important parameters, and develop a conceptual ordering, for further investigations. The coding of the last six interviews was more selective and critical because it served the double purpose of developing and challenging the conceptual ordering that was developed during the first five interviews. This allowed the interview study to be an iterative research process as described in (Strauss & Corbin, 1998).
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2.3 Data In this section, I account in detail for the writing processes of two mathematicians R1 and R2. The purpose is to give some insight into the writing process and workplace of mathematicians, and to describe the process that lead to the development of the conceptual ordering (Strauss & Corbin, 1998) that I use to describe the interviews. The cases represent two common but substantially different approaches to writing mathematics.
2.3.1 The workspace of mathematicians As already mentioned, all the interviews were conducted in the respondent’s. This proved to be a good choice in that I obtained valuable information about their working practice just by being there. Most of the respondents referred to their blackboard, desk or computers several times during the interview, and, not surprisingly, they often referred to working papers while talking about their writing process. Their workspaces had many similarities that are important for understanding the context of working professionally with mathematics and I will describe some of these. Every one of the offices contained a desk and a desktop computer, and in most cases the computer was on a separate desk. All of the offices contained either a whiteboard or a blackboard, and all of the researchers explained that they used their whiteboard or blackboard. There was a lot of paper in all the offices. In some cases the working papers was merely staked on shelves and in filing cabins, but in other cases also on additional tables in the office, or on the desk, and even the floor. A general feature of the offices was a substantial desk-space and a working space that was directed at least as much towards working with pieces of paper and books as towards working with the computer. It is of course important to note that these data where collected in the fall of 2002; the level of electronic versus paper based media could be changing rapidly. All of the respondents used a desk top computer. Most common was a UNIX terminal but several respondents used a Windows PC and one respondent used a Macintosh computer.
2.3.2 The writing process of R1 R1 uses three different media for writing mathematics (two paper-based and one electronic) and he clearly classifies his work according to the medium in use. The media are blank scrap paper (the
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flipside of old printouts) for handwriting, a lined pad, also for handwriting, and his computer with an email client and LATEX. When I went to see R1, he explained that he had recently been thinking about a problem he was working on together with a collaborator. He had the main idea worked out and had just begun the process of writing it down in detail. I received a photocopy of his detailed draft three days later and a week later I received a first edition of a paper on the subject, which he emailed to his collaborator.
Figure 2.1: Scrap paper: “Look at my desk, typically I have some paper that I draw diagrams and write stuff on, just very brief to see if this works at all…”
R1’s office contained a whiteboard next to a large bookshelf and across from that a small desk with R1’s computer next to the main desk. At the desk there was a pile of scrap paper, a binder, a pad of lined paper and two books. One sheet of scrap paper was placed in the middle of the desk and filled with scribblings (see Figure 2.1). The binder was filled with sheets from the lined pad, and on the lined pad there was handwritten mathematical text (Figure 2.2). R1 uses the scrap paper for personal scribblings, and he explained that these papers only make sense to him while he is working on a problem, and that most of them are thrown away almost immediately. The content of the scrap paper has no obvious linear structure (in the meaning used by Morgan, 1998, p. 89); it consists of scribblings scattered all over the paper. In Duval’s terminology,
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this scrap paper is non-discursive. In chapter five, the diagrams in Figure 2.1 and Figure 2.2 are discussed.
Figure 2.2: Lined pad: “If it seems to work out right, I will take another piece of paper and write it out to see if the details are right” (my translation from Danish).
R1 explained that if something from the scrap paper seems to work out, he will take the lined pad and try to write down as many details as possible. When that is done, the scrap papers are thrown away, but the lined paper with all the details are kept carefully in a system of binders. R1 explained that the purpose of writing it out in detail is twofold: To make the work accessible later, and to check in detail if his ideas are correct.
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Figure 2.3: LaTeX version: “Then, while I am trying to write as many details as possible in order to make sure that I got it right, then when I am satisfied I put it into the computer, but much shorter to fit a paper”.
When R1 thinks he has enough work for a paper, or a part of one, he will write a LATEX version of his work. He described that version as “much shorter, to fit a paper”. The LATEX version is sent to collaborators for comments and proofreading, but the main reason for making the LATEX version is to produce a paper. R1 explained that the notes on lined paper in his binders contain more information than the finished paper. He keeps the notes partly to be able to go back and investigate ideas he never published, and also so that he can always go back and check how he arrived at a given result. R1 imagines that this would be practical if confronted with questions of how he worked out a specific result, both to be able to defend his results and to more easily acknowledge if he made a mistake. R1 primarily communicates with collaborators by email. He explained that the content is often very close to the content of the notes on lined paper that he keeps in his binders. To be able to express mathematics in an email he will often use LaTeX code in the e-mail. This gives rise to some extra
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work with moving the content to another medium. If R1 communicates longer arguments and developed ideas, he will sometimes type it in LaTeX because it is easier to send, but he only does that if there is a good chance that he needs it in LaTeX later, for instance for publication. To avoid the typing in either LATEX or an email, R1 has tried to fax the notes from the binder, but he has found this to be problematic. R1 explained that if, when writing, he had to think about how it could be faxed (i.e. choose a dark pen, avoid using the margin, think about a potential reader etc.), this would disturb his thinking and he would be less able to concentrate on checking the mathematical details of his idea.
2.3.3 The writing process of R2 In the early stages of his writing, R2 uses paper and pen. He described this “first treatment” work as meaningless to others and not suitable for archiving, at least not without copious explanatory notes. What R2 will do next depends on how things proceed. If he arrives at a publishable result, he will write an early version in LATEX, and then start to work with pen and paper on a printout version of that, successively adding to and correcting the LATEX draft, which therefore evolves dramatically over time. R2 explained that he deliberately does not try to get it right from the beginning. On the one hand, R2 writes the LATEX draft to save his work and start the production of a paper. On the other hand, the contents develop over time; the creative work is not over when the first draft is written in LATEX. R2 argues that this way of working suits him better than trying to have it all figured out from the beginning. If R2 does not arrive at something publishable, he will sum up his work and save it in folders in a large metal drawer. This is done with pen and paper, the same type of paper as he uses in the first phase. The summary consists of things like “I have been working on x using the approach y; I got stuck there.” Apart from these overviews, the summary consists of annotations and re-writings of some of the early stages of his research. The purpose of this summary phase is to make a saveable version of his early scribbling.
2.4 Five functions of writing mathematically The eleven interviews have led me to consider the following five different functions of writing in mathematics explained below. By distinguishing between these five functions, I do not mean to imply that they are mutually exclusive or that they completely cover what mathematical writing is.
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Rather, I try to give a framework of categories useful for further analysis of the mathematical writing process. The five functions are: 1. Heuristic treatment consists of getting and trying out ideas and seeing connections. 2. Control treatment is a deeper investigation of the heuristic ideas. It can have the form of pure control of a proposition or be a more open-ended investigation (e.g. a calculation to determine x). It is characterised by precision. 3. Information storage is to save information for later access and use. 4. Communication with collaborators. Such communication can have various forms ranging from annotation of an existing text, comments or ideas regarding a collaborative project to suggestions of parts to be included in a paper. 5. Production of a paper, where writing is used to deliver a finished product intended for publication and aimed at a specific audience.
The conceptual ordering of mathematical writing into these five functions developed gradually during the interviews, several of these functions were already present in the interview guide for the last six interviews. Therefore these last interviews also served the purpose of verifying and challenging these functions as important markers in mathematical writing. The verification and final development of the functions was obtained in the second round of interviews, for instance in my interview with R9, who said that he “could not do anything without paper”. When asked about the distinction between getting ideas and verifying, on one side and communicating them on the other side, he pointed to pieces of paper on his desk, and after a while he summed up his use of these paper as follows: “This one is for getting ideas, this one is for checking them, and this is to communicate them”. The three sheets are shown below.
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Figure 2.4: “This is for getting ideas”.
Figure 2.5: “This is for verifying them”.
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Figure 2.6: “And this is for communicating them”.
Often R9 will omit having a final paper draft (as the one in Figure 2.6) and go directly to the computer instead. In these cases, papers like the one in Figure 2.5 are extremely important, because he will constantly go back and forth between writing on the computer (essentially to produce a paper) and verification and calculations on a sheet like Figure 2.5. To summarize the five functions and relating them to the working papers shown here one can say that the figures 2.1 and 2.4 are examples of heuristic treatment, the 2.2 and 2.5 examples of control treatment and 2.3 and 2.6 are examples of communication and saving of information. In the rest of the chapter I will use these five functions to describe and analyze the data from the interviews.
2.4.1 Media to support the functions The media that each one of the eleven respondents use to support each of the five functions are summarized in table 2.1. R1, R2, and R9 use pen and paper for heuristic treatment, and table 2.1 clearly show that this is the norm. Respondent R10 uses no media at all to support his heuristic treatment; in this sense he differs from the rest of the respondents who all are more or less dependent on handwriting to support heuristic treatment. The only one that uses computers for heuristic treatment is R4 who use a Computer Algebra System to find and check ideas for algorithms to solve numerical problems.
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R1 and R9 use one type of paper for heuristic treatment and another type for control treatment and R2 only uses one type of paper to support both these functions. In general, approximately half of the respondents (R1, R3, R4, R7, R9 and R10) maintain a sharp distinction between whether a piece of paper is used for heuristic or control treatment. The distinction is less clear for the rest of the respondents, but both functions come into play in the beginning of their work and are supported by handwriting. R1 uses papers in binders to save information while R2 and R9 use either an evolving LaTeX draft or pieces of paper from heuristic or control treatment annotated in order to be saveable. Table 2.1 shows several of the respondents uses both LaTeX files and handwritten notes to save information. R6 and R11 always use computers to save information. R6 explains that material which is not entered into the computer has a tendency to get lost, and this is his motivation for saving things on his computer. R11 uses LaTeX as a diary where he puts in all the results that he proofs and ideas he has. Several respondents (R2, R8, R9) claim however, that the number of ideas that does not make it to the publication stage is huge, and therefore it is not feasible to save everything on the computer, even though they generally prefer that form of storage.
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Respondent
Heuristic treatment Paper
Control treatment Paper
Information storage Paper
Paper (blank sheets) Paper CAS
Paper or printouts Paper CAS
LaTeX and paper Paper
R4
Paper and CAS
Math Type (word) , CAS and C++
R5
Paper
R6
Paper
Paper Math Type (word) or LaTeX Paper and LaTeX or scientific word Paper Paper LaTeX
R7
Paper
LaTeX
R8
Paper
Paper
R9
Paper
R10
No media
Paper or Paper Paper and and LaTeX LaTeX Paper Paper
Email, LaTeX, phone, and fax Paper, fax and email
R11
Paper
Paper and printouts
Email and LaTeX
R1
R2 R3
LaTeX and paper Paper
LaTeX
Communication Emails, LaTeX, fax and letter Emails Emails and paper
Math Type (word), LaTeX and fax Email and LaTeX Email and LaTeX LaTeX and email LaTeX, email and fax
Production LaTeX
LaTeX Paper draft for secretary who types in LaTeX Math Type (Word) or LaTeX LaTeX LaTeX LaTeX Paper draft, typed in LaTeX LaTeX Paper draft for secretary who types in LaTeX LaTeX
Table 2.1. The media supporting the five functions in mathematical writing. The table shows that the closer you get to publication, the more computers are used.
All the respondents communicate with their collaborators by email, sometimes attaching LaTeX files. All but R4 use LaTeX commands directly in the email in order to include mathematics. There is a tendency to send longer arguments and fully developed ideas as attached LaTeX files. Questions and less developed ideas are generally written in the body of the email using LaTeX commands. All but R7 prefer to perform annotations and reviewing with pen on a printout, and
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several of the respondents (R1, R4, R8, and R10) frequently fax handwritten comments or annotations of manuscripts. From time to time the respondents email annotations to collaborators as a list of corrections specifying page and line number and what needs to be changed. To produce papers, all of the respondents use LaTeX, except R3 and R10 who produce handwritten manuscripts that they give to a secretary who types it in LaTeX. The respondents report that the LaTeX files are important in collaborative projects both for communicating ideas and to keep track of progression, but for many of the respondents such LaTeX files are from the outset thought of as potential publications, and therefore this medium supports a more formal genre than the other media they use (e.g. emails, fax, phone, etc.). R4 and R5 use WYSIWYG (What You See Is What You Get) editors for writing mathematics, but both of them use LaTeX as well. They give two reasons for using LaTeX. R4 explains that the journals make it easier to publish papers written in LaTeX, and they both explain that their collaborators often use LaTeX. R4 explains that if he works alone, he can just write his paper in MATH-TYPE (an advanced version of the equation editor in MS Word) and then export it as LaTeX, but as the machine-generated code is difficult to work with, he often types parts of his work twice when he collaborates, first in MATH-TYPE and later in LaTeX.
2.4.2 The writing process With the five functions from the previous section, it is now possible to describe the writing processes of the respondents in some detail. In the following figures, I have tried to map a timeline of how the functions of heuristic treatment, control treatment, and production come into play for R1 and R2, and in which medium this occurs. Each box describes a specific medium used for a certain function at a certain stage, and is annotated with information regarding information saving and communication during this stage of work.
Timeline:
Heuristic
Scrap paper
Not save, not share
Lined paper
Produce
Control
Save in binder, maybe share as email Share as attachment, save as LaTeX
LaTeX
Figure 2.7: Diagram of R1’s writing process.
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Timeline: Paper
Heuristic
Produce
Control
Co…
Not save, not share Not save, not share
Printout LaTeX
Produce
Save and share (maybe as email)
Not… Save and share
Figure 2.8: Diagram of R2’s writing process in the case where he arrives at something he decides could be a paper. Note that in the last phases of his work he will switch back and forth between working with LaTeX and with a pen on a printout of his document.
Figure 2.7 shows that R1 has a very linear writing process where he moves from heuristic treatment to control treatment, both of these handwritten to production on a computer. R1 uses printouts to proofread and to review his document, but he does this to ensure that it works as a paper (is understandable for the intended audience) and to check for typing errors. Figure 2.8 shows that R2 has a more circular writing process, where he continuously review and change his work; this procedure starts after the heuristic treatment and is supported by a computer in the case where he thinks it is realistic to produce a paper based on his work, and by pen and paper otherwise. Looking at the eleven researchers, we find a large variety in how they write, but we can say that R1, R3, R8, and R10 have a very linear approach, and that R2, R6, R7, R9, and R11 have a more iterative approach. The respondents R4 and R5 are difficult to classify as either iterative or linear. They both use a WYSIWYG editor to write mathematics on a computer (R4 uses MATH-TYPE and R5 uses SCIENTIFIC WORD). They both use the computer fairly early in their writing process and they do so in an iterative manner, but they will both often use LaTeX when they start producing a paper, and this change of program marks a complete change in working style; from then on, the mathematics is considered finished and the purpose of their work is to produce a paper. R1 emails his collaborators in the control treatment and in the production phase of his work, but usually not during heuristic treatment. In general it seems that the eleven respondents mainly communicate with collaborators during heuristic treatment in face-to-face situations.
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2.4.3 Representations Both R1’s and R2’s working papers develop gradually from containing mainly representations using a symbolic register in the heuristic phase to the extended use of natural language in the control phase. But where the samples from R1 show a clear progression from being non-linear (Morgan, 1998, p. 89) or non discursive (Duval 2000a, see chapter one) in the heuristic phase towards linearity and the use of discursive registers in the control and production phase, R2’s working papers are linear throughout the writing process. If we look at all the respondents we can say that R3 and R4 in particular, and to some extent, also R1, R7, R9 and R11 use graphical nonlinear elements to support heuristic writing. Almost all of the respondents use equations and mathematical notation to support the heuristic phase. None of the respondents use a lot of natural language to support heuristic treatment. Control treatment seems to involve longer discursive arguments containing more natural language, more complicated algebraic formulas, and less use of representations in a non discursive register, such as diagrams and drawings. As mentioned above, email mediates a large part of the mathematicians’ communication with peers, and it is normal to use LaTeX commands in these emails to communicate mathematical notation. This use was denoted pseudo-LaTeX by some of the mathematicians. Since the LaTeX code written in emails is supposed to be read directly by another person rather than compiled by a computer, the writer can be less strict with the syntax as in the following example taken from a long email that one of the respondents had recently received:
p : T \mapsto n(T)
+ modulus of d(T)
where n(T) is the usual operator norm of T. Then this is an algebra norm because p(ST) \leq n(S)n(T) + modulus of \phi(T)d(S) + modulus of \phi(S)d(T) …
This shows LaTeX commands used in an email, some parts omitted (e.g. the $-signs that separates normal text and mathematics), but more interestingly, the LaTeX commands are mixed with natural spoken language (“modulus of”). Furthermore, when listening to mathematicians spoken discussions, you will sometimes hear that LaTeX commands are used in natural language to speak
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formulas aloud. Hence LaTeX, natural language, and standard mathematical notation bleed into mathematicians’ language as shown in Figure 2.9.
Natural Language
LateX code
Standard symbolic register
Figure 2.9: Mathematical email language.
This influence could bias communication on mathematics towards using the kinds of representations that are most easily written (and most readable) in LaTeX, or alternatively towards use of concepts that can be represented in natural language.
2.5 Discussion 2.5.1 The five functions The empirical work has shown that it is reasonable to distinguish between five functions in mathematical writing, but the results also show that these five functions can overlap. For instance R11 did not talk about having ideas and checking them as two different processes. Two of the functions that mathematical writing serves have a special status, namely heuristic treatment and control treatment. The meaning of the word treatment is borrowed from Raymond Duval (2001), and signifies here the use of external semiotic representations to ‘treat’ conceptual objects and thereby to support thinking. It is an interesting result that the data shows that treatment has two different sides namely heuristic treatment and control treatment, these categories appeared significantly. There are many aspects of treating semiotic representations, and therefore the question of why these two functions became dominant arises naturally. I did not pay particular attention to those categories prior to the interviews, but they came up in force in many of the interviews.
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One reason for this might be that the interviews where largely concerned with material aspects, and quite a large number of respondents used different media, or just different pieces of paper to support these two functions. For many of the respondents, control treatment is intimately related to saving information and to communication with collaborators, whereas the heuristic treatment does not seem to be connected to those functions. The production of a paper is connected to communication with collaborators since many of the respondents send drafts to collaborators for commenting, but production does not seem to be strongly connected to heuristic treatment for any of the respondents. For approximately half of the respondents, there is a strong connection between production of a paper, saving of information and control treatment because they often save their work as an evolving LaTeX document and work with developing their mathematical ideas using a printout of the LaTeX document.
2.5.2 Design and use of mathematical writing tools Pen and paper play a central role in the heuristic phase of almost all the mathematicians’ work; only R4 draws exclusively on a computer in that phase. This suggests that the computer systems that mathematicians use do not support their heuristic mathematical writing. Since none of the mathematicians connects support for heuristics directly to saving information or to communication, it may be argued that computer support for heuristic treatment is not essential for mathematical writing and collaboration. But on the other hand it seems likely that an improved digital writing tool could support heuristic treatment. Such a tool could potentially change the mathematical writing process. The control treatment function seems to be closely tied related to communication and the saving of information. Most of the respondents mainly use pen and paper to support control treatment, even though this means that they often have to transcribe a finished draft from paper to computer in order to save or email it. Therefore, development of better computer-based tools to support writing that serves a control treatment function seems to be relevant in order to support collaboration in mathematical research.
2.6 Partial conclusion In this chapter, I have begun the investigation of the nature of mathematical writing process by looking at the writing processes of professional mathematicians’. The interviews have shown that it is useful to distinguish between five different functions that writing serve in mathematics. The
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findings show that heuristic treatment and control treatment are difficult to support with existing digital writing tools, and suggest that especially the problem with supporting control treatment can be a hindrance to distributed collaboration. On the other hand technology such as digital ink (Golovchinsky & Denoue, 2002) and tablet PCs might in time make the computer the preferred writing tool for heuristic and control treatment. Such a development could in the long run change the mathematical writing process entirely and thereby have strong influence on mathematicians’ collaboration. I believe that such investigations can be of use in the design of technology to support collaboration among mathematicians. I also hope that the investigations can provide a different perspective on the problem of introducing collaborative information technology in mathematics education by describing what parts of the writing process professional mathematicians’ support with handwriting and what parts they use computers to support.
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3 Students’ collaborative writing
In this chapter I continue the empirical investigation of mathematical writing. In the previous chapter I investigated professional mathematicians’ personal writing process, but in this present chapter I focus on the social aspects of mathematical writing. I describe an investigation into undergraduate students’ collaborative work on writing thematic projects. The investigation emphasizes the informal collaborative situation by participating in, and video recording, out-ofclass meetings among students working on projects. I have followed two groups of students looking at how they work together and how writing is part of their work. The empirical part of this chapter consist of my descriptions of how the student work, and is based on my observations, diaries written by the students recording what they have done in relation to their mathematics project during a specific week in the middle of the semester, and a qualitative analysis of a conversation between three students. Based on these data, I discuss what functions writing serves the students, and what challenges and barriers this present to students’ collaborative writing. I briefly discuss implications for the design of collaborative activities and how to teach collaborative techniques. Students collaborating outside the classroom represent an interesting didactical challenge. Teachers are obviously unable to interact directly in such collaborations but this does not mean that they cannot attempt to affect the collaboration through designing the students’ conditions for collaborating. Apart from being a general investigation of the nature of mathematical writing, the investigation presented in this chapter also informs a research-based revision that redesigns the course that the students I investigate are enrolled in. The aspects that relate directly to the development project is not addressed in detail in this chapter but Appendix A contains a book chapter (Grønbæk, Misfeldt, Winsløw, to appear) that describes the development project.
