LEARNING MATHEMATICS FOR THE WORKPLACE: AN ACTIVITY THEORY STUDY OF PIPE TRADES TRAINING by Lionel N. LaCroix B. Mus., University of Western Ontario, 1983 B. Ed., University of Toronto, 1985 B. Sc., McMaster University, 2000 M. A., University of British Columbia, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Curriculum Studies) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July, 2010 © Lionel N. LaCroix, 2010 ii ABSTRACT This study examines a single pipe trades pre-apprentice within a one-on-one impromptu tutoring session making sense of fractions-of-an-inch on a measuring tape within the context of a pre-apprenticeship program for the pipe trades. The multi-semiotic analysis of this event is framed using cultural-historical activity theory and Radford’s theory of knowledge objectification. From these complementary perspectives, mathematics is considered a culturally situated purposeful activity. Specifically, mathematics learning involves a cultural-historical, socially, and semiotically mediated process of objectification, (i.e., a process in which one becomes progressively aware and conversant, through one’s actions and interpretations, of a cultural logic of mathematical objects). The analysis focuses on the pre-apprentice’s and tutor’s joint activity during this encounter, drawing on video data and various artifacts used. This entailed slow-motion and frame-by-frame analysis of the video to assess the role and coordination of various semiotic systems, actions, and artifacts. Particular attention is paid to: the semiotic system of cultural signification, norms of practice, contradictions or conflicts that serve to motivate this activity, specific objectives of or sub-goals in the learning process for this student, semiotic processes used both by the student and tutor in the objectification process, as well as changes to the subjectification of both the pre-apprentice and researcher-as-tutor in this process. This analysis informs Radford’s theory of knowledge objectification by showing, through fine-grained analysis, relevant aspects of its dynamics and by calling attention to a new form of iconicity and a process of semiotic extraction, both original contributions to research. It also shows various ways in which a learner’s subjectification is evident in the process of learning mathematics. The results have a number of practical implications for the teaching of mathematics generally, and mathematics for the workplace in particular, by drawing attention to the social, iii cultural, historical, and mediated dimensions and dynamics of mathematics learning activity. The findings also illustrate the complexity of learning to measure by identifying a number of processes and conflicts involved and practical ways these are negotiated or resolved. iv TABLE OF CONTENTS ABSTRACT .............................................................................................................................. ii TABLE OF CONTENTS .......................................................................................................... iv LIST OF FIGURES ................................................................................................................ viii ACKNOWLEDGEMENTS.........................................................................................................x DEDICATION.......................................................................................................................... xi CHAPTER ONE: INTRODUCTION..........................................................................................1 Background for this study ...........................................................................................................1 Purpose of and rationale for this study.........................................................................................5 Research questions......................................................................................................................7 Research setting and data sources................................................................................................7 Limitations of this study..............................................................................................................9 Organization of this dissertation ................................................................................................10 CHAPTER TWO: REVIEW OF THE LITERATURE ..............................................................11 Activity theory ..........................................................................................................................11 Historical roots of activity theory ..............................................................................................12 Vygotsky and the theoretical foundations of activity......................................................14 Leont’ev and activity theory ..........................................................................................18 Engeström and cultural-historical activity theory ...........................................................22 Critiques of activity theory........................................................................................................26 Radford’s theory of knowledge objectification ..........................................................................28 Mathematical thinking and the role of culture ................................................................31 The major parts of the theory and their interrelations .....................................................32 Mathematical objects and objectification .......................................................................34 vMathematical thinking as a dialectical cultural reflection and subjectification................36 Semiotic means of objectification ..................................................................................37 Methodological implications of the theory of knowledge objectification........................39 Research on workplace mathematics and mathematics learning for the workplace.....................39 Summary...................................................................................................................................43 CHAPTER THREE: RESEARCH METHOD/OLOGY.............................................................45 Methodological implications of activity theory for research ......................................................45 De/constructing activity.................................................................................................47 Study design .............................................................................................................................50 Background to the present study ....................................................................................51 The pipe trades pre-apprenticeship program, data collection, and data selection.............53 The research subject and research setting.......................................................................57 Data analysis.............................................................................................................................58 The role of the researcher in this learning activity .....................................................................62 CHAPTER FOUR: DATA ANALYSIS I.................................................................................64 The mathematics related activities within the pre-apprentice program .......................................64 C learning to measure with a measuring tape.............................................................................66 Significant mathematical threads in C’s learning throughout the tutoring session...........67 Setting the scene............................................................................................................68 Vignette one: Discovering C’s difficulty measuring with the measuring tape .................69 Vignette two: Noticing the different division patterns on the measuring tape .................72 Vignette three: Exploring of C’s understanding of the relative sizes of fractions ............81 Vignette four: Starting to read different types of fractions on the measuring tape...........82 Vignette five: Understanding fractions per inch and fractions as intervals .....................89 vi Vignette six: Using the smallest intervals on the measuring tape as benchmarks ............91 Vignette seven: Using quarters as benchmark fraction intervals .....................................92 Vignette eight: Becoming more fluent with the number of unit fractions per inch ..........94 Vignette nine: Using thirty-seconds and quarters as benchmarks....................................98 Vignette ten: Becoming proficient measuring with the measuring tape ........................100 Vignette eleven: Using the measuring tape in a new way .............................................103 A summary of the generalizations that C enacts during the tutoring session .................105 CHAPTER FIVE: DATA ANALYSIS II ................................................................................109 C’s activity system..................................................................................................................109 Mediation within C’s activity of learning to read a measuring tape ..............................113 The semiotic system of cultural signification...........................................................................117 Beliefs about conceptual objects ..................................................................................117 Conception of truth......................................................................................................118 Methods of inquiry ......................................................................................................118 Legitimate ways of knowledge representation..............................................................119 The territory of artifactual thought ..........................................................................................119 Semiotic means of objectification ................................................................................120 Iconicity in C’s process of objectification.........................................................120 L’s use of semiotic nodes .................................................................................126 C’s use of semiotic nodes and semiotic contractions.........................................130 Cultural-historical semiotic contractions and semiotic extraction......................137 C’s and L’s subjectification during the tutoring session ...........................................................139 C’s subjectification during the tutoring session ............................................................140 L’s subjectification ......................................................................................................144 vii Contradictions within this activity ...........................................................................................146 Chapter summary ....................................................................................................................148 CHAPTER SIX: CONCLUSIONS AND IMPLICATIONS ....................................................149 The nature of mathematics activity within this pre-apprenticeship program.............................149 The activity of learning to read fractions-of-an-inch on the measuring tape..................150 Contradictions within the activity............................................................................................153 Significant processes in mathematics learning not yet addressed by activity theory or TO.......154 Implications for teaching.........................................................................................................154 Implications for teaching students how to read a measuring tape or a ruler ..................155 Implications for teaching mathematics for the workplace.............................................156 Implications for education policy ............................................................................................158 Implications for further research..............................................................................................158 Further development of the theory of knowledge objectification ..................................158 Avenues for further research on mathematics learning for the workplace .....................159 Applying the theory of knowledge objectification within other cultural contexts..........159 Epilogue..................................................................................................................................160 REFERENCES .......................................................................................................................161 APPENDIX I: TUTORING SESSION TRANSCRIPT ...........................................................170 Transcription codes .................................................................................................................170 APPENDIX II: UBC BEHAVIOURAL RESEARCH ETHICS BOARD CERTIFICATE......202 viii LIST OF FIGURES Figure 1. Vygotsky’s model of a complex mediated act............................................................15 Figure 2. The structure of an activity system ............................................................................23 Figure 3. Radford’s model of the theory of knowledge objectification......................................35 Figure 4. Change in smallest increment size on the measuring tape at twelve inches.................59 Figure 5. C points to the first division past 11 inches................................................................70 Figure 6. C points to the second division past 11 inches ...........................................................70 Figure 7. C points to the third division past 11 inches...............................................................70 Figure 8. C points to the fourth division past 11 inches.............................................................70 Figure 9. C sweeps down the fifth division marking past 11 inches ..........................................70 Figure 10. L sweeps up measuring tape with his left index finger .............................................73 Figure 11. L pauses with index finger at the 12 inch mark ........................................................73 Figure 12. L points to the 12 inch mark with his left fourth finger ............................................73 Figure 13. L sweeps up from the 12 inch mark with his fourth finger .......................................73 Figure 14. C sweeps up the measuring tape with his left fourth finger ......................................75 Figure 15. C makes chopping motions with his left hand ..........................................................75 Figure 16. C points to the 12 inch mark with his left fourth finger ............................................75 Figure 17. C points at the 12 inch mark now with his right index finger ...................................76 Figure 18. C begins sweeping a wide interval up from twelve inches........................................76 Figure 19. C continues his wide interval sweep ........................................................................76 Figure 20. C grasps the measuring tape at 12 inches ..................................................................76 Figure 21. C sweeps upwards while holding his grasping gesture ..............................................76 Figure 22. C makes a very brief narrow-interval gesture...........................................................77 Figure 23. The set of rulers on transparencies............................................................................83 ix Figure 24. L aligns the eighths transparency for C....................................................................86 Figure 25. C holds out four fingers ...........................................................................................96 Figure 26. L turns his right palm upwards ................................................................................97 Figure 27. L aligns paper with the 10 inch mark .....................................................................104 Figure 28. L sweeps his hand up the measuring tape from the 10 inch mark ...........................104 Figure 29. The activity system within which C is learning to measure ....................................114 Figure 30. C points to the region of the measuring tape above 12 inches ................................133 Figure 31. C then points to the region of the measuring tape below 12 inches.........................133 Figure 32. C begins a chopping gesture ..................................................................................133 Figure 33. C ends the chopping gesture begun in the previous figure......................................133 Figure 34. C makes a chopping motion with his fourth finger.................................................133 Figure 35. C’s left hand at the apex of a small hopping gesture ..............................................134 Figure 36. C’s left hand at the bottom of the small hopping gesture........................................134 Figure 37. C makes a pointing gesture in the air .....................................................................134 Figure 38. C before raising his eyebrows................................................................................143 Figure 39. C with eyebrows raised..........................................................................................143 Figure 40. Actions and operations enacted within each minute of the tutoring session ............145 Figure 41. A redesigned tool for teaching how to read fractions-of-an-inch ............................156 xACKNOWLEDGEMENTS First and foremost, I wish to express my gratitude to my wife Lynne and daughters Celeste and Elise who not only allowed me to uproot them from our home, friends, and community to move 4500 km to Vancouver, but supported me at every step of the way. Second, I wish to acknowledge the guidance, support, and encouragement provided by my supervisor, Dr. Stephen Petrina, and the other members of my dissertation committee: Dr. Lyndon Martin and Dr. Dan Pratt, and especially Dr. Luis Radford who served as my mentor as I became an activity theorist. And third, I wish to acknowledge the support and encouragement of my dear friend and colleague Dr. Doug Karrow. The successful completion of my Ph.D. program and this dissertation would have been much more difficult, if not impossible, without the help that each of these individuals provided. Many other individuals contributed in a variety of different ways as well as I journeyed through my Ph.D. program at UBC. These include: Dr. Susan Pirie who guided me through the early part of my Ph.D. program; Sue Grecki and Lynda Fownes from SkillPlan, our collaborators on the UBC Workplace Numeracy Project; George Stenning, Graham Young, and all of the students with whom I worked in the pre-apprenticeship training program that was the focus of my research; Bill Evans, Dale Pfaff, Karl Lutsch, and the other piping department instructors at BCIT; Bob Hapke, Dr. Jennifer Peterson, Dr. Katherine Borgan, my other graduate student colleagues, the faculty members, and the other support staff within the Department of Curriculum and Pedagogy at UBC; as well as Dr. Joyce Mgombelo, Dr. Hilary Brown, and my other friends and colleagues in the Faculty of Education at Brock. Last but not least, I would like to acknowledge the support provided by my mom and dad, Margaret and Leo LaCroix, and our dear family friends Dr. Tim and Rita Schouls. Thank-you all! xi DEDICATION To my wife Lynne and my daughters Celeste and Elise. 1CHAPTER ONE: INTRODUCTION This study examines mathematics practice and learning within a pre-apprenticeship training program in the pipe trades1 through a lens of activity theory (Leont’ev, 1978; Engeström, 1999a) and Radford’s (2002, 2006a, 2007, 2008c) recent elaboration, the cultural- semiotic theory of knowledge objectification. This study grounds and extends Radford’s theory, which is ideally suited for analyzing mathematics learning as a complex and multi-layered culturally situated activity. The central focus of this study is on the activity of an individual pre- apprentice as he learns to read fractions-of-an-inch on a measuring tape, a common and essential part of workplace activity within many Canadian industries. These theories provide a comprehensive framework for understanding mathematics learning within pipe trades training generally and learning to read a measuring tape in particular as a goal-directed, semiotic and social process mediated by artifacts and norms or conventions of workplace activity. Background for this study Workplace activity, including the use of mathematics, is mediated by many things including: established workplace practices, the technology and production materials particular to different trades, industry norms, the constituents of each particular workplace community, and the distribution of labour within the workplace, to name a few (Engeström, 1999a). It is not surprising then that studies of mathematics in the workplace reveal that practice is often idiomatic and unlike mathematics done in other contexts, including school. Unlike school mathematics where the focus is often on learning decontextualized and supposedly generalizable 1 The term pipe trades in this context refers to the skilled trades of plumbing, sprinkler fitting, gas fitting, and steam fitting. The students in this common pre-apprenticeship program take more specialized training particular to one of these trades in their subsequent formal apprenticeship programs. 2mathematics content to meet various terms of assessment, the primary focus of workers using mathematics in the workplace is on completing production tasks without a need to linger on the mathematics used in the process for longer than is necessary (FitzSimons, Mlcek, Hull, & Wright, 2005; LaCroix, 2002; Noss, Hoyles & Pozzi, 2000; Williams, Wake & Boreham, 2001; Zevenbergen, 2000). Thus, an in-depth examination of the mathematics activity generally and mathematics learning in particular within a workplace training program provides an opportunity to further our understanding of the unique features of this activity. In turn, this has the potential to inform our understanding of how to facilitate the successful transition between school mathematics and workplace training for students as well as to inform the teaching of mathematics in workplace training. To date, the research related to workplace or vocational mathematics has focused, largely, on three things. First, mathematics content used, from a perspective of school mathematics, in specific workplaces or vocations for the purpose of informing mathematics curriculum design and the teaching of the mathematics needed for work (e.g., FitzSimons, 2005; Zevenbergen & Zevenbergen, 2004, 2009). Second, the unique features of mathematics practice within specific workplaces including the use of workplace specific conventions and artifacts or the connection between local knowledge and these practices (e.g., Martin & LaCroix, 2008; Williams & Wake, 2007b). And, very much related to both of these, the problematic notion of transfer of school mathematics competencies to the workplace (e.g., Williams & Wake, 2007a). Zevenbergen and Zevenbergen (2009) provide a poignant critique of the existing research literature on workplace mathematics as follows: Much of the work undertaken in mathematics education that explores workplace numeracies is premised on seeking to identify the [formal] mathematics of the workplace.... Applying a mathematical lens to observe workplace activity means that the activity can be overridden by the mathematical imperative. Skovsmose (1994) argued that this phenomenon can be seen as the formatting power of mathematics, so that it is often difficult to see events for their activity, but rather 3to subjugate the activity for mathematics. Such an approach preserves the hegemony of particular forms of knowing and doing. However, it fails to recognize and validate the processes employed by workers as they undertake their tasks and how they go about solving problems. (p. 184) Despite a rapid reform of vocational education and training systems in many English- speaking countries as a result of wider social and economic concerns, it is also the case that very little has been reported about mathematics activities and learning within workplace training in general and within formal apprenticeship training for skilled trades in particular (FitzSimons, 2000). Describing the situation in Australia regarding research in vocational education and training, FitzSimons (2000) stated: “Funded research is primarily concerned with managerial issues, and rarely focused on issues of teaching and learning—and even then only in the workplace. Mathematics is generally labeled as numeracy and often bracketed with literacy” (p. 221). A relatively small number of mathematics education researchers have conducted research for the purpose of understanding the mathematics within the workplace as culturally situated forms of practice making explicit use of cultural-historical activity theory (e.g., FitzSimons, Mlcek, Hull, & Wright, 2005; Noss, Bakker, Hoyles & Kent, 2007; Williams & Wake, 2007a; Zevenbergen & Zevenbergen, 2004, 2009). This body of research draws heavily upon the work of the contemporary activity theorist Yrjö Engeström (e.g., 1990, 1999a, 2000, 2001). Adopting this theoretical perspective serves to differentiate mathematics activity found in different contexts, such as formal school mathematics and various workplaces, by foregrounding prominent mediating roles of motives, goals, semiotic systems and specialized artifacts, and norms of practice. A significant limitation of Engeström’s popular take on activity theory is that while acknowledging the prominent mediating role of semiotic resources in activity, such as workplace mathematical activity, it does not provide for a detailed account of this. As Noss, Bakker, Hoyles, and Kent (2007) have argued, 4In much recent workplace research, tools, artefacts or instruments are broadly taken as mediating between [the] subject and object of activity. But because mathematical “tools” are often signs such as tables and graphs, we also need a specific theory of semiotic mediation, which takes account of how different types of mathematical signs are used at work. (p. 381) Radford’s (2002, 2008a, 2008b) recent cultural-semiotic elaboration of activity theory, the theory of knowledge objectification, was developed to unpack the nuances of mathematics activity and learning from a cultural-semiotic activity perspective. This approach addresses squarely Noss, et al.’s (2007) call for an activity-theoretical approach to account for the semiotic dimensions of mathematics activity and also provides a basis for delineating more thoroughly workplace mathematics practice from that of school mathematics. First, Radford’s cultural- semiotic activity theory perspective foregrounds the intimate and dialectical relationship between material culture and human thinking–including mathematical thinking–and, second, it emphasizes the central role of cultural-semiotic systems and social interactions in different mathematical forms of cultural practice and mathematical learning. Radford accomplishes these two things by introducing the concept of semiotic systems of cultural signification. This concept positions beliefs about conceptual systems and conceptions about truth (ontology), knowability (epistemology), methods of inquiry (methodology), and legitimate knowledge representation (semiotic systems) as essential elements of any form of mathematics activity. Radford also introduces the territory of artifactual thought to highlight the integral role of material artifacts in human thinking. These concepts, in turn, call our attention to the explicit recognition of mathematics within different historical or cultural contexts, including various forms of workplace activity, as distinct and entirely legitimate forms of semiotically and artifactually mediated mathematics practice. From this perspective, academic or school mathematics as we know it is only one of a number of different and legitimate forms of mathematics. While, to my knowledge, Radford’s theory of knowledge objectification has been applied only to the analysis 5of mathematics learning within school classrooms to date, it is ideally suited to the task of analyzing mathematics activity and learning in workplace training. In turn, the study of mathematics activity and learning within the context of workplace training is an ideal setting in which to ground the theory of knowledge objectification. Purpose of and rationale for this study Mathematical literacy or numeracy is widely recognized as an important requirement in the workplace (FitzSimons, 2005; Hoyles, Wolf, Molyneux-Hodgson & Kent, 2002; Ontario Ministry of Education, 2004; Parsons & Bynner, 1997; Wedege, 2002) and tradespersons in particular use a variety of mathematics skills and understandings in diverse situations (e.g., Folinsbee, 1990, 1994, 1995; Martin, LaCroix & Fownes, 2006; National Literacy Secretariat, 1997; Nicholson, 1998; Taylor, 1997). Furthermore, with ongoing technological change in the workplace, mathematical competence is becoming increasingly important for individuals to participate fully in all aspects of their workplaces (Wedege, 2000). Yet, despite the importance of mathematical competence within the industrial workplace, the mathematical demands of the workplace remain a problem, if not an obstacle, to better jobs and advancement for many trainees including students within apprenticeship training (Fownes, Thompson & Evetts, 2002). Zevenbergen and Zevenbergen (2004) reported, for example, that within the Australian context almost 50% of traineeships and/or apprenticeships undertaken by young people are not completed. There are many reasons for such non-completion but one of the most concerning reasons is that young people are unable to cope with the numeracy expectations of the formal study associated with these programs or the numeracy demands of the workplace. (p. 506) FitzSimons, Mlcek, Hull and Wright (2005) have explained that “there has been little attention ... focused on how numeracy is learned in the workplace, taking into account the complex issues which surround apparently simple calculations, and the importance of 6social, cultural, and historical contexts” (p. 26). And to date this situation has not changed significantly. Research has focused, for the most part, on wider issues relating to workplace mathematics and issues of preparing school students for the workplace with very little, if any, research providing close analyses of mathematics learning in workplace training contexts to account for these as particular forms of culturally situated activity. As Coben (2006) explained, “research in adult numeracy and mathematics teaching and learning is still in the exploratory phase of development” (p. 29) and the field “is beset by conceptual difficulties” (p. 18). Thus, theoretically grounded research of mathematics activity within workplace training that provides in-depth and situated analysis has the potential to make a valuable contribution to mathematics education in general and mathematics training for the workplace in particular. In an attempt to better understand mathematics activity and learning within workplace training, this study examines mathematics activity within a pre-apprenticeship training program in the pipe trades with an in-depth cultural-semiotic analysis of one student learning to measure in fractions-of-an-inch using a measuring tape. Within Canada, measuring in imperial units is difficult for many new students in construction trades training owing to the fact that many of them have used metric units exclusively during their K-12 schooling. As indicated, this analysis is framed broadly by activity theory and particularly by Radford’s theory of knowledge of objectification. This theoretical perspective helps transcend prevailing rationalist and constructivist orientations that configure much current mathematics education research (Radford, 2009a) including research on workplace mathematics. Together, activity theory and the theory of knowledge objectification are well suited for micro-analysis of mathematics activity and learning within various cultural contexts. 7Research questions The following research questions to examine mathematics activity and learning within workplace training for the pipe trades are shaped by and, in turn, shape the use of activity theory and the theory of knowledge objectification: 1) What is the nature of the mathematics activity within this pre-apprenticeship program? 2) Within the context of a tutoring session, what constitutes the mathematics activity of learning to read fractions-of-an-inch on a measuring tape? (a) In what ways do semiotic resources, other artifacts, workplace conventions, and divisions of labour serve to mediate this learning activity? (b) What forms of iconicity, changes in semiotic nodes, semiotic contractions, and forms of subjectification occur within this learning activity and how do these shape and reflect the student’s objectification of the fractions-of- an-inch marking pattern on the measuring tape? (c) What contradictions or tensions are there within and between various elements of this learning activity and how are these reflected within this activity? 3) What other significant processes within this mathematics activity are not yet addressed by activity theory or the theory of knowledge objectification and in what ways do they inform these theories? Research setting and data sources This research draws upon data collected as part of a larger Social Sciences and Humanities Research Council of Canada funded endeavor, The University of British Columbia Workplace Numeracy Project, led by Dr. Lyndon Martin in partnership with the British Columbia Construction Industry Skills Improvement Council (SkillPlan). SkillPlan is a joint labour union and management sponsored initiative within the British Columbia (BC) construction industry. This project examined the mathematics within a number of construction trades training programs in the BC Lower Mainland from 2003-2005 for the purpose of 8identifying and analyzing challenges faced by apprentices and pre-apprentices with the mathematics-related parts of their training. I served as the field researcher for this project, visiting and collecting data within a variety of construction trades training programs over a 21 month period, played a central role in the analysis of this data, and was a co-author for a number of research papers that stemmed from this project. The data for much of the present study comes from a 33 minute impromptu tutoring session involving an individual pre-apprentice in an eight-week pipe trades pre-apprenticeship training program conducted at a trade union school. This particular program was designed to give pre-apprentices a head start with important skills that would be addressed subsequently in the early years of their formal apprenticeship training in a number of different pipe trades and involved pencil and paper work in the classroom as well as practical work in the school workshop. The tutoring session took place after it became apparent to me that the pre-apprentice was having difficulty using his measuring tape as he worked on a practical pipe-construction task with his colleagues in the workshop. By this point in the larger research project I had already visited and collected data within a number of apprenticeship classrooms for a variety of construction trades, including an earlier section of the very same pre-apprentice pipe trades course with the same instructor. I had also worked intensively in one-on-one situations with two other students helping them to make sense of the marking pattern on their tape measures on separate occasions prior to the present study. One of these students was in the earlier session of the pre-apprentice pipe trades course mentioned above (see Martin, LaCroix & Grecki, 2004; Martin, LaCroix & Fownes, 2005a; 2005b, 2005c) and the other was an entry-level iron-working apprentice. As a result of the these interactions I developed a tool for teaching the pattern of fraction markings on an imperial measuring tape–the matched set of rulers printed on 9transparencies printed on transparencies–that figures prominently in the present study (see chapter four). Throughout the pipe trades pre-apprenticeship course that is the focus of the present study I served as a math tutor for any of the pre-apprentices in the class who wanted my help. At other times I observed pre-apprentices and engaged them in dialogue about their mathematics coursework as they worked on both pencil and paper activities in the classroom and hands-on fabrication tasks in the workshop. The activity of individuals and groups of pre-apprentices, working either on their own or with me, as well as whole class activities led by the course instructor were documented using a portable video camera. Copies of the course print materials, selected copies of pre-apprentices’ written work, and background information on individual students were also collected for analysis. The data used for the present study was selected from this larger body of data. Limitations of this study This study is necessarily limited by the available data, its scope, and the unit of analysis employed: the study focused on the activity of an individual pre-apprentice learning to measure with a measuring tape within a pipe trades pre-apprenticeship program as documented primarily using video. Mathematics and workplace educators will, however, be able to use the detailed and theoretically grounded analysis of this student learning mathematics within this authentic workplace training context to understand mathematics learning here as well as in other contexts (Lincoln & Guba, 1985). 10 Organization of this dissertation This dissertation is organized into six chapters. Chapters one, two and three contain the introduction, the review of the background literature, and the research methodology respectively. Chapters four and five contain the data analysis and findings. Chapter four includes an overview of the mathematics related learning activities within this pre-apprenticeship training program, as well as an in-depth multi-semiotic analysis of the one-on-one tutorial with the pre-apprentice learning to read a measuring tape from an activity theory perspective. Chapter five provides a more holistic analysis across the set of actions that made up this tutorial session again drawing upon activity theory as well as Radford’s theory of knowledge objectification. Chapter six provides the conclusions and implications for this study. Finally, an annotated transcript of the tutoring session involving the pre-apprentice that serves as the basis for much of the analysis is provided as an appendix. 11 CHAPTER TWO: REVIEW OF THE LITERATURE This chapter provides a review of the literature that pertains to this study. The present introduction is followed by a description of the background to cultural-historical activity theory starting with its roots in the philosophy of Kant and Marx and the socio-cultural psychology of Vygotsky. This is followed by a summary of the work of Leont’ev, the founder of activity theory and the more recent development of this theoretical perspective by Engeström and his colleagues. The next section presents Radford’s recent cultural-semiotic elaboration on activity theory—the theory of knowledge objectification that focuses specifically on mathematics learning. The final section provides an overview and critique of the recent research on workplace mathematics and mathematics learning for the workplace as it relates to the present study. Activity theory In society a man [sic] finds not simply external conditions to which he must accommodate his activity, but that these same social conditions carry in themselves motives and goals of his activity, his means and methods; in a word, society produces the activity of the individuals forming it. Of course, this does not mean at all that their activity only personifies the relationships of society and its culture. There are complex transformations and transitions that connect them so that no direct information of one to the other is possible. For a psychology that is limited by the concept of “socialization” of the psyche of the individual without its further analysis, these transformations remain a genuine secret. This psychological secret is revealed only in the investigations of the genesis of human activity and its internal structure. (A. N. Leont’ev, 1978, p. 51) Activity theory is a philosophical and cross-disciplinary theoretical framework for studying how humans purposefully transform natural and social reality, including themselves, as an ongoing developmental process (Davydov, 1999; Engeström, 1993; Kuutti, 1996). Central to activity theory is the view that “contexts are neither containers nor situationally created experiential spaces, contexts are activity systems. An activity system integrates the subject, the object, and the instruments [material tools as well as signs and symbols] into a unified whole” 12 (Engeström, 1993, p. 67). Activity theory attempts to overcome traditional dichotomies between micro- and macro-, internal and external, mental and material, individual and social, thought and action, quantitative and qualitative, observation and intervention, and agency and structure by integrating three perspectives: the objective, the ecological, and the socio-cultural (Engeström, 2000; Kaptelinin, 1996b). In practice, activity theory is used to frame human thinking and behavior as co- constituent elements of collectively organized, culturally mediated, historically situated and simultaneous activity systems. Each activity system is motivated and sustained on an ongoing basis by its own unique object that satisfies a human need or desire (Cole & Engeström, 1993; Engeström, 1987, 1999b; Engeström & Miettinen, 1999; Leont’ev, 1978). Lave (1993) stated, If context is viewed as a social world constituted in relation with persons acting [as it is within activity theory], both context and activity seem inescapably flexible and changing. And thus characterized, changing participation and understanding in practice—the problem of learning—cannot help but become central as well. (p. 5) Thus, as Engeström and Miettinen (1999) have asserted, activity theory is a powerful and adaptable tool for the study of cultural practices and practice-bound cognition. Historical roots of activity theory The lineage of present day activity theory can be traced back to German idealist philosophy of the eighteenth and early nineteenth centuries. Notions of activity were developed by prominent German philosophers such as Kant, Fichte, and Hegel (Davydov 1999). In general terms, German idealism “emphasized both developmental and historical ideas and the active and constructive role of humans” (Kuutti 1996, p. 25). This body of work remained relatively unknown to most Anglo-American readers because its early development coincided with the 13 emergence of (British) empiricism that has dominated mainstream scientific thought in the English speaking world (Kuutti, 1996). Marx and his frequent collaborator Engels drew heavily upon Hegel’s conception of dialectics in developing their comprehensive view of human life and human nature that, in turn, became the basis of activity theory. In opposition to the “metaphysical” mode of thought, which viewed things in abstraction, each by itself and as though endowed with fixed properties, Hegelian dialectics considers things in their movements and changes, interrelations and interactions. Everything is in continual process of becoming and ceasing to be, in which nothing is permanent but everything changes and is eventually superseded. All things contain contradictory sides or aspects, whose tension or conflict is the driving force of change and eventually transforms or dissolves them. (Encyclopedia Britannica, 2004) Inverting Hegel’s ideas to focus on the personal and adding a materialist emphasis, Marx asserted that human nature (consciousness and behavior) forms, develops, and resolves not within the human individual or spirit, but rather in the dynamic interplay between the individual and his or her co-evolving material world in the course of tool and artifact creation and use (Cole & Scribner, 1978; Engeström & Miettinen, 1999). This perspective became known as dialectical materialism. An important corollary of this, and a position asserted by Marx, is that “the content of his [a human’s] reason must be determined by conditions external to his reason, conditions which are strictly material” (Cropsey, 1972, p. 757). In contrast to the prevailing views of humans’ place in the world at the time, Marx’s dialectical and materialist conception of human activity and human nature overcame and transcended the classic dualism of the inner world of the individual subject and the outer world of objective societal and material circumstances (Engeström & Miettinen, 1999). 