3.1.1 Context of investigation The context of the investigation is an undergraduate course in mathematics taught at the University of Copenhagen, for a detailed description of the course see (Grønbæk & Winsløw, 2004 or Appendix A). The list of topics includes metric spaces, continuity, Hilbert spaces, Fourier analysis, and partial differential equations. The course material consist of a book written by N. L. Carothers
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(2000), and this course requires a completed course on advanced calculus (as for instance Adams, 1995) and linear algebra (Messer, 1994). It may be argued that the course is the students’ first experience with advanced abstract mathematics, in the sense that all arguments are based on axioms rather than on intuitive or geometric properties, and the course is indeed considered a ‘hard’ course and traditionally there is quite a large number of students that quits the course or fails to pass the exam. The course is currently going through a research-based revision and has been modified in a number of ways (Grønbæk & Winsløw, 2004, to appear). One of the most significant changes consisted of the introduction of a new working and evaluation form; ‘thematic projects’. The formal format of the examination is unchanged as a three-hour written examination combined with a 30 minutes oral presentation of a randomly selected question. But the oral exam has been changed so that the students no longer present theory from the book but instead present one of five ‘thematic projects’. A ‘thematic project’ is a short (5-10 pages) note prepared by the students in groups of 2-4, in response to a task or set of instruction given by the lecturer. The instructions consist of (1) a description of the learning goal of the task, (2) an introduction to the theme of the tasks, (3) a number of questions, (4) and finally an indicator of which questions are kernel questions. My investigation took place among the students as they were preparing the thematic projects. I followed the work of two groups using diaries and video-taped observations of their meetings.
3.2 Question The objective of this investigation is to explore the students’ collaborative writing process. What functions does writing serve the students in their collaborative work on the thematic projects? My focus was on the students’ face-to-face meetings.
3.3 Methodology As described in the introductive chapter, the overall approach of this dissertation is ethnographical in the sense that the aim is to empirically uncover and theoretically interpret the phenomenon mathematical writing. This chapter does indeed draw on classically ethnographical methods, as participant observation, frequent contact, interviews, and diaries. The ethnographical approach means that the researcher is caught in context and unable to work from an entirely objective position (Hastrup, 1988). On a practical level this means that I tell the story about how I experience
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the students working processes. The story is told on the basis of my observations and interactions with the students. I use several methods that in different ways provide detailed and objective information about the students’ working processes. Video data reveals information about the details of the students’ communication during their meetings and the diaries reveal data about the overall structure of their working processes. The analysis comprises filtering and interpreting all the data, both video, diaries, notes and memories from my participation in group meetings. This filtering and interpretation is subjective and will contain value judgements. The choices I have made, for instance the choice of which episode to analyse in depth, are of course motivated by my desire to answer the research question, but a few words on objectivity and my relation to the field site and are in order. The Institute for Mathematical Sciences is a familiar place to me, since I did my undergraduate and Masters’ degree at this institute. Hence many aspects of the students’ activities are well known to me. This allows me to easily enter the field site and orient myself in it. One could of course object that several years of experience with the practice of being a mathematics student might lead me to some fast conclusions not rooted in data. The simple answer to this objection is that it is crucial that the conclusions I draw is rooted in video and diary data. This should insure that my analysis and interpretations are open to be reviewed by others and hence meets scientific criteria. Nevertheless the general descriptions and choice of episode to analyse are of course value judgments that I have done, and as such not open to be reviewed by others. But this is an unavoidable aspect of a qualitative ethnographical investigation (Hastrup, 1988). All in all the intimate knowledge of the field site allowed me to easily understand and follow the students’ activities, and furthermore I think that it helped me to obtain the necessary trust from the groups.
3.3.1 Methods I followed two groups of students, and investigated their work using a combination of informal interviews, video observations, and diaries kept by each of the students (see Hyldegård, 2003 for a detailed description of a diary methodology). The purpose of the diaries was to obtain an overview of how different activities connected to the group work was organized during one specific working week. During the ‘diary’ week, in the middle of the project period, all participants in the two groups where asked to keep a detailed record
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describing what tasks he or she worked with and during which timeslot. Furthermore, each student were asked to describe the type of activity, explain if he or she worked alone or collaborated (and with whom), describe how many pieces of paper they had used and kept (if they used a computer for writing they would write that here). Furthermore the diary had a reflection field that allowed for free reflection in connection to the activity reported.
Figure 3.1: A diary opening.
The diaries were accompanied by video observations of working meetings. I recorded a total of seven meetings of various lengths (about six hours of video in total). I was in frequent contact with the group participants, mainly by finding them in the area where they usually worked (a large open space in the university). This frequent contact served several purposes. The contact was necessary to gain access to the students’ working meetings, because the students usually saw each other several times a day, and they were therefore able to change schedule for meetings with a very short notice, and they often did that. The contact also gave me a better longitudinal picture of the activities by allowing me to do frequent informal interviews with the groupmembers, which proved to be a valuable source of information, and finally it helped to build trust between me and the group which is crucial in order to participate with a camera in these intimate meetings. The videos were summarized, interesting parts identified and transcribed using the Transana program for video analysis (Fassnacht & Woods, 2003).
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3.4 The students’ working process 3.4.1 The field site The Institute for Mathematical Sciences is situated at the H.C. Ørsted’s institute, a building that contains institutes for Mathematical Sciences, Chemistry and Physics. Many of the University’s science departments: Computer Science, Biology etc. are situated close to the H.C. Ørsted’s institute. The H.C. Ørsted institute contains a number of large lecture halls and a very large open cafeteria area. This area is called Vandrehallen (“the hall were you walk”), and it was here the students I followed met and worked together. The hall is used for a large number of activities, from eating lunch and having coffee to meeting place for a Friday afternoon beer. A large number of students, mainly undergraduates, used the hall to read and work. The reason might be that the undergraduates go to different classes at different institutes. At a later point in their studies the students will typically be more physically bound to one specific institute or laboratory, and meet there.
3.4.2 The meetings The students generally met once or twice a week to discuss their projects. The meetings were to some extend concerned with dividing the labour and coordinating their activities. This articulation work was typically followed by the students quickly discussing strategies for solving parts of the task given. When I asked the students why they met, they pointed to the articulation work as one of the important reasons for meeting. A short meeting could end when the articulation work was done, but in the majority of the meetings observed, the students also engage in mathematical problemsolving during the meeting. Working papers, drafts and books are often referred to during these problem-solving conversations, and the amount of pointing is very substantial, mainly in the sense of deictic words or phrases as “this one here” supported by indexing gestures as pointing with a hand, finger or pencil. In many cases, the pieces of paper on the desk seemed to function both as a personal tool for example for heuristic treatment or control treatment and as a conversation tool pointed to during the conversations. The switching between private and social use of writing showed to be a surprisingly delicate matter, and the episode analysed later in this chapter is chosen exactly because it is a typical example of the difficulties in switching between these ways of using writing.
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3.4.3 Diaries The following two tables are distilled versions of the students’ diaries. The tables show the amount of time that each person declares to have used on the thematic project each day. G. signifies groupwork and P. personal work. In order to explain what parts of the work that is done at the university and at home, I have placed the work done at the university above a horizontal line, and the work done at home below the line. Person
Monday
1a
G. 1:25h
Tuesday
Wednesday
Thursday
Friday
G. 0:15h
G. 0:30h
G. 3:00h
P. 1:30 h 1b
1c
G. 1:25h
P. 0:20h G.0:15h
G. 0:30h
P. 0:50h
P. 0:30h P. 0:30h
G. 1:25h
Saturday
Sunday
P. 3:30h
P. 2:30h
G. 3:00h P. 1:00 h G. 3:00h
G. 0:30h P. 2:00h Table 3.1: Overview of the diary week for group no. one.
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P. 0:30 h
Person
Monday
2a
G. 1:45h
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
P. 1:00h 2b
G. 1:35h P. 1:25 h
2c
G. 1:45h
P. 0:15h G. 0:45h
P. 0:45h 2d
P. 0:30 h
G. 1:35h
Table 3.2: Overview of the diary week for group no. two.
3.5 A Conversational Episode In the following episode we are half an hour into a meeting between three male students from group 2, 2d was absent. The episode shows that the students were unable to maintain a healthy conversation on the topic. I chose to look deeper into this example because it shows the challenges with switching between social and personal use of writing. In this episode, the students had increasing problems with maintaining a healthy conversation throughout their collaborative work with solving mathematical problems.
3.5.1 Task The students work on a task about double sequences. They are given a number of hypotheses and are asked to determine if they are true, furthermore they should either prove the statement or give a counter-example. The task that they discussed was to determine whether the following five propositions (7-11) are true for a double sequence (xnm) with the additional property that B
B
lim n lim m x nm = lim m lim n x nm = λ .
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{
}
(7) sup xnm n, m ∈ N < ∞ (8) for all m, n we have that lim k xnk = lim k xkm (9) the diagonal sequence converges towards lim i xii = λ
(10) there exist a sequence of double indicies (nk , mk ) such that lim k xnk , mk = λ
(11) there exist a sequence of double indicies (nk , mk ) such that lim k xnk mk = lim k xmk nk = λ
These five questions are embedded in a larger set of problems that constitute a thematic project. In the formulation of the thematic project the students are encouraged to work with a “matrix representation” of the limit of a double sequence:
Figure 3.2: the matrix representation of limit as it is presented in the task (the text is in Danish).
3.5.2 Summary of episode The transcript of the episode is shown next to the flowchart below. The following is a brief summary of the episode. We enter the conversation when the students discuss proposition seven (see the transcript ‘node’ 19), and identifies a counter-example in the form of a matrix representation: ⎛1 ⎜ ⎜0 ⎜0 ⎜ ⎜0 ⎜M ⎝
0 2 0 0 M
0 0 3 0 M
0 0 0 4 0
L⎞ ⎟ L⎟ L⎟ ⎟ 0⎟ O⎟⎠
Figure 3.3: Matrix representation of a counter example to proposition seven.
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They agree that they have a solution to proposition seven and later (node 13-16) they agree on seeking a counter example as a strategy to proposition eight. Student A attempts to formulate an idea, talking and gesturing. But before he is has formulated a clear idea, student B proposes a counterexample consisting of the double sequence represented by the matrix: ⎛0 ⎜ ⎜2 ⎜0 ⎜ ⎜0 ⎜M ⎝
1 2 1 1 M
0 2 0 0 M
0 2 0 0 M
L⎞ ⎟ L⎟ L⎟ ⎟ L⎟ O⎟⎠
Figure 3.4: A counter example to proposition eight.
Student C then realizes that the counterexample to proposition seven is also counter example for proposition nine (node 17). This makes all of them suspicious since this is the third negative result in a row, and thee question was formulated as prove, or find a counter example to the following propositions. They review their previous results critically (node 18 to 24). The students then discuss if proposition number ten is true, and student B starts constructing the proposed sequence. Student B withdraws to a long and silent calculation (node 34, 38, 46, 48, 52). Student A attempts to introduce a few ideas but student B is very involved in his own calculation. As opposed to student B, student A does not calculate on paper, and he is unable to get trough to student B with his ideas. Even though student B is involved in his own calculations, he does not forget the other participants. He approaches them several times and explains his ideas using the calculations that he already has written out on a piece of paper in an ostensive2 way (node 31, 36 TP
PT
and 49), pointing towards the paper while he talks. Despite his attempt to involve the others in his calculations, the social mathematical process ends as student C leaves (node 47) and Student A’s comments gets less relevant (node 44), and the conversation ends in a non-mathematical discussion about who should type their solution (node 62).
TP
2
‘Ostensive’ use of writing designates in this dissertation writing used, during a conversation, to demonstrate something. I do not attempt to imply that the writing is used merely to impress your conversation partner. PT
47
3.5.3 Flowchart analysis In this episode, the students’ communication is challenged. One student leaves during their attempt to collaborate and up until this point of the conversation student B works more and more on his own. In order to analyse how the students interact during the conversational episode, I have applied the flowchart method developed by Anna Sfard and Carolyn Kieran (2001a; 2001b). Sfard and Kieran describe an instance where the conversation between two 13 year old boys working with mathematics is very counterproductive and discuss to what extend the competent involvement of teachers in students’ conversations can help such conversations to stay productive. The flowchart demonstrates what an utterance refers to. In Sfard’s and Kierans’ version of the flowchart, they distinguish between utterances based on how much the utterance requires and invites for an answer, I have not done that, because of the complexity in applying the method to a conversation with three parties rather than two. Hence invitations for answers, or pro-active arrows as Sfard and Kieran describe them, are not visualized, hence I only uses the top row of the schema presented in Figure 3.5. Private
Interpersonal
Reactive Proactive
Figure 3.5: Proactive and reactive arrows in the flowchart method. In this analysis only reactive arrows are used (Sfard & Kieran, 2001b).
Sticking to the reactive arrows gives a simpler and less fine grained analysis, but still one that is strong enough to trace the flow of the conversation. In the flowchart shown on the next seven pages I have presented the transcript on the right side of the page. To the left is the flowchart where each column represents one of the students. Furthest to
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the left student A, student B in the middle and next to the transcript is student C. The transcript is translated from Danish by the author.
Student A
Student B
Student C 1
2 3 4 5 6 7
1 student C: I sat and messed around with this one and was very confused, I might be wrong: The supreme … this should be correct, right? 2 student A: Well it is right, because it converges for all these, so we should be able to… 3 student C: For every column and row we should be able to say that that … this is true. 4 student B: “Are we sure, can’t we imagine that it somehow has a bulge somewhere in the middle (draws an iconic bulge on his paper) and that this bulge is placed askew in the scheme (iconic ‘bulge like’ gesture pointing towards the paper) so that it does not disturb anything? What if we take the example from before, your example, one, two three, four upwards down the diagonal (writes “1,2,3,…” diagonally on paper), then the sup is infinity, but if we then chooses zeroes down here then…” 5 student C: That is the same question. 6 student B: So do we have a counter example? 7 student C: Yes, that is a counter example. 8 student A: Yes it is.
8 9
9 student C: But that is what this matrix explains, just that? 10 student A: yes
10 11
11 student C: But the next must be very straightforward. 12 student B: yes 13 student C: It can’t be right that just because you have convergence in the two [incomprehensible]
12 13
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13 14 15 16 17
13 student C: It can’t be right that just because you have convergence in the two … [incomprehensible] 14 student B: We should be able to construct such a sequence. 15 student A: Oh yes, you could have one that goes from minus something towards zero and something that goes from the positive side and… 16 student B: One with the number one here and two there and then zeros for the rest. 17 student C: Well the diagonal sequence, isen tit convergent in…??? 18 student B: Shouldn’t any of this be true?
18
19 student A: Yes
19
20 student C: This can’t be right, I think this is a little to easy.
20 21 student B: we were sure about this one, yes that was the argument from before.
21
22 student B: could it be that these are the ones Jens Christian (instructor) talks about?
23 student C: Hmm I don’t think so.
22 23
24 student B: Yes it must be them.
24
50
25
26
26
26
25 student C: Here I had myself convinced that the diagonal sequence actually would work, an then ten and eleven were very easy. (Laughs)
26 All Laughs
27 student A: We have a convergent subsequence…
27 28 student A: I would say that number ten was false too. 29 student B: Actually I don’t think so.
28
30 student C: None of the examples we have worked with so far points it out as false.
29 30 31 32 33
31 student B: couldn’t you imagine that you choose something that is very close to the values down here? If we have infinety down here and then chooses something that is close, and then let them go to the right… 32 student A: I have to remember it. We assume this to be true. Yes of course you can do that, that is what I thought of. Of course it will converge, yes it must be true. 33 student B: Well how are we going to construct it?
34 student B calculates
34
51
33
34 35 36
37
38 39 40 41 42 43
33 student B: Well how are we going to construct it? 34 student B calculates 35 student A: What if we, couldn’t you just take one for each, we have a lot, for 1 and 2 we have convergent subsequences, we have some convergent sequences for every n and m, if you assumes,… 36 student B: I think that if you take this and puts a-n equal the limit down here and then constantly choose something that is epsilon close to a_5, then you can chose something in here, epsilon close. An then for every a_n you can choose such epsilon intervals and then choose a number t inside, then you have a sequence that converges and then, I don’t know, well you can of course chose epsilon as one divided by n, then you should get it. 37 student B: Because the sequence will get infinitesimally close to every of the a_n’s that converges towards our limit. 38 student B: Let me write it down so we see what we get. 39 student C But the essence of it is that you say: Ok first we go downwards as long as needed in the one direction, we have n_1, 2, 3 , a billion. Then we start to move to the right. That is what you think, right? 40 student B: Well, yes 41 student C: It is down there and then straight ahead. 42 student B: That’s fine.
43 student A: Or we can go and say, now that we can find an n, we should be able to find some n large enough, and gives an epsilon, the distance… well hmm... 44 student A: these chi’s are they x’s?
44
52
45 student B: They are k’s.
45
46 student B calculates
46 47 student C leaves
47 48 student B calculates
48
49 50
49 student B: I think that you can at least construct the sequence. I am 100 % convinced that it works, I just have to show it. You have to show it converges and that it converges to the right thing – lambda. 50 student A : No
51 student B: It is slightly boring to do that in two steps.
51 52 student B calculates
52 53 54
53 student A: The question is whether you could use some of the conditions we have here? 54 student B: Yes, yes it converges to a, that is for a certain step this minus a. but if we… but a_n minus x less than epsilon. Then you can do a triangle inequality on it. 55 student A: Ok 56 student B: If a_n is close to x_n then all we need is to expand one of them, then we are home.
55 56
53
56
56 : student If a_n is close to x_n then all we need is to expand one of them, then we are home. 57 student B: a_n minus x_n [mumbles] yes, no wait a minute oops.
57 58 59
58 student A: How about using this number five to say, if we should show this one then we could to the nessesary condition, if this is true then that is true to…. 59 student B: But number five isn’t a necessary condition. 60 student A: Oh no it is a …
61 student B No this one works, there is just a little…
60 61
62 student B: Yes we can show that, should I write it down. No we should let Student C do that.
62
3.6 Flow of conversation and conversational breakdown The flowchart describes the flow of the students’ interaction. In the beginning of the discussion (node 1- 13) the flowchart show a conversation where all three participate and contribute. From node 14 and forward we see an increasing tendency that student B is the one that speaks the most and that he refers mainly to his own utterances. Of course the other students A and C speaks too but, particuary towards the end of the conversation, these utterances are not referred to by any of the following utterances (e.g. 43, 44, 50, 53 and 55). Hence the flowchart reveals that the utterances from student A and C are often conversational dead ends, either because they are comments to the line of thoughts presented by someone else, or because the idea student A or C propose is not taken up by the others. Student B referring increasingly to himself and the other two mainly contributing with dead ends to the conversation, is a degenerated situation where no real collaboration occurs. I will describe this
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degeneration of the conversation as a ‘conversational breakdown’. Of course nothing breaks in this breakdown, but the nature of the conversation changes from having, in this case three, parties to having just one, namely student B. The two other students more or less play the part of an audience.
3.7 Functions of writing in collaborative settings It might not be a coincidence that student B both is the one who leads the conversation and the one who writes on paper. Writing on paper means having the power to discard and follow ideas since mathematics is so dependent on semiotic representations. The notion of the person writing as the most powerful participant in a mathematical conversation can be applied in our example here. Student B is definitely the dominant one in choosing the direction of the conversation, and on several occasions he does not seem to hear the other students’ contributions: In node 22-24 where student C disagrees with student B, but student B keeps referring mainly to his own line of thought anyway, or node 48-56 where student A attempts to take part in the conversation, but his utterances for instance node 53 is not actually heard by student B.
3.7.1 Ostensive and private writing A very concrete way in which the mathematical power relations between the students are realized through the use of writing is when student B explains his calculations to the rest of the group (for instance node 36). This ostensive use of paper, to show something to someone, was typical in all of the meetings I observed, and the ability to focus everyone’s attention is a powerful aspect of written representations. Writing enables the whole group to focus on the same written representation, but the episode analysed here shows that writing also enables one person to work privately within the group. When student B works alone during the episode presented here, it might not be due to lack of interest in the others or an ambition to feel powerful while explaining the solutions to the rest of the group members, it is of course mainly because it helps him solve the task. Sitting with pen and paper helps him create a private space for his work in the middle of the group conversation.
3.7.2 The five functions Apart from ostensive use of writing and the use of writing to create a private space, the functions that were found in chapter two can be described in this collaborative setting.
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Heuristic treatment and control treatment functions are often in play in the students’ conversations. In the episode described here, it is student B’s personal calculations that support heuristic and control treatment. Student B writes for himself, both in order to generate (for example node 33-34) and verify (for example node 51-54) his ideas. Heuristic treatment seems to also come into play in the conversations, for instance in node four where student B says: “Are we sure, can’t we imagine that it somehow has a bulge somewhere in the middle (draws an iconic bulge on his paper) and that this bulge is placed askew in the scheme (iconic ‘bulge like’ gesture pointing towards the paper) so that it does not disturb anything? What if we take the example from before, your example, one, two three, four upwards down the diagonal (writes “1,2,3,…” diagonally on paper), then the sup is infinity, but if we then chooses zeroes down here then…” The drawing on paper that student B performs here, is an example of ostensive use of paper, because student B communicates his idea to the rest of the group. The group members think while student B is talking. Therefore this drawing of a bulge and later referring to it can be considered as heuristic treatment. The function of communication is in a sense obvious in this episode because the conversations are instances of the group communicating internally. This is different from the researchers’ use of ‘writing for communication’ (i.e. to write something in order for another specific person to read it later). This way of combining writing and talking is exactly what I have describes as ostensive use of writing. The combination can have several forms. A good example can be found in node 36, where student B shows his calculations to the others explaining his approach using the already made inscriptions in an ostensive way. Communication as it was defined in chapter two is less present in the meeting, but the diaries reveal that the students in both groups communicate in written form internally. In group one, they meet and exchange handwritten drafts and comments among the group members several times during the ‘diary week’, and group two exchange emails with drafts and comments. Information storage is done by the students, both individually and collectively. For individual saving of information, the student tends to use handwritten notes and for collective saving of information LaTeX document seems to be preferred (see appendix A).