14 Vygotsky and the theoretical foundations of activity Activity theory has grown out of the socio-cultural school of Russian psychology established by Vygotsky during the 1920s and early 1930s. Vygotsky’s students and collaborators, A. N. Leont’ev and Luria, have played a prominent role in developing and extending his ideas throughout much of the twentieth century (Davydov, 1999; Engeström, 1987, 1999a, 2001). Vygotsky began his career as a psychologist within a politically charged post-revolution Soviet society dominated by Marxist doctrine. Rather than seeing the obligation of honoring this perspective within state controlled academia as an encumbrance, as did many of his peers, “Vygotsky saw in the methods and principles of dialectical materialism a solution to key scientific paradoxes facing his contemporaries” (Cole & Scribner, 1978, p. 6). Furthermore, Vygotsky brilliantly extended this [Marx’s] concept of mediation in human- environmental interaction to the use of signs as well as tools. Like tool systems, sign systems (language, writing, number systems) are created by societies over the course of human history and change with the form of society and the level of its cultural development. Vygotsky believed that the internalization of culturally produced sign systems brings about behavioral transformations and forms the bridge between early and later forms of individual development. Thus for Vygotsky, in the tradition of Marx and Engels, the mechanism of individual developmental change is rooted in society and culture. (Cole and Scribner, 1978, p. 7) Vygotsky’s (1978, 1979c) concept of a complex mediated act or the instrumental method of psychology is central to his view of mediation. This concept is crystallized within his famous triangular model shown in Figure 1. According to Vygotsky, the process of mediation changes the simple stimulus-response process that characterizes elementary forms of behavior. Specifically, cultural mediation creates a new and additional relationship between stimulus and response and, by implication, an object and subject of an activity, as indicated by an X in Figure 1. This new link is formed whenever the subject makes use of a psychological tool such as a sign, or makes use of a technical tool such as a physical artifact when responding to something 15 Figure 1. Vygotsky’s model of a complex mediated act This model of a complex mediated act replacing a simple stimulus-response process (Vygotsky, 1978). in his or her experience. While mediation by technical tools creates a new relationship between a subject and tool by extending the subject’s influence on the object of activity, mediation by psychological tools acts in reverse—it is internally oriented, acting on the individual rather than on the environment (Cole & Engeström, 1993; Vygotsky, 1978, 1979c). Vygotsky (1979c) identifies all forms of conventional signs, as well as various systems for counting; mnemonic techniques; algebraic symbol systems; schemes, diagrams, maps, and mechanical drawings; and works of art as examples of psychological tools. A brief example central to this study is that of the imperial system of inches and fractions-of-an-inch that is common to pipe-trades practice. The regular use of this system of linear measure leads workers to think exclusively in terms of binary fractions-of- an-inch (e.g. halves, quarters, eighths and sixteenths) on the job. (More on this later.) Vygotsky (1978) elaborated on the importance of psychological tools in human thinking implied by this model: “Because this auxiliary stimulus possesses the specific function of reverse action, it transfers the psychological operation to higher and qualitatively new forms and permits humans, by the aid of extrinsic stimuli, to control their behavior from the outside” (p. 40). Thus humans’ use of signs marks a break from purely biological development resulting in new forms of artifactually mediated and “culturally-based psychological processes” (p. 40). Engeström (1987) added that while “both technical tools [artifacts] and psychological tools mediate activity, Stimulus Response X 16 … only psychological tools imply and require reflective mediation, consciousness of one’s (or another person’s) procedures” (p. 56). Engeström (2001) explained the significance of Vygotsky’s view of mediation with cultural artifacts for understanding of the relationship between culture and thinking as well as the development of an activity-theoretical perspective as follows: Vygotsky’s idea of mediation of human actions by cultural artifacts was revolutionary in that the basic unit of analysis now overcame the split between the Cartesian individual and the untouchable societal structure. The individual could no longer be understood without his or her cultural means and the society could not longer be understood without the agency of individuals who use and produce artifacts. This meant that objects ceased to be just raw material for the formation of logical operations in the subject as they were for Piaget. Objects became cultural entities and the object-orientedness of action became the key to understanding human psyche. (p. 134) Furthermore, the notion of humans controlling their own behavior “from the outside” through the use and creation of artifacts offers an optimistic view of human self-determination (Engeström, 1999a). At the same time, Vygotsky’s view of mediation provides an internally consistent and complementary perspective for understanding the collective and self-reproducing aspects of culture—through the shared use of historical artifacts (Kaptelinin, 1996a). Wartofsky (1979), commenting on the profound significance of mediation by cultural artifacts, asserted, “the artifact is to cultural evolution what the gene is to biological evolution” (p. 205). For Vygotsky (1978) the process of internalization, whereby external operations amongst individuals are reconstructed internally within an individual, also plays a central role in the genesis of the individual’s culturally framed mental processes as well as in the very continuity of culture itself. Furthermore, “the internalization of socially rooted and historically developed activities is the distinguishing feature of human psychology, the basis of the qualitative leap from animal to human psychology” (p. 57). Vygotsky (1979b) calls this process the general genetic law of cultural development. 17 Complementing the process of internalization is the process of externalization. Lektorsky (1999), a prominent Russian commentator on Vygotsky and modern activity theory explained that, “Humans not only internalize ready-made standards and rules of activity but externalize themselves as well, creating new standards and rules. Human beings determine themselves through objects that they create. They are essentially creative beings” (p. 66). This passage illustrates the dialectical relationship between the individual and his or her material world—that we are formed through our interaction with the objects around us, as we in turn re-form these very objects on an ongoing basis—a central tenant of Vygotsky’s work as well as activity theory. In other words, internalization relates to the way that culture is reproduced, while externalization provides opportunities for the transformation of culture through the possibilities afforded by the creation of new artifacts. And, these inseparable processes operate continuously at all levels of human activity (Engeström and Miettinen, 1999). Another of Vygotsky’s important contributions to our understanding of learning (that remains part of cultural-semiotic activity-theoretical analyses of learning) is the concept of the zone of proximal development. This is familiar to many educators as the “discrepancy between a child’s actual mental age and the level he reaches in solving problems with assistance” (Vygotsky, 1986, p. 187). The key point here is “that instruction leads the course of development and that such a course depends on the kind of relationship that is created between the student and her context” (Radford, 2009a, p. 13). This is an important insight when analyzing learning processes because it foregrounds the important role of the teacher or tutor in mobilizing resources to establish context that create and maintain zone of proximal developments for the learner so that he/she is able to function at a higher level than he/she would working individually. 18 Leont’ev and activity theory As mentioned earlier, Leont’ev was one of Vygotsky’s students and colleagues until Vygotsky’s premature death in 1934. While Vygotsky introduced a great number of the ideas that are central to activity theory, it was Leont’ev, over the four decades following Vygotsky’s passing, who emphasized social activity over individual activity and was instrumental in developing this into a theoretical framework we now call activity theory (Engeström, 2001; Nardi, 1996b; Wertsch, 1979b). It is worth noting that Leont’ev’s activity based approach was the most widely accepted theoretical approach in Soviet psychology until the collapse of the Soviet Union in the early 1990s (J. V. Wertsch, personal communication September 6, 2004 & December 1, 2004) and his work continues to hold prominence within the international activity theory community as evidenced by the number of references to his work in the academic literature (Roth, 2004). While Vygotsky’s unit of analysis for psychological analysis was tool-mediated human behavior, Leont’ev (1981)—prompted by his reading of Marx—elaborated Vyotsky’s idea of mediation by other individuals and social relations. Leont’ev constructed the concept of activity that is central to activity theory by differentiating between collective activity and individual action (Engeström, 1987; Engeström & Miettinen, 1999). This move that “took the paradigm a huge step forward in that it turned the focus on complex interrelations between the individual subject and his or her community” (Engeström, 2001, p. 134). Leont’ev (1981) used the oft-cited example of a team of hunters, with one member of the team (a bush beater) serving to frighten the prey towards others waiting in ambush, to explain the need for an elaboration of Vygotsky’s basic unit of analysis. Here the bush beater’s actions (scaring animals away) seems irrational if one fails to take into account the overall collective activity within which the bush beater is 19 involved. This distinction between activity and action became the basis of Leont’ev’s three-level model of activity (Engeström & Miettinen, 1999). The defining characteristic of an activity is its unique object that motivates it towards meeting a particular human need (Leont’ev, 1978). Kuutti (1996) explains that the object of an activity can be a material artifact, something less tangible such as a plan, or something completely intangible such as a common idea (e.g., a mathematical object). The essential characteristic is that the object is available for manipulation and transformation by the activity participants. And, within an activity, the subject—the individual or sub-group whose agency is chosen as the point of view in the analysis—and the object of activity are connected and transform one another as parts of a dialectic unit (Center for Activity Theory and Developmental Work Research, 2004; Roth, 2004). This subject-object relationship stems from Leont’ev’s (1978) two-part conception of the object of activity that emerges, “first, in its independent existence as subordinating to itself and transforming the activity of the subject, [and] second, as an image of the object, as a product of its property of psychological reflection that is realized as an activity of the subject and cannot exist otherwise” (p. 52). Leont’ev’s two-part conception of the object of activity, and thus of activity itself, stems from Marx and Engels’ efforts “to fuse into one concept the object of practical activity (German, Object) and the object of thought (German, Gegesnstand) to overcome the problems of idealism, on the one hand, and materialism, on the other” (Roth, 2004, p. 3). Engeström and Escalante (1996) described further the nature of the object of activity as slippery, multifaceted, constantly in transition and under construction, animate, and transitional; and point out that the object of an activity appears differently for the various participants within the activity. And, Nardi (1996b) added that, “Objects do not, however, change on a moment-by-moment basis. There is some stability over time, and changes in objects are not trivial” (p. 74). 20 Activities realize and reproduce themselves through the generation and undertaking of conscious, goal-directed, individual and cooperative actions or chains of actions—discrete processes that are aligned towards meeting the objective of an activity (Engeström, 1987, 2000; Kuutti, 1996; Leont’ev 1978; Nardi, 1996b). But, as Engeström (1999b) stated, “being a horizon, the object (of an activity) is never fully reached or conquered” (p. 381). In comparison to activities that evolve through extended and recurrent historical cycles, actions have clearly defined beginnings and ends and are relatively short lived (Engeström, 1999b). And, to reiterate the point exemplified earlier by Leont’ev’s (1981) bush beater example, a particular action cannot be fully understood without reference to the context of its corresponding activity (also see Kuutti, 1996). Engeström (2001) described object-oriented actions as, “always, explicitly or implicitly, characterized by ambiguity, surprise, interpretation, sense making, and potential for change” (p. 134) and even “the most well-planned and streamlined actions involve failures, disruptions, and unexpected innovations” (Engeström, 1999a, p. 32). Furthermore, “the creative potential [of an activity] is closely related to the search actions of object construction and redefinition. This situation-specific reconstruction and instantiation of the object of an activity takes the form of problem finding and problem definition” (Engeström, 1999b, p. 381). Implicit in this description and a point raised elsewhere by Engeström (1987) is that conscious actions taken by the subject of activity are the site of thinking and he also defines specific learning actions, as “actions where the subject is consciously aware of the object of the action as an object of learning” (p. 87). A final note on actions within an activity. As mentioned earlier, multiple actions may be undertaken to meet the same goal (Nardi, 1996b; Kutti, 1996). But the opposite relation is also possible. A single action can be instrumental in realizing multiple activities (Engeström 1987; Kaptelinin, 1996a; Leont’ev, 1978) although, as Kuutti (1996) points out, the action will take on 21 a different personal sense for the subject within the different context of each activity when this occurs. The level below individual action is that of automatic operations, the third level of activity identified by Leont’ev. Operations arise out conscious actions that become automatic and the way individuals respond on a subconscious level to the tools and conditions of the action at hand. They are the way that actions are actually carried out (Leont’ev, 1978). Leont’ev (1978) illustrated how operations develop from actions, as well as the subconscious and routine nature of operations, with the following explanation of the process of learning to drive a car with a standard transmission: Initially every operation, such as shifting gears, is formed as an action subordinated specifically to this goal and has its own conscious 'orientation basis'. Subsequently this action is included in another action, ... for example, changing the speed of the car. Now shifting gears becomes one of the methods for attaining the goal, the operation that effects the change in speed, and shifting gears now ceases to be accomplished as a specific goal-oriented process: Its goal is not isolated. For the consciousness of the driver, shifting gears in normal circumstances is as if it did not exist. He does something else: He moves the car from a place, climbs steep grades, drives the car fast, stops at a given place, etc. Actually this operation [of shifting gears] may, as is known, be removed entirely from the activity of the driver and be carried out automatically [as is the case in cars with automatic transmissions]. Generally, the fate of the operation sooner or later becomes the function of the machine. (p. 66) And, as Kuutti (1996) affirmed, the formation of operations from actions is a ubiquitous feature of activity. All of the constituent levels of an activity—activity, actions, and operations—can change in response to changing conditions (Kuutti, 1996; Leont’ev, 1978; Nardi, 1996b). Davydov, Zinchenko, and Talyzina (1982) provided a more detailed description of this dynamic: An activity can lose its motive and become an action, and an action can become an operation when the goal changes. The motive of some activity may become the goal of an activity, as a result of which the latter is transformed into some integral activity…. The mobility of the constituents of activity is also manifested in the fact that each of them may become a part of a unit or, conversely, come to embrace previously relatively independent units (for example, some acts may be 22 broken down into a series of successive acts, and, correspondingly, a goal may be broken down into subgoals.) (p. 36) Leont’ev (1981) described how the converse might occur, that is, how an action might become an activity in its own right: “It is a matter of an action’s result being more significant, in certain conditions, than the motive that actually induces it…. A new ‘objectivation’ of its needs comes about, which means that they are understood at a higher level” (pp. 402-403). And, elsewhere, Nardi (1996b) citing Leont’ev (1974) explained that “An operation can become an action when conditions impede an action’s execution through previously formed operations” (p. 75). This inter-level dynamic within activity theory makes activity theory well suited for describing and understanding developmental processes over extended periods of time. However, this also makes it, “impossible to make a general classification of what an activity is, what an action is, and so forth because the definition is totally dependent on what the subject or object in a particular real situation is” (Kuutti, 1996, p. 32). Engeström and cultural-historical activity theory While Leont’ev identified the crucial distinction between individual action and collective activity, as well some of the interrelations between the individual subject and his or her community, he did not develop the implications of these ideas for the structure and analysis of activity, nor did he elaborate upon Vygotsky’s triangular model of a mediated act in order to depict the fundamental and dialectically-related elements of activity (Engeström, 1999a, 2001). This task was taken up by Engeström (1987) who formalized and systematized many of Leont’ev’s ideas, formalized the idea of the activity system schematized in Figure 2, and popularized the term cultural-historical activity theory and its acronym CHAT to foreground the essential situated nature of activity. Engeström (1993) explained his activity system model as follows: 23 Figure 2. The structure of an activity system Each of the relationships between elements of the activity system, indicated by the straight lines between them, is mediated by other elements of the system which form mediating triangles like that within Vygotsky’s model of a complex mediated act. (Engeström 1987, 2001) In the model, subject refers to the individual or subgroup whose agency is chosen as the point of view in the analysis. The object refers to the “raw material” or “problem space” at which the activity is directed and which is molded or transformed into outcomes with the help of physical and symbolic, external and internal tools (mediating instruments and signs). The community comprises multiple individuals and/or subgroups who share the same general object. The division of labor refers to both the horizontal division of tasks between themembers of the community and to the vertical division of power and status. Finally the rules refer to the explicit and implicit regulations, norms and conventions that constrain actions and interactions within the activity system. Between the components of an activity system, continuous construction is going on. The human beings not only use instruments, they also continuously renew and develop them, whether consciously or not. They not only obey rules, they also mold and reformulate them—and so on. But the life of activity systems is also discontinuous. Besides accumulation and incremental change, there are crises, upheavals, and qualitative transformations. An activity system is not only a persistent formation; it is also a creative, novelty-producing formation. (pp. 67-68) The activity of doing activity theory is, by definition, itself an evolving and dynamic process! For the purposes of this study of culturally situated and mediated mathematics learning a number of the key principles from Engeström’s (2001) more recent summary of activity theory are significant. These include: 24 • that a collective, artifact-mediated and object-oriented activity system, seen in its network relations to other activity systems, is taken as the prime unit of analysis…; • [that] an activity system is always a community of multiple points of views, ... and interests….; [and] • the central role of contradictions as sources of change and development [of an activity], …; (pp. 136-137) As stated at the beginning of this chapter, “for activity theory, contexts are neither containers nor situationally created experiential spaces. Contexts are activity systems” (Engeström 1993, p. 67). Therefore, while Engeström’s activity system model invites the analysis of individual components and individual relations within it, it is essential that the activity system model be treated as a fully integrated whole when using it to analyze human practice and/or learning (Engeström, 1987, 1993, 2000, 2001; Engeström & Miettinen, 1999). The reason for this is that “goal-directed individual and group actions, as well as automatic operations, are relatively independent but subordinate units of analysis, eventually understandable only when interpreted against the background of entire activity systems” (Engeström, 2001, p. 136). This point was illustrated earlier in Leonte’ev’s (1978) example of the bush beater participating in the activity of hunting. The multivoiced nature of activity systems results from the division of labour within them. This positions participants of an activity differently with each one bringing the perspective of his or her own personal experience and interests. Activity systems also contain remnants of their histories within their artifacts, conventions, and rules. And, this plurality is compounded further by the network of interacting activity systems within which any single activity system is located (Engeström, 2001). The inevitable multivoiced nature of activity “is a source of trouble and a source of innovation, demanding actions of translation and negotiation” (Engeström, 2001, p. 136). As a result, “activity systems are best viewed as complex formations in which equilibrium is an 25 exception and tensions, disturbances, and local innovations are the rule and the engine of change” (Cole & Engeström, 1993, p. 8). Specifically, as the participants within an activity system work to resolve contradictions, new forms of activity emerge in response to the contradictions of the preceding forms resulting in ‘invisible breakthroughs’ or innovations from within the activity itself (Centre for Activity Theory and Developmental Work Research, 2004). It follows then, that understanding the inherent contradictions and tensions within an activity system is an important part of understanding its internal dynamics and its development over time. Roth (2004), in the introduction to a special issue of Mind, Culture, and Activity focusing on narrating and theorizing activity within educational settings, commented that many academics have been unjustifiably critical of Engeström’s (1987, 2001) activity system model by pointing out that this representation is “fundamentally static, underlining its structuralist dimensions”(p. 2). Roth argued that: such a characterization misses the mark because it fails to recognize that the model is inherently dynamic. The dynamic aspect arises from two features. First, subject and object form a dialectic unit, which is the epitome of an engine of change (citing Il’enkov, 1977). Second, human praxis and self-change coincide with change in life conditions (citing Marx & Engels, 1978); that is, the very notion of activity (Marx and Engel’s Tatigkeit, practical action) at the heart of Engeström’s representation embodies change. (pp. 2-3) Elsewhere in this same piece, Roth explained the origins of this misconception by asserting that, “Anglo-Saxon scholarship has appropriated activity theory in a particular way, grafted a dialectical theory onto a fundamentally dualist epistemology” (p. 7) with fundamental aspects of activity no longer making sense within this new and inappropriate context. This misconception highlights the need to be very clear about the dialectical complexity of activity that is fundamental to Engeström’s activity system model. 26 Critiques of activity theory The present overview of activity theory would be incomplete without acknowledging criticisms that have been raised within the activity theory literature. These include criticisms that activity theory is under theorized in its present form with a number of significant unanswered questions (e.g., Davydov, 1999; Kaptelinin, 1996a; Nardia, 1996a) and criticisms of some of the paradigmatic assumptions associated with activity theory (e.g., Latour, 1996a, 1996b). In the edited volume Perspectives on Activity Theory (Engeström, Miettinen & Punamäki, 1999), Davydov (1999) devoted most of a chapter to identifying and discussing a number of important points that activity theory had yet to address. These include: (a) how objects are transformed over time; (b) the relationship between collective activity and individual activity, especially in light of the acknowledged role of the process of internalization within activity theory; (c) the place of other processes and structures within activity theory, for example the means for solving problems, and traditional psychic processes such as perception, imagination, memory, thinking and feelings; (d) how to differentiate between different kinds of activity given the different basis for the classification of types of activity with the various disciplines that make use of activity theory; (e) the interrelation between activity and human interaction in general, and activity and communication in particular; (f) the connections and relationships of activity theory with other theories and approaches to human conduct; (g) the relationship between the social and biological aspects of human existence; and finally, (h) how to coordinate and organize the interdisciplinary study of human activity. Nardi (1996a) raised similar kinds of concerns about activity theory. These include the need to develop an understanding of how activity theory deals with the complex interaction amongst multiple activities and conflicting objects both for individual subjects and collective 27 subjects. She also raised concerns about the inability of activity theory to deal fully with a number of significant aspects of human experience. On this matter she stated, Activity theory excels at describing object-related activity but says little about how we are diverted, distracted, interrupted, seduced away from our objects, subject to serendipity and surprise… there is still a need for a theoretical concept that defines the relationship of object-related activity to the very real and sometimes substantial impingements of events outside the scope of our object- related activities, … a way of explaining the articulation and coordination of object-related activity and impinging events external to objects. (p. 378) Kaptelinin (1996a), echoing one of Davydov’s (1999) concerns, stated that “it remains to be clarified to what extent the conceptual apparatus of activity theory is applicable to collective subjects” (p. 63). Kaptelinin supported Nardi’s (1999a) view regarding the inability of activity theory to address fully human experience. On this point he stated, While culture, values, motivation, emotions, human personality, and personal meaning are embraced by the conceptual system of activity theory, the theory does not aim at giving a conceptual description of all these phenomena. It captures only some of their aspects: those related to rational understanding of human interaction with the world.... Activity theory cannot completely substitute for an anthropology that defines and understands culture. (Kaptelinin, 1996a, p. 63) He also asserted that the prominence of tool mediation within activity theory can impose limitations on the potential application of this perspective in research on human computer interaction. Using the example of human activity within a computer based virtual reality environment, Kaptelinin (1996a) explained that it is problematic for activity theory to deal with the computer program or artifact that produces the virtual world within which this activity occurs for the subject. Latour’s (1996a, 1996b) actor-network theory challenges a number of paradigmatic assumptions upon which activity theory is based. Drawing from sociobiology, Latour (1996a) asserted that human social structure—a central element within activity theory—can be understood as interactions within networks of collective animate and inanimate actors spread 28 across time and place. This perspective challenges, for example, the view that a single or collective subject is the origin of action, that tools and subjects can be considered as separate types of entities when examining human praxis, and the view that culture as a knowledge base to be transmitted exists, let alone, is mediated by artifacts and language. Latour (1996b) also argued that the dialectical perspective central to activity theory fails to bridge the divide between individual and collective action. He asserted that “dialectical reasoning is the way invented to entrench even deeper the dualism between individual and collective action and to behave as it had been triumphantly overcome” (p. 269). Latour (1999, 2005) has since critiqued and revisited actor-network theory. In any event, it provides a useful reminder that activity theory is one of multiple perspectives for analyzing human agency and practice, each with its own epistemological and ontological assumptions, utility, biases, and limitations. Radford’s theory of knowledge objectification Cultural-historical activity theory provides a broad framework and set of general principles for analyzing human activity as well as change in activity over time. It situates any particular human activity both culturally and historically, in large part by drawing attention to the mediating role within an activity system of cultural tools (including artifacts and semiotic resources), cultural-historical norms or rules, and customary divisions of labour as inextricable elements in dialectical relationship with the subject, community, object of activity, and goal(s) of the activity, and as well as with one another. To date, researchers working strictly from a CHAT perspective have addressed learning largely in terms of changes in activity over time. The thinking and learning of individuals, it should be noted, has been the focus of a considerable body of work that has been carried out within a broad Vygotskian socio-cultural tradition from which activity theory has emerged. 29 The mathematical thinking and learning of individuals within activity is the primary foci of Luis Radford’s cultural-semiotic theory of activity—the theory of knowledge objectification (TO).2 Rooted in his reading of Vygotsky’s semiotics and Leont’ev’s activity theory as well as the more recent work of Felix Mikhailov and Evald Ilyenkov, Radford addresses head-on a number of the undertheorized aspects of activity theory identified earlier. In the TO, learning is conceptualized as an interactive and creative acquisition of historically constituted forms of thinking. Such an acquisition is thematized as a process of objectification; that is, as a process of making sense of and becoming critically conversant with the cultural-historical logic with which systems of thought, such as mathematics, have been endowed (see also Radford, 2008b, 2009b). Radford’s concept of objectification is a refinement of Vygotsky’s notion of internalization in that it emphasizes the dialectical relationship between the subject and the cultural object being attended to. The question of objectification is not only the manner in which personal and cultural meanings become tuned, for personal meanings can only arise and evolve against the backdrop of forms of activity. Here the TO departs from other perspectives on mathematics learning (e.g., those used by Cobb, 2002; Kieren, Pirie & Calvert, 1999). The problem is precisely the very social formation and evolution of personal meanings as they evolve within goal-directed activity and are framed by the cultural meanings conveyed within socio-cultural contexts. One of the aspects that makes the idea of objectification distinctive is the close relationship that it bears with 2 Radford has articulated his theory of knowledge objectification a number of times in recent years as he has developed and refined his ideas (e.g., Radford 2006a, 2007). Radford’s most recent and most comprehensive formulation of the theory to date is found in The ethics of being and knowing: Towards a cultural theory of learning (Radford, 2008c) and the present summary of the theory of knowledge objectification draws heavily from this. To avoid constant repetition, references to this work will omitted in this section of the literature review, with the exception of references for direct quotes. References to Radford’s other publications in all other parts of this section are included where they lend additional support for or embellish the points made in the 2008c paper. The only exception to this is the sub-section titled, Semiotic means of objectification. References in this sub-section are written in the conventional manner. 30 the Vygotskian concept of consciousness and its mediated nature (Vygotsky, 1979a; also Leont’ev, 1978). From this perspective, consciousness is formed through encounters with other subjects and the historical intelligence embodied in artifacts and signs with which we mediate our own actions and reflections As is true of the relationship between the subject and object of any activity, the relationship between a mathematics learner and a mathematical-cultural object within activity is a dialectical one. Thus, the mathematical object being objectified, itself, evolves as the subject comes to reflect the object through her/his participation within the cultural activity. Within the TO, Radford draws attention to this latter transformation with his notion of subjectivity. In Radford’s words, “Objectification is not only the appropriation of cultural ideas by an individual or the insertion of the cognizance of the individual into cultural life. It goes the other way around [as well]. That is to say, objectification is also a process of becoming, of agency, and the growth of subjectivity” (Personal communication, September 16, 2009). The TO contributes to activity theory by putting forward the concept of the territory of artifactual thought. As the name implies, this concept foregrounds the integral mediational role of signs, and material objects in thinking, or as Radford (2008a) has described it, “the technology of semiotic mediation” (p. 454). The role of tools and artifacts in human practices has been a distinctive characteristic of theorizations within activity theory, of course, but the TO makes an explicit attempt to avoid reducing tools and artifacts to their technological dimension. They are not merely material or technical instruments to accomplish things but constitutive parts of what we are, feel, and think. The TO also extends beyond the existing boundary of activity theory with its concept of semiotic systems of cultural signification. This encompasses the cultural conceptions that relate to notions of truth and the way the world is, the nature of mathematical objects and their relation to the material world, the manner in which mathematical and other 31 objects of knowledge are considered to be knowable, mathematical and other methods of enquiry, and the way in which mathematical knowledge can be represented. Radford originally formulated the TO to address problems with existing rationalist, idealist, and individualist-adaptive views of cognition and social interaction (Radford 2006a, 2007). These widely held perspectives in education fail to account for, among other things, the full complexity of thinking and the vital role of culture and social environments in the ways that individuals come to know and be. The TO addresses these issues by relying “on a non-rationalist epistemology and ontology, which gives rise, on one hand, to an anthropological conception of thinking, and on the other, to an essentially social conception of learning” (Radford, 2008c, p. 217). The main features of the theory to be discussed in the brief overview that follows include: its view of mathematical thinking, the role of culture in mathematical thinking and learning, major parts of the theory and their interrelations, mathematical objects and the process of objectification, mathematical thinking as a dialectical cultural reflection and its connection with the process of subjectification, and semiotic means of objectification including iconicity, semiotic nodes, and semiotic contractions. Some methodological implications of the theory for the investigation of mathematics learning will also be addressed. Mathematical thinking and the role of culture Central to the theory of knowledge objectification is its non-mentalist perspective on mathematical thinking—that “thinking is an interpretive and transformative reflexive social praxis encompassed by a cultural rationality and oriented from the outset towards a cultural system of ideas” (Radford, Bardini, Sabena, Diallo, & Simbagoye, 2005 p. 119). The reflexive nature of thinking means that the individual’s thinking is neither the simple assimilation of an external reality (as the Empiricists and Behaviorists suggested) nor an ex nihilo construction (as certain constructivist schools claim). Thinking is a re-flection, that is, a dialectical movement between a historically 32 and culturally constituted reality and an individual who refracts it (as well as modifies it) according to his/her own subjective interpretations, actions and feelings. (Radford, 2008c, p. 219) More specifically, the starting point of this alternative approach to classical mental-oriented views of cognition is a multimodal ‘material’ conception of thinking. The very texture of thinking, it is suggested, cannot be reduced to that of impalpable ideas; it is instead made up of speech, gestures, and our actual actions with cultural artifacts (signs, objects, etc.)…. Thinking [including abstract mathematical thinking] does not occur solely in the head but also in and through a sophisticated semiotic coordination of speech, body, gestures, symbols and tools. (Radford, 2009b, p. 111) In other words, “intellectual and sensual activities are different sides of the same coin. They constitute the dialectical unit of thinking” (Radford, 2005a, p. 115). Culture plays a significant role in thinking and learning by suggesting ways of perceiving reality and phenomena through established patterns of working, means of communication, artifacts, etc. As the anthropologist Clifford Geertz (1973) asserted, “the human brain is thoroughly dependent upon cultural resources for its very operation; and those resources are, consequently, not adjuncts to, but constituents of, mental activity” (p. 76). Thinking and the objects that thinking creates originate from individuals but at the same time are, inescapably, shaped and subsumed by their cultural reality, including the historically shaped concepts that individuals meet within their environment. As Radford (2008c) stated, “This is why we originate thinking, but at the same time become subsumed by it” (p. 219). The major parts of the theory and their interrelations Within the TO, Radford (also 2003b, 2008c) explicitly acknowledges the role of culture in mathematical thinking explicitly with a concept that he terms semiotic systems of cultural signification or semiotic superstructures. These refer to the “cultural conceptions surrounding mathematical objects (their nature, their way of existing, their relation to the concrete world, etc.) 33 and social patterns of meaning production” (Radford, 2008c, p. 219). In other words, these refer to the cultural terrain within which mathematical activity, thinking, and learning occur, including epistemological commitments, sanctioned means of knowledge production and ways of working, rules of validation and other cultural norms, etc. Radford uses the idea of mediation with artifacts and semiotic resources in precisely the same way as it was used by Vygotsky and it is used by other activity theorists—to refer to the role of tools (e.g., artifacts, instruments, sign systems, etc.) in carrying out social practices including thinking. Echoing Geertz’s view, Radford (2008c) asserted that, “Artifacts are neither merely aids to thinking nor simple amplifiers, but rather constitutive and consubstantial parts of thinking. We think with and through cultural artifacts” (p. 218). And, we regularly use artifacts to think and co-think with others, that is, in a social plane—a region that Radford calls the territory of artifactual thought. Radford introduced this key concept in the TO to draw attention to the subjective, social, historical, as well as the material tool-mediated dimensions of thinking, learning, and feeling. Here subjectivity and cultural objectivity intersect and “the mind extends itself beyond the skin” (Radford, 2008c, p. 219, citing Wertsch, 1991). The TO posits that activity, the territory of artifactual thought, and semiotic systems of cultural signification work in concert and in dialectical relationships with one another to significantly shape the process of mathematics thinking and learning. Thinking is not merely generated in the course of human activity. The form of the activity imprints its mark on thinking and its product—i.e., knowledge. Now, the form that all activity takes depends on symbolic superstructures [semiotic systems of cultural signification].... In their interaction with activities (their objects, actions, division of labour, etc.) and with the territory of artifactual thought, the Semiotic Systems of Cultural Signification give rise, on the one hand, to forms or modes of activities, and, on the other hand, to specific modes of knowing or epistemes (citing Foucault, 1966). While the first interaction gives rise to the particular ways in which activities are carried out at a certain historical moment, the second interaction gives rise to specific modes of knowing which allow for the identification of ‘interesting’ situations or problems and the methods, reasoning, evidence, etc. that will be considered culturally valid. (Radford, 2008c, pp. 219- 34 220) These relationships are highlighted in Figure 3. A direct implication for understanding mathematical thinking and learning highlighted in Figure 3 is that it cannot be understood simply in terms of the kinds of problems being addressed, the artifacts used, or the context by themselves. The ways that we think about and come to know mathematical objects, the ways that we approach mathematical problems and judge them solved, are all under-girded by the form of the particular activity and corresponding cultural episteme (also Radford 1997, 2003b, 2003c). This is why we are obligated to accept various historical and contemporary forms of mathematical activity, including mathematics practice within different workplace contexts, as genuine forms of mathematics in their own right and avoid the ethnocentrism of regarding anything other than the current form of formal academic mathematics as primitive or imperfect. Mathematical objects and objectification The theory of knowledge objectification posits that mathematical objects “are fixed patterns of reflexive human activity incrusted in the ever-changing world of social practice mediated by artifacts” (Radford, 2008c, p. 222). These objects bear the embodied historical intelligence or patterns of reasoning from the goal directed, mediated, and reflective mathematical activity of previous individuals (also Radford, 2003a, 2006b). The theory also posits that the objectification of mathematical objects (in other words, mathematics learning) is a process of becoming conscious of, and critically conversant with, the cultural-historical logic with which mathematical and other objects have been endowed through encounters with other voices as well as artifacts and signs. This is not an all or none proposition but instead “an active and creative process of noticing and making sense of the conceptual object that is the goal, in 35 Figure 3. Radford’s model of the theory of knowledge objectification The arrows in the figure show the interaction between a semiotic system of cultural significations, activity and the territory of artifactual thought. Their interaction generates the forms of the activity and the modes of knowing on the base of the specific historic-economic dimension. In a dialectic process, forms of activity, modes of knowing, and the historical- economic dimension [of the activity] alter the triangle’s vertices. (Radford, 2008c, p. 220) Leont’ev’s sense, of activity” (Radford, 2008b, p. 87). Mathematical objects from this perspective are neither static nor homogeneous, but comprised of layers of generality that depend on the “characteristics of the cultural meanings of the fixed pattern of activity in question” (Radford, 2008c, p. 226). The learner notices and becomes fluent with different layers of generality of a mathematical object and their empowering forms of action in a subjective and progressive manner as he or she carries out activity around it in a reflective and critical manner, provided that the activity makes apparent the mathematical object’s essential features (see also, Radford, 2006a). The process of objectification within the TO is considered an active, multi-semiotic, and mediated process that also includes making sense of the cultural logic behind the mathematical object(s) being learned (also Radford, 2005a, 2005b; Radford, Miranda, & Guzman, 2008). This requires that the individual is open to and actively seeks out meaning in the mathematical 36 activity that he or she engages in. Mathematics learning, thus described, depends on two important things that are external to the individual. The first is involvement in mathematical activity with mathematical artifacts, including semiotic systems that bear, in crystallized form, the mathematical knowledge of those who have come before (also Radford, 2003a). The role of artifacts is more than materializing thinking and making it thinking- with-and-through-artifacts. Artifacts, indeed, are bearers of historically deposited knowledge from the cognitive activity of previous generations…. Furthermore— and this is a fundamental element of human cognition—unlike animals, the human being is profoundly altered by the artifact: by making contact with it, the human being restructures his/her movements (citing Baudrillard, 1968) and new motor and intellectual skills are formed such as anticipation, memory, and perception (citing Vygotsky & Luria, 1994). (Radford 2008c, pp. 223-224) The second important source for mathematics learning is social interaction as part of mathematical activity (also Radford 2003a). Specifically, working with others who are able to “read” the mathematical meaning embedded within mathematical artifacts, enact the related cultural forms of thinking in the course of activity, and can help the learner to objectify it. This is not simply a process of appropriation and/or assimilation of the mathematics. The process of knowing in common or knowing-with-others is the very process through which human cognitive capacities, including consciousness, are formed (Leont’ev, 1978, Radford 2008c). Mathematical thinking as a dialectical cultural reflection and subjectification As mentioned earlier, thinking “is a re-flection, that is, a dialectical movement between a historically and culturally constituted reality and an individual who refracts it” (Radford 2008c, p. 219). This process is mediated within activity by the norms of practice and conventional roles or division of labour. From this perspective, mathematics learning or objectification is seen as “the elaboration on the part of the student [learner] of a reflection defined as a communal and active relationship with his/her cultural-historical reality” (Radford 2008c, p. 226). And, as the learner comes to objectify mathematical knowledge in the course of reflexive mathematical 37 activity, he or she inevitably finds him/herself changed as he/she learns to be in particular mathematical activities, a process that Radford terms subjectification. In short, learning involves both a process of coming to know and a process of becoming. Semiotic means of objectification In the course of mathematical activity we regularly use semiotic means to draw and sustain the attention of others and our own attention to particular aspects of mathematical objects in an effort to achieve a stable form of awareness, to make apparent one’s intentions, and/or to carry out actions to attain the goal of one’s activity. We draw upon a wide variety of semiotic means, often in concert, such as: gestures, signs, graphs, formulas, tables, drawings, words, calculators, rules, and so on (also Radford 2003, 2005b, 2006b, 2008c, 2009b), a process that Radford calls semiotic means of objectification. The first feature of semiotic-mediated learning activity to be discussed is iconicity. Iconicity is a link between past and present action: it refers to the process of noticing and re- enacting or re-voicing significant parts of previous semiotic activity for the purpose of orienting one’s actions and deepening one’s own objectification (Radford, personal communication, September 29, 2008). Radford (2008b) explained this in more detail: It is the process through which the students draw on previous experiences to orient their actions in a new situation. In other words, iconicity is based on the projection of an earlier experience onto a new one—a projection that works on the progressive identification of the similar and the different and that makes possible, through a back and forth movement, the emergence of the second conceptual form (here [in the cited paper] a generalizing procedure). … But iconicity is not a mere matter of logic. It is first of all a matter of making apparent the relevant similar. (p. 94) Radford’s research with school students (2002, 2008b) provides examples of two forms of iconicity, 1) the re-enactment of a mathematical explanation using the same semiotic system as it 38 was originally presented and 2) re-enactment using a different system. In both of these cases the initial semiotic acts being re-enacted originate in the actions of another individual. Another prominent feature of semiotic-mediated learning activity is a theoretical construct that Radford and others refer to as semiotic nodes (Radford, 2005b, 2008b, 2009b; Radford, Demers, Guzmán, & Cerulli, 2003). These are places in mathematical activity where multiple semiotic systems are used together and in a coordinated manner to achieve knowledge objectification. “Since knowledge objectification is a process of becoming aware of certain conceptual states of affairs, [changes in] semiotic nodes are associated with the progressive course of becoming conscious of something. They are associated with layers of objectification” (Radford, 2005b). To illustrate the dynamic nature of semiotic nodes Radford (2009b) provided an example of grade ten students working together to make sense of a graph. Over the course of a mathematics lesson, the students’ actions—their gestures and language became more meaningful as they move towards achieving a stable awareness of the meaning of the lines on the graph that the teacher has given them. And, by the end of the session the semiotic node has shifted even further towards the use of language, with fewer actions and gestures required. In Radford’s words, “ The centre of gravity [of the semiotic node] leans towards language” (p. 122). The example above also highlights a third feature that is central to semiotic-mediated learning activity, namely semiotic contraction. This is the process of coming to recognize and attend to the essential elements within an evolving mathematical experience; and making one’s semiotic actions compact, simplified, and routine as a result of this acquaintance with conceptual traits of the objects under objectification and their stabilization in consciousness. This is, indeed, necessary for attaining knowledge: semiotic contraction is the means by which we limit our attention to the relevant details. As Radford (2008b) explained, “In general, contraction and 39 objectification entail forgetting. We need to forget to be able to focus. This is why to objectify is to see, but to see means at the same time to renounce seeing something else” (p. 94). Methodological implications of the theory of knowledge objectification The use of semiotic means of objectification is both central to and ubiquitous within mathematics learning. These actions are also readily noticeable both by participants of and outside observers to the activities within which these occurs. It is for these reasons that the process of mathematics learning can be accounted for readily by the analysis of social interactions and the use of semiotic means of objectification within an activity. The theory of knowledge objectification, in Radford’s (2008c) words, “provides a broad, but sufficiently specific, frame with which to track students’ progressive acquisition of cultural forms of mathematical being and thinking” (p. 227). Research on workplace mathematics and mathematics learning for the workplace As mentioned earlier, the existent research related to workplace or vocational mathematics has focused, for the most part, on 1) the use of mathematics in specific workplaces or vocations for the purpose of informing school mathematics curriculum design and the teaching of the mathematics needed for work (e.g., Eberhard, 2000; FitzSimons, 2005; Hogan & Morony, 2000; Hoyles, et al., 2002; Masingila, 1994; Millroy, 1992; Ridgway, 2000; Smith, 1999a & 1999b; Wake & Williams, 2001; Williams, Wake & Boreham, 2001; Zevenbergen & Zevenbergen, 2004, 2009); 2) the unique features of the mathematics practices within specific workplaces including the use of workplace specific conventions and artifacts and/or the connection between local workplace knowledge and these mathematics practices (e.g., LaCroix, 2002; Martin & LaCroix, 2008; Martin, LaCroix & Fownes, 2006; Mercier, 2000; Noss, Hoyles 40 & Pozzi, 2000; Pozzi, Noss & Hoyles, 1998; Williams & Wake, 2007a, 2007b, Williams, Wake & Boreham, 2001) and, very much related to both of these, 3) the problematic notion of transfer of school mathematics competencies to the workplace (e.g., Evans, 2000; Noss, Hoyles and Pozzi, 2000; Williams & Wake, 2007a; also see Carreira, Evans, Lerman & Morgan, 2002). The predominant focus in this work has been on workplace practice and when activity theory has been cited, it has often been used simply to frame descriptions of the features of the mathematics practice (norms, tools, division of labour, etc.) and illustrate how these are different from features of school mathematics practice (e.g., Fitzsimons, 2005; FitzSimons, Mlcek, Hull, & Wright, 2005; Pozzi, Noss, & Hoyles, 1998; Wake & Williams, 2001; Williams, Wake & Boreham, 2001). FitzSimons, Mlcek, Hull and Wright (2005) have provided the following concise summary of the existent international research findings on workplace mathematics practice. These studies show that mathematical elements in workplace settings are subsumed into routines, structured by artefacts like workplace texts and tools, and are highly context-dependent. The mathematics used is intertwined with technical expertise at all occupational levels, and judgements are based on qualitative as well as quantitative aspects. Unlike students in the majority of traditional school mathematics classrooms, workers are generally able to exercise a certain amount of control over how they address the problem-solving process. Finally, because the focus is on task completion within certain constrains (for example, time, money), mathematical correctness or precision can be somewhat negotiable, depending on the situation at hand. (p. 20) In comparison to the research on mathematics use and practice in the workplace, very little research has been reported that focuses explicitly on mathematics learning in workplace training. Martin, LaCroix and Fownes (2006) and Martin and LaCroix (2008) (as part of the UBC Workplace Numeracy Project mentioned in chapter one), for example, analyzed the mathematics thinking of a group of ironwork apprentices as they work collaboratively to complete a single workplace simulation task within the classroom component of their formal apprenticeship training at a technical college. The task that these apprentices worked on involved 41 the determination of the sling size needed to lift an assembly of iron beams and, therefore, the weight of this assembly based upon the specifications of the components provided on a standard technical drawing. Using the Pirie-Kieren theory for the dynamical growth of mathematical understanding (Pirie & Kieren, 1994; Kieren, Pirie & Gordon Calvert, 1999) as the basis for analyzing the mathematics understanding of the apprentices as they worked on the problem, the authors concluded that proficiency in mathematics-for-working depends upon having the right kinds of mental images related to the task at hand and applying them in a coordinated and dynamical way. Three discrete kinds of images used by apprentices in their collaborative problem solving effort were identified as essential parts of this process. They are 1) abstract and general images from formal mathematics that are assumed to be transferable, 2) an image of the problem task, and 3) images of workplace practices that include common ways of working, materials, and mathematical conventions. The identification of these three different kinds of images was an elaboration of the Pirie-Kieren theory. Martin and LaCroix (2008) also described the problem-solving process evident within the spoken dialogue of the three apprentices as a trajectory that moves back and forth between these three different kinds of images. In another set of papers from the UBC Workplace Numeracy Project (Martin, LaCroix & Fownes, 2005a, 2005b, 2005c; Martin, LaCroix & Grecki, 2004) Martin et al. examined the mathematical understandings of a pre-apprentice at a trade-union run school in British Columbia as he worked on a pipe fitting calculation and then attempted to locate a particular measurement on his measuring tape with this researcher serving in the role of math tutor.3 Here again the Pirie-Kieren theory was used to frame the analysis. The focus in this set of papers was on the 3 The tutoring session referred to in these papers took place during the first of the two sessions of a pre-apprentice program in the pipe trades that I visited as part of the UBC Workplace Numeracy Project. The episode that I have use for my dissertation comes from the second session of this course that I visited. This will be described in detail in chapter three. 42 student’s mathematical image or, more precisely, the limitations or inadequacies of his images. And, while mention is made of the importance of folding back—the process, according to the Pirie-Kieren theory, through which understanding is observed to grow—no details were provided of the specific actions undertaken by the student and the tutor as they worked through this process while navigating the layer of image making posited by the theory. The unit of analysis in the research by Martin et al. described here was the images constructed within the minds of the students as inferred by the researchers from their actions and discourse. An idealist and rational view of formal mathematics served as the basis for interpreting the mathematics involved and the Pirie-Kieren theory was used to interpret the level of the students’ understanding and categorize their thinking into two discrete categories, image making or image having. From the standpoint of a cultural-semiotic activity theory perspective, the result is an interpretation of mathematical thinking and learning that is inattentive to relationships between culture and mathematical thinking that, consequently, results in a severely restricted view of specific processes through which mathematical meaning is developed, enacted, and sustained by human activity within a cultural and material world. Another study that includes a focus on mathematics learning for the workplace— FitsSimons, Mlcek, Hull and Wright’s (2005), Learning Numeracy on the Job: A Case Study of Chemical Handling and Spraying—examined mathematics practices and learning within the Australian horticulture and agriculture industries. The intent of this study was to inform government policy on the mathematics needs of the workers in this sector of the workforce as well as the design of mathematics instruction for them. These authors concluded that within this context: Learning on the job is largely experiential, with opportunities provided for workers to become ‘encultured’ into communities of practice through interrelationships with other employees. Supervisors are often involved in initial training and check regularly on work practices. Most workplaces place a strong 43 emphasis on ensuring that workers are prepared to check before acting, of being unafraid to ask a ‘dumb’ question. (p. 17) The primary focus of this report was on the mathematics practices within this workplace sector. And while the study also considered workers’ mathematics learning on the job, it did so only very general terms. Summary Much of the existing research on workplace mathematics in the mathematics education literature has been at a general level, focusing for the most part on the formal mathematics used in the workplace, on the particular features of workplace mathematics practice, or the problem of transfer of mathematics learning from school to work. The research that has addressed mathematics learning in workplace training has been somewhat limited in scope either due to a more general focus on the nature of mathematics training practice, as was the case in FitzSimons et al. (2005), or due to limitations of the theoretical perspective employed, as is the case for the papers by Martin et al. (2005a, 2005b, 2005c). The research, to date, that has used cultural- historical activity theory as the basis for analysis of workplace practice or mathematics learning in the workplace has been limited by a lack of theoretical tools for attending to nuances of cultural meaning and meaning-making at a level of individual learners. Radford’s recently developed theory of knowledge objectification, an elaboration of activity theory, provides a conceptual framework for moving beyond limitations of the existing research relating to mathematics learning in workplace training. It provides a comprehensive and well defined set of concepts and methodology for understanding, in precise ways, different forms of culturally mediated mathematics practice and learning within different contexts (such as various workplaces), each as a distinct and legitimate form of mathematics activity. Thus, the use of activity theory along with Radford’s theory of knowledge objectification to analyze 44 mathematics practice and learning within workplace training has the potential to provide valuable insights for understanding mathematics practice and learning in these contexts. 45 CHAPTER THREE: RESEARCH METHOD/OLOGY This chapter provides a description of the research methodology for this study of a pre- apprentice learning to measure with an imperial measuring tape. It begins with a description of the implications of activity theory and the theory of knowledge objectification for research methodology generally and the analysis of mathematics learning in particular. This is followed by a description of the larger study of mathematics learning in workplace training from which the data for the present study were drawn. A description of the design of this study follows. This includes: the process used in selecting the episode from the larger study that serves as the primary data for the present study along and a description of the research subject and context. The chapter concludes with a detailed description of the multi-semiotic analysis used to reflect the sensual4 and mediated view of mathematical thinking and knowledge objectification that underpins this work. Methodological implications of activity theory for research Both parts of the term activity theory, referring to the Soviet-originated cultural- historical research tradition, are slightly misleading because the tradition is neither interested in activities in general nor is it a theory, that is, a fixed body of accurately defined statements. (Kuutii, 1996, p. 25) The term activity theory encompasses a complex and evolving family of theoretical perspectives on human practice that have direct methodological implications for research (Engeström, 1993). The ontological, epistemological, and therefore methodological commitments of activity theory stem from its roots in Marx and Engel’s conception of dialectical materialism, which suggests that human nature (consciousness and behavior) forms, develops, and resolves not within the human individual or spirit, but rather in the dynamic interplay 4 The term sensual refers here to employing the senses as in the process of objectification. 46 between individuals and their co-evolving material worlds in the course of tool and/or artifact creation and use (Cole & Scribner, 1978; Engeström & Miettinen, 1999). Nardi’s (1996a) description of activity theory highlights the connections that activity theory has with its dialectical materialist heritage as well as a number of features of its subsequent development: Activity theory incorporates strong notions of intentionality, history, mediation, collaboration and development in constructing consciousness. Activity theorists argue that consciousness is not a set of discrete disembodied cognitive acts (decision making, classification, remembering), and certainly it is not the brain; rather, consciousness is located in everyday practice: you are what you do. And what you do is firmly and inextricably embedded in the social [and cultural] matrix of which every person is an organic part. This social matrix is composed of people and artifacts. Artifacts may be physical tools or sign systems such as human language. Understanding the interpenetration of the individual, other people, and artifacts in everyday activity is the challenge activity theory has set for itself. (p. 7) In practice, activity theory provides a set of powerful conceptual tools and general methodological principles for the analysis and comparison of human practices as situated, mediated, and developmental processes within a wide variety of disciplines such as education, sociology, technology studies, mental health, workplace practice, and organizational learning (e.g., Engeström, 1993, 2000, 2001; Engeström, Miettinen & Punamaki, 1999; LaCroix, 2009; Nardi, 1996b; Radford, 2008c; Radford, Bardini & Sabena, 2007; Roth, 2003; Williams & Wake, 2007a; Williams, Wake & Boreham, 2001). The task of applying these tools and principles in any particular research context depends upon the particular activity being investigated and is left to the researcher (Engeström 1993; Kuutti, 1996; Nardi, 1996a). Engeström (1993) explained that “methods should be developed or derived from the substance as one enters and penetrates deeper into the object of study” (p. 99). And, Vygotsky (1978), who established the foundations of activity theory, asserted that, The search for method becomes one of the most important problems of the entire enterprise of understanding the uniquely human forms of psychological activity. 47 In this case, the method is simultaneously prerequisite and product, the tool and the result of the study (p. 65). This description of search for method reflects the dialectical worldview that is central to an activity theory perspective. De/constructing activity Like ethnography, research from an activity theory perspective focuses attention on people interacting together within a cultural context that can include the use of various forms of language and other semiotic systems, particular conceptions of the world, the use of physical artifacts, and other features of situated practices. Activity theoretical research differs from ethnography in its foregrounding of consciousness and how it situates its interests in the motives of individuals, and thus also easily accommodates time series analysis in the analysis of learning. When analyzing activity, activity theorists generally identify or deconstruct participants’ conscious goal directed actions, and subconscious ways of responding to their immediate circumstances called operations, as well as the physical tools and semiotic resources, rules or norms of practice, and divisions of labour that mediate these processes. From patterns in observed actions and operations, researchers identify an individual or collective subject enacting a particular set of goal directed actions and operations which, as part of a larger and orchestrated set of actions and operations carried out within a community, are oriented collectively towards an outcome producing object that satisfies a particular human need or desire. It is the object/objective seen as an historical form of human practice that defines an activity. The subject, community, mediating tools, rules or norms, division of labour, and the object towards which these elements are oriented as well as the interactions of these elements, each constituted in dialectical relationships with one another, make up an activity system. Different activity 48 systems share elements and interact within dynamic and evolving networks of multiple activity systems (Engeström, 1987, 2001; Engeström & Miettinen, 1999). As mentioned in chapter two, Engeström (2001) summarized a number of principles of activity theory, three of which have direct implications for conducting research on the study of mathematics practice and learning as a culturally situated and a thoroughly mediated process. These principles will now be reviewed and elaborated upon in terms of their relevance for conducting research into mathematics learning. The first of these principles is that a collective activity system is taken as the unit of analysis for research. In practical terms, the researcher’s own objectification of an activity system requires sustained observation and interpretation of actions and operations over time in order to ascertain the constituent interacting elements of the activity and their attributes, while at the same time identifying the object/objective that defines the particular activity system. The second principle recognizes that an activity system is always a community of multiple points of views, traditions and interests. Consequently, the researcher is obliged to understand an activity system from the perspectives of various participants in order to develop an understanding of its complexity. The third principle is that contradictions within and between the dialectically constituted elements of an activity system and between activity systems play a central role in its/their change and development. These contradictions are manifest as problems, ruptures, breakdowns, and clashes within activities (Kuutti, 1996). Given that activity systems are constantly evolving, it is therefore essential to analyze these contradictions in order to understand this dynamic. Radford’s (2008c) elaboration of activity theory—the theory of knowledge objectification—elaborates the concept of activity just discussed and, in turn, adds further methodological implications for the analysis of this type of activity. To start, Radford positions the concept of semiotic systems of cultural signfication as an essential part of mathematics 49 practice and learning. As outlined in the previous chapter, this concept includes, but radically expands, the traditional notions of rules and norms within activity theory to include cultural conceptions of ontology, epistemology, sanctioned means of knowledge production and rules of validation—as they apply to mathematics practice and learning. Radford (2008c) also introduces the concept of territory of artifactual thought to the activity of mathematics practice and learning. This positions mathematics thinking, and thus mathematical activity, squarely within the arena of a subject acting, thinking, and feeling through his or her enactment of cultural-object oriented actions and operations mediated by semiotic systems and material artifacts. A ubiquitous feature of mathematics communication and learning highlighted by Radford is the concept of semiotic means of objectification. This refers to the active and multi-semiotic means of bringing the cultural logic behind a mathematical object into one’s own consciousness, one’s consciousness through the consciousness of another, or into the consciousness of one another. Attending to the semiotic means of objectification used within a mathematics activity provides a clear and readily accessible means for tracking the changing mathematical consciousness and thus the mathematics learning of individual participants within the activity. For the researcher, this includes attending to: (a) forms of iconicity—the process through which individuals notice and re-enact significant parts of previously experienced semiotic activity for one’s own objectification as well as for the objectification of others; (b) semiotic nodes—those places in mathematical activity where the coordinated use of multiple semiotic systems is used to achieve knowledge objectification as well as the changes in these over time; and (c) semiotic contractions—the process of making semiotic actions compact, simplified, and routine as one becomes more and more familiar with their meaning. Lastly, Radford’s concept of subjectification—the process through which learners find themselves 50 changed as they learn to be part of particular mathematical activities—also directs researchers to attend to the changing relationship of individuals to the activities they are part of. It is not uncommon in activity theory research that researchers are recognized as participants within the activities they are investigating, either as part of the collective subject of the activity or as a member of the community within which the activity occurs (e.g., see Engestrom, 1993). This stems from the view that all of the participants connected to an activity, including a researcher, are unavoidably related in a dialectical way with the other parts of the activity system. This recognition allows for activity theory to be used for analyzing and understanding events as they occur without the need to engineer situations for data collection in ways that enable the researcher to argue that his or her presence has been controlled or rendered insignificant as the case in other research traditions. This ecological validity provided by activity theory is indeed one of its strengths, provided that the researcher is up front in addressing his or her involvement as part of the analysis. Study design This qualitative study employs an emergent design to examine the measuring activity of a pre-apprentice within a pipe trades training program with a particular focus on a single impromptu tutorial session with the researcher serving as his mathematics tutor. It focuses on the manner in which the pre-apprentice attempts to make sense of and become fluent with the mathematics embedded in a measuring tape marked in feet and inches—an essential skill for the pre-apprentice’s chosen vocation. 51 Background to the present study The present study uses data collected as part of a multi-year study of mathematics practice and learning within workplace training—The UBC Workplace Numeracy Project. This was a Social Sciences and Humanities Research Council of Canada funded research project on mathematics learning within workplace training in the construction trades under the direction of Dr. Lyndon Martin, in partnership with The BC Construction Industry Skills Improvement Council (also known as SkillPlan). My role in this project included management of the fieldwork logistics and conducting all of the fieldwork over a two-year period. Subsequently to this, I worked with Dr. Marin analyzing students’ mathematical understanding evidenced within selected episodes from the data (albeit using a different theoretical perspective than that of the present study). It was understood from the outset of the project that I would use data from it as the basis for this dissertation. It should be noted that all of the individuals from whom video data or artifacts were collected had given their informed consent for inclusion in this research project and this research project had been approved beforehand by the UBC Office of Research Services. In the early stages of this project I worked closely with SkillPlan staff to learn about issues relating to mathematics teaching and learning within the apprenticeship programs for a variety of construction trades and current teaching practices within these programs. The staff and SkillPlan also used their contacts to provide me with access to a number of these programs located within the Lower Mainland region of British Columbia for conducting field work. I visited a variety of apprenticeship programs from May 2003 until March 2005 and throughout the fieldwork portion of the project I consulted with SkillPlan staff regularly regarding questions that arose from my work. Shorter visits spanning one to a few days were made to each the following apprenticeship programs either at trade union run schools and at a local technical 52 college: operating engineers—crane operation, heat and frost insulation, HVAC—heating, ventilation and air conditioning, and electrician. Two full days were also spent in the field shadowing workers in the HVAC5 service industry. This included discussing the mathematical aspects of their work as my hosts carried out their regular duties. Extended visits were made to a pre-apprenticeship pipe trades course (for two separate sessions of this course) offered at a trade union run school, a level four plumbing apprenticeship course (that focused largely on gas fitting) offered at a technical college, and all levels (combined entry-level/year one, year two, and year three) of an ironworker apprenticeship program also offered at the technical college. These visits spanned the duration of these courses (between five and 24 weeks) and were organized to coincide with the occasions when the students were doing mathematics related tasks as part of their regular classroom work or in their practical work outside of the classroom. While my primary role during the extended visits to workplace training classes was to observe whole class, small group, and individual student’s mathematics related activity, I also observed much of the other classroom and practical work that the students did as well. I took field notes relating to the mathematics practices that I observed both in the classroom and workshop, worked through many of the assigned mathematics-related tasks myself, video recorded students doing or talking about mathematics on their own or with me, and made myself available to anyone who wanted help with their assigned coursework. My video taping initially targeted a variety of students in each class and I always made an effort to capture substantial parts, if not all, of students’ work on the mathematics tasks they were completing or explaining to me. I also spent time on a regular basis during coffee breaks and lunch breaks with a variety of 5 It is not unusual in the heat, ventilation and air-conditioning (HVAC) industry for a crew to include members from different trades and this was the case for the crew that I visited. This particular service crew included credentialed tradespersons from: HVAC, plumbing, gas fitting and steam fitting. 53 the students and some of the course instructors to help orient us to one another, to learn about their related experiences outside of the training program that I was observing, as well as anything else that might inform my understanding of these training programs. This served to develop my own subjectivity as a participant and intersubjectivity with other participants within the activity that I was investigating. During my visits to these workplace training programs I documented much of the mathematics related activity in the classrooms and practical work areas using a video recorder. In total, over 70 hours of video recordings were made of students engaged in mathematics related activity. Copies of classroom pencil-and paper materials, copies of some students’ written work and other print resources used (both class handouts and published course materials) were also retained for future analysis. Back at the university all of the video recordings were converted to MPEG format and saved, reviewed by another graduate research assistant or myself, and a summary of the contents each video file documented in a database. During this process rich episodes of mathematics activity by the participants were also flagged for further analysis. All copies of the classroom print materials and copies of students’ written work collected were also catalogued so that they could be cross-referenced with the video data as needed. An additional 15 hours of one-on-one informal interviews with a variety of apprentices, course instructors, and course administrators where we discussed a variety of issues related to mathematics practice and learning within workplace training courses and in the workplace were audio or video recorded as well. The pipe trades pre-apprenticeship program, data collection, and data selection The set of data that served as the primary date source for this study was collected from the second session of the pipe trades pre-apprenticeship program visited over an eight-week 54 period in May and June of 2004. This program was designed to give the pre-apprentices a head start with important skills that would be addressed subsequently in the early years of their formal apprenticeship training in a number of different pipe trades. It involved an intensive program of pencil and paper work in the classroom as well as practical work in the workshop. All but one of the 15 students in the class were high-school graduates and most had full-time work experience before entering the program. Upon successful completion of this pre-apprenticeship program the students would be placed in an entry-level position to begin their formal apprenticeship in one of the pipe trades mentioned previously. After approximately four years of workplace experience, four levels of formal apprenticeship coursework (six to eight weeks per year beyond the target pre- apprenticeship course), and successful completion of a written exam set by the British Columbia Industrial Training Authority, the apprentices would become credentialed tradespersons. The course instructor was himself a qualified steamfitter with many years of experience in the field and he and I had established a good working relationship during the previous session of the pre- apprenticeship course that I visited. While this course was not part of the formal apprenticeship credentialing process, the intent was to address some of the more challenging content covered in years one and two the formal apprenticeship curricula for pipe trades apprenticeships set by the British Columbia Industrial Training Authority. This situation provided flexibility for the instructor to place additional emphasis on the mathematics content, which he did to enhance my opportunities for data collection within his course. The selection of the second pipe trades pre-apprentice class for the present study was based on convenience from the six class sets of data that were available from the larger study. In comparison with the activity in the other classes that I investigated, this particular class provided both a higher proportion of class time spent on mathematics related work and considerably more 55 video data of individual students and small groups of students working to make sense of the mathematics-related tasks assigned to them. The pre-apprentices in this particular class also engaged much more readily with me than did the apprentices in any of the other classes I observed. This was, in large part, because they were new to this program and had yet to establish an identity for themselves as a member of their trade. As a consequence, they did not view me as an outsider. Furthermore, given that I had the experience in the course (having worked with pre- apprentices in a previous section of this pipe trades pre-apprenticeship program) and had an established working relationship with the course instructor, the pre-apprentices were very receptive to my suggestions to help them with their work. In stark contrast, I was clearly regarded as an outsider by the apprentices in each of the level two and three ironworking classes and the level four plumbing class when I began my observations there. My modus operandi in this pre-apprenticeship pipe trades class was to observe and video- tape the mathematics-related whole-class sessions led by the course instructor, to video-tape the activities of small groups of students and individual students as they completed the mathematics- related tasks assigned to them in both the classroom and workshop, to engage students in discussions about their mathematics-related work and their thinking as they worked through these tasks, and to assist any student who sought me out for this. For the purpose of this study, a mathematics-related task or set of tasks was considered to be any task assigned to students that required a multi-step mathematical process or calculation. Throughout this session of the pipe trades pre-apprenticeship course I was particularly active as a mathematics tutor at a help table that I had established at the back of the classroom for any student(s) who wanted my help or who wanted to work with the other students who were seated there. During much of the in-class work time between one and eight pre-apprentices worked at the help table and I recorded much of their activity using a tripod mounted video 56 camera and a tabletop microphone. Occasionally, I was joined by Sue Grecki, a veteran workplace educator from SkillPlan, who also assisted pre-apprentices with their work. At other times I left the pre-apprentices to work on their own while I continued to video record their efforts. Approximately 35 hours of the mathematics-related activity from this class were documented on video. Copies of all of the print materials used in the course, field notes that I recorded during many of the classes, copies of some students’ written mathematics-related coursework, and background information on individual students were also retained for analysis. After the initial review and summary of the data collected from this class, the print materials handed out to students (which included all of the mathematics related tasks assigned to them during the course) were reviewed to identify the different kinds of mathematics related tasks that were worked on during the course. This analysis revealed a set of relatively well- defined mathematics-related tasks. These tasks were then categorized according to the type of mathematical objects involved and then all of the episodes of mathematics activity on the video recordings were coded according to this set of categories. The initial analysis of this coding revealed a largely fragmented collection of different individuals and small groups dealing with different aspects of the mathematics related tasks assigned to them, often using very different approaches of their own choosing. At this point I decided that the single 33 minute episode of the impromptu tutoring session with a pre-apprentice (who will henceforth be referred to as “C”) provided a uniquely rich, intensely focused, and sustained episode of mathematics learning for analysis. Furthermore, the focus of this episode—on understanding the cultural logic behind an imperial measuring tape—reveals the complexity of and the difficulties in learning the mathematics behind reading a measuring tape, a ubiquitous part of measurement activity in the construction trades generally, and the pipe trades in particular. 