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3.8 Factors influencing conversational breakdowns The analysis of the functions of writing in face-to-face collaboration points to at least one important reason for the conversational breakdown in the episode presented above, namely the way that the ostensive writing and private writing interact. But it seems reasonable to ask what, if any, other factors that influence the conversational breakdown, and particularly how these factors can be influenced by the tasks and the general didactical milieu (Brousseau, 1997) established by the teacher. Sfard and Kieran show that students’ different interest can create an unhealthy collaborative environment (Sfard and Kieran, p. 201), but the interactions studied by Sfard and Kieran are didactical situations, and the students are expected by the teacher to engage in a mathematical conversation. This is only partly the case in this present example, since the students’ collaboration takes place outside class, and the result is actually that one of the students decides to leave and attends something else rather than continuing the fruitless conversation. But what are the reasons for the communicative breakdown and what can course developers do to help students to establish fruitful conversations during their collaboration? The students meet in order to discuss and develop their collaborative thematic project, and the only indications that they do not show interest in collaboration show up very late in the process (node 44 and 62-64) and might very well express a frustration with the fruitless communication, rather than a lack of interest in communicating. The task that the student attempts to solve is an important factor for the students’ collaboration. It is interesting that on the occasion where fruitful communication is threatened by one student communicating primarily with himself, the students are engaged working with a task (task number ten) that, on the structural level, was different from the ones before. Task number ten was: given that lim n lim m x nm = lim m lim n x nm = λ . (10) there exist a sequence of double indicies (nk , mk ) such that lim k xnk , m k = λ
The most immediate suggestion why the students’ work with this task is less fruitful than their work with the prior tasks is that the students, for various reasons, are not able to solve the problem proposed in this task as easily as they solved the previous ones. Nevertheless, it is still interesting to
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see that from the very beginning the students approach this problem differently than the previous ones. The propositions in task seven, eight and nine are wrong and the students construct very simple counterexamples using the proposed matrix representation (see Figures 3.3 and 3.4). Task number ten concerns the existence of a convergent subsequence, and working with concrete examples does not lead to a solution. To solve this problem one most work with an abstractly formulated double sequence. In order to obtain the desired result, one need to go through several logical steps, including the reference to an important theorem from the book, and this seems to lead student B to ignore the other students and work on his own. When the students work with tasks seven, eight and nine, they refer again and again to a specific matrix representations, such as the ones from Figures 3.3 and 3.4, and they uses them to discuss potential candidates for counter-examples. When they talk about task number ten, they also use this matrix representation (see for instance node 36) but there are also longer periods (for instance node 45 to 48), where student B works on his own using an algebraic register in the form of longer calculations. It is difficult to draw firm conclusion on the basis of this single example, but there seems to be several semiotic parameters that influence the students’ ability to work towards a solution together. The use of algebraic register and the use of paper to support the generation of a private space, and hence disrupt collaboration, seems to go hand in hand, whereas the more iconic matrix representation seems to support discussion and generation of examples. The data presented here are of course not conclusive but it demonstrates the relevance of investigating the relation between types of tasks and fruitful collaborative strategies. If future research can point to types of tasks that are more well-suited for some collaborative strategies than for others this can be of value when designing tasks for collaborative learning. Furthermore, it might be a good idea to think about how to teach collaborative strategies to students. knowing how and when to interact and retreat in order to use conversation partners effectively when doing collaborative writing or problem solving, is of course extremely valuable. It might be possible to teach students about typical problems and dynamics through collaboration on mathematics, and hereby create awareness among students about the challenges inherent in communicating and collaborating on mathematics.
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It might also be of value to look at the students’ working environment. Creating an environment where it is possible to move spontaneously back and forth between group discussion and individual work while remaining focused is always a challenge, but at the university where these particular students studied, this was given very little attention, if any.
3.9 Partial conclusion This chapter has continued the empirical investigation of mathematical writing by reporting on an investigation of meetings in connection with students’ collaborative writing. I have presented an episode where three student’s out-of-class collaboration on a writing task is hindered because they are not able to keep a conversation going since one of the students retreat from interacting with the others, and exclusively refers to his own utterances. The discussion has shown that the writing serves two important functions in relation to mathematical conversations, namely an ostensive function and a support for the creation of a private space. I have discussed the role of design of tasks for collaborative learning, the collaborative strategies taught and the physical environment that the students are offered for their collaborative writing activities. In the following three chapters, I will attempt to conceptualise mathematical writing from different theoretical perspectives, namely as a writing process, as a semiotic activity and as a social activity. These discussions will take both the present and the previous empirical investigation into account.
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4 Mathematical work as a writing process
In the last two chapters, I have described how writing plays a role in two different mathematical settings. The main findings are that mathematical writing serves cognitive (heuristic and control treatment), communicative (saving of information, communication and production), and social (ostensive and private space) functions. In this and the two the following chapters, I will discuss these results from three different viewpoints. In this chapter, mathematical writing is described and analysed as a writing process; in the following chapter, the role of the semiotic representations for mathematical thinking is described; and chapter six comprise a discussion of the role of writing in group-work and social interaction about mathematics. To study mathematics as a writing process means that I use two existing conceptualisations of writing processes to understand the way external representations are used in mathematical work. The two conceptualisations, described briefly in chapter one, looks at writing either as a process of solving rhetorical problems or as a process of discovery. The choice of framework is important for what a mathematical writing process is. Both of the existing conceptualisations attempt to explain the creative potentials of writing, in each their way and the purpose of this chapter is to compare these two explanations and see what the explanations would mean if we considered mathematical use of written representations as a writing process. For example, the idea of writing as a process of discovery operates with a concept of ‘private writing’ as an important aspect of writing and a source of creativity, whereas the other conceptualisation of writing sees the main creative potential of writing as a meeting between two psychological ‘problem spaces’, namely a rhetorical space and a content space. The question of ‘private writing’ is treated very different by the two conceptualisations, and in the end of this chapter I present a case of ‘private mathematical writing’.
4.1 Two conceptualizations of the writing process One of the important questions for theories about writing processes is to what extent one can talk about ‘discovery through writing’.
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I will present two views on this question. On the one hand is the idea of discovery writing, which states that the actual process of writing generates fundamentally new ideas in an unpredictable manner. The other view is that ideas and arguments are produced mentally without any significant influence of writing, which main function is to communicate and record these ideas. The theories that follows this later idea (mainly the theory of Bereiter & Scardamalia, 1987) do acknowledge that knowledge is generated in the process of writing, but sees this as an effect of the writers’ attempt to put herself in the place of the reader in order formulate herself in an understandable manner, rather than a ‘spontaneous’ interaction with written representations. Hence I have labelled this approach to the writing process ‘rhetorically driven problem solving’.
4.1.1 Writing as rhetorically driven problem solving The idea of considering writing processes as a psychological phenomenon grew out of an attempt to help children write better that began in the United States in the sixties (Norris, 1994). An important idea proposed in a report from US department on health and education (Rohman & Wlecke, 1964) was to divide the writing process into steps such as planning, writing and rewriting. Following the idea of planning and rewriting, Flower and Haynes (1980) conducted a central empirical study in the research on writing processes. One of their goals was to demystify the discovery potential in writing. They describe discovery as a metaphor for the amount of hard work that goes into generating new meaning while writing and say: “Discovery, the event and its product, new insights, are only the end result of a complicated intellectual process. And it is this process that we need to understand more fully.” (Flower and Haynes p. 21). Their theory is that this process consists of (mentally) setting rhetorical goals and solving rhetorical problems in order to achieve these rhetorical goals. A rhetorical goal is a goal that has to do with obtaining an intended effect on the reader. One should not think of rhetorical goals as restricted to problems of persuading or seducing the reader into taking the authors’ personal position on a moral or political issue (even thought this endeavour would be a rhetorical goal). A rhetorical goal may just as well be a matter of organising a text in such a way that it is easy to understand for the intended reader. What makes a goal as a rhetorical goal is that it concerns thinking about a potential reader.
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Flower and Haynes show empirically that the concept of goal setting is important in distinguishing good and poor writers, and it is a central element in their investigation to describe how these rhetorical goals and rhetorical problems are represented mentally and how they affect the writing process. A very interesting result of their work is a mapping of the large effect that rhetorical problems seem to play on the idea-generating process. Flower and Haynes report that approximately two thirds of the new ideas that are developed by the writers in their investigation are found as a response to a rhetorical problem whereas only one third of the ideas are generated only as a response to the topic itself. The idea of writing as a rhetorical driven problem-solving process has been developed further by Bereiter and Scardamalia (1987). They define writing as “The composition of written text intended to be read by people not present.” Bereiter and Scardamalia (1987) distinguish between two basically different modes of writing: knowledge telling and knowledge transforming, the knowledge transforming being the more advanced mode of writing. Furthermore, they develop psychological models describing these two modes of writing. In a writing-process following the knowledge telling model the writer uses his discourse knowledge (knowledge about creating texts, about different genres, etc.) together with already existing content knowledge to create a text. The composition process starts with a mental representation of an assignment (“write an assay on my summer holiday”) that splits up in two components, a discursive (write an essay), and a content oriented (my summer holiday). When an initial idea of form and content is established, the writer searches his memory for knowledge, all knowledge is tested for appropriateness (and the search criteria, or ‘memory probes’ as they are called, are updated accordingly) and finally written text is generated. The generated text feeds back to a new search for relevant content.
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Mental representation of assignment
Knowledge telling process Content knowledge
Locate topic identifiers
Locate genre identifiers
Discourse knowledge
Construct memory probes
Retrieve content from memory using probes
Run test of appropriateness
Fail
Pass Write (notes, draft, etc.)
Update mental representation of text
Figure 4.1: The knowledge telling model (Bereiter & Scardamalia, 1987).
An important feature of this model is that even though the writer’s content knowledge and discourse knowledge are both brought into play, these two types of knowledge do not interact. Therefore, this model is not very suitable for describing even simple calculations made with pen and paper as an act of writing, because the writer uses the exact rules of algebraic discourse to obtain new content knowledge (e.g. the value of x). Berieter and Scardamalia do not consider algebraic calculations as writing but they do consider writers that use the written text actively in knowledge production: “[…] they are used to considering whether the text they have written says what they want it to say and whether they themselves believes what the text says. In the process, they are likely to consider not only changes in the text but also changes in what they want to say. Thus it is that writing can play a role on the development of their knowledge.” (Bereiter & Scardamalia, 1987, p. 11). To explain such a transformation of knowledge, Berieter and Scardamalia uses the ‘knowledge transforming’ model. In this model, Berieter and Scardamalia consider writing as imbedded in a
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cognitive problem-solving process where the knowledge telling process plays just one small part. The problem-solving goes on in two different psychological ‘spaces’, one concerned with content related problems and the other concerned with rhetorical problems.
Mental representation of assignment
Problem analysis and goal setting Content Knowledge
Content problem space
Discourse knowledge
Rhetorical problem space
Problem translation Problem translation
Knowledge Telling process
Figure 4.2: The knowledge transforming model (Bereiter & Scardamalia, 1987).
The purpose of the knowledge transforming model is to explain the way writing support knowledge generation. Their explanation relies on problem translations between the two psychological spaces related to content knowledge and to discursive knowledge. As an example of this interaction, Bereiter and Scardamalia mention that a writer working in the rhetorical problem space with an issue of clarification may end up deciding that she needs to clarify the concept of ‘responsibility’ and work on this concept in the problem space leading to an insight that will go back to the rhetorical space and effect the entire text. Bereiter and Scardamalia (1987) provides empirically evidence that the writing process can follow each of the two models. All writers occasionally experience a writing process that follows the knowledge-telling process, but the more advanced writers tend to use the knowledge-transforming model more often than novices. Both these models of writing operate with a ‘mental representation
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of the assignment’ that is established prior to the writing activity. This mental representation is in the knowledge-transforming model followed by an explicit planning phase and a problem solving phase in order to figure out what to say and how to say it. One might question the value of looking at writing merely as a rhetorically driven activity if one wants to account for the mathematical writing process. Think once more about the example of using pen and paper to do a calculation to find x. This activity can actually hardly be considered just a rhetorically driven activity. One can definitely do a calculation to find x for other reasons than to have other people read the calculations. So even though the knowledge-telling and the knowledgetransforming processes together constitute a coherent framework for analysing rhetorically driven writing, it does not attempt to analyse the generation of meaning that goes on in personal notetaking and, for instance, calculations. In the next section, the notion of ‘discovery writing’ is considered as a relevant alternative to the rhetorically driven writing.
4.2 Discovery writing Scardamalia and Bereiter (1987) and also Flower and Haynes (1980) view writing as the process of finding out how to say what you know and mean on a subject. In this process it sometimes happens that you uncover gaps or inconsistencies in your knowledge on a given topic. According to David Galbraith (1992), these theories do not fully explain the experience of generating new knowledge while writing. Galbraith argues that the potential for generating new knowledge during writing is what has lead to pedagogical methods like discovery writing and process writing (Elbow, 1998). Galbraith (1992) develops what he describes as a ‘romantic position’ to writing; the purpose is to produce a model that explains discovery writing without accommodating it to the models of Scardamalia and Bereiter or Flower and Haynes. The romantic position considers writing a process of figuring out both what we mean and what we know on a topic. As soon as a sentence is externalized in writing, another sentence may ‘pop up’ disclaiming or contrasting the first sentence. Viewed in this way the composing process becomes a process of constant negotiation between viewpoints trough rewriting until some balance (or deadline) is reached. The basic claim is that during a writing process, the author continuously generates new knowledge. This knowledge generation is more due to the continuing changing stimuli that the developing text provides, than it is due to setting and archiving rhetorical goals, as claimed by Flower and Haynes (1980), or due to an interplay between rhetorical knowledge and content knowledge, as proposed by Berieter and Scardamalia (1987).
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The romantic position does acknowledge that we remember all sorts of things, but in reality these memories and other knowledge that we posses does not a priori make a coherent story, this story is found or created trough negotiation and association, and would not be possible (at least for some writers) without externalizing ideas trough writing. Galbraith states his notion of knowledge that underlies this view on writing as: “For the romantic position, the writer’s knowledge is contained in a network of implicit conceptual relationships which only becomes accessible to the writer in the course of articulation.” (Galbraith 1992, p. 50).
Mental system Provoke Generate Text
Figure 4.3: A graphical representation of the romantic position concerning writing; the generation of meaning occurs between the internal and external representations. Notice that this position does not attempt to distinguish between content space, rhetorical space and memory.
The writer might have a lot of ideas about the topic she writes about, and these ideas do not always fit into one consistent argument, they may even be contradicting. Also, new ideas are generated in the writing process. The role of articulation on paper is therefore to bring forth the various positions in order to figure out not only what to say and how to say it, but also to develop ones knowledge on the topic. This process fundamentally changes the “mental representation of the assignment”, which the theory of Bereiter & Scardamalia operates with. This constant feedback is not represented in either the knowledge telling or the knowledge transforming model, because they both work under the assumption that the writer starts writing after he has a good mental representation of the task, and that this basic task-representation usually is stable throughout the writing process. Galbraith uses his ideas about a “romantic position” to writing to empirically distinguish between different types of writers. He distinguishes two categories of writers; ‘low self-monitors’ and ‘high self-monitors’. High self-monitoring persons are very sensitive to the way they are perceived by
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others, and generally have good self presenting skills. Low self-monitoring persons have less developed self-presenting skills, and are generally less influenced by how they are perceived by others. Using these categories, Galbraith studies how ideas are generated during writing processes from early note-taking to finished essay. He shows that while high self-monitors have a lot of ideas early in the writing process, the number of ideas that low self-onitors have increases the closer they get to a finished essay. Galbraith suggests that there are two fundamentally different ways that knowledge is produced while writing. Firstly ‘rhetorical planning and execution’ can give rise to new knowledge as a result of the author’s initial ideas being re-shaped in order to fit into written form, maybe respecting a specific genre. This knowledge production is characteristic for the high self monitors and in coherence with the knowledge transforming model proposed by Scardamalia and Berieter. Secondly, ‘discovery writing’ is an equally important mode of knowledge production. Here, knowledge is produced, not because of an attempt to put already existing ideas into writing, but also because what you write triggers new ideas as well as develop existing ones. This way of producing knowledge is coherent with the way that Galbraith found that the low self-monitors wrote.
4.3 Writing and mathematics The two models of knowledge production distinguished by Galbraith highlight different aspects of the writing process, and it is therefore obvious to ask whether mathematical writing has a tendency to favour one of these models. In chapter two, we saw five functions that writing serves professional mathematicians in their personal research work distinguished. The five functions are; heuristic treatment, control treatment, saving of information, communication and production. These functions allow us to rephrase the question to ask more specifically if the different functions that writing serves in mathematical work has a tendency to follow one of these models. Heuristic treatment is a function of writing that has to do with idea generation, and with seeing connections. Control treatment also in concerned with supporting conceptual activities, but has more of a flavour of verification than of idea generation. The two conceptualisations of writing suggest two different sources of new ideas in writing processes; the interplay between rhetorical space and content space and the discovery writing potential. One may ask how the distinction
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between heuristic and control treatment can be related to these two conceptualisations of knowledge production. In chapter two, heuristic treatment was defined as the use of written representations to get ideas and see connections. Control treatment on the other hand, was defined as deeper investigation of the heuristic ideas characterised by ‘precision’. Heuristic treatment is the use of mathematical representations to challenge and support thinking, in the line of “coming up with and trying out ideas and seeing connections”. I have provided examples of heuristic treatment writing in figures 2.1 and 2.4. Heuristic treatment writing is both private and temporary. The view of this type of writing as rhetorically driven problem solving might therefore not be meaningful. The way the author thinks that his heuristic treatment writing will be looked at by others is simply irrelevant, because he would typically not show it to others. Discovery writing, on the other hand, seems a reasonable conceptualisation of the heuristic treatment. The view that as soon as a sentence is externalized through writing a contrasting (or maybe logically equivalent) sentence will ‘pop up’ in the authors’ mind, would certainly explain why heuristic treatment is an important function in mathematical writing. The details of how the process of discovery writing or heuristic treatment goes about can be very different from case to case, but the constantly changing text generates the environment for changing thoughts. The functions of production and communication belong in the rhetorical problem space, and the connection between these functions and heuristic and control treatment seems to be sparse. The function of information storage does not belong in the content problem space, but to consider it as belonging entirely in the rhetorical problem space does not seem correct either since it is not concerned with communicating to another person. In chapter two, it was demonstrated that control treatment was strongly connected to saving of information for at least five (R1, R3, R4, R8 and R10) out of the eleven respondents (those who mainly save paper drafts). In order to explore the interplay between control treatment and information storage as a either a direct interaction between external representations and thinking or as an interplay between two psychological spaces, I will present a case of private mathematical writing, where a mathematician (R3) uses one kind of notational systems for personal writing and another kind for communicating to others.
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4.4 Private writing in mathematics One of the big differences between the two ways of conceptualizing writing processes is evident from the way these standpoints consider private writing. Private writing can be described as writing that not is intended to be seen to by others. Scardamalia and Berieter considers private writing a borderline phenomenon outside the scope of their work, and they specifically define writing as writing intended to be seen by others. From the perspective of discovery writing, private writing is a typical knowledge producing activity. Whether or not there is private writing in mathematics is in a sense obvious. We saw cases in chapter two that some calculations were performed or some diagrams drawn solely for personal writing, only one of the respondents (R10) from the investigation presented in chapter two did not use writing to support his heuristic treatment, which in almost all cases is private. In chapter three, we saw that writing was used to create a private space during mathematical face-to-face conversations. In these two senses at least, private writing does exist in mathematics, but what about longer private writing, such as personal archives? This case shows how information storage can be of a very private nature and still serve control treatment function. The respondent R3 is about 50 years old. He described his writing process as consisting of two main parts, the black notebooks and a paper draft. The paper draft is in a sense the final stage for him, when it is finished he gives it to a secretary who types it in LaTeX. The black notebooks have the purpose of saving ideas, proofs and other types of information for him to access later. The black notebooks are kept in a chronological order. R3 stresses that it is important for him to keep all “the garbage” out of the black notebooks. Therefore, he uses other pieces of paper and his blackboard to support heuristic treatment, and to some extent also control treatment. He uses a computer algebra system to support heuristic treatment and control treatment. He showed me no records of these working papers because he did not keep such records. When I asked him about how the black notebook and the paper draft differed, he showed me a page in the notebook (Figure 4.4) and explained that the stair shaped graphs would typically not be included in a research paper, because they were of an informal and personal nature. To R3 however, these graphs were a very useful and precise way to store his ideas.
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Figure 4.4: The notebook, R3 would usually not write these representations in a paper.
R3 explained that the mathematical community has developed a strong tradition of presenting research in a very formal way. This is obviously an advantage, because it insures the validity of the knowledge produced in the field. Nevertheless, this also means that journal articles gives only little insight into what R3 describes as “the ideas” behind the work, and also conference presentations, informal discussions and personal relations are still very important. When I asked him to describe the difference between what he saves for himself and what he would write for someone else, R3 he showed me a page from the notebook (Figure 4.5) and two pages of a pen and paper draft of a publication (Figure 4.6), he was currently working on. He was in the process of writing the paper draft taking outset in the notebook (written at least a year ago), and he stressed that these two samples represented the same content as he would write it for himself and for the community.
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Figure 4.5: The notebook.
Figure 4.6: Two pages from a draft.
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R3 explained that these pages show the same calculations but that the notation differs, and that many more words are needed in the paper draft, than in the black notebook. The case shows that R3 writes for himself in the black notebook mainly in order to save information, but also to support control treatment. This notebook does not consist of scattered calculations and drawings, it is definitely fair to describe the content as mathematical writing. The writings in R3s black notebooks is private. R3 has no intention to show this writing to others. One may ask whether the writing that R3 does in his notebook is discovery writing or rhetorically driven problem solving? The first answer to this would be that the notebook is essentially private. Hence the writing in it is not intended to be viewed by others; therefore it can not be rhetorically driven. But in order to classify the notebook, it is important to look at the notebook as a part of a long process hopefully leading to publication. So even though the sheets from the notebook are not supposed to be seen by others, it does represent the beginning of a more formal writing process. A very clear indication that the black notebook also represents rhetorical issues is given by R3’s comment about keeping all the garbage out of the notebook. The notebook represents thorough calculations that R3 performs to check his ideas and to save them. For both these purposes it is very important to write down calculations very accurately. Hence the knowledge generation that happens while working with the black notebook is a result of a commitment to the standards for writing in the mathematical genre (rule following, showing every step of the calculation etc.), and can be viewed as a result of a rhetorical problem solving process. On the other hand, the notebook contains representations of a nature that is not meant to be included in formal reports or research papers. The step-shaped graph shown in Figure 4.4 is not a type of representation that R3 will typically include in a research paper. A comparison of Figures 4.5 and 4.6 shows that the one suited for publication contains many more words and explanations.
4.5 Partial Conclusion In this chapter, I have tried to describe mathematical work as a writing process. In order to do that I have reviewed some of the most typical frameworks for describing writing processes, and attempted to use these for describing the data presented in chapters two and three as instances of writing processes. We have seen that the cognitive frameworks for describing writing (Bereiter & Scardamalia, 1987; Flower & Haynes, 1980) can be used for describing the writing that serves
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communicative purposes. That is the saving, communication, and production purpose. Bereither & Scardamalia does suggest that knowledge is developed when writing follows the knowledge transformation model. Nevertheless, the very concrete use of external representation to support thinking which many of the respondents described in chapter two refer to, does not seem to be exhaustively described as an interplay between content knowledge and rhetorical knowledge, because the theory leaves only little room for a direct and continuous interaction between content knowledge and external representation. Stated differently; we need to look carefully into how written or semiotic representations affects mathematical thinking. This is an area where a lot of research has been done over the years, and in the next chapter I will attempt to look at how this research can shed light on what makes mathematical writing special.
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5 Semiotic representations and mathematical thinking: the case of commutative diagrams
In the previous chapter, I described the mathematical writing process. I did that by describing two competing ideas about the how new ideas are generated while writing. One of these ideas highlighted the rhetorically driven problem-solving process as the most important source for knowledge generation and the other focuses on the direct feedback between the generating and perceiving written representations. In this chapter, I will focus on the later of those aspects. I ask whether semiotic representations play a role in mathematical thinking and what that role might be. To be more specific, I investigate how various types of semiotic representations support mathematical thinking. I will outline how these questions have been conceptualised by Luis Radford, Heinz Steinbring and especially Raymond Duval. This theoretical review is exemplified and discussed on the basis of an analysis of commutative diagrams. The purpose of this example is twofold; to describe and compare different theoretical perspectives on the use of semiotic representations in mathematics, and to understand how this specific kind of representation support mathematical thinking.
5.1 Introduction One might think that the role of semiotic representation in mathematical work is first and foremost communicative, because mathematical work is mental work having to do with structures and relations. Nevertheless, the archetypical image of mathematical work is a person doing calculations using pen and paper. In this chapter, I draw upon Raymond Duval (2000a; 2001), Heinz Steinbring (2001) and Luis Radford (Radford, 2002a, 2002b) to describe what role semiotic representations play in mathematics. I base my discussion on a special kind of representations, namely commutative diagrams. The reason for focussing on this type of representations is that they combine iconic aspects of drawings with a precise syntax and hence a priori occupies a place between discourse and drawings, a place that could be particularly interesting in connection with mathematical writing.
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5.2 Commutative diagrams Commutative diagrams are used to denote mappings between sets in mathematics. So for instance the diagram in Figure 5.1 denotes that f is a mapping from X to Y, g a mapping from A to B, ϕ one from X to A and ψ a mapping from Y to B. Stating that the diagram is commutative is saying that g○φ= ψ○f. X
f
Y
ϕ A
ψ g
B
Figure 5.1: A commutative diagram.
Commutative diagrams serves as a convenient way to denote systems of maps and spaces for many mathematicians. In some mathematical disciplines (e.g. homological algebra, algebraic topology and other functorial disciplines) this notation is used extensively, not only for establishing an overview of a system of maps and spaces, but also to introduce mathematical objects and to perform calculations. See Hatcher (2002) or Bredon (1993) for further introduction to commutative diagrams.
5.3 Semiotic representations in mathematics One very important task for the study of mathematical writing processes is to examine the influence of semiotic representations on mathematical thinking.
5.3.1 Registers, treatments and conversions Raymond Duval describes “the paramount importance of semiotic representations” (Duval, 2001). His main argument is that the only access we have to mathematical objects is through semiotic representations, but that these objects should not be confused with any semiotic representation of them (Duval, 2001, p. 7), because any mathematical object has several different semiotic representations, and these representations are qualitatively different. Duval points to the fact that many learning difficulties in mathematics can be analysed by looking at the transformation of
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semiotic representations. The theory is briefly described in chapter one, but the central aspect of the theory is transformations of semiotic representations, particularly treatments and conversions. •
Treatments are transformations inside a semiotic system, such as rephrasing a sentence or isolating x in an equation.
•
Conversion is a transformation that changes the system, maintaining the same conceptual reference, such as going from an algebraic to a geometric representation of a line in the plane.
A semiotic system that allows for treatments inside the system and for conversion of sign to signs in another system referring to the same conceptual object is designated a ‘register’. Duval shows empirically (Duval, 2000a figs. 4 and 5) that conversions can bevery difficult for students. The conversions that seem to be easiest for students are the ones that are congruent, meaning that the representation in the starting register is transparent to the target register (Duval, 2000a, pp. 1-63). A congruent conversion is for example when a sentence in natural language can be translated into an algebraic expression while maintaining the order of signs, and just translate each word to its similar algebraic symbol. The relation between semiotic representations and conceptual understanding can then be described with the following figure from chapter one:
Concept
Treatment Register A
Register B
conversion
Figure 5.2: From Duval (2000) figure 6.
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Duval divides registers into four categories depending on if they are discursive, non-discursive on one hand and multifunctional or monofunctional on the other. Discursive registers are registers where it is possible to perform valid deductions, and for all practical purposes these types of registers are linear in some form. Non discursive registers are typically geometrical. Multifunctional registers are registers used in many fields whereas monofunctional registers are more technical registers used for a narrow purpose. The following table, taken from Duval (2000a, figure 7) exemplifies these types of registers.
Multifunctional registers (processes cannot be transformed into algorithms)
Discursive representations
Non-discursive representations
Natural language
Plane or perspective geometrical (configurations of 0,1,2 and 3 dimensional forms)
Verbal (conceptual) associations Reasoning: -arguments from observations, beliefs… -valid deductions from definitions or theorems
Monofunctional registers (most processes are algorithmic)
Operatory and not only perceptive apprehension Ruler and compass construction
Notation systems:
Cartesian graphs
Numeric (binary, decimal, fractional…)
Changes of coordinate systems
Algebraic
Interpolation, extrapolation
Symbolic (formal language)
Table 5.1: Types of registers (Duval, 2000, fig. 7).
5.3.2 Empirical grounding of mathematical signs One could view Duval as controversial in the sense that mathematical concepts seem in his work to be connected only to semiotic representations. Duval does not discuss references of a non semiotic nature. Michael Otte (2001) explains that “The ultimate meaning or basic foundation of a sign cannot be a sign itself; it must be of the nature either of an intuition or of a singular event” (Otte, 2001, p. 1), but he also stresses that “A mathematical object, such as ‘number’ or ‘function’, does not exist
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independently of the totality of its possible representations, but it must not be confused with any particular representation, either”. (Otte, 2001, p. 3) This conflict is addressed by Heinz Steinbring (2001) with the introduction of what he describes as the ‘epistemological triangle’ which connects concepts, not only to the signs that represents them, but also to a ‘reference context’. The reference context consists of the actual processes and concrete objects, but Steinbring (p. 8) notes that there in some cases are exchangeability between the reference context and the sign/symbol. This is obvious because many/most advances mathematical problems and results are formulated purely symbolically since the objects in play are only perceivable through semiotic representations. What Steinbring points out is that signs can serve as objects constituting a reference context in some cases and as representation for a mathematical concept in other.
Object, reference context
Sign/ symbol
concept
Fig. 5.3: The epistemological triangle (Steinbring, 2001).
It is an important point that signs never exist in a vacuum. Signs, defined as a relation between a material signifier that designates something, a signified, are affected by the actual context, empirical or symbolic that the signs occur in. The folowing section describes how the contextualisation of mathematical symbolism can influence mathematical work.
5.3.3 Background and focus in mathematical symbol manipulation Luis Radford (2002a) considers the interplay between the grounding or designation of mathematical concepts and concrete operations carried out on the symbols designating these objects. The context of his investigations is children in lower secondary education, working on word problems.
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He describes the flexible thinking that is needed to designate meaning to mathematical symbols and how this designation of meaning can be an obstacle and help when working with these abstract symbols. He describes this process: “Indeed, in the designation of objects, the way signs stand for something else is related to the individuals’ intentions as they hermeneutically unfold against the background of the contextual activity. In the designative act, intentions come to occupy the space between the intended object and the signs ‘representing’ it. In doing so, intentions lend life to the marks constituting the corporeal dimension of the signs (e.g. alphanumeric marks) and the marks then become signs that express something, and what they express is their meaning. The possibility to operate with the unknown thereby appears linked to the type of meaning that symbols carry.” (Radford, 2002a). In order to compare these ideas with the ideas from Raymond Duval, we can represent them in a diagram similar to Figure 5.1. The point with the diagram is that the designation of meaning is connected to the entire process consisting of both the translation from natural language to algebra and the work in the algebraic register.
Meaning
designation
Hermeneutically unfold
Natural language
Algebra Treatment
conversion
Fig. 5.4: An attempt to describe Radford’s idea of designation of meaning, compare to fig. 5.2.
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Luis Radford (2002a) describes how students interact with word problems of the type: “Kelly has 2 more candies than Manuel. Josée has 5 more candies than Manuel. All together they have 37 candies.” (Radford 2002a). Given this narrative, the students should express the number of candies that each of the children have mathematically. The students solve the task three times designating x to the number of candies the three different children have. In his essay, Radford uses the narrative metaphor of ‘heroes’ to describe the focus point of the students activities, and how it evolves when the students move from working with a story about three children that have some candy to working with solving equations. In the story, the a priori heroes are the three children. People are usually heroes in stories, and in this case what we are interested in is connected to the children in the sense that it is the amount of candies assigned to each of them. Nevertheless there is a problem in the move to algebra. The “hero” changes from being the persons (or their candies) to being the relations between the numbers of candies described in the story. While the symbol, x, is assigned to one of the old ‘heroes’. Radford makes several interesting observations. Some students translate the problem to algebra directly more or less maintaining the people as heroes. This leads to misunderstanding or miscalculation. Radford describes one example where a student erroneously translates “Kelly has two more than Manuel” to x+2=Manuel, where x is supposed to describe the number of candies for Kelly. As a narrative it actually makes some sense to translate “Kelly has two more” to “x+2” if x is Kelly. In the language of Duval this misunderstanding can be described as the student’s attempt to do a conversion not respecting the non congruent nature of this specific conversion. The narrative viewpoint, taken by Radford, reveals other problems. For instance that the designation “x is Manuel” (introduced by the teacher) strongly influences some of the children’s abilities with respect to treatment in the algebraic register. An example is one student having very strong emotional response to the innocent treatment going from: (x+2)+(x+5)+x=37 to 3x+7=37. This treatment totally collapses the narrative. In the first formula, one can see the relations between the number of candies the children have, but in the second formula this is entirely gone. (x+2), (x+5), and x becomes signs designating each of the three children. Of course it is an important and unavoidable aspect of mathematical work that you do this transformation (in this case from question to answer), but what Radford’s analysis points out is that in order to use the flexibility of the
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mathematical system, one has to be able to temporarily let go of the meaning assigned to symbolic expressions, or collapse the narrative, while maintaining the mathematical goal proposed by this narrative. The point here is that the cognitive struggle with the many representational forms in mathematics is not only an inherent feature with the semiotic systems that is in play. The meaning that students assign to symbols, and the narrative framing of the problem they work on are both crucial for understanding students’ struggles with mathematics. Radford’s work explains us several things; first and foremost that the heroes changes when the representation is changed, for instance in a conversion of register, and secondly the meaning that different persons designates to semiotic representations in mathematical discourse affects what treatments and conversions that are possible for that person.
5.4 Mathematical thinking and commutative diagrams In this discussion, I will use the framework of Raymond Duval together with some ideas from Luis Radford to describe the cognitive advantages, and challenges, of using commutative diagrams to support thinking. It should be mentioned that the use of commutative diagrams is considered hard to master in the mathematical community, at the same time these diagrams are considered very powerful for some types of mathematical work.
5.4.1 The register of commutative diagrams It is worth noting that commutative diagrams actually constitute a register, as defined in the section “Semiotic representations in mathematics”. This implies checking whether it is possible to perform treatment inside the system of commutative diagrams, and if conversions to another register reveal qualitatively different aspects of the conceptual objects involved. One example of a treatment taking place inside the systems of commutative diagrams is to state that there exists a ‘lift’ in a diagram like fig 1. That is; there exists a mapping h from A to Y, such that the new diagram commutes.
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f
X ϕ A
Y ψ
h g
B
Fig 5.5: a lift in the diagram.
That commutative diagrams reveal qualitatively different aspects of the systems of maps they describe is actually evident from the same example, since the property that a mapping is a ‘lift’ is not a natural property in a purely algebraic description of the system of maps. Actually, the lifting property has a geometric origin (see for instance Bredon, 1993) but the nature of the problem is still reflected in the more formal diagrams, and this is an example of the strength of this register. In order to study the conversions between commutative diagrams and other registers we can attempt to express the mathematical content of the diagram in another way. The relations in the diagram in figure 5.5 can be expressed as: f:X→Y, g:A→B, φ:X→A, ψ:Y→B, and h:A →Y, such that g○φ= ψ○f, and h○ φ=f. The target register for a conversion from a commutative diagram is a mixture between logical uses of natural language and symbols, and I do not think that the conversion between these two registers is congruent in either direction. The main reason is that the diagrams uses the non linear nature of a two dimensional figure very actively, and for instance there is no obvious order of the propositions in the linguistic/symbolic register, the target for the conversion could just as well be: f:X→Y, φ:X→A, g:A→B, ψ:Y→B, and h:A →Y, such that h○ φ=f and g○φ= ψ○f. Conversely, the above linguistic/symbolic expression does not show exactly how to create the diagram in fig 5.5. If commutative diagrams constitutes a register, it should be possible to express whether this register is monofunctional or multifunctional and also if it is discursive or non-discursive. Commutative diagrams comprise a register that is only used in the discipline of mathematics. In that sense, the register is monofunctional. Duval states that in a monofunctional register, most processes can be transformed to algorithms, but in the example of commutative diagrams it is not easy to see
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how. Of course, the processes we consider here can only be those that do not depend on details about the spaces and mappings that are symbolized in the diagram. The question whether the register is discursive or not is not easy to answer either. On the one hand, commutative diagrams have many of the properties of discursive registers; there are rules and syntax, and they can definitely play an important part in valid deductions, but nevertheless this register is definitely not linear. Commutative diagrams use two dimensions like algebraic notations do (fractions is one example of that) but in commutative diagrams, this two dimensional nature is in a sense more radical. A fraction can be ‘spoken aloud’ or written in a linear form in a simple, congruent form. As we saw above, conversions from commutative diagrams to a linear register tends to be non-congruent.
5.4.2 The heroes of commutative diagrams The diagrams are designed to show relations between sets or spaces. This means that this notation introduces a new hero in the study of mappings between sets or spaces. This new hero is not an element of the sets, as they are transformed by the described mappings, not f, φ, g, ψ or the sets A, B, X and Y, but the entire amount and shape of relations between the sets in the diagrams. A formal theory for describing the ‘new hero’ was developed in the middle of last century (Eilenberg & Maclane, 1945). This new theory is called “Category Theory” and takes its outset in diagrams – ignoring what the elements and arrows in the diagrams designate (Maclane, 1971; Marquis, 2004). Luis Radford (Radford, 2002a) made the good point that the experienced educational challenges from introducing commutative diagrams could be due to this change of hero. It is necessary to move back and forth between many different heroes; sometimes it is crucial to know and focus on concrete aspects of a mapping (e.g. to calculate what is f(a)), but at other times it is equally important to be able to forget such issues.
5.4.3 Commutative diagrams as a grounding context for mathematical objects Having a way to designate systems of mappings provides a new symbolic context. Following Steinbring (Steinbring, 2001) this new context should allow for new concepts to develop. Without going into a throughout historical analysis, this definitely seems to be the case. In the years after the introduction of notation based on diagrams, there was an enormous development in algebraic topology. The ability to express a large, or even infinite amount of relations, for instance between
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maps and spaces, allows new phenomena to be explored. One example is the lift from above, another the introduction of Category Theory, but there are many other examples.
5.5 Partial Conclusion In this chapter I have looked at commutative diagrams as an example of a semiotic representation used to support mathematical thinking. I have shown that commutative diagrams constitute a register as defined by Duval. This register seems to be monofunctional and discursive, but not linear. Furthermore, conversions between commutative diagrams and linear registers generally seems to be non-congruent, and therefore difficult. Using the ideas of Luis Radford, I have also discussed to what extent commutative diagrams gives a new focus in working with relations between sets and maps between the sets, and found that this new focus allow for other objects to be designated and explored.
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6 Common ground and private space; coordinating mathematical cooperation
This chapter concerns collaboration in mathematics. In chapter three we saw that one of the functions of mathematical writing in face-to-face situations was to generate a private space for one of the students to work in. In the episode analysed in chapter three, the phenomenon of conversational breakdowns was observed as a result of one student working increasingly more on his own during a meeting. Using data both from this investigation and the interview study with professional mathematicians, I attempt to go deeper into the conflicting needs to be face-to-face, and to use writing ostensively and to work privately, maybe distributed, in mathematics. I discuss issues of common ground versus the need to work alone, and I discuss whether explicit division of labour can be useful in mathematical work. Furthermore, I discuss how different media support different collaborative situations. In chapter two, I discussed to what extent computers support the wide range of representations used in mathematics. Interface-related issues are obviously an important parameter for the success of computer-supported collaborative learning in mathematics. In this chapter I want to raise another issue concerning mathematical collaboration; the issue of interplay between working together and working privately. The video-based investigations of students’ collaboration presented in chapter three showed that conversational breakdowns often occurred in connection to collaborative writing on mathematics: One student would start to work on his own leaving the others behind. In this chapter I focus on the impact of being face-to-face or not when one is performing various types of mathematical work.
6.1 Coordination and common ground Pragmatic concepts for describing coordination of collaborative processes are developed in the literature on computer support for cooperative work (Schmidt & Simone, 1996; Yoneyama, 2004, p. 59). The overall division of labor, i.e. the way that tasks and work are coordinated, is called the ‘cooperative work arrangement’. The subject of work is called ‘common field of work’ and the
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work performed to establish and maintain the cooperative work arrangement is called ‘articulation work’ (Yoneyama, 2004, p. 60). The physical situation of work affects the cooperative work arrangement, and the setting has traditionally been described as a two by two matrix (Carstensen, 1996, p. 29; Christiansen, 1997) highlighting the issues of proximity in two dimension concerned with time and place (Figure 6.1).
Synchronous Asynchronous Non-
Distributed
distributed Figure 6.1: Communicative conditions for cooperative work.
6.1.1 Common ground The theory of common ground was developed by Clark and Brennan (1993) to describe the challenges that various media poses to communication and collaboration. Common ground signifies the mutual knowledge, shared beliefs, and mutual assumptions that collaborators can share (Clark & Brennan, 1993, p. 222). In order to communicate effectively, individuals that collaborate need to assume a large amount of shared knowledge in order to be able to refer to the same concepts. Clark and Brennan describe how different media support grounding, and identify central constraints on grounding that are different for different media, and they describe how the “costs” of communication is different in different media. Using these concepts, Clark and Brennan can describe the choice of media for communication as cost tradeoffs (Clark & Brennan, 1993, p. 232), and they stress that the purpose of the act of communication plays a crucial role for this tradeoff. Generally speaking face-to-face interaction is very important for establishing common ground. For a description of the constraints and costs see (Clark & Brennan, 1993).
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6.2 Private space and conversational breakdowns The investigation in chapter three showed that writing was used both as an ostensive tool and to create a private space for mathematical work. The analysis showed that the use of writing as a private tool could challenge the students’ conversations, because one student would work more and more on his own, leaving the others behind. I described this phenomenon as a ‘conversational breakdown’. In this chapter, I ask if such conversational breakdowns can be avoided or minimized by coordinating the activities in collaborative mathematical writing. The problem in the conversational breakdown described in chapter three is that there is essentially one person working and the two others looking at this one person. I will use the interview data from professional mathematicians to determine why they consider being face-to-face valuable and what they do to coordinate their activities. It is not a straightforward task to compare the nature of collaboration across the investigations among undergraduates and professional mathematicians. The research was conducted with two very different methods; interviews, in the case of professional mathematicians, and participating in meetings in the case of students. Also the professional mathematicians have more experience than the students, and finally the physical environments of the collaborative activities are very different across the two settings, the researcher’s work in offices and the students work in an open cafeteria area. Nevertheless, the experience that the professional mathematicians typically have in coordinating collaborations on mathematics might yield useful insights into some strategies for coping with the type of problems that I have described as conversational breakdowns, assuming that the professionals experience this type of problems.
6.3 Face to face interactions and coordination in professional mathematical collaboration The interviews with professional mathematicians, reported on in chapter two, also touched on issues related to coordination of mathematical activities. In the beginning of each of the interviews, the respondent were asked to explain about a recent collaborative project, how it was coordinated and how the researchers communicated throughout this project. This narrative constituted a solid basis for the rest of the questions. The narrative was used to point out typical and non-typical features
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with collaboration and communication in mathematical research. Preferably, this narrative should focus on distributed collaboration, a criterion that originally was added to the interview guide to shed light on electronic communication throughout a collaborative project. This purpose with the interview also led most of the respondents to describe to me when and how they met face to face with their collaborators, and what purposes such face to face meetings served their collaboration. This chapter comprise a presentation and analysis of data from the narratives told by professional mathematicians. The purpose is to extract information about how the professional mathematicians coordinate their collaborative activities. What do they attempt to achieve when they meet face to face? And what do they leave to be done in a more private setting? In other words, this chapter is about the cooperative work arrangement in mathematical collaboration, focused on face-to-face meetings. What are the motives for meeting face to face? What are the professional mathematicians’ strategies to maximize the benefit of face to face meetings? Do they consider anything related to the ‘conversational breakdowns’ described in chapter three? Throughout this dissertation, we have seen how different media support mathematical work differently and this chapter ends with a discussion of how various media can support different mathematical work arrangements.