57 The research subject and research setting C, the individual whose activity is the focus of this analysis, had completed all of his K- 12 schooling within the B.C. public school system and he was a secondary school graduate. As part of his secondary schooling C completed mathematics to grade eleven in the academic stream. During the three and a half years between the time that he finished secondary school and the time he began the pre-apprenticeship program in the pipe trades C had been in the workforce and completed a small number of courses in an electronics-technician training program at a community college. Throughout the pre-apprenticeship course he regularly and eagerly sought me out for help with his mathematics related work and we had quickly developed a relaxed and productive working relationship. On the sixth day of the course, two days before the target episode took place, the students were assigned their first hands-on fabrication task in the workshop—to construct a screwed-pipe assembly based on a technical drawing provided by the instructor. This required the pre- apprentices to determine the cut pipe lengths for the straight pipe components of the pipe assembly, taking into account the fitting allowances for the pipe fittings (e.g., elbow- and tee- fittings) that were to connect the straight pipes. Starting with the specifications given for a pipe assembly shown on the technical drawing, the students were to calculate the required straight pipe lengths to the nearest sixteenth-of-an-inch, cut and thread these, and then assemble the various components. To this point, most of the students’ mathematics-related coursework had been classroom based, focusing on pencil and paper measurement calculations, conversion between measurements values in inches and fractional parts-of-an-inch and feet and decimal parts of a foot, and the calculations used in the construction of various types of pipe assemblies including those for threaded pipe. It should also be noted that this pre-apprenticeship program was the first time that C had encountered working with imperial linear measure in inches and 58 fractions-of-an-inch and to this point in the class there had been no dialogue in the class pertaining to reading fraction-of-an-inch measurements on a measuring tape. After a less-than-successful first attempt, a number of students, including C, were reviewing their calculations together at a table in the workshop in preparation for their second attempts to build their pipe assemblies as the target episode begins. While observing this discussion it became apparent to me that C was having difficulty locating 11 1/4 inches on his measuring tape. After a short dialogue with him at the workbench we moved to a table in a classroom where we were able to work more easily and without interruption. Data analysis The present study is based on a multi-semiotic analysis of the joint activity of C and L–the researcher as tutor–during their one-on-one tutoring session from the video recording of this episode as well as other artifacts retained from this encounter. The focus is on C’s process of objectification of the fractions-of-an-inch marking pattern inscribed on his measuring tape. Key features of this process are identified and explained using concepts from activity theory and the theory of knowledge objectification. As mentioned earlier, this impromptu tutoring session took place at a table in a classroom immediately after L discovered that C was having difficulty reading fractions of-an-inch from his measuring tape while working on the pipe fitting project with his colleagues in the workshop. The particular measuring tape being used was marked to thirty-seconds-of-an-inch below 12 inches or one foot, and to sixteenths-of-an-inch thereafter. These two marking patterns can be seen in Figure 4. 59 Figure 4. Change in smallest increment size on the measuring tape at twelve inches The smallest increments to the left of one foot are marked to thirty-seconds-of-an-inch. On the right side of one foot the smallest increments are marked to sixteenths-of-an-inch. (C has inscribed a line across the measuring tape with his pencil at 11 1/8”, partly obscuring the measuring tape inscriptions, and another short line over the marking at 11 5/32”.) Once this target episode was selected, a verbatim transcript of the session was written with corresponding time references from the video recording documented on a regular basis throughout to allow for cross-referencing. The transcript also includes detailed descriptions of the other actions carried out by both participants, such as other utterances, gestures, body position, as well as the use of artifacts. At times the process of analyzing the video involved slow motion and frame-by-frame analysis to determine the precise ways that spoken language, artifacts and gestures were used and coordinated. Annotated screen shots of the action were included throughout the transcript to help provide a detailed account of the use of artifacts and gestures that were integral parts of the events taking place. The completed transcript was then reviewed thoroughly from start to finish for accuracy and revised as needed, with further adjustments for accuracy made throughout the analysis process. A copy of this transcript is provided as an appendix. At the outset my formal analysis involved an initial and tentative formulation of the activity-theoretical elements of C and my shared activity. This involved writing descriptions of the corresponding elements of the activity including: the participants, their motive and goals, the object of the activity, the tools used to mediate the activity, the norms enacted, and divisions of 60 labour. This formulation was reviewed and refined on an ongoing basis throughout the entire analysis process. In the next phase of the analysis, I interpreted the goal or purpose of each utterance in the dialogue relative to the goal of learning to measure with the measuring tape and documented this in writing.6 At this stage of the analysis I also identified all instances of semiotic means of objectification used, either by C or myself, and coded each according to the semiotic system involved. I also identified instances where multiple semiotic systems were used together or between the two of us; as well as the use of artifacts and the enactment of, or reference to, divisions of labour and enacted norms or rules. These norms were also categorized as either norms of plumbing practice, norms of apprenticeship training, or norms of general mathematics practice. Throughout the entire analysis process, as my own objectification of activity theory and data analysis using activity theory developed, I went back, reviewed, and revised earlier analysis on an ongoing basis to ensure the consistency and quality of all parts of it. The next step of the analysis involved the identification of the ways that C’s and my actions related to different aspects of reading the mathematical patterns within the measuring tape. This analysis was based upon the original video recording, the retained artifacts from the tutoring session, as well as the transcript. From this analysis, the entire tutorial session was subdivided into a sequence of 28 goal-specific sets of actions. These episodes lasted anywhere 6 The analysis of the goals or purposes served of each of my verbal responses and other actions within the context of the activity of the tutorial session (documented in the 440 lines of the entire transcript) indicates that all of these served my goal of helping C to measure with the measuring tape. On only six occasions in my line-by-line analysis of these actions, did I interpreted specific utterances as possibly serving a secondary purpose as well, that being the purpose of documenting our activity as part of my simultaneous activity as researcher. In all six instances I sought to makde C’s actions and/or his intentions more explicit to me and, in turn, for the video recording that I was making of the tutorial session. Given the small number of actions related to this second activity, as well as their compatibility with and inclusion within the analysis of C’s learning activity (the primary focus of this study), the analysis of my activity as researcher during the tutorial session will not be addressed here. 61 from six seconds to over three minutes. The goals of the participants within the episodes were then analyzed and the sequence of episodes were reconceptualized as a series of eleven distinct action sets that were then written up as a series of vignettes or narratives. These vignettes, which are detailed in chapter four, account for all of the mathematical activity that took place during tutorial session and highlight the significant elements of this mathematics learning activity. Ten different mathematical relationships that make up layers of, or are aspects of, the mathematical object—the fractions-of-an-inch pattern on a measuring tape—were also identified and tracked to determine where they were referred to during the tutorial and the most prominent of these were recast and are summarized as three mathematical threads that run through the eleven vignettes. The second major phase of the analysis, presented in chapter five, draws upon the micro- analysis provided in chapter four. It starts with the detailed formulation of C and L’s joint activity system from a cultural-historical activity perspective including analysis of mediating relationships within this activity. It then draws upon concepts from the TO to analyze this mathematics learning process during the tutoring session in greater detail. Specifically, this includes analysis of the semiotic system of cultural signification; the territory of artifactual thinking and semiotic means of objectification including: iconicity, semiotic nodes, and semiotic contractions; and the process of subjectification for both C and L. During this part of the analysis attention is also paid to processes evident within the data that are not yet accounted for by the TO. Finally, the analysis returns to a significant component of any activity from the perspective of cultural-historical activity theory by focusing on conflicts within and between various elements of the activity during the tutoring session. 62 The role of the researcher in this learning activity As mentioned earlier, I served in the role of mathematics tutor for all of the students in the class throughout this pre-apprenticeship course and, in particular, for C during the one-on- one tutoring session that serves as the basis for this study. From an activity theory perspective my involvement within the activity being investigated is not considered to be problematic as it is in many other forms of research. I address my involvement as tutor within this shared activity with C by including analysis of my own actions throughout the entire tutorial session and by including analysis of my own enactment of various semiotic means of objectification and my own subjectification over the duration of the session. It should be noted that during the fieldwork for the larger research project, including the tutoring session from which the data for the present study was taken, my modus operandi was guided by a constructivist perspective on learning generally (e.g., Cobb, Yackel & Wood, 1992) and the Pirie-Kieren theory in particular (e.g., Pirie & Kieren, 1994; Kieren, Pirie & Gordon Calvert, 1999).7 This included attending patiently and carefully to students thinking, attending carefully to breakdowns in their work as places with the potential for revealing their mathematical understandings and seeking clarification or elaboration from them as needed to help both me and them to become clearer about their mathematical thinking. At the same time I made a point of inviting students to elaborate on their thinking in detail, with a minimum of leading from me prior to offering them any suggestions or direction. And, when I did offer help, I worked to provide students with a minimum of direction so that they would be encouraged and allowed to make sense of the mathematics for themselves. For the purpose of data collection, this approach is compatible with both the activity theory perspective of the present study and the 7 The constructivist perspective of the larger research project is reflected clearly in the published papers relating to early studies from this project, e.g., Martin & LaCroix, 2008; Martin, LaCroix & Fownes, 2005a, 2005b, 2005c, 2006; Martin, LaCroix & Grecki, 2004. 63 constructivist perspective of the larger UBC Workplace Numeracy Project in that both perspectives encourage mathematics learning activities in which students proceed as far as they can on their own before the tutor or teacher intervenes. Where these perspectives differ, however, is in terms of the epistemological, cognitive and ethical interpretation of such interventions (Radford, 2009a). 64 CHAPTER FOUR: DATA ANALYSIS I This chapter is the first of two that present the analysis for this study. It begins with an overview of a single class’s mathematics activity in a pipe trades pre-apprenticeship program. It then provides a detailed description the actions that take place during the one-on-one tutoring session along with in-depth multi-semiotic activity analysis of the development and coordination of mathematical meanings relating to measurement. This activity is mediated by C (the pre- apprentice) and L’s (the researcher serving as tutor) social interaction, a number of different semiotic systems and artifacts, and the cultural context of the activity. The mathematics related activities within the pre-apprentice program The mathematics related tasks assigned to the students in this pre-apprentice program are similar to those that they will see subsequently in the early part of their formal apprenticeship training. These tasks reflect common workplace competencies required of tradespersons in the pipe trades. The use of mathematics in serves workplace processes including the production of: bent pipe, threaded pipe, and welded pipe assemblies; rolling off-set pipe assemblies; and calculations relating to the specifications of different types of pipes and fluid storage tanks. The mathematics involved in these tasks includes: conversion between fractions and decimal numbers; conversions between various units of imperial and/or U.S. measure; the use of fractions, mixed numbers, decimal numbers and simple algebraic formulas in calculations involving the determination of length, perimeter, circumference, surface area, volume, capacity, weight, force, and pressure; the use of primary trigonometric relationships and the Pythagorean theorem in calculations involving right triangles; the interpretation of technical drawings, tables, and graphs as well as the inscription pattern on an imperial ruler or measuring tape. Regular use is made of industry standard reference tables found in The Pipe Fitter’s and Pipe Welder’s 65 Handbook (Frankland, 1984), a reference book that students are expected to purchase for the course, as well as other reference materials provided by the course instructor. The students are permitted to use calculators throughout the course with no restriction on the type of calculator used.8 The focus in doing the assigned tasks throughout this course, including the mathematics parts of these tasks, is on instrumental competency, that is, learning to complete the assigned problems and tasks in an efficient and reliable manner so that these same skills can be performed in the workplace. The routine used by the course instructor to teach students how to do the mathematics containing tasks in the course consists of: working through and explaining the steps taken in examples for the whole class at the blackboard or on the overhead projector, answering questions from students that result from these examples, and then assigning similar sorts of practice tasks on worksheets. Getting the right answer quickly and reliably is the goal and the right answer means getting an answer that matches or is within acceptable tolerances to the answer arrived at by the instructor e.g., to the nearest sixteenth-of-an-inch for length calculations; with little or no concern for the problem solving or computational processes used. The assigned tasks on the worksheets given to students are usually sequenced from simple to complex, the students are encouraged to help one another to resolve difficulties, the instructor is available for assistance as needed while the students completed their work, the answers are taken up as a class, and issues raised by students discussed. In contrast to school mathematics—where the goal is to develop students’ skill and understanding with topics chosen for the purpose of developing conceptual understanding of, and general competency with, mathematics topics that 8 This is noteworthy because many newer models of pocket calculators allow values to be entered directly as fractions or mixed numbers and they can display values in these forms as well. Thus, the use of this type of calculator allows a student to bypass the need to be proficient with fraction computation. This has traditionally been a source of difficulty for some students within apprenticeship training. 66 can, in-turn, be applied in a wide variety of contexts (which, of course, includes the workplace)—the goal in this particular workplace training context is on learning to perform a limited and context-specific set of mathematics processes needed for a narrow range of common and well defined pipe trades production activities. C learning to measure with a measuring tape The impromptu tutoring session with C that serves as the basis for this study occurred after the researcher discovered that C was having difficulty reading fractions-of-an-inch from his measuring tape while completing a pipe fitting assignment with his colleagues in the workshop. During the tutoring session (that moves to a table in the classroom and lasts for a total of 33 minutes) C learns to measure fractions-of-an-inch to sixteenths-of-an-inch using a measuring tape. This is an essential competency for this course and his future role in the workplace. In this process, C and L make use of a variety of formal semiotic systems including: spoken and written language; mathematics notation; the pattern of fractions-of-an-inch divisions printed on the measuring tape and on a set of rulers prepared on transparencies for use as a teaching aid by the researcher; and counting.9 A number of informal or impromptu semiotic means of objectification—semiotic systems in their own right—are also employed such as: indexical or pointing gestures; sweeping or hopping gestures through or over intervals on the measuring tape respectively; chopping gestures in reference to the fraction-of-an-inch division lines on the measuring tape; indexical inscriptions such as circling or underlining existing inscriptions; a line drawn to represent a length of five-eighths-of-an-inch; the physical positioning, orientation and 9 Attention to the particular semiotic systems identified here is sufficient for substantiating the claims made within the present study using cultural-historical activity theory and the theory of knowledge objectification. It is fully expected that attending to other semiotic systems that C and L use in the further analysis of this data (e.g. interactional resources from discourse analysis) has the potential to yield additional insights. 67 alignment of the physical objects being discussed; the use of rhythm in voice and gestures; and voice inflection and changes in volume when speaking. Each of these forms of semiotic expression were used as semiotic means of objectification by L and C to foreground the cultural meanings relating to the fractions-of-an-inch divisions on the measuring tape. The analysis here is organized into eleven vignettes from the tutoring session presented in chronological order, each framed around one or more places where C’s objectification of some aspect of measuring with his measuring tape becomes evident. In each vignette, analysis of the learning activity and the semiotic means of objectification used by C and L will be discussed along with a description of C’s developing sense of the cultural meaning of the fraction division pattern on his measuring tape and measurement practice, and his growing sense as an able pipe trades worker. Further analysis of significant elements of C and L’s work together across the vignettes will be presented in chapter five. Significant mathematical threads in C’s learning throughout the tutoring session Analysis of L and C’s shared goals and actions throughout the tutoring session reveal a number of distinct mathematical elements or threads. These mathematical threads are essential aspects of reading fractions-of-an-inch on a measuring tape and figure prominently in C’s objectification of the cultural meaning of the divisions on the measuring tape throughout the vignettes. They are identified here to draw the reader’s awareness to them as they arise in the discussion. These threads include: (a) attending to the characteristics of the physical pattern of fraction divisions on the measuring tape including both the difference in the division pattern below and above twelve inches and the pattern of different fraction-of-an-inch markings within each one inch interval, (b) attending to units and sub-units-of-an-inch as intervals on the measuring tape and seeing the lines marked on the measuring tape as the endpoints of these 68 intervals, and (c) recognizing the different types of fractions-of-an-inch (halves, quarters, eighths, etc.) within the fraction division pattern used on the measuring tape, including the relationship between the name of a fraction-of-an-inch and its size (i.e., the number of these sub- units that span one inch), as well as the relative size relationships between these different types of fractions. Setting the scene At the start of the episode, C and a number of his colleagues are working together at a workbench in the shop on a pipe fitting construction task that had been assigned to them by their course instructor10 and L is assisting them with their work. The students’ shared and immediate goal at this point is to verify their pipe-length calculation results before they cut the pipe components that they will use to build their pipe assembly projects. The activity here is pipe fitting with threaded pipe, a common practice in all of the pipe trades. The pre-apprentices are required to calculate the straight pipe component lengths needed to construct a specified pipe assembly based on the specifications provided on a technical drawing, cut and thread the ends of these, collect the necessary prefabricated fittings (in this case 90 and 45 degree elbows, tee fittings, and a union fitting), and then put them together. The students have their copies of the technical drawing of the piping assignment in front of them on the work bench as well as their plumbing reference manuals, calculators, measuring tapes (one provided for each student to use as part of the course), a ruler, pens, and pencils. After the students explain their processes for determining the various pipe lengths they require and their uncertainty with their calculations for one particular measurement, L reassures them that their answer of 11 1/4 inches is, indeed, the 10 The discourse between C and L that serves as data for this study begins at 24:27 of the video recording (line 51 of the transcript of the video recording). 69 correct length for this a particular component. Then another student, S, marks 11 1/4” on his measuring tape with a permanent marker as he explains that he wants to make sure that he doesn’t “screw up” when he cuts the corresponding pipe length for his project. This had been his fate with his earlier attempt with the same pipe fitting task. L then turns his attention to C who is following S’s lead by marking what he thinks is 11 1/4” on his own measuring tape. Vignette one: Discovering C’s difficulty measuring with the measuring tape The discourse between C and L that forms the basis of this study begins at a table in the workshop with C telling L that he had just “screwed up” while attempting to locate 11 4/16” or 11 1/4” inches on his measuring tape (line 51).11 He explains that he had simply counted one too many increments past eleven inches (five instead of the four he requires) and, having realized this error, C had inscribed a line across his measuring tape with a pencil and ruler at the point that he interpreted as 11 4/16 inches. C then reenacts his counting of five division lines past 11” for L, pointing to each of the individual lines on his measuring tape in turn with the tip of his pencil. He voices the corresponding number for each line as he counts (line 57). This is shown in the sequence of Figures 5 to 9. While C has presumed that the smallest increments on his measuring tape between 11 and 12 inches are sixteenths-of-an-inch (the subunits that he requires) it is important to note that the correct cultural interpretation of these increments (see his measuring tape below in figure 5) is thirty-seconds-of-an-inch. It is also significant to note that C 11 Numbers by themselves in brackets that appear within the excerpts from the transcript of the tutoring session, within the descriptions of the the events that took place, or with the captions for the photographs taken from the video recording refer to the corresponding time [minutes:seconds] on the video recording of the tutoring session. Where multiple video frames have been taken from the same one second interval on the video recording, their order is indicated by the inclusion of a letter or letters (i.e., a, b, c) following the video recording time. The complete transcript of C and L’s discussion is provided as an appendix. 70 Figure 5. C points to the first division past 11 inches C points near the tip of the first division marking past 11” (touching the measuring tape) [25:05b]. Figure 6. C points to the second division past 11 inches C points to the tip of the second division marking past 11” [25:06]. Figure 7. C points to the third division past 11 inches C points to the tip of the third division marking past 11” (hidden underneath the tip of his pencil) [25:07a]. Figure 8. C points to the fourth division past 11 inches C touches his pencil tip at the top part of the fourth division marking past 11” [25:07b]. Figure 9. C sweeps down the fifth division marking past 11 inches C at the end of his precise sweep with his pencil down the entire length of the fifth division marking past 11” (partially obscured by the tip of his pencil) [25:08]. touches the first three of these divisions, which show 1/32”, 1/16”and 3/32”, precisely at their end points (the ends of the lines that are of varying distances away from the edge of the tape). He finishes his counting and series of gestures by moving his pencil tip precisely down the fifth division line (5/32”) as if tracing or re-inscribing this particular marking. To this point, it has been apparent that C knows how to read whole inch values on his measuring tape. He has correctly identified 4/16” as being equal to 1/4”—the fractional amount that he is attempting to locate on his measuring tape, and he knows that four sixteenths-of-an- inch are represented by four lines that can be counted on his measuring tape. C has, however, 71 presumed incorrectly that the lines separating the smallest increments on his measuring tape below one foot indicate sixteenths, the subunits-of-an-inch that he requires. His use of the series of indexical pointing gestures when referring to the lines he has counted, by pointing to the tips of the divisions for 1/32”, 1/16” and 3/32” which are of different lengths on the measuring tape, and his iconic gesture of tracing precisely the marking on the measuring tape for 5/32” line indicate that C perceives that these lines are of varying lengths although this appears to have no mathematical meaning for him at this point. It is unclear from the data thus far if C is consciously aware of fractions-of-an inch as intervals spanning the distances between the different combinations of division lines on his measuring tape, recognizes that different kinds of fractions can be read directly from the measuring tape (e.g., intervals of eighths, quarters, etc. instead of only counting the smallest increments and using equivalent fractions to determine these other amounts), or recognizes the relationship between the sign used to identify the type of fraction (e.g., “eighth” or “1/8”) and the number of subunits of this type of fraction needed to make one whole inch on the measuring tape. These are all essential parts of the thinking needed to read the cultural meaning that is crystallized within the pattern of divisions on a measuring tape. L responds to by calling C’s attention to a number of different aspects of the fraction pattern on his measuring tape. L explains to C that he has located one-eighth-of-an-inch and that the intervals he had counted were thirty-seconds-of-an-inch. L explains that four thirty-seconds are equal to two-sixteenths or one-eighth-of-an-inch while pointing out these intervals on the measuring tape (lines 58 and 60). Then L calls C’s attention to the correct location of eleven and one-quarter inches by pointing to it with his finger and then elaborates with an indexical sweeping gesture using his pencil, that is, moving the tip back and forth between 11” and 11 1/4” and saying, “From there, there to there, four of those intervals will fit in an inch” (line 64). L 72 elaborates further on this essential characteristic of the fraction one quarter by pointing out the four quarter-inch intervals that span the inch between 11 and 12 inches in-turn for C. In doing so, L uses a series of indexical hopping gestures to emphasize these as intervals rather than as a series of points on the measuring tape, moving his pencil tip from endpoint to endpoint over each interval (i.e., 11 1/4”-11 2/4”, 11 2/4”-11 3/4”, 11 3/4”-11 4/4” or 12”). As he does this, L also speaks in a steady rhythm while counting each interval in-turn saying, “[from] there to there’s one, from there to there’s two, there to there’s three, …” (line 64) to call C’s awareness to this pattern of these mathematically similar intervals. C, acknowledges what L says, by restating for himself the goal of the measuring action, “… And we’re looking for one quarter, okay, okay,” (line 67) and then rehearses what L has just shown him by recounting these intervals on his own while pointing to the endpoint of each successive quarter inch interval with his left thumbnail. Vignette two: Noticing the different division patterns on the measuring tape The tutorial session moves from the shop to a table in the classroom and C stares intensely at his measuring tape which is extended on the table top in front of him for more than 20 seconds and points to a number of division markings on it in the vicinity of 11 and 12 inches while he waits for L to begin. The dialogue recommences with L asking C what difference he notices between the pattern of spaces on his measuring tape below 12 inches (where it is marked to thirty-seconds-of-an-inch) and above 12 inches (where it is marked to sixteenths-of-an-inch). [26:43] 75 L: … What do you notice here between the spaces here up to twelve, (G12-uses his index finger to sweep up from the zero end of the measuring tape and pauses at 12” just before saying “up to twelve”.) 12 The following abbreviations are used in the annotations of actions within the transcript of the video recording: G-gesture, LH-left hand, RH-right hand. Also note that the photographs included from the tutoring session are positioned within the body of the written transcript 73 Figure 10. L sweeps up measuring tape with his left index finger L starts sweep of index finger up from the zero end of the measuring tape [26:48]. Figure 11. L pauses with index finger at the 12 inch mark [26:49] 76 C: Yeah its, 77 L: and the spaces after twelve? (G-changes to 4th finger to sweep through the exposed interval of the measuring tape above 12”.) Figure 12. L points to the 12 inch mark with his left fourth finger L replaces his LH index finger at 12 inches with his fourth finger as he the proceeds to sweep up the measuring tape beyond 12 inches [26:50]. Figure 13. L sweeps up from the 12 inch mark with his fourth finger L at the end of his sweeping gesture well above 12 inches [26:51]. Here L uses spoken language to ask C to explain what he notices while making two distinct sweeping gestures, one below and the second above twelve inches on the measuring tape, separated by a static pointing gesture at the twelve inch point all in a coordinated manner. The use of different pointing fingers as L sweeps through each of the intervals and the use of a contrasting static pointing gesture at the end of his sweep up to 12 inches, the boundary point between them, calls attention to these regions of the measuring tape as distinct. And, as experienced educators know, posing a question like this is an effective means of having a student immediately beneath that text that they coincide with and the actions conveyed in each photograph are those carried out by the speaker of the preceding line of text. 74 engage in a more critical way with an object at hand. In reply, C uses very few words but, most significantly, enacts an elaborate series of nine gestures within a span of a few seconds and then a final gesture a few seconds thereafter that indicates that he is indeed noticing the particular way that the division patterns below and above one foot on the measuring tape are different from one another. This, in turn, indicates clearly that L’s use of spoken language and gestures is, indeed, serving as semiotic means of objectification for C by drawing his attention to mathematically significant features of the object of this activity. Gestures dominate C’s response to L’s question and his words taken on their own provide only a vague and incomplete response. C’s use of an elaborate and coordinated sequence of ten gestures, each positioned in a precise way relative to the measuring tape taken together with his use of spoken language indicated clearly that he is, indeed, becoming conscious of the precise way in which the division patterns on the measuring tape are different from one another.13 78 C: There’s, (G[Figure 14, 26:52]–sweeps up through the first few inches of the tape measure with the fourth finger of his left hand in a manner similar to the gesture just enacted by L) there’s more. (G[Figure 15, 26:53]–makes two chopping motions aligned with the divisions on the tape measure with his left hand, the first significantly larger than the second just before he says “there’s more” in reference to the divisions inscribed on the measuring tape. G[Figure 16, 26:54]–then points to the 12” mark with the fourth finger of this left hand before withdrawing it from the measuring tape). 13 The spoken words in the particular excerpt of the transcript being discussed here are printed in bold to assist the reader to differentiate these from the descriptions of the accompanying actions. 75 In line 78, C begins his description of the difference between the two division patterns. He starts by sweeping the fourth finger of his left hand upwards through the first few inches of the measuring tape (Figure 14). This is the same type of one finger indexical sweeping gesture that L had just used (albeit using a different finger) to draw C’s awareness to this region of the measuring tape (Figure 10). C embellishes this sweeping gesture by including a chopping gesture midway up this interval. This chopping gesture is oriented in the same direction as the series of parallel divisions inscribed on the measuring tape and reflects the familiar action of physically dividing or chopping up the interval on the measuring tape in the same way as is indicated by the inscribed measuring tape divisions (Figure 15). Immediately following this gesture C says “there’s more,” confirmation that he is, indeed, referring to the closely packed divisions inscribed on this region of the tape measure. C resumes and finishes his sweep through this region of the tape measure by pointing with the same finger of his left hand to the 12 inch point, the endpoint of this interval (Figure 16). Then C takes this hand away from the measuring tape and replaces it with a pointing gesture aimed at the same point using the first finger of his right hand (Figure 17). This use of a static single-finger pointing gesture at the 12 inch point Figure 14. C sweeps up the measuring tape with his left fourth finger C sweeps the fourth finger of his left hand up through the first few inches of the measuring tape [26:52]. Figure 15. C makes chopping motions with his left hand C makes two chopping motions aligned with the markings or divisions on the measuring tape with his left hand in reference to the markings below 12 inches [26:53]. Figure 16. C points to the 12 inch mark with his left fourth finger [26:54] 76 separating the two regions of the measuring tape is the same type of gesture that L used a few seconds earlier to separate his own sweeping gestures at the 12 inch point as well (Figure 11). (line 78 continues) It’s like it’s more spread out (in reference to the divisions on the tape measure after the 12 inch mark.) (G[Figure 17, 26:55a]–points briefly to the 12 inch mark on the tape measure, now with the first finger of his right hand, replacing the previous pointing gesture that had been expressed by the fourth finger of his left hand; G[Figure 18, 26:55b and Figure 19, 26:56a]–starting with his thumb positioned at the 12 inch point, sweeps his right hand up the measuring tape a short distance while holding an approximately 2.5 inch wide interval between the thumb and first finger.) Figure 17. C points at the 12 inch mark now with his right index finger Here C continues the pervious indexical gesture he had made with his left fourth finger [26:55a] Figure 18. C begins sweeping a wide interval up from twelve inches C begins to sweep an approximately 2.5” wide interval up the measuring tape starting with his right thumb at 12” [26:55b]. Figure 19. C continues his wide interval sweep [26:56a] (line 78 continues) when (G[Figure 20, 25:56b]–grasps the tape measure with his right thumb and first finger on opposite edges at the 12” point and G[Figure 21, 26:57]–sweeps his hand in this configuration upwards a short distance from 12”) you pass one, 79 L: Yeh, 80 C: one foot (G[not shown]–while maintaining the same grasping position, repeats this sweep upwards for a second time) Figure 20. C grasps the measuring tape at 12 inches [26:56b] Figure 21. C sweeps upwards while holding his grasping gesture C sweeps his hand in this configuration upwards a short distance from 12” and then repeats this motion a second time [26:57]. 77 C then forms a wide-interval gesture using his right thumb and first finger and without hesitation sweeps this up the measuring tape with his right thumb starting from the 12 inch point (Figures 18 and 19). As he does this he says, “it’s more spread out” (line 78), reflecting the wider intervals between adjacent fraction divisions inscribed here, C grasps the measuring tape at 12 inches with his right thumb and first finger in a position that looks like he is grasping or pinching it (Figure 20), sweeps his hand up the measuring tape from 12 inches (Figure 21), and then repeats this a second time (not shown). This series of three sweeps up the measuring tape from the 12 inch point (one wide-interval sweep and two grasping sweeps) serves, at the very least, to sustain both his own and L’s attention on this region of the measuring tape. (line 80 continues) And when you’re before one foot its more um. (G[Figure 22, 27:01]–makes a very brief and narrow-interval gesture with the thumb and first finger of his right hand with this hand now positioned above the region of the tape measure between 0” and 12”.) Figure 22. C makes a very brief narrow-interval gesture C makes a very brief narrow-interval gesture with the thumb and first finger of his right hand with this hand now positioned above the region on the tape measure between 0” and 12” [27:01]. 81 L: Okay. 82 C: [silence] C’s explanation comes to an end as he says “And when you’re before one foot its more, um.” (line 80) while making a very brief but distinct narrow-interval gesture with the thumb and first finger of his right hand (Figure 22). This gesture is positioned above the region of the measuring tape between 0 and 12 inches and reflects the narrower intervals between adjacent divisions on this region of the measuring tape in comparison to the intervals above 12 inches that C had described using a wide-interval gesture five seconds earlier. In his response to L’s question in lines 78 and 80, C enacts a coordinated series of semiotic actions—iconic gestures and spoken language—that reflect and help to sustain his 78 attention to the division patterns on the tape measure. It is clear that C has indeed noticed differences in the inscription pattern on the measuring tape below and above one foot. In particular, C indicates that he recognizes that there are more divisions per unit of length (the great linear density of markings) on the measuring tape below one foot that above and that the spaces between adjacent divisions below one foot are smaller in comparison with those above one foot. C’s sense of these physical patterns on the measuring tape is clear even though he communicates them, for the most part, through the use of gestures. The first of these is his iconic chopping gesture (Figure 15) in reference to the greater density of division lines on the measuring tape below one foot, followed by his two iconic interval gestures—the first indicating wide intervals between the divisions above one foot on the measuring tape (Figure 18) and the second indicating narrow intervals in reference to the intervals below one foot (Figure 22). Slow motion playback of the video recording reveals that C’s first chopping gesture precedes his verbalizing “there’s more” and his narrow-interval gesture stands alone was the only form of C’s expression of the pattern of the intervals between division markings below twelve inches. In the next part of this episode, L directs C to locate five-eighths-of-an-inch, first on the region of the measuring tape below 12 inches where the divisions are indicated to thirty-seconds- of-an-inch by locating 7 5/8 inches and then above twelve inches where the divisions are marked only to sixteenths-of-an-inch by locating 13 5/8 inches (lines 83 and 89). This requires C to attend to the difference between these marking patterns. To assist C in remembering the measurements that he is has been asked to locate, L writes them on a piece of paper for him to refer to. C’s two responses for five-eighths-of-an-inch and his subsequent explanations of how he arrived at each indicate that he simply counted five of the smallest divisions on the measuring tape past the corresponding whole inch marks to arrive at his measurements apparently without 79 regard for the physical differences between the marking patterns on these two regions of the measuring tape in both cases. This is the exact same approach that he had used earlier when locating 11 4/16 inches on his measuring tape in the shop. Here again, he presumes that the smallest increments on his measuring tape are the fraction sub-units that he requires. This suggests strongly that C has not yet begun to objectify, in a substantial way, the meaning behind L’s earlier explanation in the shop of reading thirty-seconds-, sixteenths- and eighths-of-an-inch on the measuring tape below 12 inches (line 64) nor his own more recent explanation to L (lines 78 and 80) of the differences in the divisions on the measuring tape below and above 12 inches. L then asks C to compare more closely the two “five-eighths-of-an-inch” intervals that he has just identified on the different regions of the measuring tape. L does this by drawing a straight line, using a pen and paper, between thirteen inches and the point that C has identified on the measuring tape as thirteen and five-eighths inches and labels this line with the fraction “5/8” (line 91 and 93). L then repositions this piece of paper so that one endpoint of this line aligns with the seven inch point on the measuring tape and extends along side of the measuring tape between seven and eight inches (line 93 and 95). The creation and use of this artifact—the inscription of a short line—serves as a permanent record of the distance that C had measured and helps to create of a zone of proximal development within which the interaction is embedded. This line, along with L’s active intervention, helps to draw C’s awareness more easily and more closely to the difference between the two intervals of the measuring tape that he had identified as five-eighths-of-an-inch. C begins speaking as L is in the process of repositioning the inscribed line at the seven-inch mark of the measuring tape. [28:16] 96 C: Yeah, it looks the same. Actually (surprised), no it looks bigger. 97 L: Yeah. 98 C: Way bigger. 99 L: Okay. 100 C: It’s not right then 80 At first C expects the two intervals that he has identified as “five-eighths-of-an-inch” to match (see the beginning of line 96). He says “it looks the same,” before he is able to verify this empirically. Once the inscribed five-eighths-of-an-inch line from 13 5/8” is repositioned at seven inches for comparison with the point that C had identified as 7 5/8”, he becomes conscious of the discrepancy between these two intervals that he has identified as “five-eighths” (see lines 96, 98 and 100). L then asks C to “take an educated guess” about why the intervals on the measuring tape above twelve inches are bigger than those below (line 101) in an apparent effort to deepen his sense of the difference between the two division patterns on the measuring tape. C responds in lines 102 and 104 saying, “They might be going by twos rather than (G-points to the region of the measuring tape above 12 inches with his right fourth finger while he speaks) by ones (G- switches pointing finger, now points briefly to the region below 12 inches with his right index finger)”. C’s conjecture is, indeed, appropriate and his response to L’s request to put his thinking into words sustains his attention and appears to further his objectification of the differences in the division pattern on the measuring tape. It should be noted that elsewhere during the pre-apprenticeship course C reported that he had only ever used metric units for measuring, both in and out of school, prior to this piping assignment. When measuring lengths of the order of those being measured in the present context with a metric ruler or measuring tape there is only one type of subunit of a centimetre on a standard metric ruler to deal with, that being millimetres or tenths-of-a-centimetre. The strategy that C has used to read fractions-of-an-inch on the measuring tape (i.e. to simply count the smallest increments on it) is the same approach that he would have used and been successful with in his earlier measurement experiences before the pipe trades pre-apprenticeship course measuring in metric. 