6.4 The importance of being face to face - or not All of the professional mathematicians reported that at it is important to be able to get together face to face. In the following section, we look at the way they argue why this is so. The interviews also show that the importance of being face to face is complemented with a need to sometimes work more privately. For R7, it is important to meet face to face in research projects; he describes his recent collaborative efforts mainly in terms of visits. The ideal situation for R7 is to work in the same building as his collaborator but that is often not possible. It is most important to be together at the time you initiate a collaborative project, and R7 explains that this is to create a common understanding of the topic:
R7: In my opinion it is difficult to initiate a research project without seeing each other face to face. I: Can you explain why that is?
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R7: Because when you do something, you first start your work with someone who are with you, right? I: Ok. R7: […] That is much more convenient, because anytime you want to see him you can see him, talk to him, therefore you start your work on purpose with someone who are with you […] But you can choose to have someone from someplace else, but you eventually meet sometime. I: Apart from the convenience of being able to talk, what kind of role do you think the meetings play? R7: Just by watching each other; I can explain by writing down on a blackboard to you very immediately, then he or she can catch what I want to say, but on email, he or she doesn’t get the main point of what I am going to say. So by writing is less effective than talking directly. I: Ok R7: That is what most people do. Well sometimes it happens accidentally. So for example I have seen one case, where someone sends an email with his research, completed research. And then another person in another place sees that email and then immediately realise that this result his own idea will solve. So he can immediately prove the result. In that case they don’t even see each other. But this is not really collaboration; it is a kind of accidentally initiated collaboration. But I have never seen some people try to collaborate [without meeting face to face], there is no reason not to meet. They have research grants so anytime they want they can meet each other. Because talking directly is much more effective. So I never start a collaborative project without meeting people. I: But on the other hand you described that you just met for two days. R7: And then we continue to work separately. I: And that works fine? R7: That works fine, because we have established an initial agreement. If it is a really long project, then that is not sufficient. But if it is a short project, then we only need a
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few days then we know what we want to do. Then we go back to each places and think further. In the first ten lines of this transcript, R7 describes the convenience of being able to collaborate with a person who works in the same building, the main reason why this is a beneficial situation is, according to R7, that it is possible to meet face to face when needed. He also describes that even in situations where you start working with someone from someplace else, you know that a successful collaboration requires you to meet eventually. Furthermore, R7 explains that to be co-located can be an important motivation for initiating collaboration. This means that if a person in a closely related field of study visits R7’s institution he may attempt to initiate a collaborative project. Also, if R7 finds himself in a situation where he needs a collaborator for a project idea he has, he will search for partners in his own institution first and only if he does not succeed with that he will search for external partners. R7 explains that sometimes a project can be initiated in a distributed setting, if the involved researchers already work on the same problem by coincidence. But this is the only reason he can imagine for initiating a collaborative project without meeting face to face. R7 gives several reasons why it is important for to get together face to face. First R7 mentions a communicative reason. In mathematical discussions, several media are used simultaneously, R7 phrases this nicely by saying “then he or she can watch what I want to say”. He specifically states that email communication is not effective for delivering “the main point” of a mathematical statement. The second point that R7 mentions is the need to establish an initial agreement. If R7 and his collaborator have such an agreement they are able to continue on their own. R4 also explains the importance of being together, particularly in the beginning of a collaborative effort. He describes one occasion where he wrote a paper with two people he did not meet until after the paper was written. He explained that this was possible in this case because they had already independently created the model that they investigated collaboratively. R4 explains (my translation from Danish):
R4: There is a lot more feeling each other out and discussion when you build a model that you want to analyse. Much more than when you already know exactly what problem you are attempting to analyse.
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I: And this part had been done? R4: Yes, and then we worked on, what can you say, the more mathematically hard part of the job, to do the proofs and so on. We had like, we knew how things should look, and what needed to be done. This was what we worked on. And independently we had various ideas to how to improve the models, and got a lot further than we hoped for. This was all communicated through email, mainly as exchange of LaTeX documents.
It is important for R4 to meet with his collaborator when they “build the model”. R4 explains that you need to feel each other out in order to agree on a model to analyse. The collaboration that was initiated without a face-to-face meeting was possible only because the involved parties had already worked on the same model independently. He also described that they already knew what to do, what results they wanted to prove and how, which he also mentioned as a reason for the possibility of starting a project without meeting face to face. To build the model and get an idea of results and methods, are to a large extend what in chapter two has been distinguished as ‘heuristic treatment’, but the purpose of being face to face is also to establish an agreement of what to work on and how. R7 describes a communicational advantage of being face to face. This advantage can stem from several sources. Being face to face allows for simultaneously talking and writing on a blackboard, and R7 explicitly mentions this way of communicating as useful. Furthermore, we know from chapter two that handwriting is superior to computers for generating the informal and visual representations that are used for heuristic treatment. These are good arguments that collaborative mathematical activities should be done faceto-face. We know from chapter three that it is not always unproblematic to collaborate during mathematical conversations. And the professional mathematicians do reflect on the shortcomings of face-to-face meetings in mathematics, for instance R4 goes on (my translation from Danish): R4: We have worked together since 1996, actually mostly in the same way, typically we plan to meet, especially when we start something new. I: So you will attempt to get together when you are about to begin a new project? R4: Yes, because there is a lot of messing around in the beginning when you take a problem from engineering and search for a version of that problem that allows for
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mathematical analysis. There will be a lot of throwing everything away and restarting every half hour in the beginning of such a project. When that part is over and is about to begin analysing what kind of dualities you can do, and what you can prove, can you prove convergence of algorithms, existence of solutions and so on. There is not a lot that changes here. I: And then it is ok to… R4: Then it is actually better to have some time alone and not feel stressed by someone breathing down your neck. Then there is more time to let it rest for a while. Typically it is like this; you know, I mean by experience, that when you have worked for a while together and has reached the final formulation, the one you want to work with, then you have a picture of what it is possible to do.
So for R4, the point is that when you have a good feeling about what it is possible to do, you are able to go ahead on your own, keeping a more sporadic asynchronous contact via email, fax and letters. R4 goes on to explaining that sometimes during their private puzzles, they will get to a point where there initial idea is proven wrong or just given up upon. Then they will have to find a new week and meet again. For R4 and his collaborators, travelling in order to meet face to face is a crucial part of the cooperative work arrangement in the beginning of each project. The purpose is to build the model and establish an agreement on what needs to be done. What is more surprising is that he also explains that it has great value to work alone for a substantial amount of time. He describes a feeling that someone is breathing down his neck, if he is face to face with his collaborators in a situation were he would rather work alone. The distinction is very clear to R4; together they will build the model and agree what mathematical results to aim for, and they know from experience that when they have reached the final formulation, they should split up and start working alone. In the last part of the transcript from the interview with R7 presented above, R7 talks about the need to establish an initial agreement. To get to a point were you “know what we want to do” is one of the main purposes of being together. This is not only a case of clarifying the articulation work, it is
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also a case of understanding the common field of work, and share goals and hopes for mathematical results. Almost all the respondents highlights the importance of being face to face at some point during mathematical research. Only a few say out loud that it is important to be able to work in private too. Of course, being able to work in a private setting is not a problem for these researchers, and several of the respondents points to the fact that you can easily make short informal talks as an important advantage of working within a close distance to your collaborators.
6.5 Collaboration, heuristic and control treatment Is it possible to balance the two conflicting needs of being face to face and working separately in relation to mathematical work? Taking our outset in the model describing the communicative conditions for cooperative work (Figure 6.1), we can conclude from chapter two and three that the synchronous distributed setting is typically reduced to phone conversations in the case of researchers, and is practically non-existent in the case of students. The distributed asynchronous collaboration is central to the researchers, and, as we saw in chapter two, email is a dominant medium to support this type of collaboration. In the non-distributed setting that characterises the students’ cooperative work arrangement of the students, there is still an issue of proximity. In Figure 6.1 this proximity is time wise and described as synchronous versus non synchronous collaboration. In the case of the students, the issues of synchronous versus non synchronous collaboration degenerates to a matter of being together face to face or not. And in the case of researchers we have seen convincingly that being face to face or not is an important aspect of mathematical work.
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Synchronous Asynchronous Distributed
Non distributed
Figure 6.2: Communicative conditions for cooperative work, the arrow represent the important distinction between being face to face or not.
Being face to face is of course not the same as being distributed or not, even though a distributed setting has a bearing on the ability to be face to face. In a co-located setting, the opportunity to go and sit in another room or just stop the conversation and agree on doing personal work are simple ways of supporting private work. But what I argue above is that being face to face (or not) represents a very important distinction in mathematical work. The question is, then, how this distinction relates to the functions of writing in mathematics.
6.5.1 Relations to the functions of writing In connection to the functions of writing that were proposed in chapter two and three, it is interesting to note which of these functions seems best supported by personal work or face-to-face. In a problem-solving perspective, the most interesting of these functions is the distinction between heuristic treatment and control treatment. In order to map out the relationship between these two functions of writing and the important distinction between being face to face and not, different mathematical activities are placed in a conceptual diagram having on the vertical axis the issue of proximity described as being face to face or not being face to face and on the horizontal axis the type of function - heuristic treatment or control treatment.
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Face to face Brainstorm
Critical review
Ostensive use of blackboard
Collaborative verification Ostensive calculation
Heuristic
Control Calculating
Enlightment
Reviewing
Puzzling
Not face to face
Figure 6.3: A model for collaboration.
In order to describe this model I have placed some words describing typical mathematical activities in the diagram above. The words are placed there purely for illustrative purposes, and do not represent the result of research.
6.5.2 Collaborative idea generation One thing that stands out from the interview with R4, is the need to spend time together while generating what he describes as “the model”. The model is a very fundamental thing in the research of R4 and his colleagues. The model is the object of the mathematical analysis that constitutes their research. R4 explains that the model needs to be built in a face-to-face situation because “there is a lot of throwing everything away every half hour”. The creative ping-pong that goes on when they build the model is a good reason for being together face to face. The concept of common ground does provide some reasons why it is important to get together for building the model. An important reasons that R4 gives for being face-to-face, is that there is a tendency to starting all over again, completely changing perspective etc. while building the model, this implies that common ground is established and challenged again and again.
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Face to face
Idea generating
Heuristic
Control
Not face to face
Figure 6.4: R4’s interpretation of idea generation.
R7 explains that in face-to-face situations the communication is much more immediate, which makes it much easier to communicate the “main idea”, which again can be interpreted as an issue of establishing common ground. It seems that one of the reasons for being face to face is to establish common ground (Clark & Brennan, 1993), This means to make sure that you share the same perception of the problem, not only in a formal sense, but also to share ideas, rationales and intuitions on which direction or strategy that might prove useful. Clark & Brennan (1993) has shown that it is very important to be face to face when you establish common ground, and this result fits very well with what the mathematicians say. Whether heuristic treatment works best in face-to-face situations or not is not a simple question to answer, but during heuristic treatment, the perception of the problems involved and the directions to take to solve them can change rapidly and there is usually no well-established idea of what direction one should take to solve the problem. Accepting that a well-established common ground is central for effective collaboration means that if you want to collaborate in a heuristic phase where no there is no stable common ground, then frequent face-to-face meetings are important for maintaining common ground.
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6.5.3 Need to have time to work alone In several of the interviews, the issue of being able to work more in solitude came up frequently. According to their diaries, the students did work on their own at least as much as they worked face to face. This need for individual work is not necessarily specific to mathematics. One of the most important pedagogical arguments for introducing asynchronous co-located collaborative tools (conference systems etc.) to augment classroom teaching is that these tools support personal reflection to a much larger degree than normal classroom education (Mason, 1994, Guzdial et al. 2000). The argument is that when students are given time on their own to reflect on the question at hand, they will be able to participate in a dialogue with teacher an classmates in a way that endorse learning to a much larger extent than usual classroom interaction. As mentioned in the introduction, such systems seem to work poorly in connection to mathematics. This is interesting, because the need for personal reflections is very explicitly stated in some of the interviews, and because of the problems with conversational breakdowns described in chapter three. One obvious potential explanation that was proposed in chapter two is that access to the various types of representations used in mathematics is not as seamless on computers as handwriting. R7 explains that once the initial agreement is reached, it is all right to work alone for a while. But he stresses that a longer project would require several face to face interactions. After R4 and his collaborators have built the model and agreed on what to aim for in their analysis of the model, he actually prefers to continue his work alone. He says that he sometimes feel that someone is breathing down his neck, if he attempts to do more rigid analysis while another person is present. Time is an important factor, R4 stresses that he sometimes needs to let the problem lie for a while before he can continue progress. The need to work privately seems to be more connected to control treatment than with heuristic treatment. The time-consuming and rigid work that can go into control treatment might be easier focused on when working privately, which can be visualized as in Figure 6.5.
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Face to face
Heuristic
Control
Work alone
Not face to face
Figure 6.5: The need to work alone.
It is important to realise that the need to be face to face when collaborating and the need to work in a more private setting can very well shift rapidly. In chapter three we saw that pens and paper were used by the students to establish a private mathematical working space in a face-to-face situation. And, as R7 explains, collocated researchers will have informal meetings whenever needed. These are two different strategies to cope with the conflicting needs of working face to face and privately. The need for establishing common ground does provide a relevant theoretical framing of the dilemma of being face to face and not. When common ground is established the need to be face to face is not as persistent, and a more private working environment will suit some persons better.
6.6 Collaborative aspects of different media It is one thing to determine whether heuristic treatment and control treatment is best done in face-toface meetings, but what are the requirements of the physical environment for collaboration on mathematics? In the students’ collaborative work presented in chapter three, their diaries shows very clearly that computers are only used in a private setting. In the conversational breakdown presented in the case, the ostensive use of writing that student B in particular employs is supported by pen and paper, which a priori seems like a cumbersome choice. The diagram deployed above can illustrate how the students use various media when they collaborate. The figures 6.6 and 6.7 is of a qualitative nature, but it is based on a combination of video observation and students’ diaries. The video data show that paper and, to some extent, printouts are used during the meetings. Printouts seems to be mainly used for control, while paper is used for a vide range of purposes. The diaries
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described fairly consistently that paper printouts were used in face-to-face settings and alone, whereas a computer was used solely when the students were not face-to-face.
Printout
Face to face
Control
computer
Paper
Heuristic
Not face to face
Figure 6.6: Students media use.
Figure 6.6 can be compared to a similar one based on the descriptions given by the professional mathematicians. The figure below (Figure 6.7) is derived from their reports on which media support the five functions. I have added information concerning whether it was blackboards and whiteboards or regular pen and paper that were used to support handwriting. As described in chapter two, all of the professional mathematicians had a blackboard or a whiteboard in their office, and when they reported on their working situations, at least half of them (R3, R4, R6, R7 R10 and R11) pointed, without prompting none provoked, to the blackboard as an important medium.
Printout
Control
Paper
Heuristic
Black and Whiteboards
Together
computer Apart
Figure 6.7: Researchers media use.
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The blackboards represent an important difference between the students and the professional researchers working environment: Blackboards play a dominant role for several of the researchers when they describe their collaboration, and yet this tool is virtually absent from the students’ working environment. The reason for this difference is in a sense simple; where all the researchers I talked to had a blackboard (or a whiteboard) in their office, none of the places that the students chose to work where equipped with such a device. This difference can also give a partial explanation to the ‘ostensive’ use of paper explained in chapter three, because pieces of paper serve as a blackboard substitute. Paper, and to some extent blackboards, have multiple uses. In chapter three we saw that paper was used to create a private space in face-to-face situations, and we saw that the same pieces of paper could a few moments later be used ostensively. Paper can be more flexible than blackboards. It is easy to establish a private space with pen and paper and information is in a sense stored instantly just by keeping the piece of paper. These are aspects where paper is superior to blackboards. Nevertheless, the problems with conversational breakdowns might be easier to handle if students had easier access to blackboards. First of all, blackboards would allow every one to see the written representation at once. Secondly, one could imagine that having two different media to support the private and the ostensive function would make it more transparent what the function of writing is, and hence support the articulation work.
6.7 Partial conclusion It is not at all new that face-to-face communication is important for collaborative work. But the importance of being face-to-face in mathematical work stems from a combination of the articulation work and the need to establish (and re-establish) common ground. To some extent the common ground can consist of well-established mathematical theorems, particularly in transdisciplinary projects. The main objective will often be to develop a shared idea of, and goal for, the project that they work on. In the heuristic phase of a project, face-to-face interaction seems to be very important. One possible explanation might be that in this phase there is no well-established idea of what the project is about, and a shared understanding of the project can therefore constantly be renegotiated. The potential for personal reflection are proposed by several computer supported collaborative learning (CSCL) systems does definitely seem relevant in connection with mathematical work. From the diaries presented in chapter three, we know that the students work a significant amount of
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time alone. Furthermore, the conversation presented shows problems with coordinating different pace and the need for some students to retreat to a more personal workspace. The more experienced professional researchers acknowledge that working with the more technical aspects of mathematics requires solitary work. The professional mathematicians maintain that the need to get together face to face is at some point crucial to mathematical work. With the simultaneous need to be together face to face and to be allowed to work alone one would think that a sensible implementation of conference systems, or similar computer support for communication, in connection to mathematical work and education would be highly valued. Nevertheless, some data (Guzdial et al., 2002) point to the fact that CSCL systems are not used in mathematics. There may be a complex set of reasons for that. One obvious reason is that if we look at the diagrams Figure 6.6 and 6.7, we find that both the students and the professionals use the computer solely in settings where they work alone. On the contrary handwriting seems to support the mathematical conversations in both setting, which implies that the representational flexibility that paper offers is still way beyond what computers offers. Even though tablet PCs and similar technology may soon change that, the tangibility and real estate offered by a number of sheets of paper is still superior a computer (Sellen & Harper). Stevens (2002) explains how a number of paper sheets (with architectural sketches) can be organised and reorganised continuously to support an evolving discussion. The resemblance to the discussion among students is obvious; there are very strong references to pieces of paper during the conversations among students and the paper plays a role both when the students work face to face and not. As seen in chapter three, paper is used to create a personal space in a group situation, and immediately share these personal calculations with the group. The issue of conversational breakdowns described in chapter three can be discussed in the light of the results from this present chapter. The researchers generally prefer to collaborate with people that work in the same building. The most important reason for this preference is the ability to have short informal meetings. This means that they can rapidly change physical location between being face to face and not. The ability to change physical setting rapidly is not very well supported in the work environment of the students. They often worked alone in their home and held their meetings at the university. Developing a workplace culture where the students are encouraged to do a large amount of work at the university, maybe by offering them a desk to work from, would be an interesting way to support their collaboration.
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7 Computers as media for mathematical writing: a model for semiotic analysis
This chapter presents a semiotic approach to the problem of how different media influence the writing of mathematical signs. A model for describing mathematical writing through turn-takings is proposed. The model is applied to the ways mathematicians use computers for writing and to how writing is used in connection with face to face meetings during group projects among undergraduate students of mathematics.
7.1 Introduction In chapter four we considered theories of writing processes and found that the direct interaction between mind and externalised representations is an important aspect of mathematical writing. In chapter five we saw an example on how the shapes of these external representations are crucial for the development of mathematical thoughts. In this chapter, the aim is to go deeper into the dialectics between externalizing thoughts and developing thoughts using external representations, see Figure 7.1.
Writing
Mind
Figure 7.1: writing as an interplay between internal psychic system and external representations.
On one hand the author may think something (in his internal psychic system) and these thoughts are later communicated to others through writing. And on the other hand writing, one’s own and others’, is an important part of the environment that catalyse the development these thoughts. This duality is the focal point of this chapter. I will consider the interplay between writing and thoughts as a dynamic semiotic system (Andersen, 2002). The idea is to view Figure 7.1 as a turntaking
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between two very different agents, a person (or mind) and the written representation he or she creates to support his or her thinking.
7.2 Dynamic semiotics and a model based on turntakings In heuristic treatment, control treatment and information storage, written signs are used to support the cognitive system, whereas communication has to do with communicating a message to another person and production concerns delivering work typically to a broader audience. It is a point in the studies presented in chapter two and three that writing is used to support many functions simultaneously, but that some are more in focus than others in different parts of the writing process. Furthermore, different media supports these functions differently. In order to study writing in a way that can capture this complexity of functions and media, I have looked at Peter Bøgh Andersen’s (2002) dynamic semiotics. In connection to analysing the learning potential of computer algebra systems this model has been applied by Winsløw (to appear). Peter Bøgh Andersen (Andersen, 2002) develops a dynamic semiotics in order to take into account that the meaning that individuals designate to signs are not static. He proposes a simple model for semiosis (the creation of signs) as a recursive function ‘F’ that takes input from the individuals psychological system ‘P’ and the signifiers ‘S’ that has previously been uttered. P=Minds, bodies and physical environment
F=Conversational Rules
S=Signifiers
Figure 7.2 Semiosis as perturbed recursion (similar to Andersen, 2002, figure 4.7).
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Peter Bøgh Andersen expresses this as: According to Fig. 4.7, the basic mode of operation of language is a recursive loop where previous utterances give rise to new utterances. The environment does not enter into the communicative loop, only signs can do that. (Andersen, 2002, p. 14) Peter Bøgh Andersen develops a process-oriented communication model that shows the way semiosis develops as interplay between an individual psychic system and a social syntactic system.
Individual psychic system
Se
Se
Se
Se
Se
Social semiotic system
Sr
Sr
Se
Se
Sr
Se
Sr
Se
Se
Individual psychic system
Figure 7.3: Standard turn-taking communication. ‘Se’ designates the signified and ‘Sr’ the signifier (similar to Andersen, 2002, figure 4.12).
The basic message in the communication model in Figure 7.3 is that every interpretative process builds on the interpreting individuals psychic system and the social syntactic system, and that every verbalisation process builds on the psychic system of the talking individual and the public syntactic system. This is the same as saying that the meaning of an utterance is not a fixed entity but negotiated throughout the conversation. Interestingly the exact same thing can said about semiosis in a writing process:
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Individual psychic system
Se
Se
Se
Se
Se
Social semiotic system
Sr
Sr
Sr
Sr
Figure 7.4: A monologue or writing process (similar to Andersen, 2002, figure 4.16).