81 Vignette three: Exploring of C’s understanding of the relative sizes of fractions In an apparent effort to make explicit some of the more general aspects of C’s thinking about fractions, L asks him to identify the number of eighths in one whole inch. C responds, “How many eighths in one inch? (looking at his measuring tape) … Won’t there be sixteen? Sixteen, sixteen” (line 110). In order to answer this question correctly, one needs to be able to be conscious of the precise and significant relationship between the type of fraction (i.e., halves, quarters, eighths, etc.) indicated by its English name or its denominator when it is written using digits and the number of these fraction subunits spanning a whole inch. C’s incorrect and tentative response in the form of a question indicates that he has not yet achieved a level of stability in his consciousness with this particular aspect of the mathematical object. In contrast, when presented with a series of common unit fractions written using digits on paper, C correctly judge their sizes relative to one another. This is evident when L asks him to arrange the following mixed numbers, which differ only by the denominator of their unit fraction component, from smallest to largest: 5 1/8, 5 1/2, 5 1/16, and 5 1/4 (line 113) and writes the names of the fractional parts of these values in words on a piece of paper (i.e., “eighths, halves, sixteenths, and quarters”) for C to refer to while he does this. C completes this task successfully (lines 116 to 122) while L provides direction to help him focus on one step at a time with the following prompts: “Which would be the biggest amount?” (line 113), “Which would be the smallest?” (line 117), and “If you were to put the other ones in order, here and here (while pointing with the tip of his pen to the corresponding spaces on the page), where would they go?” (line 119). L also writes down C’s responses in their appropriate positions in a row on the paper as he provides them. While it cannot be determined from the video data if C would have been successful responding to L’s request without his assistance, L’s efforts—using the fraction words that he has written on the page in front of C and his verbal prompts—appear to help to create a 82 zone of proximal development that allows him to focus his attention more easily on the essential attribute that differentiates these types of fractions (i.e., the denominator) and on the individual elements in this sequencing task instead of the task as a whole. Vignette four: Starting to read different types of fractions on the measuring tape L introduces a set of five ruler transparencies to C, a teaching tool that he had created earlier for the purpose of drawing awareness to the various fractional subunits-of-an-inch that can be read from an imperial ruler or measuring tape and the mathematical relationships between these different types of subunits. The transparencies consist of a matched set of rulers each printed separately on a clear transparency. The smallest increments on each transparency ruler show either unit inches or one of the subunits-of-an-inch that can be measured using a measuring tape (i.e., halves, quarters, eighths or sixteenths). The actual marking patterns that appear on the individual transparency rulers are shown together in Figure 23. By limiting one’s attention to only one type of fraction-of-an-inch interval at a time (using all of the fractional divisions indicated without the need to differentiate between different types of fractional subunit divisions as is the case when using the measuring tape) the user can more easily attend to the fraction-of- an-inch intervals represented and measure in terms of the particular fraction-of-an-inch units represented on each transparency. The set of transparencies was also designed so that the individual transparencies can be superimposed directly on top of one another to illustrate how the various fraction intervals and their corresponding divisions are organized on the measuring tape and, in particular, how equivalent fractions share division lines with other transparencies. For example, the eighth-inch ruler uses the same division line to indicate 2/8” as is used on the quarter-inch ruler for 1/4”. 83 Figure 23. The set of rulers on transparencies Each row in this figure shows the division markings on one particular ruler in the set. Note that the rulers in the figure have been reduced in size here to fit the page. When introducing the set of transparency rulers to C, L tells him that the measuring tape is “five rulers in one” (line 124). L then employs a sequence of actions involving the coordinated use of eight different semiotic systems (see lines 130, 134 and 136). This calls C’s awareness both to the pattern of different fraction intervals that can be read off of the measuring tape and to the general relationship between each fraction name, the corresponding numerical symbols used in representing them, and the relationship between a fraction denominator and the number of these subunits that span a whole inch. [30:24] 128 L: …it’s a, it’s a ones ruler (places the units transparency to top of the paper already in front of C on the table top.) Okay. 129 C: (G-starts nodding his head up and down) Okay. 130 L: It’s a half ruler at the same time (superimposes the halves transparency ruler on top of the units transparency ruler already on table top) Okay. Now the half ruler, it’s called half or one over two (G-sweeps pen tip through the first half inch interval at the zero end of the transparency ruler, pauses briefly, and then through the second half inch interval marked on the transparency in front of C. Then writes one half using conventional fraction notation on the paper and circles the denominator of this fraction.), because there’s two, the whole, each inch (G-sweeps pen through the first whole inch interval on the transparency ruler.) is split into two equal parts. (G-sweeps the tip of his pen again through each of the two adjacent half-inch intervals within the whole inch previously indicated on the half transparency.) Okay. Now it’s kind of complicated because they’re on top of each other. Okay. Now, next, next comes splitting it in, splitting each of the halves in half (G-sweeps his pen tip through an interval of a half-of-an-inch on the halves 84 transparency and then G-moves his pen in a downward motion at the midpoint of this half inch interval as if inscribing a one quarter inch division), which would be quarters, ... (pulls out quarters transparency and places it on the table.) One, two, three, four (G- points to successive quarter inch intervals within a single inch on the transparency as he counts them) equal parts, 131 C: Okay, okay, got that, 132 L: and, 133 C: four equal parts (says aloud but to himself). 134 L: that overlays as well (positions and aligns the quarters transparency over the units and half transparencies already of the table). Lets look at what we were measuring (referring to the original 5/8” measuring tasks on the measuring tape above and below 12” that led to the recognition of C’s problem locating fractions on the measuring tape). We were measuring five eighths (Removes the stack of whole inch, half inch, and quarter inch transparencies from table top). Now on this one (referring to the eighths transparency that he is putting out on the table in front of C), lets just double check that there are eight intervals, (G-points to divisions on the eighths-of-an-inch transparency ruler). There are eight-eighths in an inch (“eighth eighths” spoken with increased volume and more emphatically) (G-extends first finger of RH in air for emphasis while speaking). Okay, so there’s, there’s sixteen sixteenths in an inch, (“sixteen sixteenths spoken more emphatically for emphasis, circles the fraction 1/16 written previously as part of 5 1/16 on sheet 040519e in front of C on the desktop) 135 C: Okay 136 L: There’s eight eighths in one inch. (Circles the denominator 8 in the number 5 1/8 written previously on the paper in front of C.) There’s four quarters in one inch (circles the denominator 4 in the fraction 1/4 written previously), there’s two halves in an inch (circles the denominator 2 in the fraction 1/2 written previously). (In this utterance each of the fractions per inch relationships, e.g., “eight eighths” are spoken emphatically with increased volume.) 137 C: an inch, Okay. 138 L: Okay. So, and it’s the spaces, (G- pointing at each successive eighth-of-an-inch intervals on the eighths transparency) 1, 2, 3, 4, 5, 6, 7, 8 (says eight louder than the numbers before it) 139 C: Okay 140 L: So that’s eighths (spoken at an increased volume.) (G-sweeps pen over the interval- of- an-inch just counted), … The semiotic systems that L employs here in a coordinated manner in order to draw C’s awareness to the cultural meaning crystallized in the divisions on the measuring tape include: (a) spoken language; (b) the fractions-of-an-inch pattern, printed on measuring tape and the transparencies; (c) gestures—including a series of sweeping gestures, pointing gestures, and an iconic inscribing gesture when L moves his pen as if inscribing a quarter-of-an-inch division at the midpoint of a half-inch interval; (d) conventional fraction notation—the fraction one-half 85 written on the page; (e) indexical inscriptions—drawing circles around the denominators of the fractions written in numeric form to draw awareness to this part of the fraction; (f) physical positioning—the alignment of the transparency rulers on top of one another; (g) the whole number system—counting the number of intervals spanning one inch to confirm that these intervals represent eighths-of-an-inch; and (h) differences in the volume of his speaking voice (e.g., when he says the words louder for emphasis). In addition, the episode also includes C rehearsing what L has just said regarding quarters-of-an-inch, “four equal parts” (see line 133) (an example of C using semiotic means of objectification for his own objectification) and L explaining in a systematic and explicit way the general pattern of fractions per inch when he says that “there’s sixteen sixteenths in an inch, … eight eighths in an inch, … four quarters in an inch, and two halves in an inch” (see lines 134 and 136) while simultaneously circling the corresponding denominators in a sequence of fractions that had been written previously on the paper in front of him. L then directs C to locate five-eighths-of-an-inch using the eighths transparency (line 140). This is the same fraction-of-an-inch value that he worked with earlier on his measuring tape when locating 7 5/8” and 13 5/8”. Here the division pattern on the eighths-of-an-inch transparency appears to scaffold this task for C and help his to succeed with this task. When C holds the eighths transparency against the measuring tape at the point he had identified earlier as 7 5/8” C recognizes for himself that his earlier determination was incorrect saying, “Yeah, its different, totally.” (See line 145 in the excerpt from the transcript provided below.) Another aspect of the process of C’s objectification of the relationship between the intervals and fraction divisions on his measuring tape is evident in this same part of the session. At first, C focuses on the division lines on his measuring tape when describing the difference between the pattern of divisions on his measuring tape and the pattern on the eighths 86 transparency (line 147) and L interjects with the word “intervals” to call his attention to this essential attribute of a fraction-of-an-inch (line 148). And, by the end of this episode, C speaks in terms of both intervals and spaces on his measuring tape. [31:11] Figure 24. L aligns the eighths transparency for C L aligns the eighths transparency beside the measuring tape for C to compare with his measuring tape [32:11]. 145. C: (…) Yeah its different, totally. 146. L: Yeah. How would you explain the difference, if you were to try to put your thinking into words? They are different (said emphatically) (G-still pointing at measuring tape with pen).… How do you, how do you make sense of that? 147. C: Um, well the lines (G-points at the divisions on the measuring tape between seven and eight inches with the fourth finger of his right hand) indicate, there’s three 148. L: intervals 149. C: lines, three lines, like there’s four, it looks like it goes by four (referring here to four thirty-seconds to each eighth). (G-pointing to 1/32” divisions with the finger nail of his right fourth finger as he talks about them.) 149. L: Each, for each eighth 150. C: For each eighth 151. L: Yeah. 152. C: Four 153. L: Yeah. 154. C: (inaudible) 155. L: There it is there (G-pointing to the transparency with the tip of his pencil) 156. C: … Okay. (stares intensely at the measuring tape) … 1, 2, 3, 4, 1, 2, 3, 4. (G- pointing at the 1/32” intervals as he counts them) Yeah, it goes by intervals of four. 157. L: Okay, so each eighth has four little spaces (G-points to four spaces within one eighth on the measuring tape below 12”) 158. C: Four (G-points very briefly to the measuring tape and transparency on the table with his right index finger) little spaces (becomes inaudible). 159. L: Let’s look over, what was the other one? We had five and an eighth and (G-points to that general location on the measuring tape) 160. C: (G- points to 13 5/8 written earlier on the top left of the page) thirteen and five eighths. 161. L: and thirteen and five eighths. (Lines up the eights transparency with 13” on the measuring tape where the smallest divisions indicate sixteenths-of-an-inch.) There’s five eighths again. (G-moves pen back and forth repeatedly as if tracing/extending the 5/8” mark off of the measuring tape onto the transparency ruler.) 162. C: (G-nods his head slightly four times) 87 163. L: So what do you notice about that? 164. C: Now it’s different. Now it’s just the two interval (referring here to 2 intervals on the measuring tape per eighth-of-an-inch as indicated by the eighths transparency). Early in this episode (line 147) C says, “the lines indicate there’s three,” in describing how the pattern of the smallest increments on his measuring tape between seven and eight inches (thirty-seconds-of-an-inch) compare with the smallest increments on the eighths transparency.14 At this point L intervenes with the word “interval” (line 148) and C revises his response saying “it goes by four” (line 149). Twenty-six seconds later (line 157) C uses the word “interval” for the first time in the tutoring session while explaining the number of the smallest fractions shown on his measuring tape per eighth-of-an-inch. In line 159, C repeats L’s use of the phrase “little spaces” (from line 158) again in reference to the intervals between divisions on the measuring tape. (Note: This marks the first time in the entire session that C uses the term “spaces”.) And, by the end of this excerpt (line165) C uses the term “interval” on his own, away temporally from L’s earlier prompting and modeling of this way of this way of seeing and describing fractions-of- an-inch. L then poses the following question to C. If this is eighths (referring here to the increments on the eighths transparency and emphasizing the “ths” at the end of the word eighths) and there’s two spaces for every eighth, (G-points to two of the 1/16” spaces on the measuring tape making one eighth) what fractions are shown by each line here (on the measuring tape) between 13 and 14 inches? (G-sweeps his pencil tip between 13” and 14” on the measuring tape as these are mentioned). (line 166) By calling C’s attention explicitly to eight eighths per inch and two spaces per eighth on the measuring tape, L invites C to use these numeric values to calculate the type of fraction indicated by these smallest intervals. C responds immediately by counting the individual intervals between 14 Note that there are three markings separating the four individual thirty-second-of-an-inch intervals on the measuring tape between each pair of the eighth markings. These appear distinct to the eye between each pair of eighth-of-an-inch interval markings on the acetate. 88 thirteen and fourteen inches on the measuring tape and arrives at an answer of 18 (line 167). L responds by saying, “Actually 16”, counting aloud the eight intervals spanning the one inch interval on the transparency and then saying, “each is split into two” (line 168) at which point C corrects himself saying, “so yeah, it’d be 16” (line 169). When L makes explicit the possibility of using multiplication in this situation for determining the required value by stating, “two times eight” (line 170) C repeats what L says and finishes this statement by saying, “two times eight is sixteen” (line 171). L poses a similar question in line 172 in an apparent effort to keep C’s attention focused on the use of multiplication for determining the number of an increment per inch on the measuring tape based on the number of these increments per eighth-of-an-inch: And this one over here would be (repositioning the eighths transparency next to the region on the measuring tape below 12 inches), be what, from between seven and eight inches, if there’s four spaces (on the measuring tape as discussed a very short time earlier) for every eight, every eighth, how many little intervals has the inch (G-sweeps pencil tip over the interval between seven and eight inches) been split up to? C’s response, “It’d be four, four times, it’d be eight” suggests that he recognizes multiplication as the process to use to arrive at the number he requires, but his statement, “four times” and his answer of eight indicate that if he did, indeed, use multiplication he multiplied four times two instead of eight times four as required (line 173). I speculate that C may have used the previously discussed product—eight times two—as a model for this new task, and incorrectly (and instrumentally!) changed the factor eight in the initial product to a four instead of changing the factor two to a four as required. In other words, C’s arithmetic model for this task (four times four) did not reflect the appropriate cultural logic for the situation at hand. L then reiterates, “Every eighth (on the measuring tape) has four (smaller sub-intervals spanning them), so between seven and eight inches (G-sweeps pencil tip over this interval) how many little spaces do we have?” (line 174) at which point C abandons the use of multiplication and reverts back to 89 his reliable counting strategy by counting (by twos) the number of intervals spanning the distance between seven and eight inches for himself and answering with “thirty-two” (lines 175 and 179). At this point in the tutoring session, C has begun to objectify an important aspect of measuring with a measuring tape. Specifically, that it is the intervals between divisions on the measuring tape that need to be attended to when dealing with fractions-of-an-inch. Vignette five: Understanding fractions per inch and fractions as intervals L continues to emphasize fractions-of-an-inch as intervals and use sweeping gestures when referring to specific fractional measurements on the transparency rulers and on the measuring tape with C. While focusing with C on the half-inch transparency, L poses the question, “How many half inches does it take to make two inches?” (line 194). C responds by looking closely at the transparency in front of him and saying “one, two, three halves” (line 195). He then points to the three lines that separate the four successive half-inch intervals between zero and two inches on the transparency ruler (line 197). In retrospect, this response is not a surprise because the discourse had just been addressing the way that the division lines on the half inch transparency split each of the whole inch intervals into two equal parts when superimposed on the whole inch transparency (line 188-192). However, it does suggest that C’s objectification of fractions-of-an-inch as intervals that was evident in the previous vignette has not yet reached a level of stability within his consciousness. L attempts to redirect C’s attention to the half inch intervals that span two inches by saying, “actually its spaces,” (line 198) after which C responds with the correct answer of four (line 199). Then, L sweeps his pencil accross each successive half inch interval on the measuring tape between zero and two inches in-turn, pausing at the end of each, as he counts them using a steady rhythm, “1 space, 2 space, 3 space, 4 space” (line 200). This calls C’s awareness again to the idea of half-inch fractions on the measuring tape as a series 90 of equal intervals bounded by particular division marks and C responds by saying “four spaces” (line 201). A short time later L poses two questions that require C to attend to the number of fractions-per-inch generalization for quarters. L starts by asking, “How many quarters fit into two inches?” (line 222) and C immediately begins to count to eight aloud while pointing with his pen to each interval in turn. L emphasizes the general pattern by saying, “eight, four every inch.” (lines 224 and 226) and then asks C, “And in seven inches, how many quarters?” (line 228) At this point C does not count the spaces on his measuring tape, nor point to the measuring tape with his pen as he successfully he calculates the product mentally and without hesitation saying, “twenty-eight” (line 229). L’s use of these questions relating to quarters appears to expand C’s zone of proximal development in an incremental way by facilitating C’s move towards a more generalized way of working. And, by the end of this sequence of questions, C is objectifying the multiplicative relationship relating to the number of a specified unit fraction needed to span a given interval on the measuring tape, for quarters-of-an-inch at least. L poses the question, “So, an inch and a half is an inch and how many quarters?” (line 230) in an apparent effort to get C to apply the idea of four quarters per inch and equivalent fractions in a slightly different context. C says, “Wait, an inch?” in line 231 to get clarification. When L says, “an inch and a half” (line 232) once more C repeats this aloud and writes it down without being asked to do so. C appears to have considerable difficulty understanding what he is being asked to do here. It is important to note that this marks the first time in the tutorial session where C asserts some control over the course of the discourse by requesting that L waits and provides clarification for him. Immediately thereafter, C also begins to make a written record (in other words, create a semiotic tool) for himself of the mixed number value that he is being asked 91 to consider and the answers using standard fraction notation. C then attends carefully to and acknowledges the things that L is saying, at times utterance by utterance (lines 243-265). Vignette six: Using the smallest intervals on the measuring tape as benchmarks L leads C through an explanation of the interval patterns and different division line lengths used for signifying each of the types of fractions on the measuring tape from halves to thirty-seconds. L uses the transparencies in an attempt to draw C’s attention to the process of counting the number of intervals within an inch to verify for himself the type of fractions represented by the various interval patterns and then positions the transparencies on top of one another with their inscriptions aligned in an attempt to draw C’s awareness to the ways in which different subunits-of-an-inch share division lines on the measuring tape (lines 266-274). As L finishes explaining the set of transparency rulers, C focuses on the last of these presented—the one showing sixteenths-of-an-inch, and asks, “Oh, so then, little ones, the little ones, all the little ones (referring to the smallest increments), if I have three sixteenths I would just count. If I have one inch and three sixteenths, I’d just count the little ones—one, two, three.” (line 277). Here C is verifying this strategy that he came up with on his own—to simply count the smallest increments on the measuring tape to measure sixteenths-of-an-inch. L proceeds to tell him that below one foot he needs to modify this strategy because every pair of the smallest intervals (thirty-seconds-of-an-inch) indicates one sixteenth-of-an-inch. L also tells him this strategy can be applied directly above one foot where the smallest increments are, indeed, sixteenths (lines 280 to 290). L also reminds C that he can verify the size of the smallest increments on any measuring tape by counting the number of them that span one inch (lines 290- 300). C then confirms with L his revised strategy of counting “the little small ones (referring here to the divisions below twelve inches)” (line 301) if he needs thirty-seconds-of-an-inch and L 92 tells him that in plumbing the norm is to measure only to the nearest sixteenth-of-an-inch (line 306). The strategy of using only the smallest increments on the measuring tape without regard for the lengths of the various division lines being used, something that C has apparently noticed for himself, is very much like the action of measuring to tenths of a centimetre or millimetres on a metric ruler or measuring tape. (Recall, that prior to this pre-apprenticeship course, C had only ever worked with metric units when measuring length.) This approach, of focusing on the smallest increments on the measuring tape, simplifies matters greatly for C at a perceptual level in that he can ignore the complex pattern of different division line lengths which represent, simultaneously, the various types of binary fractions-of-an-inch. This approach is also very reliable in that C needs only to attend to the one-to-one relationship between the individual increments marked on the measuring tape and the fractional units indicated by these divisions (i.e., either sixteenths above 12 inches or thirty-seconds below). Vignette seven: Using quarters as benchmark fraction intervals C takes the lead in the session with L for the first time at this point by setting, for himself, a number of fraction-of-an-inch lengths to locate on the measuring tape. C’s appropriate choices of fraction values that can be located on his measuring tape, themselves, provide evidence that he is, indeed, objectifying the cultural practice of measuring with an imperial measuring tape. C first confirms with L that he has identified the four quarter-of-an-inch intervals between 12 and 13 inches on his measuring tape correctly while pointing at each of them in succession with the fourth finger of his right hand (line 309). L again draws his awareness to the rationale used to identify quarter-of-an-inch intervals—four per inch (lines 310 to 314) and C 93 repeats what he says very closely, at times word for word, on the verge of co-speaking with him (lines 310 and 311; 312 and 313). [43:04] 309. C: Okay. I’m getting. So say 12 and one quarter, right (grasps the extended measuring tape with two hands while L is still holding the measuring tape case), so if I was looking for 12 and one quarter, just like on the (pipe fitting) project, these are all quarters you said, right? (G-pointing at 12 1/4” 12 2/4” and 12 3/4” in turn on the measuring tape above 12” with his right fourth finger.) These are all quarters. 310. L: Because there’s four (slight pause) (G-points to the end points of each of four successive quarter inch intervals between 12” and 13” with the tip of his pen) 311. C: (G-finishes his counting of quarter inch intervals by pointing to 12 4/4” when L points to the fourth quarter inch interval and speaks the word four) four 312. L: spaces (slight pause), 313. C: spaces 314. L: per inch. 315. C: So it’d be, right, this would be 12 and one quarter. (G-points at 12 1/4” on measuring tape with his fourth finger.) 316. L: Yep. C then sets another task for himself, to locate 12 1/8” on the measuring tape, which he does promptly and successfully by sight (line 317) after which L directs him to locate 12 7/8” (line 318). C locates this measurement by counting seven individual eighths-of-an-inch intervals from 12 inches. L follows up by asking C how many eighths there are in a whole inch and C once again counts these same spaces on his measuring tape, now employing a series of indexical hopping gestures with the tip of his pen (lines 325 and 327). Two things are significant here. First, C relies on his ability to identify quarters by sight on the measuring tape above 12 inches where is it marked to sixteenths-of-an-inch and then his, what appears to be tacit, understanding of the relation of eighths-of-an-inch intervals to quarters (2/8 = 1/4) to identify the eighth eighths-of-an-inch within a one inch interval on the measuring tape in front of him in order to determine the number per inch, instead of using the number per inch indicated by the name of the fraction, here the eight in eighths. And second, unlike his previous fraction counting with gestures (e.g., lines 57, 88, 141, 197, 223) where his utterance of each counting number coincided with touching the pen or his finger tip to a particular division on the measuring tape, in 94 this instance and for the first time since the start of this discussion, C attends to each eighth as an interval by making a distinct hopping motion over each one before he utters the corresponding number in his count (line 325). This indicates that C is, indeed, perceiving eighths-of-an-inch as intervals on the measuring tape. Vignette eight: Becoming more fluent with the number of unit fractions per inch After C successfully determines that there are eight eighths in the inch between eleven and twelve inches by counting them (lines 325 and 327), L asks him to articulate explicitly the cultural meaning of “eighths.” By calling attention explicitly to this, L’s action has the potential to further C’s objectification of the very important general relationship between the name of a fraction and the number of these sub-units per inch so that he can use this relationship for identifying different types of fractions on his measuring tape. [44:11] 330. L: Now what is it about eighths that … gives you a shortcut so you don’t have to count them? How many eighths in an inch? 331. C: aah. How many eights in one inch? (laughs and shakes his head from side to side (no)) … Four, right? Wait (G-points to the measuring tape with the tip of his pen as he counts) 1, 2, 3, 4, 332. L: Remember eighths (G-pointing to 1/8” intervals on measuring tape) are from here to here. 333. C: Okay 334. L: You just counted them. 335. C: Yeah. 336. L: What is it about the way that we write that and say it, that (takes paper from in front of C and writes “1/8” in the middle of the page) that tells you how many of those are in one inch? 337. C: … (no response) 338. L: Not sure of that? 339. C: Not sure, no. 340. L: Okay. 341. C: Not sure. After C’s difficulty responding to the initial question in line 331, L attempts unsuccessfully to expand C’s zone of proximal development by sweeping his pen through an eighth-of-an-inch 95 interval on the measuring tape, identifying this interval as an eighth, and then by reminding him that he had just counted them. L continues by asking C explicitly to identify what it is about the way that “eighths” is written and spoken that indicates how many of them are in one inch, while also writing the fraction 1/8 on the page in front of him. After getting no response from C, L changes his strategy and calls C’s attention to the relationship between a fraction name and the number per inch relationship using the fractions on the measuring tape, one type of fraction at a time in a systematic manner starting with halves-of- an-inch. [44:57] 342. L: (Writes one half on the paper at the left side near the bottom of the page.) How many halves are in one inch? 343. C: How many halves in one inch? 344. L: Yeah, how many half inches? (G-waves RH to emphasize each syllable in “half inches for emphasis) Show me half, … show me half-an-inch on your ruler. (G-leans forward to look at measuring tape in front of C.) 345. C: Okay, so half an inch would be right here (G-points with pen to the 7 1/2” point on the measuring tape and holds it there) 346. L: Okay, that would be seven and a half (G-sweeps pencil through interval between 7 and 7 1/2” on measuring tape.) 347. C: seven and a half 348. L: From here to here’s half-an-inch. (G-sweeps back and forth through interval with the tip of his pencil between 7” and 7 1/2” on measuring tape) 349. C: Okay 350. L: From here to here’s half-an-inch (G-sweeps through interval between 7 1/2” to 8”), so how many half inches in a whole inch? 351. C: (G-sweeps pen tip between 7” and 8” and then points to and counts divisions on the measuring tape in the decreasing direction—what appear to be evenly spaced quarters) … Four. 2, 3, 4, 5 … (counting off what appears to be 1/4” intervals on his measuring tape in one direction and then the opposite direction) 1, 2, 3, 4 In the middle of line 344, after C’s apparent hesitation with the question regarding the number of halves in an inch, L changes his approach and asks C to point to half an inch on the measuring tape. C points to the seven and a half inch mark. L then directs C’s attention to the two equal half inch intervals between seven and eight inches, each bounded by the mark on the measuring tape at seven-and-a-half inches, using a pair of sweeping gestures through the two adjacent intervals 96 in-turn (lines 348 and 350). This attempt to draw C’s attention to the meaning of half-an-inch as an interval rather than a particular point on the measuring tape was not immediately successful as evidenced by C’s incorrect response when L repeats the question, “How many half inches in one whole?” (line 350). C counts, what appear to be quarter inch intervals within a one-inch interval on the measuring tape and responds that there are four. In his continued attempt to make the pattern of fractions per inch apparent for C, L positions the half-inch transparency against the measuring tape and tells C that the intervals on the transparency are half-inches. [45:34] 352. L: I think I might not have done a clear job for you. (Positions the half-inch transparency up against the measuring tape aligning the transparency divisions with the corresponding divisions on the measuring tape.) Those are the half-inches there. 353. C: (G-points at transparency with his pen and then moves the pen to the right as if counting and without hesitation he corrects his earlier response.) Oh, two. Okay. 354. L: So there’s two in an inch. (Writes the word “two” below the fraction 1/2 written on the paper, and then writes the fraction 1/4 next to the fraction 1/2 written earlier.) How many quarters are there in an inch? 355. C: (G-holds out four fingers and waves his right hand a number of times just off of the table top and then puts his hand down. This gesture starts just before C speaks and continues while he speaks.) There’d be four then. Figure 25. C holds out four fingers C holds out four fingers and waves his hand as he says “four” in response to L’s question, “How many quarters are there in an inch?” [45:51] 356. L: Yeah. (Writes the digit four on the sheet beneath the fraction 1/4 written previously on the page.) 357. C: Four. (G-the previous gesture continues and then ends as C says the word “four”) 358. L: How many quarters in a dollar? 359. C: (G-as L mentions the word quarter C raises his right hand off the table.) 360. L: (G-lifts his right hand up off of the table, turns his palm up—to imply it’s common sense and invite C’s response—and then puts his hand back down.) 97 Figure 26. L turns his right palm upwards L turns his right palm upwards to invite C’s response [45:55]. 361. C: Four (G-just after L begins his gesture C repeats his previous gesture raising his right hand off the table while speaking the word “four” in unison with this gesture.) 362. L: Yeah 363. C: Exactly! Four. The use of the transparency ruler in this sequence appears to expand C’s zone of proximal development in that he responds correctly and without hesitation that there are two half-inches in one inch (line 353). Here L uses the familiar relationships of two halves in one whole, and four quarters per whole along with the fraction “1/2” and the word “two”, and then the fraction “1/4” and the digit “4” written in close proximity to one another on the paper in front of C while these fractions are being discussed as a means for helping C to notice the relationship between any particular unit fraction name and number of this type of fraction per inch. The discourse continues: 364. L: That’s why they call. (writes 1/8 in the row of fractions started on the paper) How many eighths? 365. C: There’d be eight (G-raising hand one time off of table top and stretching out his fingers in unison while speaking.) 366. L: Eight 367. C: Eight 368. L: (writes 1/16 in the row of unit fractions on the page) How many sixteenths? 369. C: (G-raises hand off of the table just starting before speaking) Sixteen. 370. L: Yeah, the size, the name of the fraction (circles the denominator 16 in the last fraction written) tells you how many are in (G-motions with right hand in the rhythm of his speech to emphasize the point being made.) 371. C: in (affirming what L is saying, while glancing up at L, G-taps table top twice with the second finger of his right hand) 372. L: one whole, (G-raises one finger as he says “one whole”) whether it be an inch, foot, or a dollar. (G-continues right hand-waving gesture) 373. C: (nods slowly) Okay … 98 By line 365, C’s responses indicate that he is objectifying the fraction name/number-per- whole relationship for eighths and sixteenths without having to refer to the measuring tape. L then models the use of the fraction intervals per inch relationship to locate 12 7/8” by counting down one eighth-of-an-inch from 13 inches for C after which time he successfully applies this counting back strategy completely on his own to locate six and five eighth inches (by counting back three one-eighth inch intervals from seven inches) (lines 375), again using this generalization. Vignette nine: Using thirty-seconds and quarters as benchmarks After a quick review of the strategy of counting intervals on the measuring tape to determine the type of fraction represented by any particular size of interval (lines 376-381), L asks C to put into words his newly developed sense of the division pattern on the measuring tape. When C starts to speak he reports that his is “all confused now” (line 383). L then prompts him to help get him going. [47:59] 386. L: What did I say? Each, (G-points to a 1/32” interval on the measuring tape with pencil) each little interval 387 . C: Each little interval (G-brings his pen to the measuring tape at the point where L is pointing as soon as L begins his pointing.) 388 . L: is how much? 389 . C: Its, you said it’s thirty-seconds. From, like if you count, if I were to count (G-sweeps pen over a one inch interval on measuring tape then sweeps pen again over this interval touching down on a few successive divisions in rapid succession during the sweep) every one, every one would be thirty-second. 390 . L: Yeah. 391 . C: And then you said the next one would be every sixteen. (G-pointing to 1/16” divisions in succession on his measuring tape.) 392 . L: Yeah. 393 . C: So every two, every second one, sixteen. 394 . L: Yeah. A number of things are significant here. C’s utterance in line 389, “Its, you said it’s thirty-seconds…. if I were to count every one, every one would be thirty-second” indicates, 99 albeit using less than precise language, that C has a stable sense of the strategy of counting intervals on the measuring tape as a means of verifying what type of fractions they are. This utterance also suggests a turning point in the way C sees himself in relationship to L and to the cultural object that he is objectifying—specifically, he starts in response to L’s request, by saying, “you said it’s …”, a reference to what L had told him earlier, but moves within the same utterance (in line 389) to speaking in terms of what he can do himself, “if I were to count every one…” as a means to verify for himself the type of fraction marked on the measuring tape. His utterance in line 393, that every two (thirty-seconds) would be one sixteen(th) is also the culturally appropriate interpretation of these divisions on his measuring tape. This marks the start of C taking the initiative to use the smallest intervals—thirty-seconds-of-an-inch—as a benchmark for identifying larger fractions-of-an-inch, namely sixteenths and eighths, which he does on his own without having been prompted or directed to do so here by L. The only reference during the tutoring session that was made to two thirty-seconds being equal to one- sixteenth occurred more than six minutes earlier in line 290. When C then tries to use thirty-seconds as a benchmark for identifying eighths he runs into trouble. He conjectures, “So, wouldn’t every third one, be eight?” while lifting his pen off of the measuring tape and making an exaggerated hopping motion (iconic gesture) in the air (line 395). L immediately corrects him, sweeping his pencil through an interval of two sixteenths on the measuring tape and saying, “Not every third one, every pair of sixteenths, … every four thirty-seconds would be a (pause)” (lines 396 and 398). L pauses here to prompt C to finish the sentence, which C does appropriately in line 399, by saying “one eighth.” L then leads C through a process of verifying this assertion by multiplying eight (eighths per inch) by four (sub- intervals on the measuring tape per eighth) and pausing for C to finish it (line 400), which he does readily and correctly by saying “is 32” (line 401). 100 L then steers the discourse to focus briefly on the meaning of the various lengths of the fraction division lines on the measuring tape. In line 404 while pointing to the different types of division lines in turn, L says, “So the littlest ones are … thirty-seconds, the next ones are sixteenths, the next ones are eighths, (and) the next longest line is?” At this point C extends this pattern by saying “quarters” (line 405) and proceeds spontaneously—after an expression of apparent sudden realization (i.e., saying, “Oh!” in a quiet and reflective manner and tapping his pen at a 1/4” mark on the measuring tape), followed by a pause while he stares intensely at the measuring tape and counts aloud the four individual quarters within a particular one inch interval on his measuring tape, “one quarter, two quarters, three quarters, four quarters” (line 407). This is the first time that C recognizes and counts quarters-of-an-inch on his measuring tape without employing artifacts (e.g., the transparencies, his pen etc.) or gestures (e.g., pointing as he counts) thus indicating that his level of objectification of this is moving to a deeper level. C continues to stare intensely at the measuring tape, nods his head indicating the affirmative, and finishes by saying “Okay, yeah, I think I got it now” (line 407). This indicates that C perceives that his relationship and confidence with the system of binary fractions on the measuring tape—the object of this activity is indeed growing. Vignette ten: Becoming proficient measuring with the measuring tape This part of the dialogue begins with L asking C what is different about his thinking now (line 408) in comparison to the start of the tutoring session. In response, C talks about being a “visual person,” and not “clueing in” when L was explaining things to him verbally, as well not knowing that the measuring tape was “three rulers in one” (line 411). L then identifies all the fractional sub-units on an inch represented on the measuring tape above twelve inches for C (whole inches, halves, quarters, eights, and sixteenths) and begins to introduce a strategy for 101 simplifying the process of measuring with the measuring tape (lines 412-420). C ignores L completely, cutting him off mid-sentence, and explains, now in more specific terms, his newly developed sense of reading fractions on the measuring tape below twelve inches. 421. C: (interrupting L) Like, like now I understand it because every two would be sixteenths (referring here to the smallest divisions marked on the tape measure below twelve inches while pointing at these with his pen), every four would be eighths (while pointing now at eighths division lines on his tape measure with his pen), every space would be quarters (C makes a distinct sweeping gesture with his pen through a quarter-of-an-inch interval), um, every two spaces would be halves (while pointing, in turn, at each of two quarter-inch intervals within a half inch on the measuring tape). It is significant here that C has developed and is expressing his own strategy, based on his objectification of thirty-seconds and quarters-of-an-inch on the measuring tape, for identifying all of the fraction values from thirty-seconds to halves, thereby deepening his objectification of the related generalizations between these fractions, without being directed to do so by L. In this process C describes, for the first time, quarters in a distinct way—as “spaces”—in his account of how he is making sense of these on his measuring tape. Furthermore, C is recognizing and using quarter inch intervals as a benchmark for identifying half-inch intervals. This description and use of quarters-of-an-inch confirms that C is, indeed, objectifying this interval on the measuring tape in a significant and sensual way (here through the sense of sight). C’s strong need to express himself, as evidenced by the abrupt manner with which he interrupts L and takes the lead in the dialogue in line 419 (for the first time in the tutoring session) also suggests that his newly objectified sense of the pattern of fraction divisions on the measuring tape and ability to measure is something that he senses and values strongly. However, as the following analysis will reveal, C’s understanding depends on using the division pattern on the measuring tape below 12 inches. L then asks C to locate 16 3/8” (line 422) in an apparent effort to ensure that his is attending to the difference between the fraction division patterns on the measuring tape above twelve inches (where it is marked to sixteenths-of-an-inch) and below (where it is marked to 102 thirty-seconds). C starts by pointing to 16 3/16”. After getting no response/a response of silence from L, C begins this process again by first identifying the quarter inch interval between 16” and 16 1/4” by sweeping his pen back and forth over this interval and asking L to verify that this is, indeed, a quarter inch (line 423). He then points to 16 3/16” on his measuring tape once again as his answer for 16 3/8” (line 425). L responds this time by counting aloud the number of these fraction intervals (sixteenths) that span one inch for C (line 426). After L reminding him that he is looking for 16 3/8”, C immediately points to 16 1/8”, seeks verification from L that he has it right, and then counts aloud two and three eighths past 16” correctly (lines 429 & 431). C’s error here in line 425 while attempting to locate 16 3/8” would not have resulted from the application of the process that he had just described in line 421 for finding sixteenths and eighths based upon the smallest increments on the measuring tape below twelve inches (2/32 = 1/16 and 4/32 = 1/8). And earlier in the dialogue (vignette seven), C had identified quarters successfully by sight, both below and above twelve inches (lines 407 and 309 respectively). Furthermore, in line 317 he identified eighths using a quarter inch interval as a benchmark on the region of the measuring tape above 12 inches. Therefore, based on the available data we can only speculate on the understanding being reflecting by C’s actions here in line 425. It is safe to say, however, that C’s conscious awareness of fractions on the different parts of the measuring tape is not yet stable and his objectification of this remains a work in progress. To summarize, to this point in the tutorial session C is systematically identifying larger fraction intervals (sixteenths, eighths and quarters) on his measuring tape based on the size of the smallest increment on the measuring tape—provided he has identified this correctly, he can also recognize quarter-of-an-inch intervals by sight both below and above the twelve inch mark, and he can use quarters as a reliable benchmark to find halves and, at times, eighths-of-an-inch. The difference in the smallest increments on the measuring tape between the region below twelve 103 inches and that above is something that C has not yet objectified to a point that allows for him to identify the size of these intervals automatically and reliably (at the level of operations). In the final vignette, L changes his approach with C from a strategy of seeing the different kinds of fractions on the measuring tape as successive subdivisions of intervals to a practical strategy that avoids having to deal with the thirty-seconds-of-an-inch divisions on the measuring tape altogether. Vignette eleven: Using the measuring tape in a new way In the final part of the discussion L leads C through an example of measuring 10 3/16” using the ten-inch mark on the measuring tape as the initial or starting point of this interval and, therefore, the 20 3/16” mark as the terminal or end point (lines 440-456). The intent here is to draw C’s awareness to a general strategy for measuring that avoids having to deal with the thirty- seconds-of-an-inch divisions below twelve inches on the measuring tape by selecting an initial or starting point for the lengths being measured other than zero. To help C visualize this measuring process and thereby extend his zone of proximal development for this, L aligns one edge of the piece of paper on the desk top with the ten inch point on the measuring tape, directs C to imagine putting the end of his pipe where the paper is positioned, and adds a slicing gesture to emphasize this location. L then sweeps his hand up the measuring tape from the ten inch point to show where the other end of the pipe would be positioned for measuring, as well as the region of the measuring tape being used in this case (line 442) (Figures 27 and 28). C immediately begins to count inches from the initial ten inch point on the measuring tape and, after L corrects a minor counting error (C counted the ten inch mark on the measuring tape as one instead of zero), C easily identifies 10 3/16 inches as the endpoint for this interval (line 451). C expresses his high level of satisfaction with this approach saying, 104 Figure 27. L aligns paper with the 10 inch mark L aligns the paper with the 10 inch mark on the measuring tape and with his left hand makes slicing motion as he says, “put one end of your pipe here” [52:38a]. Figure 28. L sweeps his hand up the measuring tape from the 10 inch mark L then sweeps his hand up the measuring tape to show where the pipe would be positioned for measuring [52:38b]. Good one. Yeah, that’d be better. (Picks the measuring tape up right off of the table top for the first time during the tutoring session, sits back in his chair, extends it in front of himself and stares at it intensely.) Like in my case it’d be better because it’s, I’ll know automatically, this is sixteen (G-sweeps his left thumb back and forth over division markings on the measuring tape above 10 inches while staring intensely at it.) after a foot. … I’m probably going to end up doing that. It’s way easier to just, cause this (G-points to the interval below one foot on the measuring tape), this is going to confuse me like crazy (G-sweeps his thumb over the divisions below 12 inches on the tape). Cause I have to remember, okay thirty, thirty-seconds, and this way I know it’s sixteenths, (G-sweeps this thumb over a small interval above 12” on measuring tape) and you can see it clearly, right, like the spacing-wise. (G-Moves the measuring tape to place it more directly into L’s view as he says this.) It’s a lot easier to understand. (G-waves the extended measuring tape in the air.) (lines 461-467) C again takes the lead in the dialogue by explaining to L some of the difficulties that other students in the class were having with their piping projects and staring at his measuring tape, breaking his gaze only occasionally while speaking (lines 463 through 469). After C finishes speaking, he continues to stare intensely at, and actively point to various points on his measuring tape for more than a minute (54:16-55:20), thereby enacting his newly objectified way of seeing the measuring tape. 105 The silence is broken when C asks L to verify that he has located 3 3/8 inches correctly, using the ten-inch mark as his starting point (line 473). After indicating to C that he is correct, L directs him to find 13 5/16”which he also does successfully (line 485). C then asks L to verify that he is locating quarter inch intervals correctly between 13 and 14 inches while he points to each in-turn and counts them aloud (line 487). In these three cases, C is recognizing the smallest intervals on his measuring tape above 1” as sixteenths-of-an-inch, using these as a benchmark for identifying eighths, and continuing to recognize quarters by sight. The dialogue ends with C telling L that he will use this new strategy for measuring, thanking L for his help, and returning to the workshop to finish his piping project. A summary of the generalizations that C enacts during the tutoring session In the early part of the tutorial session C enacts a number of generalizations related to fractions and measuring in fractions-of-an-inch with the measuring tape based upon his prior knowledge. Specifically, C has some sense of equivalent fractions in that he is able to come up with 4/16 as an equivalent fraction for 1/4 when locating 11 1/4” at the start of vignette one. Although, as was revealed in vignette nine when C makes conjectures about the number of thirty-seconds-of-an-inch required to make 1/16 and 1/8 (as every second and every third 32nds respectively), his objectification of equivalent fractions is unstable and of limited applicability for working with other equivalent fractions on the measuring tape until much later in the tutoring session. Another generalization that becomes evident early on, relates to the meaning of the smallest division lines on the measuring tape. C expects these to be the ones he needs to use regardless of the type of fraction he is enumerating, for example, when attempting to locate 4/16” below 12” on the measuring tape in vignette one and when attempting to locate 5/8” both below and above 12” in vignette two. Clearly, C is not differentiating between the different sizes 106 of these fractions-of-an-inch nor the difference in the division patterns on the measuring tape below and above one foot. It was suggested earlier in this chapter that C may have been expecting that the pattern of subunit divisions on the imperial measuring tape is read in the same manner as the pattern on the metric ruler (which is marked to tenths of a centimetre) that he was familiar with from his experience in school mathematics. Lastly, it is evident that C has some prior understanding of the meaning for the denominator of a fraction in that he is able to judge the relative magnitude of unit fractions presented to his in digit form, for example the fractions 1/16, 1/8, 1/4 and 1/2, early on in the tutoring session. L and C’s efforts attending to the mathematical meaning crystalized within the division pattern on the measuring tape begins in vignette two when L draws C’s attention to the difference in the pattern of the division lines on the measuring tape below and above twelve inches using a variety of gestures and C, in-turn, expresses these using the combination of an elaborate sequence of gestures and a few spoken words. C’s objectification of the measuring tape pattern deepens in vignette four when he recognizes that neither of the intervals he had identified as 5/8”, at 7 5/8 inches and at 13 5/8 inches, corresponds with the culturally accepted interpretation of 5/8” that he found using the eighths transparency. In this same vignette C begins to refer to fractions-of-an-inch as intervals between the divisions on the measuring tape instead of as the end-points of these intervals alone. C’s objectification of this is seen to continue as, in vignette seven, he points to eighths-of-an-inch on the measuring tape and, for the first time during the tutoring session, uses a distinct hopping gesture over each one with the tip of his pen while enumerating them. In vignette five we also see C moving away from his exclusive reliance on counting in combination with the use of indexical gestures when enumerating fraction-of-an- inch intervals within a given interval on the measuring tape. Here he relies upon mental 107 multiplication alone to correctly determine that there are twenty-eight quarters in seven inches based on the generalization that there are four quarters per inch. C begins to objectify particular aspects of the measuring tape that he can use strategically for measuring in different fractions-of-an-inch in vignette six. Here, C notices that the smallest increments on the measuring tape below twelve inches are thirty-seconds-of-an-inch. This is so, I argue, because C appears to follow L’s explanation of this closely, and subsequently in vignette nine without any further mention of this within the intervening discussion, C raises the point himself, that the smallest intervals on this region of his measuring tape are, thirty-seconds-of-an- inch. In vignette seven C points to and counts the four quarters on his measuring tape between 12 and 13 inches, while touching the end point of each one with his finger as he goes. His objectification of quarters on the measuring tape continues in vignette nine when he recognizes and counts four quarter-inch intervals within a one inch interval on his measuring tape below twelve inches by sight alone without the need for indexical gestures while doing so for the first time. While in the process of objectifying thirty-seconds and quarters-of-an-inch C is also objectifying relationships between these benchmark fraction intervals on the measuring tape and other binary fraction-of-an-inch values. Again in vignette seven, C identifies one-eighth-of-an- inch based on his initial recognition of one-quarter and in vignette nine he uses thirty-seconds (two of them) to identify an interval of one-sixteenth-of-an-inch. Subsequently, in vignette ten, C explains a strategy that he can use to identify sixteenths and eighths based on his recognition of thirty-second-of-an-inch and quarter-of-an-inch intervals; specifically that 2/32 equals 1/16 and 4/32 equals 1/8; and based on his recognition of quarters, that two quarters equal one-half-of-an- inch. 108 Throughout this activity C also objectifies the meaning of the denominator of a fraction, which indicates how many of the particular sub-units-of-an-inch span one whole inch. In vignette five C understands that there are four quarters in an inch and by vignette eight C is generalizing the this pattern this for eighths and sixteenths as well. Furthermore, by this point he successfully uses a strategy of counting down 3/8 from seven inches to locate 6 5/8” on his measuring tape. By the final vignette C is using a strategy, suggested to him by L, of measuring exclusively on the part of the measuring tape above twelve inches to avoid having to deal with the thirty-second-of-an-inch divisions below one foot. At this point C is recognizing the smallest increments above twelve inches as sixteenths-of-an-inch. He is using these as a benchmark for identifying eighths-of-an-inch, and along with his now consistent visual recognition of quarters as a benchmark, he is identifying halves as well. These values cover all of the fraction-of-an-inch sub-units that C needs to use within in his pipe trades training, as well as in the workplace when he became a fully qualified tradesperson. 109 CHAPTER FIVE: DATA ANALYSIS II This chapter presents further analysis of C’s mathematics activity in terms of cultural- historical activity theory (CHAT) as well as the theory of knowledge objectification. It begins with an analysis of the activity system within which C and L were working during the tutoring session, including the constituent elements of the activity, and illustrative examples of some of the mediating relationships between elements. This is followed by an analysis of the semiotic system of cultural signification within which this activity is situated and a discussion of various ways that C’s thinking is mediated by his use of semiotic artifacts or what Radford terms the “territory of artifactual thought.” This includes an analysis of iconicity, semiotic nodes, and the related processes of semiotic extraction and contraction; all significant processes that relate to the semiotic means of objectification used within this activity. It should be noted that the concept of semiotic extraction presented in this chapter is an original contribution to the theory of knowledge objectification. It involves a knowing subject (here L as tutor) elaborating on the cultural meaning of a conventional mathematical sign or set of signs for the purpose of drawing the attention of a learner to the way they reflect the corresponding conceptual object. The development of C’s subjectification during the learning process as well as my own subjectification as the tutor (L) is then analyzed. The chapter concludes by returning to a CHAT perspective with an examination of conflict and change within this activity system over the course of the tutoring session. C’s activity system As mentioned in chapter two, activity theory takes the activity system as the unit of analysis. This includes consideration of the subject, object, motive, objectives, community, 110 division of labour, rules or norms, and tools including artifacts and semiotic systems as co- mediating constituents of a single integrated system. Hence, a mathematics activity system involves a set of interacting elements beginning with subject, object, and tools. In this study of C learning to read a measuring tape, C is taken as the primary subject of the activity. He is motivated within the context of the pre-apprenticeship program by his desire to become a fully qualified plumber. At the level of the tutoring session, C is learning how to measuring using a measuring tape marked in fractions-of-an-inch so he can successfully complete his pipe fitting project, a requirement of the pre-apprenticeship training program. The object of C’s mathematics activity is the imperial system of linear measure including binary fractions-of-an-inch represented in physical form by the divisions on his measuring tape from his pipe trades school-issued toolbox.15 The community of this activity system includes multiple levels. The immediate community includes L as tutor, C’s colleagues within the class with whom he collaborates on a regular basis and the pre-apprenticeship course instructor who leads the pre-apprentices through their course work, evaluates their performance, and ultimately decides if they pass or fail. On a broader level, the community also includes those involved in administering the pipe trades training program that C is taking; those involved with the formal pipe trades apprenticeship credentialing system at the provincial level who determine what topics are addressed within pipe trades apprenticeship training, the standards to which these are learned, how students’ learning is evaluated, and how the apprenticeship training is delivered; and as well as the wider community 15 It is worth noting that the pattern of inscriptions on this artifact stems, no doubt, from its cultural-historical development as a tool for measuring in, rounding-off to, and converting between various subunits-of-an-inch as dictated by the demands of material workplace production. The alternative to having multiple fractional subunits-of-an-inch represented by a single complex inscription pattern would be to have a different measuring tape for each of the sub-units-of-an-inch that one might be required to measure. 111 of members of the pipe trades and others who interact with them in their work. It is the latter group who form the workplace community within which the apprentices will gain their on-the job training–an essential part of apprenticeship training–and whose workplace practice determines what knowledge and skills need to be addressed within apprenticeship training. There are divisions of labour between C and L specifically within the tutoring session. One example is the role of taking the lead in the discourse. For most of the first half of the session L leads the discourse by posing questions and setting tasks for C to do. Then in line 231 (in minute 14 of the 33 minute tutoring session) C asks something of L for the first time saying, “Wait, an inch?” to get L to repeat what he had just said about the relationship between the halves and quarters transparencies (in line 230). Then in lines 277 and 279 of the transcript (vignette six–minute 18 of the tutoring session) C asserts himself further in the discourse by asking L for clarification with the following question: “Oh, so then, little ones, the little ones, all the little ones, if I have three sixteenths I would just count. If I have one inch and three 16ths, I’d just count the little ones–one, two, three? Right?” And, after asking for clarification once again (a minute later in line 301) C takes control of the course of the dialogue for the first time in line 309. Here C identifies a measurement of his own to locate on the measuring tape and seeks L’s verification that he has it right: Okay. I’m getting. So say 12 and one quarter, right (grasps extended measuring tape with two hands while L is still holding the measuring tape case), so if I was looking for 12 and one quarter, just like on the (pipe fitting) project. These are all quarters you said, right? (G-pointing at 12 1/4” 12 2/4” and 12 3/4” in turn on the tape measure above 12” with his fourth finger.) These are all quarters. (line 309) By the final four minutes of the tutoring session C has taken over much of the role of leading. Here C sets for himself a variety of measurements to locate on his measuring tape and seeks L’s assistance to verify that he is doing this correctly (lines 463-469). And, in minute 56, C terminates the session itself (lines 489 and 491). Another example of the division of labour 112 within the tutoring session is the role that L plays in verifying that C’s work is correct—L does this throughout the entire tutoring session. The functional requirements of workplace production carried out by members of the pipe trades, namely installing, maintaining, and repairing pipes and related equipment, and the technology they use in their work influence the cultural-historical norms of their measurement practice. These have, in turn, been adopted as norms within the pre-apprenticeship course and within the tutoring session. For example, it is an established norm in the pipe trades that imperial measurements of length are made to the nearest sixteenth-of-an-inch. The pipe trades also use a number of conventions common to school mathematics as well. An example is that of expressing fractions-of-an inch, both when speaking and in written notation, in simplest or lowest terms and expressing amounts greater than one inch as a mixed number; although in the construction trades (including the pipe trades) the binary fractions-of-an-inch indicated on a measuring tape are the only fractions used for length when working in imperial measure. During the tutoring session, it also became customary for both C and L to justify their reasoning to one another for decisions made while working with the measuring tape. L initiated this by prompting C to explain his reasoning for the decisions he made and by explaining his own reasoning relating to the measuring tape as he worked with C. The artifacts that C and L use during their work together include the imperial measuring tape, a pencil, pen, paper, and L’s set of rulers printed on transparencies. In addition to the pattern of fraction divisions on the measuring tape and set of rulers on transparencies just mentioned, the semiotic resources that C and L employ in their activity include (in no particular order): • spoken language including mathematics vocabulary; • voice inflection and changes in volume; • mathematics notation; • three forms of gesture—pointing or indexical, sweeping, & chopping; 113 • a line drawn to represent five-eighths-of-an-inch; • indexical inscriptions such as circling or underlining existing inscriptions; • counting; • written text; • rhythm in speaking or gestures; and • the position, orientation, alignment of physical objects. These semiotic resources belong to different semiotic systems and it is apparent that their coordination is a central feature of C’s process of objectification. The use of these semiotic resources was described in detail earlier in chapter four and will be elaborated on here in terms of the theory of knowledge objectification. A diagram of the C’s activity system is provided in Figure 29. Mediation within C’s activity of learning to read a measuring tape Mediation within C’s mathematics activity refers to the ways that the various constituent elements of an activity shape or otherwise influence relationships between other elements of the system. The following examples are provided to illustrate these relationships keeping in mind that from an activity theory perspective it is presumed that all of the constituent elements play a mediating role in the relations between other elements within any activity system. Imperial measuring tapes are produced in a variety of different formats. And, as mentioned earlier, the tape that C was using from his toolbox was inscribed to thirty-seconds-of- an-inch below twelve inches (one foot) and to sixteenths-of-an-inch thereafter. This particular feature of the inscription pattern influenced C’s relationship with the object of activity, the pattern of binary fractions-of-an-inch, by necessitating that he attend to the difference between, and make sense of, these two division patterns. For example, had the measuring tape been marked to sixteenths-of-an-inch both below and above 12 inches, C’s approach at the very start of the session with L for determining 11 1/4”–by counting four of the smallest increments on the 114 Figure 29. The activity system within which C is learning to measure As mentioned earlier, the relationship between any two elements of the activity system, indicated by the straight line between them, is mediated by each of the other elements of the system. These form multiple and overlapping triangles on the model. Subject of analysis: C the pre-apprentice Object of the Activity: The imperial system of linear measure and the measuring tape Tools (artifacts and signs): a pencil, a pen, paper, and L’s set of rulers on transparencies Semiotic resources used include: • spoken and written language including mathematics vocabulary • written mathematics notation • the pattern of fraction divisions on the measuring tape and the set of rulers on transparency • gestures, e.g., pointing, sweeping, and chopping gestures • a line drawn to represent 5/8” • indexical inscriptions e.g., circling or underlining existing inscriptions • counting • rhythm when speaking and/or in gesturing • physical position, orientation, and alignment of physical objects Outcome: C’s objectification of the system of fractions marked on a measuring tape Community: • L as tutor • other students in the class • the course instructor • administrators of the pipe- trades training program and pipe trades apprenticeship • members of the pipe trades • persons in the construction industry who work with pipes-tradespersons Rules or cultural-historical norms: • measuring only to the nearest sixteenth-of-an-inch • expressing fractions in simplest or lowest terms when speaking and in writing • justifying reasoning for work done on the measuring tape during the tutoring session Division of labour: • leading the discourse • responsibility for determining if work is correct 115 measuring tape–would have been correct. In this case, C may not have needing L’s help to make sense of the measuring tape at all, or he may not have gotten help if he needed it because his difficulties might not have become apparent. And, had C and L engaged in discourse to make sense of the fraction division-pattern under these circumstances their joint activity would certainly have unfolded in a different way. Furthermore, the design of the fraction-of-an-inch divisions on C’s particular measuring tape draws his attention to particular aspects of the imperial system of linear measure (i.e., binary fractions to thirty-seconds-of-an-inch) on the one hand, while having the effect of limiting his attention only to these kinds of fractions. The pattern of divisions on this particular measuring tape does not, for example, draw C’s attention to binary fractions beyond thirty-seconds such as sixty-fourths-of-an inch-that, while not used in the pipe trades, are used when measuring in other trades such as metal machining. These examples illustrate the way that a particular pattern of fraction divisions on the measuring tape mediate the relationship between C and object of activity, characterized by his shifting relationships with the imperial system of inches and binary fractions-of-an-inch. C became aware of the mathematics represented within the pattern of divisions on his measuring tape and a particular way to use it (e.g., starting to measure from a point above zero to avoid having to deal with the thirty-second-of-an-inch divisions below 12”) over the course of his tutoring session with L. Had C not worked with L then his relationship to the system of binary fractions-of-an-inch inscribed on his measuring tape would likely have been different. This is an example of the subject–object relationship (C and the fraction pattern on his measuring tape respectively) mediated by a member of the community of the activity (L). The practice of starting to measure on the measuring tape away from the zero end, introduced by L to avoid dealing with the thirty-seconds-of-an-inch divisions below 12 inches, also highlights the way that a norm in the pipe trades, specifically that of measuring to 116 sixteenths-of-an-inch, mediated C’s relation to his measuring tape and the binary system of fractions-of-an-inch. This is an example of the subject–object relationship being mediated by a norm of measurement practice in the pipe trades. The last example of mediation to be discussed here focuses on the particular design of the fractions-of-an-inch division pattern on an imperial measuring tape, the complex pattern of inscriptions, and how this draws attention to the endpoints of the intervals measured rather than the intervals themselves. And, this is precisely how intervals measured on a tape are normally referred to by using words or gestures in North American measurement practice16—by referring to the numeric value of the endpoints of the intervals alone rather than as intervals between the zero end of the measuring tape and these endpoints. Had the pattern of divisions on the measuring tape evolved within the cultural-history of imperial measurement practice in a different way (e.g., as a system of inscriptions emphasizing measurements as intervals, rather then then endpoints of these intervals) it is conceivable that the way we would customarily refer to them while measuring with a measuring tape or ruler might be different than present practice.17 At the start of the tutoring session C refers to fractions-of-an-inch on the measuring tape using indexical or pointing gestures. By the end of the session, and as a direct result of L’s actions, this has changed completely. Following L’s lead, C starts to refer to fractions-of-an-inch as intervals using sweeping gestures on the measuring tape instead of pointing gestures during the course of the tutoring session. This is an example of both a tool (the semiotic system of fraction-on-an-inch divisions on a measuring tape in a material form of the object) as well as a 16 I refer here only to North American practice because of my first hand experience as a mathematics educator and my personal industrial work experience within this community. The key point is that the practice is widespread. 17 Note: An alternative design of the fractions-of-an-inch pattern on the measuring tape based on inscriptions of the fractions-of-an-inch intervals rather that their endpoints is discussed in chapter six. 117 member of the community (L in his role as tutor) mediating the relationship between C (the subject of the activity) and the measuring tape with the system of binary fractions-of-an-inch represented on it (the object of activity). The semiotic system of cultural signification As mentioned in chapter two, semiotic system of cultural signification refers to the cultural conceptions that relate to the nature of mathematical objects, mathematical methods, and means of representing mathematical knowledge. It relates both to the form of the activity and the mode of knowing or the episteme that relates to the activity. This will now be discussed in relation to the tutoring session with C. To help make the analysis more meaningful for the reader, some mathematics practices from the pipe trades will be contrasted with those of school mathematics. Beliefs about conceptual objects The conceptual object in this activity is the imperial system of linear measure in feet, inches, and fractions-of-an-inch. Given that the use of fractions within the present context is limited to halves-, quarters-, eighths-, and sixteenths-of-an-inch, the mathematical conceptual object is, mathematically speaking, a set of discrete numbers. In other words, no other numerical values matter when measuring in the pipe trades. In contrast, the concepts of number customarily dealt with in secondary school mathematics include the system of rational numbers–the entire set of numbers that can be represented using decimals and the infinite number of different kinds of fractions as well as the continuous (as opposed to discrete) set of real numbers that includes both the rational numbers and the non-rational numbers such as radicals (e.g., square roots, cube roots, fourth roots, etc.). One effect that this has on the form of mathematical activity in the pipe 118 trades is that when performing calculations with measurement values one needs to use methods that ensure that the results are accurate only to the nearest sixteenth-of-an-inch, without additional steps that provide any greater degree of accuracy. These additional steps serve no practical purpose and are therefore an inappropriate use of time in the workplace. (As the instructor reminded the students in this pre-apprenticeship class, “Time is money!”) Conception of truth Truth or, in the context of measuring in the pipe trades, fit deals with matters of equality and inequality of linear measure. Within the pipe trades and most, if not all, forms of material production, qualitative and empirical methods provide the basis for establishing mathematical fit. In other words, does a particular measurement get the job done? This message is conveyed to pre-apprentices by the course instructor who regularly indicates that the mathematical methods used to arrive at their results do not matter as long as their results are “close enough,” which means to within a sixteenth-of-an-inch. This has the effect of drawing attention to the physical objects being measured for validation of measurement results rather than to the calculations involved. Compare this again to the activity of school mathematics where the equality or correctness of results is often based solely upon the strict use of rational methods provided by the teacher or textbook. Methods of inquiry The pre-apprentice’s method of inquiry in the context of this training course is empirical trial and error (i.e., Does the pipe-assembly that they build in the workshop fit together as expected?). In keeping with this practical approach, L’s guidance of C during the tutoring session includes drawing C’s attention to contradictions that he can recognize in his own work through 119 the empirical comparison of physical measurements. This includes the comparison of a line drawn to represent a length of five-eighths-of-an-inch with a number of intervals that he identified on the ruler transparencies and his measuring tape. Legitimate ways of knowledge representation The mathematical knowledge represented using various semiotic means in C’s tutoring session with L dealt primarily with lengths in inches and fractions-of-an inch. The methods used for representing quantities of length within the pipe trades are the same as those taught in school (i.e., traditional mixed fraction notation, e.g., 13 7/8”). Fractions in the pipe trades are always rounded off to and expressed using a denominator that is a power of two (two, four, eight or sixteen) and they are expressed in simplest or lowest terms. For those of us who have grown up immersed within North American culture and who are fluent with this practice, it might seem odd to think that there could be any other or even a better way of doing this. But, consider for a moment the possibility when measuring or referring to the value like 13 7/8 inches, for example, alternatively as fourteen inches less an eighth. As the analysis of C’s work with the measuring tape in chapter four reveals, expressing 13 7/8 inches in this way is indeed a useful and, arguably more practical way to think about this value when locating it on a conventional measuring tape. The territory of artifactual thought Within C and L’s activity of making sense of the measuring tape, the territory of artifactual thought refers to the manner in which thinking becomes shaped by and enmeshed within the range of tools and signs that mediate it while tackling the questions or tasks at hand. Here it involves the pattern of feet, inches, and fractions-of-an-inch on the measuring tape and the ruler transparencies, fractions written using digits using standard mathematical notation, 120 spoken language, and fractions written in words. It is difficult, if not impossible, to even conceive of doing linear measurement to any degree of precision without the use of an tool, whether it be a simple artifact whose length we designate arbitrarily as the unit of measure, a commonly used cultural artifact with standardized units marked on it such as a ruler or measuring tape in imperial or metric measure, or some other measuring instrument. The particular measuring tool that one uses or has available within any particular activity often depends on the history, means of production, and corresponding degree of accuracy required within one’s particular cultural context. Semiotic means of objectification As mentioned in chapter two, semiotic means of objectification refers to the use of a wide variety of semiotic resources to draw the attention of others or oneself to some aspect of a mathematical object for the purpose of achieving a stable form of awareness, to making one’s intentions known, or carrying out actions to meet the objective of one’s activity. And, a number of examples have already been identified in chapter four. The analysis here will now focus on aspects of semiotic means of objectification that are particularly significant in mathematics learning activity generally and during the tutoring session in particular, namely C’s enactment of iconicity, semiotic nodes, semiotic contraction, as well as a new process identified through this analysis, that being the process of semiotic extraction that L enacts. Iconicity in C’s process of objectification To recap, iconicity is the process of noticing and re-enacting or re-voicing significant parts of previous semiotic activity for the purpose of orienting one’s actions or deepening one’s own objectification. Iconicity, however, does not refer to a form of knowledge representation but 121 to a form of objectification where consciousness of the cultural logic of thinking is being attained. C and L’s use of iconicity permeated the entire tutoring session in the classroom as well as their encounter in the workshop prior to this. Specifically, there are 60 instances when C re- voices or re-enacts something that L has said or done relating directly to the task at hand, expressing it using either the same or a different semiotic resource. There is also an example of C re-enacting a unique form of gesture that he, and only he, had enacted just seconds earlier in the tutoring session: a previously unreported form of iconicity. There are 14 instances where C or L repeats back a simple word like “okay” or “yeah” that had been just spoken by the other–suggesting C’s and L’s orientation toward one another L and the object at hand. There are also a few occasions when L re-enacts or re-voices what C has done or said and, prior to the start of C and L’s interaction, there is an example of C following the example of a classmate when he inscribes a line across his tape to help him remember what he thinks is the location of 11 1/4 inches as needed for his piping project. This frequent and varied use of iconicity illustrates the intensity of C’s and L’s sustained and active engagement together within the activity of C learning to read the fractions-of-an-inch divisions on the measuring tape. We now turn to the analysis of some particular instances of these various forms of iconicity in the process of C becoming consciously aware of the cultural meaning of the fractions-of-an-inch division pattern on the measuring tape. Some of the richest examples of C’s using iconicity occur during C and L’s encounter in the shop and the classroom at the very start of the tutoring session. This begins with C re- enacting for L how he counted division lines to arrive at what he thought was 11 4/16” by carefully pointing to the end points of these lines and then tracing the last division line with the tip of his pencil. C’s gestures, which are similar to an act of positioning or marking these particular inscriptions on the measuring tape draw L’s attention to and keep his own attention 122 focused on these tiny markings. When L shows C the correct location of 11 1/4” and indicates that the divisions that he has counted are thirty-seconds-of-an-inch he immediately repeats “thirty-seconds-of-an-inch” and then places his own thumbnail at the correct location of 11 1/4” on the measuring tape as L takes his pen tip away. During this brief episode, lasting less than a minute and a half, we see C using iconicity in the following ways: by following the example of another student and making a line on his measuring tape (across the measuring tape at the point he thinks is 11 1/4”), by re-enacting a process of inscribing or positioning individual division lines on his measuring tape with the tip of his pencil while counting them for L, by repeating what L had explained to him about the intervals he counted being thirty-seconds- of-an-inch, and by re-enacting L’s indexical gesture by pointing at the 11 1/4” point on the measuring tape himself. All of these actions serve to orient or attune C to the pattern of inscriptions on the measuring tape. Under L’s direction, C beings to notice the complexity of the division pattern on his measuring tape in vignette two using three different forms of iconicity. The immediate goal here is for C to begin noticing differences and similarities in the division patterns on the measuring tape below and above 12 inches. We see both forms of iconicity that have been identified to date in the research within this brief and intense exchange between L and C as well as a third form that, to my knowledge, has not been reported elsewhere. To review, this part of the dialogue begins in line 75 with L asking C to explain what difference he notices between the division pattern on his measuring tape below 12 inches and that above 12 inches. In the process of coming to see these two parts of the measuring tape as distinct C notices and re-enacts all of the hand gestures and corresponding hand positions that L had used while posing the question to him. These included his use of different fingers for pointing at the two different regions of the measuring tape in line 78 (see figures 14 and 17), a 123 sweeping gesture when referring to the region of the measuring tape below 12 inches in line 78 (see figure 14), and a static one-finger pointing gesture directed at the 12 inch point separating the two regions in line 78 (see figure 16). The re-enactment of gestures performed earlier by someone else is one of the forms of iconicity reported by Radford. While attending to and articulating the difference that he notices in the relative linear- density of divisions as well as the difference in the amount of space between adjacent division markings on the regions of the measuring tape below and above 12 inches, C re-enacts the inscription pattern on his tape albeit using a different form of semiotic resource then that of the printed divisions on the measuring tape. By employing an elaborate series of hand gestures along with a few spoken words, C expresses clearly the difference in the inscription patterns on his measuring tape below and above 12 inches. This includes C’s chopping gesture to characterize the closely packed pattern of division lines below 12 inches in line 78 (see figure 15), his wide- interval gesture to characterize the relatively wide intervals between adjacent division lines above 12 inches also in line 78 (see figures 18 and 19), and, immediately after this, his