In the word of Peter Bøgh Andersen, this diagram does explain why meaning can be generated in a writing process: Fig. 4.16 could for example model the writing process: The writer perturbs the genre system he is using, generating new sentences out of an old ones. He reads the new sentence, this perturbs his mind, and a new psychic state results. This explains why the process of writing can generate new ideas that would not have come forth by merely thinking. (Andersen, 2002)
The process-based communication model can be used to show the different functions of mathematical writing that are developed in this dissertation. In chapter two, five functions were distinguished; heuristic treatment, control treatment, saving of information, communication, and production, and in chapter three an ostensive function was added. In the following I have omitted the horizontal arrows to keep the model as simple as possible. The horizontal arrows highlights that the meaning of signs is dynamically generated and negotiated. If we remember this point we can omit the horizontal arrows without losing any information. If a person use a medium for heuristic or control treatment, it will look as figure 7.5. Similarly information storage could look as figure 7.6.
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Person A
medium
Figure 7.5: Turntaking in heuristic and control treatment, the media is used only to support the ongoing thinking process.
Person A
Medium
Figure 7.6: Turntaking in information storage.
The standard communication model can easily be shown as in Figure 7.7, but it is important to keep in mind that writing for communicative and production related purposes may differ.
Person A
media
Person B
Figure 7.7: The standard communication model.
The strength of this model is that it is easily able to handle several media and persons in a way where it highlights how the medium and social constellations influence the cognitive abilities of the involved. For example the ostensive function of writing can be shown as Figure 7.8:
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Person A
written
oral
Person B
Figure 7.8: Ostensive use of writing.
In this chapter, this model is used to analyse some of the data discussed in the previous chapters. I will examine and compare how formulas and diagrams are written in LaTeX and use the model to distinguish between the interactions with LaTeX that each of these representations allow. Furthermore, I will use this model to describe the conversational breakdows in mathematical conversations, described in chapters three and six.
7.3 Mathematical typesetting Most mathematicians use the LaTeX typesetting program for writing mathematics. All the respondents from the interview study in chapter two seem to rely on pen and paper or blackboards for cognitive support when it comes to diagrams or iconic types of notation. As an example, R7 wrote everthing directly into the computer, using LaTeX, but he explicitly noted that drawings and diagrams were first done by hand. In this section, I will take a closer look at how LaTeX handles commutative diagrams and compare this to how normal algebraic notation is handled. In order to do that, I will use the turntaking model to look at the way different media are involved in mathematical writing.
7.3.1 LaTeX and algebraic notation In LaTeX, you write a source code in a text editor and process it to a print file typically shown in an on screen previewer (Oetiker et al., 2002). Hence there are a priori two registers involved, namely LaTeX code in the text editor, such as the following: $$ \int_a^b f(t) dt = F(b)-F(a) $$
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and standard mathematical notation in the print file, typically shown in a previewer: b
∫ f (t )dt = F (b) − F (a) a
We can apply the idea from the last section of showing the interaction between the psychic system and the syntactic system as a turntaking diagram. In the interaction with LaTeX, we will consider the previewer and the text editor as different media and note that the text editor supports LaTeX code and the previewer gives feedback in standard mathematical notation. Hence, the interaction will look as shown in Figure 7.9, where person A thinks of some mathematical utterance, writes it in LaTeX and then relies on the previewer for cognitive support in standard mathematical notation.
Person A
Text editor LaTeX code LaTeX Previewer Mathematical code
Figure 7.9: Writing in LaTeX.
7.3.2 Diagrams In this section, we shall look at an example of mathematical notation where the translation between mathematical notation and LaTeX is complicated, namely the case of commutative diagrams. The register of commutative diagrams is described in chapter five. One of the findings of chapter five is that the conversion between the more or less linear algebraic register and the register of commutative diagrams is not congruent. If we take a closer look at R1’s working papers, it is obvious that he uses diagrams a lot in the two handwritten samples, whereas the diagrams are less dominant in the LaTeX manuscript. R1 explains that these diagrams is a type of notation that he finds it particularl complicated to write in LaTeX. Diagrams are interesting because they, as shown in chapter five, are important for
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mathematical thinking and crucial for the development of certain mathematical concepts. In this section, I will look at how commutative diagrams are expressed in LaTeX.
7.3.3 Diagrams and LaTeX The diagram in the top left corner on R1’s scrap paper (Figure 2.1) is repeated here: I
ln ∞
l* ∞
lp Figure 7.10: The diagram from R1’s working papers, and in a print version.
the diagram can be expressed in LaTeX3 with the following code: P
P
\xymatrix{ {l_\infty^n}\ar[rr]^I \ar[dr] &
& s^{\star}_\infty }
// &
l_p \ar[ur]
&
}
Here the \xymatrix command creates a diagram (in this case as a 3×2 matrix), l_\infty^n specifies the upper left entry ‘l∞n’. The arrows are determined by commands like \ar[dr] meaning arrow starting B
PB
P
here pointing one step down and one step right in the matrix structure. The commands will result in the diagram to the right in Figure 7.10: R1 explained that in the case of diagrams, he finds it cumbersome to use the LaTeX program, and if we look at the LaTeX code and the diagram in Figure 7.10 it seems probable that the diagram provides a cognitive support that the LaTeX code is unable to provide. This means that the dotted arrow in Figure 7.9 does not really provide a lot of feedback for the author. This is not just a superficial matter of layout. For instance, it is not obvious what arrows goes into a matrix entrance
TP
3 PT
Using the special package Xy-pic (Rose, 1995).
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because in the LaTeX code, you can only see the arrows going out from a matrix entrance, not the ones coming in. Bearing in mind that the translation between algebra and diagram is not exactly trivial, we see that R1 will have to think of a diagram and then translate this diagram to LaTeX and wait for the print file in order to get the visual feedback.
7.3.4 The translation between LaTeX and standard mathematical notation We may consider the translation between standard mathematical notation and LaTeX a conversion (Duval, 2000a) between a mathematical register and a LaTeX register. Figure 7.9 shows the turntakings involved in this process. Of course this translation does not occur in a vacuum; I will explain here two media that indirectly influence how mathematical signs are created in LaTeX: handwriting and speaking. Handwriting is, to many of the respondents, the preferred way to create mathematical signs for heuristic and control treatment. Handwritten mathematics is generally written in standard mathematical code. The previewer program in LaTeX also uses standard mathematical code. Another type of code it might be relevant to consider is spoken mathematics. If we look at the formula b
∫ f (t )dt = F (b) − F (a) a
and try to speak it out loud we would have something like: “The integral from a to b of f of t d t
equals capital f of b minus capital f of a”, and if we compare this sentence to the LaTeX code we see that, at least in this case, the conversion is congruent since the order of symbols is exactly the same in the LaTeX code and in the spoken mathematics. Another strong indicator of the connection between spoken mathematics and LaTeX code is the fact that LaTeX commands are sometimes used in spoken mathematical language to communicate things that does not have a precise word in normal language (see chapter two, Figure 2.9). The writing process might resemble figure 7.11. The idea is that both the spoken mathematical code and the standard written mathematical code is activated (maybe implicitly) when person A uses LaTeX for writing, and the more graphical and less linear the signs are, the stronger and more explicit is the connection to the standard mathematical code, and hence the dependence on handwritten drafts and the previewer function is stronger.
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Often congruence
Handwritten mathematics Standard mathematical code Verbal Language Oral code Person A
Text editor LaTeX code LaTeX Previewer Standard mathematical code
Figure 7.11: A model for writing in LaTeX taking spoken mathematics and handwritten mathematics into account.
In chapter five, we have seen that the spatial organisation of symbols into mathematical diagrams is crucial to mathematical work. Nevertheless, this dependence is also an important aspect of the problem with supporting mathematical writing. Since the oral register per default lacks graphics, it may be used as an inspiration for computer mediated mathematical communication. Talking is linear, just as written language. This is of course also an enormous deficiency in mathematical communication. The possible contribution in this chapter is not to suggest to leave graphics out of computer-based communication about mathematics, but the empirical investigation shows that in the case of mathematical symbols and simple formulas, users can express themselves effectively with a keyboard using LaTeX. The analysis of the interaction shows that congruence between the way formulas are spoken aloud and LaTeX code (if we view these two as registers) provide an explanation of this phenomenon.
7.3.5 Design of mathematical writing tools The analysis based on the turntaking model indicates congruence between the way formulas are spoken aloud and how they are written. This congruence allow for a linear and linguistic, and hence keyboard-friendly, way of expressing mathematical formulas. A keyboard-friendly access to mathematical symbols and formulas can be valuable for design of interfaces, because mouse based systems tends to be very slow, in practice. LaTeX is an example of a system that uses this congruence actively. But LaTeX is not the perfect mathematical writing tool. There is, for instance, no direct visual feedback in the standard mathematical register.
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It might be possible to create a more user friendly interaction with mathematical formulas that takes advantage of the congruence with spoken mathematics. One example of a system that takes the oral code seriously is the way Japanese signs are written on a Windows computer using phonetic signs. After a word is written phonetically, the computer searches for corresponding signs, and the user chooses from a list of good matches. Transferring this interaction to mathematical signs would mean something along the line of the following interaction: In order to typeset
1
∑n
in such an new interface, you first type “sum [alt]+[space]” and the menu
presented in Figure 7.12 appears as a result of a search process.
sum
∑_ ∑_
sum sum_
_
sum_^
_
∑_ _
sup
supremum
Figure 7.12: An interaction with mathematical formulas.
then press [ Æ ] and the sum sign will appear. To get the fraction structure you type frac [alt]+[space]:
frac
fraction
_ _
forall
∀
function
f(x)=
Figure 7.13: interaction with mathematical formulas.
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chose the fraction structure and type 1 and n as you would do normally, using the keyboard. The advantage of a phonetic search based system is that the mouse is not used and don’t have remember a lot of commands 100% correctly in order to achieve your goal. The analysis also pointed to types of representations where the way mathematics is spoken aloud would serve very little help, namely diagrams. My guess would be that all non-discursive registers would be difficult to access in a command based system (phonetic or not), and so a more graphical interface would be preferable for these registers.
7.4 Student collaboration and conversational breakdowns A close look at the turntakings might also capture aspects of the delicate interplay between writing and speaking that occurs in the students’ meetings. In chapter three, two important functions of writing during face-to-face meetings were observed, namely writing that serves an ostensive purpose, and writing used to create a private space for mathematical work. Both of these functions have to do with a communicational setting that simultaneously draws on writing and speaking, as shown in Figure 7.8. Ostensive use of writing can take many forms, and the turntaking model can help us distinguish between them. In Figure 7.14, two different ways of using paper while talking are identified. To the left, we see an ostensive use of paper where person A writes signs on paper both for cognitive support and in order to communicate something to person B. This ostensive use will often be supplemented by oral messages, but the paper-based representation is in focus. The second typical use of paper-based representations is for referring to mathematical objects during conversations. Here the oral conversation is supported by written representations that are referred to either by deictic words or gestures.
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Person A
paper
oral
Person B
Figure 7.14: Two ostensive ways of using paper-based representations in mathematical conversations.
The other function of writing during mathematical conversation is to constitute a private space for mathematical work, in this case the conversations degenerate to a monologue (as shown in Figure 7.4). The distinction between the ostensive calculation (the left section of Figure 7.14) and a more private working space can be unclear as for instance in node 33 to 38 in chapter three, where writing is first used privately (node 34) and explained ostensively just afterwards.
Student B
paper Calculates (34)
oral
Well how are we going to construct it (33)
What if we, couldent you just take on for each, we have a lot , …. (35)
I think that if you takes this and put a_n equal the limit down here … (36)
Student A
Student C
Figure 7.15: A turntaking model of the students’ interaction.
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Because the sequence will get very close to every one of the a_n (37)
Let me write it down (38)
Figure 7.15 gives some explanation as to why mathematical conversations can be challenged by the private space generated in node 34. In node 36 and node 37, student B attempts to explain his idea to the two others, but he never really succeeds with this and gives up in node 38; the conversational breakdown seems to be a reality soon after. The above example shows that it is not a simple matter when a student is working in a private space, and wants to include the rest of the group in his findings. The reason proposed using the turntaking model is that the meaning generated in the private space (node 34) is difficult to communicate to the rest of the group. In the case above, student B unsuccessfully tries to communicate his findings, maybe because he is more interested in moving on with his own idea. This shows that it is not always easy for the student that has worked in the private space to include the others in his thoughts afterwards.
7.5 Partial conclusion This chapter has presented a semiotic approach to the problem of using computers for creating mathematical signs. The focal point has been how turntakings can provide insights to the use of LaTeX to write mathematics and to how written representations influence mathematical conversations. The semiotic approach has proven valuable in the sense that it has provided a possible partial answer to why mathematicians find LaTeX appealing, even though the LaTeX code looks so different from standard mathematical notation; the appeal seems to stem from a proposed congruence between verbal mathematics and LaTeX code. This connection also provides insight into why some types of notation are more conveniently written in LaTeX than others: The types of notation that are difficult to ‘utter’ because of their structure are also difficult write in LaTeX. Even though LaTeX is the dominant typing system among researchers and college students, it is not realistic (or advisable) to use this program in primary and secondary education, but I believe that the model for describing mathematical writing in different media could be useful in analysing other programs for writing mathematics. Furthermore, I believe that the analysis presented here points to an interesting, but so far neglected, potential in connection to mathematical writing on computers. If the connection to spoken mathematics provides a linear, and hence keyboard-friendly, way of interacting with computers, it
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could be interesting to investigate whether such ideas could be embedded in graphical user interfaces for mathematical software.
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8 Conclusion
This thesis has investigated mathematical writing empirically and theoretically. In chapter one I argued that a closer look at the activity mathematical writing might give valuable insights on to how to design better computer systems to support mathematical writing. In chapter two we saw empirically that mathematical writing serves five functions; heuristic treatment, control treatment, information storage, communication and production. Heuristic treatment and control treatment are concerned with supporting mathematical thinking, in the process of generating and verifying ideas. Heuristic treatment is a function that is concerned with generating ideas and proposing solutions and connections between abstract entities. Control treatment is concerned with verification and is characterised by precision and rigid mathematical deductions. Information storage is a matter of keeping ideas and arguments in order to access them later and communication is conveying a message to someone (typically a collaborator). Production is the function of writing as a way to deliver the end product of ones’ mathematical work, as a report or a research paper. In chapter three, we saw that two functions can be added during mathematical conversations, namely the use of writing to generate a private space and the use of writing as an ostensive tool. During conversations, writing can be used to generate a private space, which is not part of the conversation. This way of using writing is typical for the mathematical conversations I have studied. It is very practical to concentrate on this private space that consists of oneself and ones’ writing, for instance during control treatment. When parts of the mathematical insights are generated in a private space, it may be difficult to maintain the flow of the mathematical conversation. In the case analysed in chapter three, the students’ conversation degenerates in such a way that only one student contributes to the conversation. The rest of the students are eventually unable to participate in the conversation, and hence in the problem solving process, even though the leading student attempts to include them. Ostensive use of writing in mathematical conversations is writing used to show something to someone while talking to that person. Ostensive use of writing happened very often in the conversations I observed. The fact that all of the professional mathematicians that I have interviewed had a blackboard, and used it, also support the idea that writing is often used ostensively in mathematical research.
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These seven functions of mathematical writing are a result of an open investigation and are as such grounded in data. Of course the functions are not to be considered a final and disjoint classification of the mathematical writing process. The functions are results of my research and signify important clusterings that I have found in the broad landscape of what writing is used for in mathematics. All of the individual writing processes that I have investigated shows a movement from heuristic treatment and control treatment towards production, but throughout the dissertation we haw seen quite a few ways that these functions affect each other. In chapter two the data highlights close connections between some of the functions. Six of the eleven respondents consider heuristic treatment and control treatment separated processes. For the rest of the respondents, the distinction is not as sharp, and for one respondent the two are not different processes. Both information storage and communication are connected strongly to control treatment for some of the respondents. This connection mainly goes for respondents who either share or save their work as handwritten paper files. Others of the respondents have a strong connection between production and information storage or communication, they tend to save their work as electronic files. In chapter four, the description of mathematical writing that these functions provide is compared to alphabetic writing processes. Two important trends in theories of writing processes are discussed, namely writing as a rhetorically driven problem solving-process, and writing as a process of discovery. One might say that writing that follows the idea of rhetorically driven problem-solving relies on the writer’s ability to put himself in the place of the reader as a source for generating new insight. Hence an important parameter for the knowledge generation is the cultural code and rules for expressing thoughts in writing that the writer subscribes to. On the other hand, the potential that discovery writing provides is related to a very direct interaction between paper based representations and the author’s internal cognitive system. You simply ‘see’ connections and associations in a different way when you are in the process of writing. It is a reasonable hypothesis, proposed in chapter four, that the empirically generated categories of heuristic treatment and control treatment follow these two types of writing processes. Heuristic treatment is a matter of generating new ideas and seeing new connections. Discovery writing does precisely attempt to explain how writing can support spontaneous knowledge generation. Control treatment is characterised by precision and strongly dependent on the cultural mathematical code, in the form of rules and notation. To what extent the empathic ability to put oneself in the place of the
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reader is important for the control treatment remains unanswered by this dissertation, but the close connections that were shown empirically to exist between control treatment and communication or information storage could be explained by control treatment being (partly) a rhetorically driven problem-solving process. In chapter five, it was demonstrated how the nature of mathematical thinking is influenced by the signs that one uses to represent the mathematical concepts. The shift between different semiotic representations of the same mathematical object is a central transformation in mathematical work, entitled conversion of registers. In such conversions, not only the visual appearances of a mathematical object changes. The scope of possible treatments (in Duvals sense of the word) can change dramatically; to provide an example, I show that the lifting property of a commutative diagram of maps between spaces is not as natural a property if the set of mappings between spaces is presented in a (linear) standard algebraic register. Following an idea from Luis Radford, I use the metaphor of hero to describe the special aspects of mappings between spaces that commutative diagrams endorse. This new hero is the entire system of possible mappings between the spaces involved. Historically, one can trace the development of commutative diagrams to occur alongside a number of mathematical theories that has systems of mappings either as object or central tool. Hence I argue that the semiotic representations can influence, not just the way one works with and learn mathematics, but also the very development of mathematical disciplines. In chapter six, the role of face to face interactions in mathematical work is discussed. Looking at the reasons that the professional mathematicians give for being face to face, we find that there is a tendency to plan for being able to meet for initiating a research project. Furthermore, the researchers describe that later in a project it is typically all right to work more alone. The ideal situation for researchers seems to be co-located because it then is possible to have spontaneous meetings whenever needed. I examine to what extent the need to be face to face correlates with heuristic treatment, and conversely if control treatment typically is done more privately. The concept of common ground is introduced, and using this concept it is discussed whether the need to be together face to face is mainly a result of a need to establish, re-establish or negotiate common ground. It make sense to assume that the need to establish common ground is more pertinent in heuristic treatment than it is in control treatment.
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The collaborative aspect of different media is discussed. Empirically, paper and blackboards seem highly useful for face-to-face collaboration, whereas computers are not really used face to face. Two conclusions are drawn concerning media; blackboards would be relevant for the students in order for them to be able to avoid conversational breakdowns, and handwriting is superior to computers when it comes to supporting the face to face discussion on mathematics. In chapter seven I take outset in the concept of dynamic semiotics and a turn taking model for communication to describes mathematical writing. The model provides a simple framework that allows me to analyse situations where several media and persons are involved in simultaneous acts of communication. Using this model, I analyse mathematical writing with the LaTeX program. Considering the shifts between LaTeX code, standard mathematical code and the way mathematics is spoken aloud as conversions of registers, the model provides a tool for describing the difference between writing formulas and writing diagrams in LaTeX. Chapter two have provided empirical indications that diagrams are much less convenient to write in LaTeX than formulas, and the analysis shows that a likely reason for this difference is that formulas are much more congruent with the way mathematical representation is spoken aloud. LaTeX uses this congruence to provide convenient access to mathematical symbols and formulas, something which is not possible in the case of diagrams. The turntaking model is also used to look at how ostensive use of writing and writing to support the generation of a personal working space are intertwined in a way that can create conversational breakdowns. The complexity of the turntaking structure in face-to-face meetings provides a reasonable explanation why this kind of communication seems better for establishing common ground than for instance email or phone conversations. Using an example from the episode presented in chapter three, I show how it is difficult for the student who has just worked in a private space to explain his line of thinking to the others. This difficulty should be compared to this student’s desire to continue his work in the private space, that he finds promising. The main contribution from this dissertation has been to develop empirically grounded categories that describe the various functions writing serves in mathematics. The categories have made it possible to discuss a number of aspects of mathematical writing. An important insight from these discussions is the flexibility that pen and paper provides for mathematical writing and the strong
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dependence that mathematical writing continues to owe this medium. Several of the functions of mathematical writing were almost exclusively supported by pen and paper or blackboards. Another insight is the fact that most of the researchers considered there to be a substantial difference between heuristic treatment and control treatment. The dissertation provides convincing results that these two ways of supporting mathematical thinking are relevant to consider. For instance, more than half of the respondents were be able to classify a typical piece of working paper according to which one of these functions it supports. The fact that writing is used ostensively in mathematical conversations may not be exactly surprising, and neither is the fact that people write things for themselves while discussing. But the fact that these two functions merge so intensely and the amount of communicative problems, exemplified with conversational breakdowns, and possibilities, for instance for establishing common ground, that this merging creates is surprising. On the theoretical level, the main contribution of the dissertation rests in my attempt to view the mathematical writing process as a creative writing process and as a mathematical problem solving process simultaneously. Considering the ideas of discovery writing together with the framework on conversion of semiotic registers in the turn taking model does provide a new theoretical perspective on the design and use of digital mathematical writing tools.
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1 Appendix A: Assessment and Contract-like Relationships in Undergraduate Mathematics Education Niels Grønbæk, Morten Misfeldt and Carl Winsløw
1.1 Introduction. It is commonly acknowledged that assessment procedures influence university students’ activities in many ways. At a very global level this influence can be seen as simultaneously inevitable, necessary and regrettable. Inevitable because universities need to deliver credible diploma and hence must assess and declare their holders’ competencies in depth; the fact that this declaration may be decisive for the students’ future career opportunities is, at a global level, an important reason for students to direct their study activity towards maximising their ‘declared competency’ (or, more modestly, to just get their diploma). It is necessary much in the sense that (outside paradise) explicit reward is sometimes necessary to get people and hence society to work. This may be true not least for studies with many technical and difficult parts located more or less necessarily at the beginning of the curriculum: the necessity to do these parts (in order to achieve the target diploma) acts as a default motivation for students who do not themselves acknowledge their attraction or necessity. Finally it is regrettable because the existence of such a default incentive to study may seem to suspend the need for other rationalities for teaching and study, and hence reduce academic teaching and studies to something highly un-academic: work based solely on control and rule following. Obviously one would like to minimize the regrettable effects of assessment, while retaining a visible incitement for students to meet necessary work requirements as well as a credible declaration of the results of this work.