narrow- interval gesture to characterize the relatively narrow intervals between adjacent divisions below 12 inches in line 80 (see figure 22). This re-enactment of a set of signs, albeit using a different form of semiotic expression, is the second form of iconicity that has been reported by Radford. A third and new form of iconicity not reported elsewhere coincides with the second form of iconicity just described. It involves C noticing a form of gesture that he has used himself and then re-enacting this within a new context. I refer here to C’s use of a narrow-interval gesture using this thumb and first finger to characterize the relative size of the spaces between individual marking lines below 12 inches on the measuring tape in line 80 in figure 22. This occurs immediately after he used a similar wide-interval gesture using his thumb and first finger in reference to spaces between the individual marking lines above 12 inches in line 78 in figures 18 124 and 19. We can infer here that C became aware of the possibility or usefulness of utilizing this kind of interval gesture for describing the inscription pattern below 12 inches as a result of having just used it to characterize the measuring tape intervals above 12 inches because he promptly backtracked to elaborate on his previous description using this same kind of gesture. Like the first two forms of iconicity described earlier, this form of gesture also serves to deepen further C’s objectification of the difference in the system of division patterns on the measuring tape. In vignette five, we find another example of C re-voicing what L has said and re-enacting what L has expressed in written form, as he works intensely to make sense of the question that L puts to him in line 230, specifically, “An inch-and-a-half is an inch and how many quarters?” This episode follows immediately after L explained to C how the half-of-an-inch intervals are related to the quarter-of-an-inch intervals on the measuring tape using the transparency rulers. Upon hearing L’s question, C immediately seeks clarification of the amount by restating part of L’s question when he says, “Wait, an inch?” in line 231, and then repeats L’s response in line 233 saying, “Inch and a half,” while at the same time writing this value down on paper using digits. After L shows C the corresponding interval on the measuring tape C verbalizes the amount twice more by saying, “One inch and one half, okay.” in line 235 and elaborating on this in line 237 by saying “inch and a half is how many quarters?” in an apparent effort to move forward with L’s request. C then points to and counts each of the six quarters-of-an-inch within this interval on the quarters transparency and responds tentatively at the end of line 237, “Its well, six quarters, right? ” while writing this fraction in digit form on paper at the same time. While C’s responses indicate that he is objectifying only part of the original question posed by L, his responses serve to sustain the dialogue here that, in-turn, serves to deepen his objectification of the question. The episode continues with L restating the original question line 240 and C 125 writing his initial answer of six quarters in reduced fraction as three halves on the paper in digit form as well. As L restates the question twice more in slightly different forms in lines 242 and 252 C again re-voices some of the words that L has spoken (lines 247, 255 and 257). L then simplifies the question in line 258 by saying, “Or one and how many more quarters?” while gesturing intensely towards the quarter inch transparency to elaborate on this. C finally responds in the manner that L was looking for in words while he writes one and two quarters in digit form on the paper on the desktop. This episode of C and L’s activity, while not focused on one of the major generalizations that C achieves over the duration of the tutoring session, illustrates never- the-less the role of iconicity in the objectification process. Here C’s intense and sustained effort, including his use of different forms of iconicity, was directed towards making sense of the question that L was asking him. Another interesting example of iconicity occurs at the start of vignette nine. Here L invites C to explain what he is thinking (line 382) and, in response, C makes explicit reference to L having said what he then repeats back to him. What makes this form of iconicity significant and different from the other examples discussed to this point is, that it was initiated by L as a deliberate strategy to get C to reflect further upon this thinking and thereby consolidate or further his objectification of the fractions-of-an-inch pattern on the measuring tape. In contrast, all other instances of iconicity during the tutoring session were initiated spontaneously by C. In line 383, in response to a request from L to elaborate on a previous comment, C says “Okay, so you said this is...,” but is unable to go further and provide any details. L then follows- up on what C has said in lines 386 and 388 with a prompt to keep him going, “What did I say? Each, each little interval is how much?” while pointing at a 1/32” interval on the measuring tape with the tip of his pencil. This request successfully creates a zone of proximal development for C 126 as he responds in lines 387, 389 and 391 by re-voicing what L had explained to him earlier in line 290. Specifically, C responds to L’s request by saying, Each little interval (while bringing the tip of his pen to the same point on the measuring tape as L, as soon as her begins pointing there), you said it’s thirty- seconds. From, like if you count, if I were to count every one, every one would be thirty-second. And then you said the next one would be every sixteen (while pointing to successive 1/16” divisions on his measuring tape). Here C is objectifying the thirty-second-of-an-inch intervals on his measuring tape below 12 inches in a distinct way and using these intervals as a basis for identifying sixteenths-of-an-inch as well. L’s use of semiotic nodes Semiotic nodes refer to the use of multiple semiotic systems in concert as part of activity for the purpose achieving objectification. During the tutoring session L uses semiotic nodes on at least 33 occasions to help further C’s objectification of the system of binary fractions represented on the measuring tape. Specifically, L uses various combinations of words, pointing and sweeping gestures, fractions written using digits and words, the fractions-of-an-inch pattern on the measuring tape and transparency rulers, along with other semiotic resources in a coordinated manner throughout the tutoring session to assist C with his objectification of the system of binary fractions-of-an-inch on the measuring tape. The range of L’s use of semiotic nodes during the tutoring session is exemplified by the set of episodes that relate to his efforts to draw C’s attention to the relationship between different kinds of fractional-subunits-of-an-inch on the measuring tape. A summary of this sequence of episodes follows. When it becomes apparent to L that C had made an error in locating 11 1/4 inches on his measuring tape during prior to the start of there discourse in the shop (lines 58-66, vignette one), L tells him that the point he located was one-eighth and that the smallest intervals on the 127 measuring tape that he had counted are thirty-seconds-of-an-inch. L also explains that the 4/32 he had counted makes 1/8, and points to the correct location of 11 1/4 inches. In the process of providing this explanation L points to the endpoints of the different fraction intervals with the tip of his pen and his fingernail while he is speaking about each of them. L also points to the individual quarter-inch intervals that span 11 to 12 inches on the measuring tape using an exaggerated hopping gesture. While attempting to draw C’s attention to each of these quarter-of- an-inch intervals, in-turn, L uses a repeating sequence of words in a steady rhythm, “There to there’s one, there to there’s two, there to there’s three...” (line 64). A short time later in lines 113-123 (vignette three), L asks C to compare the fractions eighths, halves, sixteenths, and quarters; and writes each of these fraction names on paper in words for C to refer to as he says them. Then L adapts this task to the immediate context for C by reconstituting and restating it verbally using mixed number values similar to the mixed number values that they had just been working with on the measuring tape, specifically as 5 1/8, 5 1/2, 5 1/16, and 5 1/4 respectively. L points, in-turn, to each of the corresponding fractions that he had just written in words with the tip of his pen as he says each of the corresponding mixed numbers and directs C to start by identifying the largest of the set. L then reiterates his request by repeating this combination of spoken words along with the accompanying indexical gestures a second time. When C responds with 5 1/2 as the greatest value of the set L repeats this while writing this down on the paper in digit form. He then directs C to identify the smallest of these values while pointing again to the original set of fraction names written on the paper. When C responds with 5 1/16, L writes this mixed number in digit form to the left of the 5 1/2 written earlier, with enough space in the row between these mixed numbers for the remaining two values to be written and says, “And if you were to put the other ones in order, here and here, where would they go?” (line 119) while pointing to the space between 5 1/6 and 5 1/2 on the paper with 128 his pen. The episode ends with C stating the remaining values in their appropriate order and L writing them in their appropriate positions in the row. In this example, the semiotic nodes that L uses are comprised of combinations of the numbers written on the paper either in words or digits, their positions on the page, and the spaces between the numbers already written on the page for the remaining numbers to be written, pointing gestures, and verbal explanations of the task. In vignette four L introduces the set of transparency rulers to draw C’s attention to the way that different fractions are indicated, and related to one-another, on the measuring tape. As explained in detail in the previous chapter, the semiotic systems that L employs together in a coordinated sequence in lines 130, 134, 136 and 138 include: (a) spoken language; (b) the fractions-of-an-inch pattern printed on measuring tape and the transparencies; (c) gestures—a series of sweeping gestures, pointing gestures, and an iconic inscribing gesture where L moves his pen as if inscribing a quarter-of-an-inch division at the midpoint of the half-inch interval; (d) conventional fraction notation—the fraction one-half written on the page; (e) indexical inscriptions—drawing circles around the denominators of the fractions written in numeric form to draw C’s awareness to this part of the fraction; (f) physical positioning—the alignment of the transparency rulers on top of one another; (g) the system of whole-numbers—counting the number of a particular size of interval spanning one inch to confirm that these intervals represent eighths-of-an-inch; and (h) variation in the volume of his speaking voice (e.g., saying the words louder for emphasis). Immediately following this (lines 140 to 144) L directs C to compare the line that was drawn earlier to document the distance between 13” and what C had identified as 13 5/8”, as well as the measurement he had identified as 7 5/8”, with the interval of 5/8 that he had just located on the eighths transparency. As L speaks to C he holds the tip of his pen to the (culturally accepted) location of 5/8-of-an-inch on the eighths transparency that C had just identified and 129 aligns the drawn line and the eighths transparency along side the measuring tape between the seven and eighth inch points so that C can view these three interval sizes all together. This coordinated set of actions helps C to recognize clearly that neither of the values that were found earlier to represent 5/8” correspond to the culturally accepted value for this measurement. The final example (from vignette five and described in detail earlier in the section on iconicity) reflects L’s persistence to help C to understand the question he has asked, initially in line 230, “an inch and a half is an inch and how many quarters?” At the start of this episode L poses this question verbally while pointing to the 1 1/2” point on the quarters acetate ruler. He then says the question once more, points again to the 1 1/2” point on the ruler, and directs C to look at the ruler (line 234 and 236). C responds by saying and writing down six-quarters in digits on the paper in front of him. At this point L repeats the task (line 240) by saying, “Or one inch” while providing emphasis with a large downbeat gesture in the air with the extended first finger of his right hand followed by “and how many quarters after the inch?” accompanied by an exaggerated sweeping or hopping gesture through the air with this same finger. C then writes “1 3/2” on his paper and L says, “Now six over four is three over two,” while emphasizing each of these fractions by making a small hopping gesture with the first finger of his right hand as he says each of them. He then points to the one-inch point on the transparency ruler while he explains, “we’re going to keep the one inch,” and then sweeps his finger tip between 1” and 1 1/2” while saying, “we’re just going to change the one-half part” (line 242). L follows this by saying, “one-and-a-half is to here,” while pointing to this point on the ruler in line 244 and, “that’s the same as one and how many quarters?” while making a pointing gesture to the one- inch point on the ruler followed by a sweeping gesture with the tip of his finger again through the interval between 1” and 1 1/2” (line 246). After C responds with an answer of “six” in line 253 L uses his pen to make an indexical circling gesture around the interval between 0” and 1” on the 130 transparency ruler while saying “or, one” followed by a sweeping gesture through each of the two quarter inch intervals between 1” and 1 1/2”, one after the other, and says, “and how many more quarters past one to make, to here” (while holding the tip of his pen at 1 1/2”) (line 258). At this point (line 259) C says the correct answer to L’s revised question by responding, “Yeah, one and two quarters,” (line 260). To summarize, L’s use of gestures and other semiotic systems serve here to draw and sustain C’s attention to/on the things that he is explaining to him. L’s use of semiotic nodes described here along with analysis of his use of these elsewhere throughout the tutoring session data reveal that the configuration of his semiotic nodes changes very little over the course of the session. Given Radford’s (2005b) assertion that the changing configuration of one’s semiotic nodes reflects the progressive course of his/her objectification of a particular cultural object, this consistency is not a surprise. These explanations simply reflect L’s own relatively stable objectification of the system of binary fractions-of-an-inch used in imperial measurement practice generally. The stability of L’s own objectification of this mathematical object reflects his expertise with it. This is the result of L having decades of experience working with the fractions-of-an-inch pattern on a measuring tape since he was in elementary school (during the pre-metric era in Canadian schools) and his extensive industrial work experience using the imperial system of linear measurement earlier in his career. C’s use of semiotic nodes and semiotic contractions C also uses semiotic nodes on numerous occasions to express his thinking to L and in his efforts to bring clarity to his own thinking as well. These include instances where he does so in response to invitations from L to explain his thinking. The way that C mobilizes semiotic resources while he is focusing on particular aspects of the mathematical object changes over the 131 course of the tutoring session as well revealing instances of semiotic contraction within his knowledge objectification process. Semiotic contraction refers to the process of making semiotic actions compact, simplified, and routine as a result of acquaintance with conceptual traits of the objects under objectification and their stabilization in consciousness. In contrast to L, C enacts semiotic nodes less frequently during the tutoring session. This is not surprising given that L directs much of the activity thus leaving less opportunity for C to explain his own thinking. The way that C’s use of semiotic nodes change as the tutoring session progresses reflects the deepening of his objectification of the system of binary fractions-of-an- inch on the measuring tape. Sets of episodes from the data that illustrate this will now be presented to exemplify this process. The first set of episodes where C enacts semiotic nodes focuses the difference in the division patterns on the measuring tape below and above 12 inches. At the start of his work with L in the classroom, as was described in great detail earlier in vignette two, C responds to a question from L about the difference that he notices between the division patterns on the measuring tape below and above twelve inches. C uses a sequence of ten gestures including sweeping, chopping, pointing, and interval gestures (see figures 14 to 22) within a span of nine seconds, each positioned in a precise way relative to the measuring tape, together with a following sentence and two sentence fragments: “There’s, there’s more (in reference to the divisions on the measuring tape below 12 inches). It’s like it’s more spread out when you pass one, one foot.” followed by, “And when you’re before one foot its more, um.” (lines 78 and 80). A minute and a half later in lines 102 and 104, C makes a conjecture in precise quantitative terms about the mathematical relationship between the intervals on the measuring tape above 12 and below 12 inches. C says, “They [the intervals above 12 inches] might be going by twos rather than by ones,” while simply pointing with a static gesture to the region above 12 inches with the 132 fourth finger of his right hand and then another to the region below 12 inches with the first finger of this same hand as he speaks, in-turn, about these parts of the measuring tape. These gestures are shown in figures 30 and 31. In these two episodes C’s mobilization of semiotic resources to explain the difference in the division pattern on the measuring tape shifts from one that is dominated by the use of gestures along with only a small number of rather vague spoken words to one where the use of gesture is relatively muted–a contraction in his use of gesture. Furthermore, C’s use of spoken language here is direct and to the point by making reference to a precise mathematical relationship between the two interval sizes that he is comparing. Another set of episodes where C enacts semiotic nodes relates to his objectification of the size of the smallest increments on the measuring tape. In line 277 during his first opportunity to articulate his developing understanding of the smallest intervals on the measuring tape during the tutoring session C has to make a concerted effort to put his thinking into words. He says, “Oh, so then, little ones, the little ones, all the little ones, if I have three sixteenths I would just count. If I have one inch and three 16ths, I’d just count the little ones–one, two, three.”18 While saying, “three sixteenths” C makes a chopping/hopping gesture three times in the air with his right hand above the table top and away from the transparency ruler, each coordinated with one of the syllables in the words “one, two, three” (see figures 32 and 33) for emphasis and then three more chops in the air with his 4th finger in unison with the numbers “one, two, three” at the end of his sentence (see figure 34). A minute and fifteen seconds later (line 301) C makes reference to the smallest intervals on the measuring tape once more, having just listened to L explain how to recognize sixteenths 18 C’s conjecture here about the smallest intervals between one and two inches, while correct when working with the smallest intervals on the ruler acetates, is not appropriate for working with the measuring tape. The correct interpretation of the smallest increments on his measuring tape below 12 inches is thirty-seconds-of-an-inch. 133 Figure 30. C points to the region of the measuring tape above 12 inches [28:34] Figure 31. C then points to the region of the measuring tape below 12 inches [28:36] Figure 32. C begins a chopping gesture C begins one in a series of chopping gestures with his right hand as he says one and three sixteenths [41:24a]. Figure 33. C ends the chopping gesture begun in the previous figure [41:24b] Figure 34. C makes a chopping motion with his fourth finger C makes three chopping motions with the fourth finger of his right hand as he counts “the little ones–one, two, three” [41:25]. both below and above 12”. This time C’s use of gestures is significantly less pronounced and he speaks more directly and to the point without repeating himself as was the case in the previous episode reported here. Here C is confirming with L that his has it right when he says, “Okay. So if it was thirty-seconds then you’d count the little small ones.” While C says this he makes six very small rhythmic hopping gestures in the air with his left hand (flexing from his wrist) as if enumerating these small intervals on a measuring tape and then a pointing gesture with the first finger of his left hand as if to an imaginary tape measure in front of him. All the while he keeps his left forearm at rest on the tabletop (see Figures 35, 36, and 37). C describes the smallest intervals on the measuring tape one last time, in response to a prompt from L to explain what he 134 Figure 35. C’s left hand at the apex of a small hopping gesture C makes one of six small hopping motions with this left hand, here at the apex of a hop [42:49a]. Figure 36. C’s left hand at the bottom of the small hopping gesture C’s hand is at the bottom of the small hopping motion begun in the previous figure [42:49b]. Figure 37. C makes a pointing gesture in the air C finishes this sequence of gestures with a pointing gesture into the air [42:50]. had told C about these. C states confidently that, “each little interval, its, you said it’s thirty- seconds,” (lines 387 and 389) and then elaborates, without being prompted to do so by saying, “like if you count, if I were to count every one, every one would be a thirty-second” (line 389). While he says this, C points, in-turn, to a series of individual thirty-second-of-an-inch intervals within the span of one inch on the measuring tape with the tip of his pen. Here C is speaking in a more direct manner, explaining how he can verify that the smallest intervals are, indeed, thirty- seconds-of-an-inch by making reference to the process of enumerating them. Now his use of gestures is coordinated with his speaking and focused in a more precise way on the particular intervals to which he is referring. In the preceding set of episodes there is a distinct shift in the way that C uses hand gestures to accompany his verbal explanations as well—they become much less pronounced and become directed more towards the particular individual division markings or intervals on the measuring tape that C is talking about. This is a contraction of his use of gesture. There is also a shift in the way that C explains the smallest increments on the measuring tape in words. Over the course of these three examples C comes to speak in a more direct manner—a contraction of speech. By the final example he provides an explanation to L of how he can verify on his own 135 that the smallest intervals are thirty-seconds-of-an-inch without being specifically asked to do so. Within the context of this tutoring session this indicates that C has achieved an unprecedented level of objectification of this aspect of the fractions-of-an-inch pattern represented on the measuring tape. A particularly rich set of episodes of C using multiple semiotic systems occur in his process of objectifying quarter-of-an-inch intervals in a distinct way on the measuring tape. We know from C’s actions at the very start of his encounter with L in the workshop that he does not recognize quarters-of-an-inch on the measuring tape by sight, nor does he identify them in the culturally appropriate way using the smallest increments on the measuring tape below 12 inches. And, later in the dialogue (both in lines 110 and 195) C is unable to apply the general relationship between the denominator of a fraction-of-an-inch and the number of that fraction spanning one whole inch when asked to do so. Together, these observations indicate that in the early going of the tutoring session C’s objectification of fractions-of-an-inch in general and quarters-of-an-inch in particular on the measuring tape were, at best, unstable and lacking depth. While discussing the idea of fractions on the measuring tape as intervals of different lengths using the set of ruler transparencies (in vignette five) L aligns the quarter inch transparency with the corresponding divisions on the measuring tape and calls C’s attention to the lengths of the lines that, along with the half-inch and whole inch divisions, are used to mark quarters. In response, C spontaneously counts the four quarter-of-an-inch intervals between one and two inches on his measuring tape while pointing to each in-turn with the tip of his pencil (lines 211 to 219). In an apparent effort to deepen his objectification of the idea that four quarters span one inch on the measuring tape, L asks, “How many quarters fit into two inches?” (line 222). C immediately begins enumerating these on his measuring tape while pointing, in-turn, to the endpoints of each interval with his pencil tip (line 223). Immediately thereafter, in response 136 to a follow-up question from L about the number of quarters in seven inches (line 228), C responds immediately and confidently with the answer “28,” without gesturing in any way. Later in the tutoring session (vignette nine, line 407), as L is explaining the relationship of the division line lengths at the end-points of unit fractions on the measuring tape (i.e., at 1/32, 1/16, 1/8, and 1/4) and the corresponding unit fraction magnitudes, C spontaneously and immediately counts the four quarters spanning an inch when the focus of the dialogue gets to the quarter-of- an-inch divisions on the measuring tape, again without the use of any hand gestures at all. In the final episode of this set, less than a minute later (in vignette ten), C articulates a clear, concise and complete description of how he is now able to recognize each of the fractions, thirty- seconds, sixteenths, eighths, quarters and halves on his measuring tape, in-turn (line 421). In the process he refers to quarters in a distinct way for the first time in the tutoring session, specifically using the word “space”19 to describe them. Like, like now I understand it because every two would be sixteenths (referring here to the smallest increments marked on the tape measure below twelve inches while pointing at these division lines on the tape measure with his pen), every four would be eighths (while pointing now at eighths division lines on his tape measure with his pen), every space would be quarters (C makes a distinct sweeping gesture with his pen through a quarter-of-an-inch interval), um, every two spaces would be halves (while pointing, in turn, at each of two quarter-inch intervals within a half inch on the measuring tape). (line 421) Over this series of episodes and even within one of the episodes we can see clear evidence of the shift in the configuration of C’s use of semiotic nodes with his semiotic actions becoming compact, simplified, and routine whenever he makes reference to quarters-of-an-inch on the measuring tape. In the first of the three episodes C determines the number of quarters within an interval of one inch and then within an interval of two inches by counting each of these sub- 19 To this point in the tutoring session L has used the terms “interval” and “space” interchangeably when referring to the quantities of distance for various fractions-of-an-inch sub- units on the measuring tape. 137 intervals aloud and pointing to each in-turn using the tip of his pencil. C then responds without any hesitation using mental multiplication alone and without the use of gestures or counting, based on the generalization of there being four quarters per inch when he is asked to determine the number of quarters in seven inches. This is an example of a verbal contraction. The way that C counts quarters within the interval of a single inch on the measuring tape changes over the course of the tutoring session as well. In the second example C counts the four quarters spanning an inch on the measuring tape one-by-one, but this time he does so without the use of any hand gestures (line 407)–a contraction in his use of gesture. Then in the final example of this set (line 421) C starts to talk about quarters-of-an-inch in a distinct way, referring to them simply as “spaces,” a unique identifier that C started to use on his own in comparison to the way he and L had been referring to the other fraction-of-an-inch intervals on the measuring tape to this point, which is another example of verbal contraction. Cultural-historical semiotic contractions and semiotic extraction A widely recognized feature of mathematics activity in many settings is the ubiquity of culturally and historically developed and highly refined/semiotically compact systems of signs. I will refer here to the conventional forms of semiotic contractions used within these systems of mathematical signs as cultural-historical semiotic contractions. Linear measurement practice in the pipe trades is no different in this regard with a number of cultural-historical mathematical conventions figuring prominently. These include: the conventional way that fractions are written using digits; the convention of expressing fraction amounts in simplest or lowest terms; the common practice of referring only to the end-point of intervals on a ruler or measuring tape as a measurement when, actually, it is the interval between the zero point on the ruler or measuring tape and these end-points that represent measurements; and the convention of using a sequence 138 of successively shorter division line lengths on an imperial measuring tape or ruler (a particularly elementary form of signs) to signify each successive set of binary sub-divisions of the inch. Not surprising, it was necessary for L to start his work of helping C objectify the corresponding aspects of measurement practice with the measuring tape during the tutoring session by making a conscious, systematic, and sustained effort to make apparent the mathematical meaning of these cultural-historical contractions. I call this process of social interaction, when a knowledgeable participant elaborates on the meaning crystalized within semiotically compact mathematical signs in a deliberate and systematic way for the purpose of helping a novice to read this meaning, semiotic extraction. This novel contribution to the theory of knowledge objectification provides a complement to Radford’s process of semiotic contraction. It draws upon and elaborates Ball and Bass’ (2003) idea that the work of teachers includes a process of “unpacking” meaning for students from highly compressed and abstract forms of representation used in mathematics.20 The most prominent process of semiotic extraction that L undertakes with C during the tutoring session relates to the fractions-of-an-inch division pattern on the measuring tape using the set of ruler transparencies that he has prepared for this purpose. This teaching tool is designed to help the learner read the distinct ways that this system of divisions can be interpreted to indicate halves, quarters, eighths, sixteenths and thirty-seconds-of-an-inch by allowing for each of these fraction-of-an-inch intervals to be represented and examined one at a time. It 20 Sarbo and Farkas (2004) have developed a general cognitive-linguistic model to explain the process of understanding language that they call meaning extraction. The key differences between meaning extraction and semiotic extraction are that the former is a posited as a text summarization process within the mind of the individual based upon unproblematic language signs as input. The latter refers the process of an informed individual teaching another how to read the (problematic) cultural meaning of a mathematical semiotic system (of any form). This embodied objectification process often involves the use of a variety of semiotic resources such as mathematical notation and gestures in addition to spoken or written language. 139 allows for these different scales to be superimposed so that their inter-relationships can be made more explicit as well. It was also necessary for L to draw C’s conscious awareness to the idea of measurements represented on the measuring tape as intervals rather that as the end-points of these intervals alone, another form of semiotic extraction. L accomplished this by making explicit verbal reference to this idea on a number of occasions and through the use of hopping or sweeping gestures when referring to specific measurements on the measuring tape throughout the tutoring session. The other two forms of cultural-historical semiotic contractions that are encountered during the tutoring session deal with conventional fraction notation. On a number of occasions, L makes explicit reference to the meaning of the denominator of a fraction sign to help C, in turn, become consciously aware of the relationship between this value, the number of intervals spanning one inch for fractions of this kind, and the size of this kind of fraction. The convention of expressing fractions in simplest or lowest terms was not encountered as problematic during the tutoring session and, while this was a cultural-historical semiotic contraction used within this activity, it did not need to be addressed explicitly by L and C in their joint activity. C’s and L’s subjectification during the tutoring session Despite the brevity of C and L’s joint activity, a face-to-face dialogue lasting only 33 minutes, it is clear that both C and L changed with/in this activity system over time. As mentioned earlier, Radford (2008c) uses the term subjectification to refer to the process within the course of mathematics activity through which a subject inevitably finds him/herself changed as he/she learns to be with/in particular mathematics activity. In the present study this includes analysis of C’s role or the division of labour (as framed from a CHAT perspective), as well as his 140 sense of self, position as a participant as a subject in the activity, and both C and L’s relationship with the object of the activity. Analyses of C and L’s subjectification during their joint activity will now be explored in turn. C’s subjectification during the tutoring session The manner in which C participated within the activity of the tutoring session evolved over its duration in a number of ways. I interpret this change in his actions and operations as reflecting the process of subjectification. The analysis here focuses on: C’s attentiveness during the session; his role in the discourse; his affective responses, including his appraisal of L’s approach to teaching him how to read the measuring tape; and finally, the changing sense of sense of agency and self-reliance that C articulates in relation to his use of the measuring tape. Throughout much of the tutoring session C sits up straight in his chair with his chest either close to or touching the edge of the table top, gazes intensely at the actions taking place on the table in front of him, listens carefully to what L is saying, nods his head21 (often in response to things that L is saying), and follows his directions closely. During the first 17 minutes of the dialogue C breaks his gaze only twice from the measuring tape and other materials on the table top to look directly at L while L is speaking to him, and then only for a total of six seconds. About 18 minutes into the session C starts to look directly at L more regularly (between one and two times per minute on average) in most cases for only a second or two and on a few occasions for between three to five seconds until the end of the session. C also breaks his gaze on the activity on the tabletop by sitting back in his chair on two occasions. The first of these occurs in minute 24 when he sits back and then immediately resumes his forward sitting position. The 21 While C’s head nodding was not part of the formal gesture analysis on the transcript of the tutoring session, it is clear for even a cursory examination of the video recording that C frequently responds to the things that L is saying by doing so. 141 second time that C sits back in his chair lasts for a total of 65 seconds during minutes 30 and 31 of the dialogue. This coincides with C appearing to be more at ease and confident with his newly developed awareness of the mathematical pattern of fractions-on-an-inch on the measuring tape. As was mentioned earlier in this chapter–in the discussion of the division of labour within this activity–C’s role in this discourse begins to change in line 231 (minute 14). Here C asserts himself in the activity for the first time by asking L to repeat what he had just said. Thereafter, C asserts himself increasingly within the activity by asking for clarification, posing questions directly to L, reflecting on his earlier difficulties and the new insights that have allowed him to overcome them, and even cutting L off mid-sentence on one occasion (line 421) to express his new understanding of the fractions-of-an-inch division pattern on the measuring tape. By the final four minutes of the tutoring session C is playing the lead role in the discourse. Here he explains, in detail, why the strategy of measuring exclusively on the region of the measuring tape above 12 inches will be simpler and more reliable for him to use (e.g., lines 463-469), sets a variety of measurements to practice locating himself on his measuring tape and asks L to verify that he is reading each one of these correctly (lines 473 and 487), and draws the tutoring session to a close (lines 489 and 491). This change in C’s assertiveness over the duration of the tutoring session reflects both his objectification of the fraction-of-an-inch pattern on the measuring tape and his changing subjectivity in relation to this learning activity as a whole, including L. C makes value statements in relation to his learning throughout the tutoring session on fourteen occasions, either directly or indirectly. During the first 18 minutes C reacts outwardly to difficulties he is experiencing on two occasions. In line 185 (minute 12 of the tutoring session), for example, C says, “That’s what screws everything up,” as L brings out the halves transparency while re-explaining the different fractions marked on the measuring tape, and responds sarcastically with, “Great” in line 275 (minute 17) as L brings out the sixteenths acetate 142 as part of the same systematic re-explanation. Two minutes later in lines 297 and 299 C’s tone changes as he responds positively to L’s explanation of how to identify sixteenths-of-an-inch intervals on the measuring tape. Here C says, “...that makes sense” and even emphasizes this by repeating this statement a second time. The remaining nine instances of C making value statements in relation to his learning all occur during the final four minutes of the session. In each case C indicates that he appreciates the method that L has shown him–using the divisions above 12” exclusively where the smallest increments are sixteenths-of-an-inch. Examples of C’s comments here include “good one” (line 461, minute 30), “it’s a lot easier to understand” (line 467, minute 31), and “its really good” (line 497, minute 33). The change in tone of C’s affective remarks over the course of the tutoring session along with his flurry of nine comments in the final four minutes of the session reflect clearly a significant change in his disposition towards the activity, specifically his level of comfort and confidence in his ability to use the measuring tape as needed for his work in the pipe trades. Another form of affective response that was evident for only brief instances during the tutoring session was C’s facial gestures. Due to the position of the video camera during data collection the available video data provides only a few instances where C’s face is clearly visible. One prominent instance comes in line 463 (minute 53) as C, leaning back in his chair, distinctly raises his eyebrows in unison four times in a row while saying, “Yeah, that’d be better. Like in my case it’d be better because I’ll know automatically, this is sixteen.” (See Figures 38 and 39).22 22 The description of this single instance of C’s facial gesture is included here to further characterize the way in which C was responding during the tutoring session and to suggest facial expressions as another category of data for consideration when analyzing the subjectification of learners in future research. 