If we acknowledge the impact of student assessment on student activity during a course, and that students’ activity is the root of their learning outcome, then we are led to conclude that the design of student activities and student assessment should be looked at as a whole. Very roughly one might think of the following pattern of influence (indicated by arrows):
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STUDENT ACTIVITIES
DIDACTICAL
LEARNING
DESIGN
OUTCOMES
ASSESSMENT Figure 1.
These arrows are not all of the same nature. In particular the influence of assessment on student activity tend to be indirect, and to depend crucially on students’ images and expectancies of assessment. In order to actually investigate the nature of the effects of assessment – and even more, to investigate how to control them – we must of course consider contexts where these effects occur. In fact, these effects are not easily observed. As we shall explain (§1), they can be modelled as arising from a specific sort of ‘contract’ between students and teachers, strictly conditioned by the context (in particular the institutional and academic context). However, in absence of interventions this contract may be largely implicit and impossible to distinguish from other factors influencing the interaction of students and teachers, such as the conditions and possibilities offered by the subject to be learned and taught. As a consequence, beyond general remarks such as those professed in the first paragraphs of this introduction, we find it necessary to consider the effects of assessment from an engineering point of view (cf. Artigue, 1994): in concrete contexts, with specific aims of control and perhaps change. It is a main point of this chapter to show how such an approach can be theoretically supported by the theory of didactical situations, with appropriate adaptations for the university level.
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After introducing the relevant theory in §1, we describe the context from which our data come (§2), in brief: a format for didactical engineering which is based on collaborative student work on written “thematic projects”, subsequently assessed through an individual oral exam. We then explore two aspects of this context: the independent work of student groups (§3) and the learning outcomes as evidenced by the written projects (§4). Both are analysed with respect to the influence of the assessment format, including the perturbation of the ‘contract’ a new format leads to. We briefly discuss (in §5) the potentials and limitations of local (single-course) changes in the contract underlying the mechanisms in Figure 1.
1.2 Didactical contracts in university mathematics The theory of didactical situations (hereafter abbreviated TDS), developed by Brousseau and his school since the 1970’s, has chiefly been used in various contexts of school mathematics, such as the famous studies of the teaching of decimal numbers (Brousseau, 1997, chap.3-4). However, the key features of TDS (ibid, chap. 1-2 and 5) seem highly relevant to university teaching as well, and it is indeed used in some studies in this context (cf., for instance, Artigue, 1994). In this section, we shall try to outline some basic points of TDS that will allow us to talk meaningfully of didactical contracts in university teaching, and to understand their special features in this setting.
1.2.1 The notions of situation and game The basic assumption of TDS is that learning is a modification of a student’s knowing which she must produce herself and which the teacher must only instigate (Brousseau, 1997, 227). This is, at
face value, just the constructivist point of view: personal knowledge (termed ‘knowing’ to translate the French word connaissance) results from various types of action in a situation, which somehow presents challenges to the learner. However, modern mathematics teaching – not least in the university context! – implies that students must learn much more than is possible through spontaneous interaction with the world. Piagetian developmental psychology is mainly interested in showing the potential of the learner, at various ages, to make sense of quantities, motion, basic logic, and so on – using just elements of his immediate experience. Indeed, Piagetian theory is not concerned with teaching or other forms of institutionalised learning. On this background, TDS presents three original features, which are also crucial for the study of university teaching. The first originality resides in insisting on the epistemological value of situations for teaching: they can and must be arranged by the teacher with didactical intentions in order to achieve learning of
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some specific knowledge. Such adidactical situations – prepared and ‘instigated’ by the teacher – are achieved when the teacher withdraws to let the student(s) act. But the epistemological assumption is still stronger: Each item of knowledge can be characterized by a (or some) adidactical situation(s) which preserve(s) meaning: we shall call this a fundamental situation (ibid.,
p. 30). A main objective for didactics is, in this perspective, to construct and investigate the use in teaching of situations corresponding to target knowledge such as the mastery of decimal numbers. The second originality of TDS is to model the highly complex interplay between the teacher, the student(s) and the various forms of situations that allow the student to construct knowings and knowledge, and so to relate the epistemological and social dimensions of learning in a concrete as well as specific manner. The adidactical situation is considered as a game of the student with a milieu for learning, arranged and rearranged in such a way that the ‘winner strategies’ successively
approach the knowledge aimed at. The teacher is not just conceiving of the milieu in order to subsequently step back and watch the students’ failure or success in the game. The teacher interacts with the students’ game while presenting it and modifying it in different phases. This interaction is
called a didactical situation (ibid., p. 31). This is a larger form of game, involving the teacher as a crucial player. The process through which the teacher ‘hands over’ a milieu to the students is called devolution; it is supposed to achieve the students’ acceptance of the adidactical situation. The
phases of the didactical situation may be roughly classified as follows (ibid., pp. 8-18, 65-71, 231235): Devolution of adidactical situations of action (more or less ‘pure play’) in which the students
explore parts of the milieu in a rather immediate way; Devolution of adidactical situations of formulation, where the students are urged to articulate
observations from their game with the milieu; the teacher may occasionally intervene in order to let the students clarify their statements; Situations of validation, where the students and the teacher consider the explicit statements
(representing knowings) about the game with the aim of deciding their validity; Situations of institutionalisation, in which the teacher emphasises the validated knowings as
common knowledge. Notice that each of these phases may be considered as different sorts of games, with different objectives and rules, and with the players assuming different roles. A classical situation illustrating
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this is the so-called ‘race to twenty’ (ibid, chap. 0), conceived for elementary school; the method (for university ‘lectures’) of scientific debate developed by Marc Legrand (2001) is clearly conceived with a similar pattern of games in mind. Notice, however, that real time presence and interaction of teachers and students are clearly assumed in the didactical situations as described. It is a model of classroom teaching in a wide sense. But it is not per se a prescriptive model, as the case of university lectures show; in their usual form, these are mainly situations of institutionalisation but do not require that the knowledge communicated is already established as validated knowings for the students. The third originality of TDS is to relate the work of the teacher to that of the mathematician, which of course takes on a special meaning in the context of university teaching, where the two may be the same person (although research and teaching does not, typically, concern the same knowledge): Mathematicians don’t communicate their results in the form in which they discover them; they reorganize them, they give them as general a form as possible. (…) The teacher first undertakes the opposite action: a recontextualisation and a repersonalisation of knowledge. She looks for situations, which can give meaning to the knowledge to be taught. (Brousseau, 1997, p. 227)
This means that we have two almost opposite transpositions of knowledge: from situation (of discovery) to formalised, official knowledge, produced by the mathematician; and from such official knowledge to didactical situations (including milieus and games) that enable students to acquire the knowledge. Of course the situations that are devoluted to the students do not by any means need to be equal or even similar to the historical situation of discovery! But it is the close connection, if not similarity, of adidactical situations and mathematical discovery, as well as these complementary processes of knowledge transposition, which is the source of Brousseau’s consistent insistence on the institutional and intellectual inseparability of mathematics and its didactics (see e.g. Brousseau, 1999). In the most immediate sense, all mathematicians are practitioners and consequently connoisseurs of didactics as applied to mathematics (ibid., p. 44).
1.2.2 The notion of didactical contract and its paradoxes As was noticed above, the didactical situation in itself may be considered as a game, where the players are the teacher and the students. In order for it to succeed, the student must learn, and to do so, must accept the responsibility in the adidactical situation devoluted by the teacher. On the other hand, the teacher is also responsible for the success of the students’ game with the milieu, by
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designing it, by intervening or modifying it if necessary, and by evaluating it. In general, any teaching-learning situation implies – and requires – a set of mutual obligations between teacher and student. In a sense they become players (teacher, students) in a didactical situation through the establishment of these obligations. Without the basic willingness of the teacher to help the students
learn, and of the student to engage in the intellectual work proposed by the teacher, there is no didactical situation. As we all know, good intentions are far from sufficient to teach and learn mathematics efficiently. And the obligations of students and teachers must be understood both with respect to the complexity of didactical situations, outlined above, and with respect to the high-stakes nature of mathematics learning in many contexts (such as university). Moreover, those obligations are mostly implicit and a considerable uncertainty about them may subsist for both parties. Nevertheless, this system of mutual obligations resembles a contract (ibid., p. 31) and indeed one does speak of a didactical contract. As many other forms of regulation of social systems, it appears only – or at
least most obviously – when it is broken. And, surprisingly, breaking the contract is also a condition for learning to take place (ibid., p. 32). While the contract is certainly necessary for devolution to succeed, it must not rule the students’ game with the milieu: the student’s answer [to problems posed by the milieu] must not be motivated by obligations related to the didactical contract but by adidactical necessities o her relationships with the milieu (ibid., p. 57). Thus, the didactical contract
is the root of certain paradoxes linked to devolution and adidactical situations: after devolution and acceptance of the situation, everything that [the teacher] undertakes in order to make the student produce the behaviours that she expects tends to deprive this student of the necessary conditions for the understanding and the learning of the target notion (ibid., p. 41). It may be necessary for the
teacher to intervene, on pain of leaving the student to fail (in which case both will have failed to honour the contract). But if this intervention reduces the students’ task to little or nothing, the formal fulfilment of the contract leads to a meaningless ‘empty’ milieu where learning cannot take place. Indeed, didactical situations – especially those devoluting a very rich milieu – are ‘instable’ systems’ in the sense that small perturbations, sometimes out of the teacher’s control, may change success into failure.
1.2.3 University mathematics contracts It is important to retain that knowings achieved in adidactical situations have to be established and made common through situations of validation and institutionalisation. So, teaching needs both to
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transform official mathematical knowledge into adidactical milieus and to transform the knowings obtained there back into shared knowledge compatible with the starting point. However, to achieve this personalised learning as a basis for institutionalisation is costly and difficult, as suggested above. Therefore: There is a strong temptation for the teacher to short-circuit these two phases and to teach knowledge directly as if it were a cultural fact, thus saving the cost of this double manoeuvre. The knowledge is presented and students make it their own as best they can. (ibid., p. 227)
Indeed, this is what happens in most university mathematics courses – at least, to some extent. More precisely, a very common format is to institutionalise new knowledge in lectures (cf. Weber, 2004 for typical case) and then devolute milieus in the form of problems and exercises that require the knowledge and which are meant to help the students “make it their own”. Subsequently, a kind of formulation and validation situation takes place during class sessions. We shall call this rough scheme of teaching the lecture-problems-class model (LPC for short). In some course contexts, the problems on which students get to work are all somewhat stereotypic applications of the theoretical material. These problems are meant to establish, for students, certain associated techniques – rather than a knowing of the full theory (cf. also Winsløw, 2005). In this case the ‘repersonalisation’ of knowledge, particularly the more theoretical parts exposed in lectures, will still largely be left to students. Notice that we want to take TDS seriously as a descriptive rather than as a prescriptive model of teaching. We do believe that the four phases of a didactical situation, described above, are meaningful also in the university context; but there is no a priori reason to claim that they should necessarily appear in the order which seems to be the most natural and efficient for elementary school teaching (according to the massive experimental work done by Brousseau and his team). In particular, university students should not necessarily be fed with meticulously arranged situations of learning, but must also learn – as part of the academic trade – to access knowledge directly in its official, depersonalised form. But this is, indeed, something to learn, rather than a starting condition which can just be assumed. As suggested by several authors (cf. Weber, 2004, p.131), there could be many other rationalities behind the use of LPC than those linked to didactical finalities. But its massive use also shows that LPC offers a certain balance and stability with respect to the interests of institutions, students and teachers. Indeed, with respect to students and teachers, the LPC scheme demands – as any other
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didactical scheme – a didactical contract to establish and regulate the didactical situations involved. Parts of it are linked to assessment – which, in a university course, is rarely more than a few months away. Here is, tentatively, what the main clauses could look like, if made explicit:
1.2.4 Didactical contract for LPC-scheme (outline). Obligations of lecturer: clearly expose and explain the theory and examples required to do the problems assigned for homework. Make sure that doing the homework assigned will substantially help students towards passing the final exam (written or oral). Obligations of students: follow the lectures attentively, read the corresponding texts. Do the assigned homework. At all times, ask questions if something appears to be unclear. Obligations of classroom instructor: validate students’ answers to problems. Institutionalise good solutions and answer questions of students. Examination requirements do not necessarily dominate the contract. On the contrary, the tacit assumption that the situations devoluted by the lecturer are ‘relevant’ also with respect to these requirements, may help to put these in the background most of the time, and hence to allow the student’s work to be, some of the time, driven by ‘adidactical necessities of her relationships with the milieu’. Of course a condition for such a silent consensus is that the contract above is actually observed, in particular that the work proposed by the teacher actually does help the students towards succeeding at exams. Now, the most common forms of examination – requiring, from the students, a few hours work on a collection of exercises, or a short oral presentation of textbook theory – may not motivate autonomous, in-depth work with mathematical theory according to the contract above. Our basic analysis is therefore as follows: if we want to improve the quality and scope of students’ adidactical work, we may need to modify simultaneously the exam requirements, the milieus worked on by students, and the role of the teacher with respect to devolution and institutionalisation. All of this is likely to require amendments of the contract. A major challenge is therefore to avoid that the logic of the target adidactical work is replaced by the logic of (searching for) a new contract. The establishment of the contract and its disappearance (in the adidactical situation) must be kept under strict control.
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1.3 A didactical engineering project: thematic projects in real analysis In the following sections we describe and discuss the evidence of contractual relationships and their evolution during a didactical engineering project, carried out in the context of a second year course in real analysis at the University of Copenhagen. The necessity to simultaneously consider assessment, the nature of student tasks, and the form of teaching was at heart of the engineering, which we will briefly introduce in this section (it is described in much more detail in Grønbæk and Winsløw, to appear).
1.3.1 A difficult course The course in question, Mathematics 2AN, is a third semester course in real analysis crediting 10 ECTS. There are two weekly double lectures (2x2x45min) and one weekly problem session of 3x45min. It has an enrolment of about 175 students. In several respects the course is central in the overall study plan. It is the first course where students meet mathematics in full rigor; it is a prerequisite for many other courses; it addresses students with a wide spectrum of ambitions, talents, and time available for the course; and the syllabus is demanding (roughly speaking, it is Chapters 1-11 and 15 in Carothers, 2000). The course is assessed by means of a written 3 hours exam and an individual ½ hour oral exam. All of above are externally imposed boundary conditions on which a course designer has very little influence. The first author had taught the course for three years. As in other courses of the programme, the didactical obligations were, roughly speaking, regulated by the LPC contract. The course was suffering from a number of functional problems, such as a large dropout and low passing rates. It appeared that some of these shortcomings were rooted in students’ amount and quality of independent work with course material and that the LPC contract was a part of the problem in the sense that it did not allow for different and more explicit demands on student activity. Some early attempts had been made previously to introduce project work and peer assessment in the course, but this had not resulted in significant improvements of student work. With hindsight, one could say that the LPC contract continued to be in force, mainly because the changes in assignment of student work were not reflected in accompanying changes of teaching and assessment, but also because the teachers did not succeed to make their motives and expectations clear to the students.
Coherent changes in student activity, teaching and assessment
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In order to address the above concerns, we developed a new format that we call thematic projects (cf. also Grønbæk and Winsløw, 2005). A thematic project may be described as a written assignment consisting of mathematical tasks, some of which are quite open, and most of which are more complex and theoretical than standard ‘training exercises’. The tasks are centred on a ‘theme’ (e.g. the Hilbert space L2) so that the solutions, together, represent a coherent piece of theory; in this sense, a thematic project is, ideally, a kind of fundamental situation. Indeed, it can be considered as an adidactical milieu, its formulation being the devolution. The adidactical work implies, due to the open nature of the tasks, situations of action and formulation as well as situations of validation (integrated in teaching to some extent, cf. below). The students work on the thematic projects in groups. The subjects of the thematic project were chosen in accordance with the progression of the course, so that the students could work with the projects successively and throughout the semester. In order to support this work, parts of the class teaching (normally used for presenting solutions to training exercises) were devoted to work with the thematic projects, with the instructor available for questions. The lectures were partly changed in order to relate the presentation of theory and examples to the projects; and the lecturer was also available for questions related to the projects. For each project, a date was set where the students could hand in their work in order to get written feed back from the instructor, mainly of the form “this is good/OK/needs to be worked on”. The thematic projects of the course (six in total) replaced the traditional textbook presentation in the oral exam: instead of drawing a random “theorem” to present (as in the book), the students draw one of their thematic projects. Notice that the exam is individual, while the projects were worked on in groups. This had important consequences for the regulation of the students’ work: each group member was required, in the end, to be able to explain and defend the common projects. It was explicitly said that the product of the work on thematic projects was not required or supposed to be an extensive report, but just 4-5 pages that could serve as background for the students’ presentation and the examinator’s interrogation (the written product had to be delivered to the evaluators at the time of examination). However, we shall in the following still refer to these written products as ‘reports’.
1.3.2 The adidactical contract of the groups A main point of all this was to change the work with theoretical parts of the course from merely acquiring the text book presentation to produce and formulate theory in projects, thus aiming
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deliberately at a repersonification-redepersonification process. This organisation of the work in groups was meant to amplify situations of action and formulation. It turns out to be an important factor that the material goal of these situations is the production of a usable manuscript for the oral examination. Also, the thematic projects include tasks, which are optional and open, in particular tasks where the students must choose a level of ambition (for instance generality) in their interpretation of the task. The purpose of this is two-fold. It gives the student the opportunity to demonstrate their competencies without the risk of ‘breaking their back’ on sophistication. It also requires the students to be conscientious about the status of their knowledge; “knowing what I know” is not exclusively an issue of metacognition but is a crucial and advanced form of knowledge in mathematics. It becomes a part of the groups’ work because this work is the basis for the oral examination. All these conditions for the groups’ work, and in particular its relation to assessment, imply that the groups must negotiate a common, internal contract for their work with the tasks. We shall call these (implicit and explicit) agreements of mutual obligations within the group of students the adidactical contract of the group. It is clearly framed by the didactical contract and the didactical situation
surrounding the adidactical work, but it is also, as we shall see in Section 3, proper to the group.
1.3.3 Modifications of the didactical contract The introduction of thematic projects, as described above, is based on a few explicit and official changes in the assessment procedure (mainly for the oral examination) and in the way the theoretical knowledge is taught. But its consequences for the didactical contract, regulating the mutual obligations of students and teachers, are more far-reaching. We shall exemplify and illustrate this in Sections 3 and 4; in the following, we summarise the general tendencies and the relation to the LPC contract. First of all, the written exam is still there, and so the LPC contract remains in force for the work on more elementary and technical tasks. The introduction of the thematic projects implies certain amendments of it, which address the more theoretical work that is assessed at the oral exam. These
amendments concern potential conflicts in the new design, mainly coming from relating students’ independent work and self-awareness so closely to the final assessment. On the one hand, the students are required to let go of the usual check marking from teachers, and rely on their own judgement (e.g. in choosing their level of ambition). On the other hand, their work will be the basis of an exam, whose criteria is beyond their influence and experience. These conflicts are in part of
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an affective nature, due to fear of being “trapped” and uncertainty about the requirements. This makes it very important that elements of the didactical contract are declared rather directly, in order to avoid uncertainty and misunderstandings. The students still assume, by default, the LPC contract. Accordingly the amendments have the nature of an allonge. In return for students’ willingness to accept the proposed new format and the working mode, the course organiser declares: willingness to change the plan when students object (reasonably); intentions to follow closely how things are developing at ground level; intentions to assess students according to the circumstances under which their results are developed (rather than comparing them to (fancier) textbook versions); detailed descriptions of formal requirements (at the beginning of the course). Obviously, these declarations must be followed by practice. As an example of 1): in the first run of the course with thematic projects, it became apparent that the workload was too heavy. This led to allowing each student to discard one project for the oral exam (thus drawing among 5 instead of 6). This is (also) reasonable because it gives the students an opportunity of adapting the examination to their individual capacities, and represents yet another task of self-assessment. The contract amendments concern the teaching directly. The lecturer must relate the course material with thematic projects in order for the course to become an integrated whole rather than a collection of isolated themes. Also, both classroom instructors and the lecturer offer guidance to students in their project work, and it is explicitly stated that they will not provide complete or partial solutions. Their task in this context is to help students in the phases of formulation and validation. The written feed back on the project notes must work as a situation of institutionalisation – it turns out to be crucial for the students that they can really regard it as such. Likewise, the final assessment must respect the genesis of thematic project outcomes, and ideally the winning strategies in the devoluted games should correspond to winning strategies in the context of assessment. In most exams the grade is obtained as the result of a synthesis of two opposing principles: a subtraction process where one counts down from an ideal, thus penalising the student for flaws an errors; and an addition process, where the student is credited for achievements. In the classical oral exam the evaluation is based on a comparison of the student’s presentation with a ‘perfect’ presentation from a textbook. This will almost automatically give priority to the
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subtraction principle. When the student is evaluated on basis of his/her own product, the examinator must focus on the addition principle. This point requires deliberate attention. Whereas the change in the lecture format and its relation with the contract are rather straightforward, the assessment aspect of the contract is much more complicated. It calls for a great deal of trust from the students and corresponding responsibility from the teacher. (Can I trust that NN does give me credit for my rather weak version of theorem XX, knowing that there are much more powerful versions?). Hence it is important that these elements of the contract are discussed during the semester. It can even help establish an ethos of the course.
1.4 Students at work: adidactical situations or contract following In this section we describe what goes on when the students work on the thematic projects between classes, without the presence of the teacher or lecturer. How do they organise their work? To what extent do the students follow a didactical contract and can their work be described as adidactical situations? These questions will be addressed by looking more deeply at the students’ collaborative activities in connection to the thematic projects.
1.4.1 Investigation method and goals The second author followed two groups of students, and investigated the students’ work using a combination of informal interviews, video observations, and diaries kept by each of the students. The purpose of the diaries was to obtain an overview of how different activities connected to the group work were situated during a specific week. During this ‘diary’ week, in the middle of the project period, all participants in the two groups were asked to keep a detailed record describing what project tasks he or she worked on, and when. Furthermore each student was asked to describe the type of activity engaged in, explain if the work was done alone or in collaboration (if so, with whom), and describe how many pieces of paper had been used and kept (if they used a computer for writing they would write that). Furthermore the diary had a field that allowed for free reflections about the activity reported on.
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Figure 2: a diary opening. The diaries were accompanied by video observations of working meetings. A total of seven meetings of various lengths (about six hours of video in total) were recorded. The researcher was in frequent contact with the two groups of students, mainly by finding them in the area where they usually worked (a university canteen). This frequent contact served several purposes. The contact was necessary to gain access to the students working meetings, because the students typically saw each other several times a day, and hence were able to change schedule for meetings with a very short notice, and they often did that. The contact also allowed a better longitudinal picture of the activities by allowing frequent informal interviews with the group members. This proved to be a valuable source of information, and finally it helped to build trust between the researcher and the group of students. The videos were summarized, and interesting parts identified and transcribed, using the Transana program for video analysis (Fassnacht & Woods, 2003). For more details about the methodology for collection and analysis of data, see (Misfeldt, 2005).
1.4.2 Overall impressions from the meetings The students in the two groups met in general once or twice a week to discuss their projects, with increasing intensity close to the deadlines for instructors’ comments. Roughly the meetings were concerned with (1) dividing the labour, and (2) discussing solving strategies, and (3) actual problem solving activity, and (4) reviewing previously written work. In interviews the students point to (1)
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and (2) as the main reasons for having the meetings but (3) and (4) also occurred during all the meetings observed. The didactical contract appeared explicitly from time to time in these activities. When dividing the labour the students often referred to an explicit contract that divided the questions into kernel questions and other questions. This distinction plays at least two roles for the students, first and foremost it provides a prioritisation of the tasks: it is clear that the students considered the kernel questions most important to solve. But there was also an underlying assumption that the questions that were not kernel questions had to be very difficult, and hence perhaps above the students’ abilities. The students seem to ignore them as a consequence of the didactical contract. The contract was also present from time to time when the students were discussing solving strategies and when they were more deeply engaged in problem solving activities. The contract can pop up suddenly as we shall see in the following episode where three students (from “group two”) are working with accepting or rejecting a number of propositions and support their claim with either a proof or a counterexample. The first three propositions have been rejected with counterexamples, and we enter the conversation when they are brainstorming heuristics for the fourth. A possible counterexample is proposed when student B gets nervous:
18 student 2b: Shouldn’t any of this be true? 19 student 2a: yes 20 student 2c: this can’t be right, I think this is a little too easy. 21 student 2b: we were sure about this one, yes that was the argument from before 22 student 2b: can it be that these are the ones “name of the instructor” talks about? 23 student 2c: hmm I don’t think so 24 student 2b: yes it must be them
There are several things in this quote that shows that the didactical contract is on the mind of the students. Firstly, the concern that “there should be some of the propositions that are true” is instantaneously supported by the two other students, and we interpret this utterance as meaning “the teacher would not have asked us to prove or disprove if we only have to disprove”. Secondly, once
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the contract is introduced in the conversation the students instantaneously reflect on what the instructor said the other day. We have seen this type of contract pop-ups many times when observing meetings amongst students. They tend to be short and do not in general guide the students’ investigations, but as this quote shows the students are aware of the fact that their mathematical work is governed by a didactical contract and occasionally they make use of the logic of the contract as a part of the heuristics for their mathematical work. In this case it contributes to their solution of the problem.
1.4.3 An example of an adidactical contract Apart from the didactical contract the groups developed what we called an adidactical contract, regulating their collaborative work. Where the didactical contract is fulfilled by students learning, this adidactical contract is fulfilled by the product the students make together. In this section we describe an example of such a contract. In “group one” there was a special division of labour implying that one of the student was responsible for writing an electronic version of the groups work. The two other students were on the other hand more or less responsible for solving the mathematical problems. If we take a look at a schematic résumé of their diaries, we see that the preparation of electronic documents is a very time consuming activity. For each person and each day during the week the schema shows how much time he or she has used at the university (above the line) and at home (below the line) and furthermore the letters P and G signify respectively that the work was done personally and with the group. The diaries and meetings showed that student 1a’s personal work had to do with preparing an electronic version of the document they where working on. On Monday in this specific week, the group collaborated for an hour and 25 minutes on a task about fixed points and, in the evening, student 1a spend an hour and a half typing the results of their efforts. Wednesday student 1a handed over his electronic version for review by the other students. Thursday student 1a received a commented version of the manuscript and the same evening he finishes the report on fixed points by doing the proposed corrections. The two other students also work at home on Thursday but they are preparing the next thematic project on homeomorphisms. Friday they discuss this thematic project for three hours and in the
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weekend student 1a works six hours alone on making sense of their discussions and preparing a preliminary electronic version of the report.
Person Monday 1a
G. 1:25h
Tuesday
Wednesday
Thursday
Friday Saturday
G. 0:15h
G. 0:30h
G.
P. 0:20h
3:00h
G.0:15h
G. 0:30h
G.
P. 0:50h
P. 0:30h
3:00h
P. 0:30h
G.
G. 0:30h
3:00h
P. 1:30 h
1b
1c
G. 1:25h
G. 1:25h
P. 3:30h
Sunday
P. 2:30h
P. 1:00 h
P. 0:30 h
P. 2:00h
Table 1: a schema of the diaries from group one. If we compare the hours that each of the students put into the group work we see that in addition to the meetings student 1a uses seven hours and 50 minutes working on the thematic projects whereas student 1b uses two hours and 20 minutes and student 1c uses two and a half hour. What kind of contract governs the students’ collaboration here? And how does it relate to the didactical contract? The diaries and our observation and communication with the group suggest that the students’ adidactical contract attributes the responsibility for the content oriented work in the thematic projects to the students 1b and 1c, whereas the responsibility for writing it up electronically is given to student 1a. One can ask if it is desirable that one of the students only does the typing, and if the didactical contract should be changed in order to avoid such a division of labour. It is worth noticing that this division of labour seems to be rooted in a shared wish to “make a nice report” rather than in a concern to ensure their learning outcome. If the work was purely dictated by the didactical contract it should have been a concern for the students to make explicitly sure that
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everyone could do all the tasks. Nevertheless this division of labour is in our opinion a nice example of the way that this new design supports adidactical situations in a very balanced way. The function of student 1a is not reduced to that of a typist. Indeed he does participate in the meetings where solutions and strategies for solutions are discussed and he does not receive a perfect manuscript for his preparation of electronic version; typically he brings partially finished drafts of solutions developed by the other students home. That the situation of formulation is a significant aspect of the mathematical work is also well known. Obviously the adidactical contract is different for different groups of students. For example “group two” (the other group that we followed) took turns on the task of writing an electronic version of a document.
1.5 Evidence of contractual understandings in students’ written reports A typical exam enrolment results in approximately 200 thematic project reports. This section is based on a perusal of 80 reports from the exam in January 2004.
1.5.1 Overall characteristic features Two characteristics of the reports are prominent: they are quite similar in form but also very diverse in detailed contents.
All of the reports (save those for one group) are more extensive than required (cf. Sec. 2), in fact they are written as a full text rather than in the style of a synopsis. The mathematical argumentation is detailed and far more extensive than what it is possible to present at the oral exam. In one case a group even included an appendix to incorporate material that was not accounted for in the body text. In a thematic project about “Homeomorphisms”, one needs to use a certain result from the textbook. About one third of the perused projects give a detailed account of this result, such that an uninformed reader in principle can start almost from scratch. In short, the students have adopted the genre of a textbook or lecture notes. Also, as observed in from Sec. 3, typesetting, graphics etc. have constituted a substantial workload. Almost all reports are carefully designed in terms of lay-out, in some cases even with a fancy graphical design of the front page. In a thematic project entitled ‘The Cantor Set’, the students are asked to give at least 2 different representations of the Cantor Set. Most project groups choose ‘removing-middle-thirds-of-intervals’ as one of these representations, and illustrate the process
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graphically in their chosen word processor (usually TEX). Such an illustration is time-consuming to type and hardly difficult to remember at an oral presentation, thus seemingly superfluous in an exam synopsis. One may speculate why the groups so massively chose to produce such elaborate reports. The most immediate hypothesis would be that the formal requirements (i.e. explicitations of a part of the didactical contract) were somehow unclear to the students. However, these were discussed with the students at several ‘question lecture sessions’, and it was repeatedly emphasised that the reports should just be a support for the students’ oral presentation. A more important cause – expressed e.g. during focus group interviews with students in another run of the course – seems to be that the groups want to demonstrate, internally and externally, that their work is really ‘complete’ mathematics of the type found in text books but elaborated by them. In fact, the internal functions may be more important. As suggested by the observations in Sec. 3, the written report is an important element in the adidactical contract: the work is organised around its production, and it is an important ‘mutual obligation’ to ensure that everyone in the group will be able to use this product at the exam. Idiosyncratic ‘notes’ or ‘memos’ would not serve this purpose. The report becomes the end and means of fulfilling the adidactical contract. The other striking feature is that the project reports are truly different from group to group. Of course, there are often just a few essentially different ways of attacking a problem, but these can be varied in the detail. For example, the uncountability of the Cantor set is proved either (1) by a direct diagonal argument on one of two representations of the Cantor set as binary sequences (left/right intervals or 0-2 ternary expansions), (2) by using a binary sequence representation and referring to the cardinality of {0,1}N, which, being a power-set, the textbook informs is uncountable, (3) by referring to the Cantor function as a surjection onto an uncountable set, (4) by proving that a nonempty perfect set must be uncountable. Only approaches (2) and (3) can be derived directly from the textbook. In fact, among the 80 reports perused, we found no two reports which are not distinct in their detailed formulation and approach. It seems fair to conclude from this that the groups have worked autonomously. While the homogeneity in form of presentation, discussed above, is related to similar adidactical contracts (and, perhaps, to misunderstandings among students about the formal requirements), the diversity in contents can only be interpreted as a result of the adidactical work in the groups. This is amply confirmed by student interviews, observation of student work (Sec. 3) etc.
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Specific features
The thematic projects were designed with several possibilities for the students to choose their own level of ambition and technicality. Different formulations of problems made it possible to conclude answers at a level appropriate for the individual student. One way to do this is asking questions in a progression: (a) give a conjecture, (b) substantiate your conjecture, (c) prove as much as you can of your conjecture. For example, in the thematic project ‘Homeomorphism’, the students are given two (among other) statements about a bijection f between metric spaces M and N: (i) f is a homeomorphism, (ii) K is a compact subset of M if and only if f(K) is a compact subset of N. One group claims that (i) is equivalent to (ii), proves that (i) implies (ii) and sets out to prove that (ii) implies (i), but concludes with the remark: MISSING!!! Have not yet proven this in a satisfactory way. Apart from the optimist tone, this demonstrates confidence in the explicit part of the contract
(partial answers will also be honoured). The collected answers to the questions (a)-(c) range from mere unsubstantiated guesses to full proofs of the statement. Another mechanism is application of a theorem or concept in various degrees of difficulty and complexity. In the thematic project ‘Fixed Points’ the students may apply Banach’s fixed point theorem in some different settings. The ‘low-level’ setting consists of finding an approximation to the fourth root of 2, and the ‘high-level’ consists of showing the existence of an implicitly defined function. Both ways require full understanding of the statement of Banach’s fixed point theorem and insight into the proof. In the thematic project ‘Interchanging limits’ the students are asked to interpret known theorems as a statement about interchanging limits. Some are rather straightforward whereas others are somewhat obscure. In both of these projects the reports are distributed among all possible levels of ambition, thus demonstrating that they rely on the contract’s stipulation that their reports are accounted for on a personal level and that this will be respected at the exam. As adidactical situations are learning situations in which the teacher has successfully hidden her will and interventions as a determinant of what the student has to do (Brousseau, 1997, 236), it is
not surprising to find that sometimes that the teacher’s intentions with thematic project work can be in conflict with the effects of summative assessment. In the project Homeomorphism the students are asked to decide whether two given metric spaces are homeomorphic. Some groups respond to this by exhibiting explicit maps, which, as a very simple inspection shows, cannot possible be homeomorphisms. Yet the groups declare boldly that these maps are easily seen to be of the desired
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type. In the project Fourier Series the students work with a version of Dini’s test. Some technical calculations are necessary for the conclusion. There are several examples of ‘innovative’ rules to achieve the goal. In other words, the logic of the contracts (demands from within and from without the group that it should produce definite results) supersedes, in these cases, the logic of the adidactical milieu.
1.5.2 Institutionalising meaning When working on thematic project the institutionalisation of knowledge is to some extent left to the groups as the teacher is not present. Moreover, certain concepts are essential for the students’ understanding but are not in themselves objects of knowledge. The student may have ‘constructed a meaning’ but can it be institutionalised? Or at least depersonalised? Sometimes one must rely on convention established by common use. The negotiations of the adidactical contract, concerning ‘how much detail should be given in our arguments’ (in the report), is an example of this. Construction of meaning may also depend on mental representations of mathematical objects. The need for institutionalisation is perhaps clearest when such a representation is wrong or insufficient. One group trying to describe the Cantor Set by means of removing middle thirds writes: Even though one cannot continue to draw the corresponding picture, this procedure can be carried forward ad infinitum.
Later, when trying to prove that the Cantor Set is uncountable by means of LR-sequences (L for choice of left interval, R for choice of right) they write: By a descent in the diagram (‘corresponding picture’ above) we imagine that we choose, step by step, between L and R. Somewhere far down in the diagram we find the points of the Cantor Set.
Even though this is a misconception, the students refer (almost) correctly to the Nested Interval Theorem, which would be unnecessary if an argument as the above was valid. One sees here a conflict between an incomplete situation of formulation and a misguided attempt to follow the contract (‘the teacher expects us to quote the theorem at this point’). The institutionalisation of knowledge has partially failed.
1.6 Conclusions In this chapter, we have tried to theorise and exemplify the relations mapped out in Figure 1. In the context of university teaching where a significant part of the construction and institutionalisation of
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knowledge is delegated to project groups, it has been shown that there are two sorts of contractual relationships in play: the usual didactical contract, concerning the relation between teacher and students, which is needed to achieve the devolution and acceptance of the assigned work; but also an adidactical contract among the students in a group, developed by them and regulating their different roles in achieving the aims of the group. The didactical contract is an arrangement between teacher and student that is successfully fulfilled by student’s learning. On the contrary the adidactical contract is between the students and in this case fulfilled by the product the students deliver, rather than their learning outcome. In this sense the contracts live side by side. The didactical contract, as it is proposed by the lecturer to the students, is explicit on the demands for the written product to be not an extensive report but merely a synopsis to support the oral examination. Nevertheless the students developed an adidactical contract that implied investing a lot of energy into the elaboration of the report, sometimes by assigning the responsibility for this to one student. We have discussed the possible reasons for this result, but one can also question whether it is something one should try to avoid. A complete text rather a synopsis can be a necessary part of depersonalisation and shared validation for the students. A shared validation of the mathematical target knowledge for the thematic project will benefit from a fuller and less personal form of written product. A very short synopsis is essentially a personal tool and the more detail the report contains the more easy it will be to share the report and gain shared confidence of its correctness. The adidactical contract governing the collaborative work, is securing that all students participate and it enforces a mutual commitment among them to accomplish the required work. The fulfilment of the adidactical contract is also needed because the students have to bring the reports to the final exam and give them to the evaluators. The strength with this specific kind of evaluation is that both these contractual aspects are evaluated by the final exam. The students have to achieve a mathematical product together, but they also have to show individually what they learned. The development of contractual relationships within one course must be seen in relation to “normal” forms of contracts with which the students are familiar. Changing the form of assessment in a single course cannot be expected to result immediately in a retreat of the LPC contract and other well-established modes of conceiving the relationship between students and teachers. But as similar forms of work and assessment are increasingly introduced in other courses within the university (cf. e.g. Rump & Winsløw, this volume), one may expect that the associated
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renegotiation of the contract will no longer be just a local problem of a particular course such as the one we have considered.
References
Artigue, M. (1994). Didactical engineering as a framework for the conception of teaching products. In R. Biehler et al. (Eds.), Didactics of Mathematics as a scientific discipline (p. 27-39). Dordrecht: Kluwer. Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer. Brousseau, G. (1999). Research in Mathematics Education: Observation and ... mathematics. In: I. Schwank (ed.), Proceedings of CERME 1, European research in mathematics education p. 35-49. Osnabrück: Forschungsinst. für Mathematikdidaktik. Carothers, N. L. (2000). Real Analysis. Cambridge: Cambridge U. Press. Grønbæk, N. & Winsløw, C. (to appear). Developing and assessing specific competencies in a first course on real analysis. Research in Collegiate Mathematics Education. Grønbæk, N. and Winsløw, C. (2004). Thematic projects: a format to further and assess advanced student work in undergraduate mathematics. Manuscript, submitted. Legrand, M. (2001). Scientific debate in mathematics courses. In D. Holton (Ed.), Teaching and learning of mathematics at university level. An ICMI study (p. 127-135). Dordrecht: Kluwer.
Misfeldt, M. (2005). Conversations in undergraduate students collaborative work. Paper presented at the Fourth Congress of the European Society for Research in Mathematics Education, Sant Feliu de Guíxols, Spain. Weber, K. (2004). Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course. Journal of Mathematical Behavior 23, 115-133.
Winsløw, C. (to appear). Research and development of university level teaching: The interaction of didactical and mathematical organisations. Proceedings of CERME-4 (4th European conference on mathematics education research, Sant Feliu de Guixòls, Spain, 2005.
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2 Appendix B: Publications During my work as a research student I have been involved in writing a number of research papers. These papers has been (or are currently being) published as conference papers, research reports, journal articles or book chapters. Many of these papers have been used as input for the dissertation and as such only chapter four and six consist mainly of non published material. The list is organised after type of publication. A brief sum up of the relation between the publications, in the list below, and the dissertation would look something like: The motivation and methodological considerations presented in the introduction, is unfolded further in [4]. Chapter one is to a large extend written for this dissertation, but contains parts of [1] and [8]. The content of chapter two has previously been published as [1], [8], [9] and [13]. Chapter three is based on [3] and [5]. Chapter four consist of unpublished material, chapter five is based on [10] and to some extend [2]. Chapter six is mainly unpublished but has a minor overlap with [8]. Finally chapter seven has previous been published as [7].
Journal Articles
1. Misfeldt, M. (2005). Media in Mathematical Writing. ‘For the Learning of Mathematics’. 25(2) July 2005. Book Chapters
2. Ejersbo, L., Misfeldt, M. (to appear) Matematik og rationalitet (Danish). Invited submission to T. Schilhab et al. (eds.), Den nye Neuropædagogik. Akademisk Forlag. 3. Grønbæk, N., Misfeldt, M., Winsløw, C. (to appear) Assessment and contract-like relationships in undergraduate mathematics education. Invited submission to O. Skovsmose et al. (eds.), University science and Mathematics Education. Challenges and possibilities. 4. Lisser R. Ejersbo, Robin Engelhardt, Lisbeth Frølunde, Thorkild Hanghøj, Rikke Magnussen, and Morten Misfeldt. (to appear) Balancing Product Design and Theoretical Insights. Invited submission to Kelly et al. (eds.) Design Research in Education.
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Conferences with full peer review
5. Misfeldt, M. (2005). Conversations in Undergraduate Students Collaborative Work. Presented at the CERME 4 conference, Sant Feliu de Guíxols, Spain. 6. Magnussen, R. and M. Misfeldt (2004). Player Transformation of Educational Multiplayer Games. Proceedings of Other Players conference, Centre for Computer Games Research, IT-University, Copenhagen. 7. Misfeldt, M. (2004). Computers as Media for Mathematical Writing: a model for semiotic analysis. Plenary address in Topic Study Group 25, on Language and Communication, ICME 10, DTU Copenhagen. Available from http://www.icme-10.dk/. 8. Misfeldt, M. (2004). Mathematicians Writing: Tensions Between Personal Thinking and Distributed Collaboration. Proceedings of COOP, the 6th international conference on the design of cooperative systems, Nice, Supplement p.49-65. 9. Misfeldt, M. (2003). Mathematician's Writing. Proceedings of the 27th Conference on the Psychology of Mathematics Education, Hawaii. vol 3. p 301-308. Other scientific publications
10. Misfeldt, M. (2005). Semiotic Representations and Mathematical Thinking: The Case of Commutative Diagrams, in Didactics of Mathematics the French way. Preeprint collection, HCØ tryk, University of Copehagen. 11. Buch, T., Magnussen, R., Misfeldt, M. (2004). Research report ITMF-projekt 460, KompetenceUdviklende Matematik Spil (Danish). Learning Lab Denmark, Copenhagen 2004. 12. Magnussen, R., M. Misfeldt, et al. (2003). Participatory design and opposing interests in development of educational computer games. Electronic proceeding supplement. Level Up, Utrecht University, DIGRA 13. Misfeldt, M. (2003). Media in Mathematical Writing: Can education learn from research practice? The Nordic pre-conference for Icme 10, Vaxjö, Distributed at Conference. Broader non scientific publications
14. Misfeldt, M. Moser, T. (2005) Matematik+krop=10-4? Er der pædagogiske gevinster ved at forene undervisning i matematik og idræt? To forskere fra Learning Lab Denmark
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stillede spørgsmålet til en gruppe seminarielærere (Danish). Learning Lab Denmark Quarterly. (2005#4). 15. Misfeldt, M. (2003). Matematik er for kompliceret til computere (Danish). Learning Lab Denmark Quarterly. (2003#2). 16. Misfeldt, M. (2003). Ph.d. projektet Computerstøttet samarbejde og matematik (Danish). Forum for matematikkens didaktik (1) 2003. 17. Misfeldt, M. (2002). Speciale i matematikundervisning: om at fundere kursusudvikling i et teoretisk begrebsapparat (Danish). Mathilde 12, Marts 2002 p. 22-23.
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