143 Figure 38. C before raising his eyebrows C’s facial expression immediately prior to raising his eyebrows [53:46a]. Figure 39. C with eyebrows raised C raises his eyebrows while explaining that using the strategy of measuring exclusively on the region of the measuring tape above 12” would be better in line 463 [53:46b]. C’s sense of agency and self-reliance reading the measuring tape also change over the course of the tutoring session. We can infer this from his reports of this that come near the end of the session. In line 407 (minute 26) C signals his recognition of a shift in his relationship with the object of the activity by saying “Okay, yeah. I think I got it now,” in reference to his ability to identify sixteenths, eighths and quarters using the smallest increments (thirty-seconds) as a benchmark. In line 463 (minute 30) C explains, “I’ll know automatically this is sixteen” in reference to the smallest intervals on the measuring tape above 12 inches. In line 467 (minute 31) C makes direct reference to his earlier difficulties with the words, “So maybe that’s why yesterday’s project, I was having such a problem with it.” Then, in this same utterance, he makes an indirect reference to his own sense of agency when he offers his analysis of the difficulties being experienced by some of his classmates. Yet another indication of C’s new found sense of agency and self-reliance is also evident at the very end of his time seated at the table beside L when he stares intensely at the measuring tape for more than a minute and practices counting 144 fraction-of-an-inch intervals of different sizes silently on his own (line 471). And, finally, in line 493 (minute 33) after having gotten up from the table where he had been seated with L, C comments on his initial difficulties with the piping project prior to his tutoring session with the words, “now, now I know why the (...).” Each one of the individual sets of actions or operations that have been tracked over the entire tutoring session reflects a marked change in C’s way of being within the activity. To make these patterns more apparent for the reader, the temporal distribution of events within each of these sets of actions or operations is indicated on a minute-by-minute bases in the frequency pictograph shown in Figure 40. Note the change within each category over the duration of the tutoring session. And, most significantly, consider the pattern of change as a whole. Compare, for example, the combined frequency and coincidence of the actions and operations during the final four minutes of the tutoring session with any previous four-minute interval of the session. L’s subjectification One significant change in L’s subjectivity is evident as the tutoring session progresses. This relates to the change in his approach to teaching C how read the system of divisions on the measuring tape. At the beginning of the session L’s approach was a general one, typical of the way that school mathematics is taught. Here L’s explanations of the fractions-of-an-inch on the measuring tape all start with the whole inch, and proceed from there to halves, then quarters, then eighths, and so on (lines 182-276). This progression allows for any number of different binary fractions-of-an-inch on a ruler or measuring tape to be identified as needed (e.g., halves, quarters, eighths, sixteenths, thirty-seconds, sixty-fourths, etc.) and is thus applicable in the widest range of possible contexts where various forms of rulers or measuring tapes marked in inches might be encountered. By the end of the tutoring session L abandons this general 145 C sits back in his chair away from the table ✽ –✽– C looks directly at L during their conversation C makes direct or indirect reference to his new sense of empowerment :-) :-) :-) :-) :-) C’s says something either negative or positive regarding his learning of the measuring tape :-( :-( :-) :-) :-) :-) :-) ☛ ☛ ☛C asserts himself in the dialogue ☛ ☛☛☛ ☛ ☛ ☛☛☛☛ video ➔ 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 min➔ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Figure 40. Actions and operations enacted within each minute of the tutoring session A frequency pictograph showing instances of actions and operations relating to C’s process of subjectification within each one-minute interval of the tutoring session. For example, the column with “min 26” in its bottom row signifies that this column represents the twenty-sixth minute of the tutoring session. This also corresponds to minute 49 on the video recording. Within this one minute interval there is one instance of C making direct or indirect reference to his new sense of empowerment (), and two instances of him looking directing at L during their interaction (). Note: The icon “:-(” signifies an instance of C saying something negative or critical about his experience of learning to read the measuring tape, while the icon “:-)” signifies him saying something positive about his learning experience. The icon “–✽–” signifies that this single event of C sitting back in his chair away from the table spans both one minute intervals. approach and promotes instead a much more pragmatic approach with C that, while meeting all of his needs for working in the pipe trades, avoids the other details that would allow for the approach to be applied more generally. Specifically, L directs C to avoid having to deal with thirty-seconds-of-an-inch altogether on the measuring tape by positioning the measuring tape so that he uses the region above 12 inches (where the smallest increments marked are sixteenths-of- an-inch) exclusively for the endpoint of the lengths that he wishes to measure. This permits C to use the smallest increments on the measuring tape here as a reliable benchmark for finding sixteenths and the larger fraction-of-an-inch intervals as needed. On the two occasions when L brings this approach up in the discourse with C (lines 420 & 438) it is clear that he is crossing a 146 cultural boundary in that he signals the breaking of a rule, or the violation of a cultural norm, by referring to this approach to using the measuring tape explicitly, on both occasions, as “cheating.” Contradictions within this activity As mentioned in chapter two, the examination of contradictions or conflicts within an activity is important from a CHAT perspective for understanding its internal dynamics as an evolving system. Specifically, contradictions are seen as the engine of change within an activity. The present section provides an analysis of C and L’s joint activity during the tutoring session on two different levels. The first level addresses contradictions within individual elements of the activity system and the second level addresses contradictions between these elements. These contradictions or conflicts take the form of misalignments, breakdowns, misunderstandings, or complications, etc. The object of the activity is the imperial system of linear measure, including the system of binary fractions-of-an-inch (the object of thought), along with the inscribed pattern of lines representing fractions-of-an-inch on the measuring tape (the object of practical activity). One complication exists between the ideas of the various binary fractions-of-an-inch and their integrated and superimposed representation by a pattern of division lines in a single row on the measuring tape. It is this feature of the design of the fractions-of-an-inch markings on the measuring tape that makes learning to read it a challenge. As a result, L and C have to do a considerable amount of work to help ensure that C is able to make sense of this. Another conflict within the material form of this object relates to the use of signs (the individual divisions on the measuring tape) to draw the reader’s attention to the end-points of the intervals being measured rather than to the intervals themselves. While this conflict is not resolved within the 33 minutes 147 of the tutoring session, it does find resolution afterwards with L’s redesign of a semiotic tool (to replace the set of transparency rulers) for teaching binary fractions-of-an-inch based, in part, on his experience of working with C in this tutoring session. This will be elaborated in chapter six. Another potential source of misalignment within this activity lies within L given that he is serving in the two roles of mathematics tutor and researcher throughout the tutoring session. Line-by-line analysis of the transcript focusing on the purpose served by of each of the actions that L takes throughout the entire tutoring session indicates that he does very little, if anything, in his role as researcher that could be seen as drawing C’s attention away from the object his learning activity. (And, of course, this was my goal!) There are numerous contradictions between elements within this activity system as well. C is unfamiliar with measuring using fractions-of-an-inch (the object of the activity) and makes several errors in the early going. It is the need to resolve this problem that serves as the motive for C and L’s shared activity. There are also a number of communication difficulties (mediated by the available semiotic systems) that reflect contradictions or conflicts between C and L. The most notable of these relates to L’s question in line 230, “an inch-and-a-half is an inch and how many quarters?” As a result, considerable effort is invested by C and L both in the process of C coming to understand what L is asking. A breakdown also occurs between C and a semiotic resource or tool that figured prominently within this discussion, that being fraction notation in digit form. L addresses this by drawing C’s attention in an explicit and sustained way to the meaning of the denominator of fractions and how this relates to the various fraction-of-an-inch intervals on the measuring tape. Other contradictions between elements in the activity system involve L in his role as mathematics tutor. As mentioned in an earlier section of this chapter, the tutoring session begins with a contradiction in the norms of measurement practice that L is enacting and those of 148 mathematical activity generally in the pipe trades. At the outset, L works to enact norms of generalizability and “conceptual understanding” typical of school mathematics. He does so by promoting strategies that will allow C to read any binary fractions-of-an-inch on any kind of imperial ruler or measuring tape. In contrast the norms of workplace practice reflect a much more focused emphasis on the situated demands of the workplace, in this case learning to read fractions-of-an-inch efficiently and reliably to sixteenths-of-an-inch. By the end of the tutoring session, however, L changes his approach by focusing with C on the smallest increments on the measuring tape as a benchmark for determining a number of the smaller subunits-of-an-inch intervals needed for measuring in the pipe-trades. Chapter summary This chapter focused on elements of cultural-historical activity theory and the theory of knowledge objectification in an analysis of mathematics activity in workplace training. Understanding of objectification and subjectification is informed through an analysis of C learning to read a measuring tape. This analysis also elaborated the processes semiotic contraction and extraction within this activity. The analysis illustrates the depth and breadth of mathematics activity in workplace training. In the final chapter I summarize the findings and draw conclusions for theory, further research, and practice. 149 CHAPTER SIX: CONCLUSIONS AND IMPLICATIONS This study examines the activity of C, a pre-apprentice within a pipe trades training program, as he learns to read an imperial measuring tape during a 33 minute impromptu tutoring session with the researcher serving as mathematics tutor. More specifically, C’s objective is to learn to measure with the system of binary fractions-of-an-inch represented by the pattern of divisions on an imperial measuring tape. Mastery of this is essential for the pre-apprentice to be successful within his training as well as in the workplace. Cultural-historical activity theory and the theory of knowledge objectification are used to frame this semiotically, culturally, and historically mediated activity, as documented through video recording as well as artifacts retained from this activity. The nature of mathematics activity within this pre-apprenticeship program This training program prepares pre-apprentices for requirements, including mathematical requirements, of formal pipe trades apprenticeship programs that follow completion of this course. Mathematics is used here in an instrumental way and relates to particular requirements of workplace activity common to these trades. To summarize, the pre-apprentices are required to think mathematically in historically and culturally constituted ways that lead them to interpret various technical documents, read an imperial ruler or measuring tape, and perform calculations needed for a well defined set of pipe trades applications efficiently, reliably, and to within acceptable tolerances. In its practical dimension, this analysis is consistent with the summary of research findings on workplace mathematics practice from FitzSimons, Mlcek, Hull and Wright (2005) cited in chapter two. However, the present analysis departs from the existing reports of mathematics in workplace training by providing a detailed and nuanced view of distinctive features of this 150 activity. The use of the theory of knowledge objectification, specifically the semiotic system of cultural significations, highlights ways in which the mathematics in this activity is distinct and a legitimate form of mathematics with its own ways of doing things, not merely a sub-standard or watered down form of school or academic mathematics. Unlike school mathematics, for example, mathematics within the context of the pipe-trades uses discrete numbers (i.e., all linear measurements are made to the nearest sixteenths-of-an-inch) and empirical methods of validation or fit as a legitimate basis for establishing mathematical fact or truth. The activity of learning to read fractions-of-an-inch on the measuring tape It is a fundamental assumption of activity theory that all elements within an activity system serve to mediate it. The activity system within which C learns to read fractions-of-an- inch on his measuring tape involves the semiotic resources that C and L employ including: • spoken language including mathematics vocabulary; • voice inflection and changes in volume; • mathematics notation; • three forms of gesture—pointing or indexical, sweeping, & chopping; • a line drawn to represent five-eighths-of-an-inch; • indexical inscriptions such as circling or underlining existing inscriptions; • counting; • written text; • rhythm in speaking or gestures; and • the position, orientation, alignment of physical objects. In addition, artifacts that C and L use include: the imperial measuring tape from C’s tool box, a pencil, a pencil, a paper, and a set of rulers on transparencies. The workplace conventions, or norms, that are enacted include the use of binary fractions to the nearest sixteenth-of-an-inch for measuring, as well as norms shared with school mathematics such as expressing fractions results in lowest terms. Last, the division of labor that mediates this activity includes the manner in which C and L each participate in leading the discourse (C starts by playing a passive role in this 151 regard and becomes a more active contributer as the session progresses) and L’s role throughout the discourse as the sole arbiter of the correctness of C’s work. While it is not possible to determine precise mediating roles of these elements throughout the activity we can see evidence of each playing a dynamic role in shaping the course of events. The various semiotic systems employed, for example, serve to draw C’s attention to particular aspects of the object of the activity and to deepen his understanding. The design of the particular measuring tape used (marked in thirty-seconds-of-an-inch up to twelve inches and in sixteenths thereafter) necessitated that this difference be attended to explicitly and negotiated during the activity. And, the conventional design of the measuring tape with the endpoints of subintervals- of-an-inch indicated by a system of signs necessitated that L draw C’s attention explicitly to the intervals between these divisions rather than the division markings themselves as the object of their discussion, in the process of learning to measure. A number of processes identified within the TO that shape and reflect C and L’s understanding within the activity of learning to read the measuring tape figure prominently within their exchanges. To recap, C repeats or re-enacts what L has just said or done relating to the task at hand on 60 separate occasions. These actions reflect an effort to deepen his sense of—literally, to deepen his sensory experience of—these statements or actions using the same means of semiotic expression used by L or other means. On one occasion C re-enacts a unique form of semiotic expression, a novel form of gesture, that he had just used himself; on a few occasions L re-enacts or repeats what C had done or said earlier; and on another occasion L creates a zone of proximal development for C to help bring coherence to his understanding of the division pattern on the measuring tape by inviting C to explain what he (L) had said earlier and then by providing him with verbal prompts to help him along. These examples of repeating or re- enacting what another has said correspond to the process of iconicity, identified by Radford as a 152 significant part of the process of attaining a cultural logic of thinking or knowledge objectification. The ways that L and C use multiple semiotic systems together (semiotic nodes) throughout the tutoring session and, in C’s case the further enactment of semiotic contractions, reflect their understandings of the object of the activity. L’s frequent use of various combinations of words, pointing and sweeping gestures, fractions written using digits and words, the fractions- of-an-inch division pattern on the measuring tape and transparency rulers, along with other semiotic resources in a coordinated manner to draw and maintain C’s attention to/on various aspects of the system of binary fractions-of-an-inch on the measuring tape. And, given L’s extensive experience working with the system of binary fractions-of-an-inch extending back to his own elementary school days, it is not surprising that his use of various semiotic systems remains relatively consistent during his explanations to C throughout the tutoring session, reflecting little or no change in his understanding in the process. In contrast, there is a marked shift over the 33 minutes of the tutoring session in the way that C expresses his understanding using various combinations of semiotic systems as he communicates with L and brings clarity to his own thinking. Early on, when C responds to L’s request for him to explain what difference he notices in the patterns of divisions below and above twelve inches on the measuring tape, C’s response is predominantly gestural, accompanied by only a single sentence and two sentence fragments. As the tutoring session progresses, C’s means of expressing himself shifts completely at times to the clear and succinctly use of words alone—an ultimate form of a semiotic contraction. The last process from the TO to be summarized here is that of C and L’s subjectification. Over the 33 minutes of the tutoring session C became more active in the way in which he participated within the activity. This is evidenced by the collective changes in the patterns of his 153 gaze and attentiveness, his role in the dialogue, his affective responses, and his own expressions of agency and self-reliance regarding his use of the measuring tape. C also nods his head or says “okay” or “yeah” on numerous occasions throughout the session acknowledging to L that he is following what he is saying. This also reflects part of C’s process of subjectivity within the activity. L changes during the tutoring session as well but in a less obvious way. Specifically, L changes in his approach to teaching C how to read the measuring tape from a more generalizable approach (intended for any form of binary measuring tape or ruler), typical of school mathematics teaching, to a much more practical one tailored specifically to the workplace demands within the pipe trades. Contradictions within the activity A number of contradictions in the form of misalignments, breakdowns, misunderstandings, and complications exist on different levels within C and L’s activity. These levels include contradictions within individual elements of the activity such as the object of the activity itself—the system of binary fractions-of-an-inch represented by the inscription pattern on the measuring tape—and with L given this dual roles as tutor and researcher. Between different elements of the activity there are contradictions relating to C’s unfamiliarity with the imperial system of linear measure to fractions-of-an-inch that he is required to use in his training and his limited understanding of the meaning of the denominator of a fraction represented in digit form at the start of the session, C and L’s communication difficulties, and L’s relationship with the norm in the pipe-trades of measuring only to sixteenths. Considerable effort is made by both C and L throughout the tutoring session to attend to and resolve these contradictions as they arise in the activity. 154 Significant processes in mathematics learning not yet addressed by activity theory or TO This analysis reveals a number of features of mathematics learning activity that are new to activity theory generally and the theory of knowledge objectification in particular. The new form of iconicity identified (that of re-enacting a form of gesture that appears initially as a novel form of gesture enacted by oneself) and the social process of semiotic extraction to complement the process of contraction identified by Radford are new contributions to the TO and social cognition. The identification of particular categories of actions and operations that provide evidence of a subjectification during mathematics learning elaborates on existing research as well. Implications for teaching Both the in-depth analysis of the process of learning to read fractions-of-an-inch on a measuring tape involving one student and a tutor provided here and the use of the TO to analyze this mathematical activity within workplace training are a departure from existing mathematics education research. This focus and approach contribute to the existing body of research on mathematics within workplace training and on mathematics learning generally by drawing attention to mathematics as a socially, culturally and historically situated, and semiotically mediated activity. This, in turn, provides new ways of understanding workplace mathematics, which until now has been limited, for the most part, to analyses of differences between mathematics in the workplace and school mathematics from the perspective of school mathematics and issues of transfer between school mathematics and the workplace. Given the connections between the process of learning to read fractions-of-an-inch on a measuring tape and learning to read the divisions on any kind of ruler (e.g., a metric ruler), as well as the central place of fractions within the school mathematics curriculum, the analysis here has the potential 155 to inform school mathematics teaching as well. Specific ideas for teaching students how to read fractions-of-an-inch on a measuring tape or ruler, rethinking the goals and methods of mathematics teaching for the workplace, as well as for further research will now be discussed. Implications for teaching students how to read a measuring tape or a ruler As mentioned earlier, the complex design of the fractions-of-an-inch pattern on a measuring tape using division markings draws attention to the end points of intervals on the measuring tape rather that to the intervals themselves and this can present a challenge for students. During the tutoring session L and C made a considerable effort to see the different forms of fractions represented by this pattern of markings and used sweeping and hopping gestures to make sense of fractions-of-an-inch as intervals on the measuring tape rather than as points indicated by divisions (marking lines) alone. As a direct result of this, I propose the following new design of a teaching artifact (shown in Figure 41) for unpacking the complex meaning of division markings on a measuring tape or ruler (the process semiotic extraction) to facilitate students’ learning. (This tool is intended as a replacement for the set of ruler transparencies that were used during the tutoring session). In this new design, to be printed on a sheet of paper rather than on a set of transparencies, the inscribed horizontal bars within each row represent the lengths of either inches or a particular fraction-of- an-inch. The intention is to use these inscriptions (the bars) to draw learners’ awareness to fractions-of-an-inch as intervals as well as the various relationships between the different kinds of fractions at the start of the learning process. Furthermore, this idea of representing fractions on a ruler or measuring tape as intervals need not be limited to imperial measure. The same principal can be applied when teaching metric linear measure, as well as when developing 156 Figure 41. A redesigned tool for teaching how to read fractions-of-an-inch An enlarged image of a revised design for a teaching artefact to introduce the fraction-of-an-inch pattern to students. Students will start by measuring to the nearest whole inch using the ruler in the top row. The next ruler below shows whole inches and introduces halves, and each successive ruler introduces the next binary fraction-of-an-inch in-turn, only to sixteenths-of-inch the smallest fraction-of-an-inch used in the pipe trades. Once students have mastered the use of the first five rulers (to sixteenths) the final form of this pattern, with the horizontal lines omitted, would be introduced. This final pattern matches that found on a measuring tape with fractions- of-an-inch marked to sixteenths. fraction ideas in general using a number line model. Of course, field testing would be needed to verify the ways in which this approach is helpful for students. Implications for teaching mathematics for the workplace Attending to the ways that mathematics activity within workplace training is similar to and different from that of school mathematics from the cultural-semiotic perspective of the TO has the potential to inform our understanding of difficulties students have when they move between these different activities as well to provide ideas for facilitating this transition. This can provide a basis for helping students to become aware of the ways in which these forms of activity differ and for instructors to develop strategies to facilitate their negotiation. For example, 157 it would be useful for students to gain experience with and understand the significance of the particular kinds of methods, tools, norms, and objectives in relation to the mathematics that they might encounter within various workplaces. In short, it would be useful to include the study of specific mathematics workplace practices along with the examination of their similarities and differences and some of the practical reasons for these being as they are so that students will be better able to better understand and adapt to whatever workplace (or workplace training) they find themselves in when they finish their schooling. As was prominent in this analysis, Radford’s concept of territory of artifactual thought focuses attention on ways that mathematical thinking is mediated by semiotic artifacts. This, in turn, highlights the need for educators to examine the important role of these artifacts in workplace activity when designing and implementing mathematics training for the workplace so that learners can develop fluency thinking with them, as well as the important role that semiotic teaching artifacts can play in helping students learn mathematics for the workplace. The new design of an educational technology for introducing fractions-of-an-inch on a measuring tape in the previous section is one such example. Lastly, this analysis demonstrates ways in which workplace mathematics activity is an embodied and intensely social a form of situated practice and much more is involved than simply getting the right answers to isolated mathematics tasks. It follows then, that mathematics learning for the workplace would be served by increased face-to-face interaction between learner(s) and (a) more knowledgeable other(s) (especially for learners having difficulty) so that the full range of outcomes relating to students’ subjectivity with mathematics can be responded to and providing mathematics learning activities that address the culturally specific features of workplace contexts. 158 Implications for education policy Two recommendations for education policy relating to the preparation of students for the industrial workplace follow from this analysis. First, school mathematics curriculum needs to familiarize students with workplace mathematics practices in ways that present them as distinct and legitimate forms of mathematics–in light of the goals of particular forms of workplace production–and as an end in itself rather than as simple applications of academic mathematics. And second, school mathematics curriculum relating to measurement needs to address imperial measure along with metric so that it reflects authentic measurement practice outside of school. Both of these recommendations will help to prepare students to make the adjustment to the mathematics practice that they may encounter when they enter the workplace or workplace training. Implications for further research The use of cultural-historical activity theory and the theory of knowledge objectification open a new window for studying mathematics in various settings as different forms of a cultural practice and provide a way of looking at mathematics learning as a social and cultural endeavor. While this study focuses on only a minute part of the multitude of different forms of mathematics activities that exist, it never-the-less points clearly to the usefulness of this theoretical and methodological approach for furthering our understanding of mathematics practice and mathematics learning within different contexts. Further development of the theory of knowledge objectification The new contributions to the TO provided in this analysis suggests avenues for further development of the theory. The examination of mathematics learning in other contexts has 159 potential to further our understanding of: a) the forms of iconicity already identified and might lead to the identification of other forms, b) the process of subjectification and ways that this is manifest within different forms of mathematics learning activity, and c) different forms of semiotic extraction. The systematic examination of learners’ efforts to orient themselves to their teachers within learning activities holds promise as another possible category or process to be included within the TO. Lastly, the examination of subjects’ conscious awareness as they traverse the processes of orientation, iconicity, semiotic nodes, and semiotic contractions in the objectification process holds potential for theorization of ways in which these processes are interrelated as well as ways that these processes relate to the concepts of operations and actions within CHAT. Avenues for further research on mathematics learning for the workplace The results from this study indicate that the examination of other forms of measurement learning specifically, and mathematics learning generally, within workplace training as historically, culturally, and semiotically mediated activity, holds promise for furthering our understanding of mathematics learning for the workplace and issues surrounding this for the students in these programs. Examination of the subjectification process, in particular, holds significant potential for addressing the negative feelings experienced by students in relation to the mathematics requirements of these programs. Applying the theory of knowledge objectification within other cultural contexts While developed specifically for the analysis of mathematics learning as a historically and culturally situated and semiotically mediated activity, the TO could be used to analyze the learning of any form of cultural practice. The possibility of using the TO to analyze music 160 learning comes to mind as one example in that within music there are different forms of cultural practice (e.g., classical music, jazz, rock, and various ethno-folk traditions); well developed semiotic systems of representation; a wide variety of tools or musical instruments particular to the different forms of music activity. Epilogue The view presented here of mathematics as culturally and historically situated and semiotically mediated activity; of mathematical thinking being located at the intersection of individuals, communities, and mathematical artifacts; and of the learner as inextricably interdependent with situated and mediated activity, represents a significant departure from much current thinking within mathematics education. 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International Journal of Science and Mathematics Education, 7, 183-206. 170 APPENDIX I: TUTORING SESSION TRANSCRIPT C and Lionel’s activity turns to that of C learning to read a tape measure once it becomes apparent that this is a source of difficulty for C. After another minute of dialogue in the shop the activity moves to the classroom where L and C work together. Transcription codes […]--indicates the omission of some words or lines [words]--indicates observable actions/gestures/tone occurring [G-…description…]--Indication of gestures within the body of the transcript … --indicates a pause of 3 or more seconds . or, --indicates a pause of less than 3 seconds. [inaud] --inaudible (words) -–indicates a comment added or an unspoken elaboration of what has been said by the researcher. Note: Whenever reference is made to gesturing with fingers, the fingering convention used by wind and string musicians will be used to identify to particular digits i.e. thumb, 1st, 2nd, 3rd, and 4th. [24:27] The focus now turns from S and the group to C working at the workbench also doing pipe fitting 51. C: [following S’s lead, marks what he thinks is 11 1/4” on his tape measure] [C then holds tape measure up to show L where he had drawn his line on it.]… Screwed up. [24:34] C marks a line where he thinks11 1/4” is on his tape measure. 52. L: You screwed up? 53. C: Yeah, this is 11 and one, four sixteenths [positions tm up off table in L’s view to show the line he has drawn on his tape measure and which is positioned in the middle of the interval on the tape measure that he is holding up with his two hands in front of L] 171 [24:56] C’s line drawn on his tape measure for 11 4/16” 54. Lionel:Where, how did you screw up? 55. C: [G-points with index finger nail to the short line he had drawn at 11 5/32”] Oh no, I just, miss, I just counted, [G-points with index finger nail to successive lines on tape measure including 11 1/32”’ & 11 2/32”] I, cause I counted, right, from where [G-points with his thumb nail indicating that he started counting 4 from 11 1/32” rather than 11”.] [24:58] C points to the small line he drew on his tape measure to indicate what he thought was 11 5/16”. 56. L: Right 57. C: This would be eleven, right? [G-starts by pointing at the 11” mark at the top edge of the tape measure with the tip of his pencil] So this would be one, two, three, four. [G- pointing sequentially to each of smallest interval markings to the right of 11 inches on the tape measure—11 1/32”, 11 2/32”, 11 3/32”, 11 4/32” touching the first three markings at or near to the end closest to the centre of the tape measure, and making a partial sweeping with his pencil down the top part of the forth marking] And I just counted five [laughs]. [G-continuing the previous sequence of gestures by moving his pencil tip precisely down the entire length of the 5/32” marking as if inscribing it as he says the word “five.”] [25:05a] C points at the 11” mark at the top edge of his tape measure (not contacting the tape measure) [25:05b] C points near the tip of the first division marking past 11” (touching the measuring tape) [25:06] C points to the tip of the second marking past 11” 172 the tape measure) measuring tape) [25:07a] C points to the tip of the third division marking past 11” (hidden underneath the tip of his pencil) [25:07b] C touches his pencil tip at the top part of the fourth division marking past 11” [25:08] C at the end of his precise sweep with his pencil down the entire length of the fifth division marking past 11” (partially obscured underneath the tip of his pencil) 58. L: That’s an eighth, that’s your quarter there (indicating to C that he is incorrect). Those are thirty-seconds on this tape measure. [G—(initially off camera) pointing to markings on C’s tape measure off camera with his pen and then finger tip.] 59. C: Ah, those are thirty-seconds, oh. 60. L: Those are 32nds, four 32nds is two sixteenths, which is an eighth. [G-just off camera but points on the tape measure] Now a quarter is right there. [G-now on camera pointing to 11 1/4 with index finger] [25:23] L on left points to 11 1/4” for C [25:25] 61. C: Oh. Oh… [G-places his own thumbnail at 11 1/4” as L removes his hand from the tape measure and then moves the tape measure towards himself to have a better look] 173 [25:25] C (replaces his own thumbnail at 11 1/4” as L removes his hand from the tape measure and then moves the tape measure towards himself to have a better look) 62. L: And the way, one way to look at that is, look at that space from [G-attempts to make an interval of 1/4” with this thumb and first finger and then abandons this attempt], let me borrow your pencil here so that I can 63. C: Yeah its okay. 64. L: Okay. From there, there to there [G-pointing with pencil at 11’’ on tape measure and then 11 1/4”], four of those intervals [G-sweeping pencil tip back and forth between 11” and 11 1/4”] will fit in an inch. There to there’s one [G-pointing first to 11” mark and then 11 1/4” mark on tape measure], there to there’s two [G-moves pencil tip in a hopping motion from 11 1/4 to 11 2/4”], there to there’s three [G-moves pencil tip in a hopping motion from 11 2/4 to 11 3/4] and there’s the next inch. [G-moves pencil tip in a hopping motion from 11 3/4 to 12”on the tape measure.] So those are quarters. [G-sweeps pencil tip from 11 to 12” once again pausing very slightly at 11 1/4”, 11 2/4”, 11 3/4, and 12”] four of them will fit. [25:38] L shows C start of the interval for 1/4”—“from there” [25:38] [continuing] “to there” 65. C: Okay. 66. L: in the inch. 67. C: Okay. And we’re looking for one quarter, okay, okay. [G-Rubs his thumb along tape measure as if to be erasing the line he had drawn on it—G-then points to successive 1/4” markings using the thumb of his LH. ] 68. L: Okay. ****************************************************************************** 174 [26: 02] [C and Lionel have moved to a classroom and are working together at a table. C is studying his tape measure, which he has out in front of himself on the desk top as this episode begins. Before the dialogue resumes C again rubs his finger back and forth over the interval between 11” and 12” when he had erroneously drawn a line for 11 1/4” in the shop.] [26:02] C studies the measuring tape intensely and without speaking upon being seated at the table in the classroom. 69. L/R: […] (makes adjustments to the video camera) So we just were in the shop and we had a little bit of difficulty with reading an eighth of a, a quarter. 70. C: Yeah. 71. L/R: You had, you were counting sixteenths? 72. C: I was counting sixteenths. 73. L: Yeah. Okay. Now, Okay. [Lionel extends his own tape measure and them puts it away.] Let’s look at your ruler here [in reference to C’s tape measure which is extended and laying on the table in front of C and Lionel and L takes tm into his right hand] Is this the ruler from the toolbox? [G-points to C’s tape measure with the 4th finger of his LH while asking this question] 74. C: Yeah, it’s from the toolbox. [26:41] L initially points to the tape measure as he asks C if this is the ruler from the tool box [26:43] 75. L: Notice that, um, [Extends C’s tape measure to expose more of it beyond 12” where it is marked in 16ths instead of 32nds] [G-uses his index finger to sweep up from the zero end of the measuring tape and pauses at 12” just before saying “up to twelve”.] well, what do you notice here between the spaces here, up to twelve [Gesture-uses the index 175 finger of his left hand to sweep up from the zero end of the measuring tape and pauses at 12” just before saying “up to twelve”] [26:46] L-Initially points at part of tape measure below one foot [26:48] L-Starts sweep of index finger up from the zero end of the measuring tape [26:49] L-pauses with index finger at 12 inch mark 76. C: Yeah, its, 77. L: and the spaces after twelve? [G-now pointing with the fourth finger of his left hand to sweep through the exposed interval of the tape measure above 12”] [26:50] L replaces his LH index finger at 12 inches with his fourth finger as he the proceeds to sweep up the measuring tape beyond 12 inches [26:51] L at the end of his sweeping gesture well above 12 inches [26:52] 78. C: There’s, [G(Video frame 26:52)–sweeps up through the first few inches of the tape measure with the fourth finger of his left hand in a manner similar to the gesture just enacted by L] there’s more. [G(Video frame 26:53)–makes two chopping motions aligned with the divisions on the tape measure with his left hand, the first significantly larger than the second just before he says “there’s more” in reference to the markings inscribed on the measuring tape. G(Video frame 26:54)–then points to the 12” mark with the fourth finger of this left hand before withdrawing it from the measuring tape]. It’s like it’s more spread out (in reference to the divisions on the tape measure after the 12 inch mark.) [G(Video frame 26:55a)–points briefly to the 12 inch mark on the tape measure, now with the first finger of his right hand, replacing the previous pointing gesture that had been expressed by the fourth finger of his left hand. G(Video frame 26:55b and Video frame 26:56a)–starting with his thumb positioned at the 12 inch point, sweeps his right hand up the measuring tape a short distance while 176 holding an approximately 2.5 inch wide interval between the thumb and first finger.] when [G(Video frame 25:56b)–grasps the tape measure with his right thumb and first finger on opposite edges at the 12” point and G(Video frame 26:57)–sweeps his hand in this configuration upwards a short distance from 12”] you pass one, [26:52] C sweeps the fourth finger of his left hand up through the first few inches of the measuring tape. [26:53 C] makes two chopping motions aligned with the markings or divisions on the measuring tape with his left hand in reference to the markings below 12 inches. [26:54] C points to the 12 inch mark with his left fourth finger. [26:55a] C points at the 12 inch mark now with his right index finger. Here C continues the pervious indexical gesture he had made with his left fourth finger. [26:55b] C begins to sweep an approximately 2.5” wide interval up the measuring tape starting with his right thumb at 12”. [26:56] C sweeps upwards while holding his grasping gesture. C sweeps his hand in this configuration upwards a short distance from 12” and then repeats this motion a second time. 79. L: Yeah, 80. C: one foot [G(not shown)–while maintaining the same grasping position, repeats this sweep upwards for a second time]. And when you’re before one foot its more, um. [G(Video frame 27:01)–makes a very brief and narrow-interval gesture with the thumb and first finger of his right hand with this hand now positioned above the region of the tape measure between 0” and 12”.] 177 [26:56] C then grasps the tape measure on opposite edges with his R thumb and index finger at approximately 12”. [26:57] C sweeps his thumb and first finger up the tape measure together (and then repeats this a second time). [27:01] C makes a very brief narrow-interval gesture with the thumb and first finger of his right hand with this hand now positioned above the region on the tape measure between 0” and 12”. 81. L: Okay 82. C: [silence] [27:04] 83. L: Okay. So lets, I’m, um, Show me on your tape measure then, where you’d measure seven and an eighth, sorry, sorry, seven and five eighths, and thirteen and five eighths. [Then writes these measurements down on paper for C to refer to—sheet PTO-040519e] …Seven and five eighths and thirteen and five-eighths. 84. C: [Grasps the zero end of tape measure with his left hand (then holds on to it continuously for a number of minutes), and G-points to a point on the tape measure with the 4th finger of his left hand] … This would be seven [G-Pointing to the 7” mark on tape measure with his 4th finger.] and then 85. L: Yeah. 86. C: … and then, right there [G-now pointing at 7 5/8”]. 87. L: Okay. So you’ve got, [G-L points with his pen tip to the location indicated by C on tape measure] right, you’re counting the little spaces? [G-points with pen at eighth inch intervals between 7” and 7 5/8”] [27:29] C initially points at the tape measure at 7” and then 7 5/8” in the process of locating 7 5/8” [27:36] L points with his pen to where C is pointing for clarification & precision 178 88. C: Yeah, I’m counting one, two, three, four, five, right there, that little one. [G-points to individual 8ths on the tape measure in turn as he counts them] 89. L: Now show me thirteen and five eighths. 90. C: Okay, so. [G-moving finger along tape measure pausing first at 13” and then coming to rest further along on the tape measure.] So, it’d be right here, thirteen and five eighths. [27:54] 91. L: Okay. I want to show you something. Lets just, I’m just going to [places paper under tape measure and draws a line the same length as the interval between the 13” mark and the point C has identified as 13 5/8”-Sheet PTO-040519e top left] So you went from there [G- points to 13” on tape measure with the tip of his pen] to where?” 92. C: Yeah. I went from there [G-pointing to 13” mark on tape measure] to this one right here, yeah, five eighths. [G-points now to what he has identified as 13 5/8”] 93. L: Okay [draws a line from 13” to the point C used for 13 5/8” and writes 5/8 beside it.] Now compare that with the seven and five eighths. [Aligns the end point of the line just draw with 7 inches so that it extends up the tape measure from there.] 94. C: … 95. L: [holds the tape measure flat on the paper to help C compare the line drawn with the markings on the tape measure] 96. C: Yeah, it looks the same. Actually (surprised), no it looks bigger. 97. L: Yeah. 98. C: Way bigger. 99. L: Okay. 100. C: It’s not right then 101. L: If you were to take an educated guess, what would you think the reason would be, that they’re, these ones are bigger [G-pointing to the region of the tape measure at 7 5/8” and then G-beyond 12” with the 4th finger of his LH] over here? 102. C: They might be going by twos rather than [G- points to the region of the tape measure above 12 inches with his right fourth finger while he speaks.] 103. L: Okay. 104. C: by ones. [G-switches pointing finger, now points briefly to the region below 12 inches with his right index finger.] 28:34 [L102] C points to the region of the measuring tape above 12 inches. 28:36 [L104] C then points to the region of the measuring tape below 12 inches. 105. L: Yeah (indicating agreement). Let me, um, … Now, let me ask you a couple of questions on it 106. C: Yeah. 107. L: I’ve got a way of actually showing this to you. [G-two hands held out in from of himself above the table with a small wave with both hands in unison done twice] 179 108. C: Okay. [28:47] 109. L: How many eighths would there be in a, in an inch? 110. C: How many eighths in one inch? [looking at his tape measure] … Won’t there be sixteen? Sixteen, sixteen. [starts turning the tape measure back and forth along it’s longitudinal axis] 111. L: Okay. Okay. And, uh, um, … 112. C: [Taps the zero end of the tm up and down on the table top and stops.] 113. L: If you were to compare eighths [writes fraction names on paper in words-- sheet 0519e (actual artifact)], […] eighths and halves, and sixteenths, and quarters [then positions the paper closer to C for him to be able to see it easily], which of these are, which is the biggest? [quickly edits something written on the page] Which is the biggest amount? So if it was five and an eighth, or five and a half, five and a sixteenth, five and a quarter, [G-pointing with pen to each of the fraction types written on the paper in turn while saying numbers with each of these kinds of fractions], which would be the biggest amount? 114. C: … Um. 115. L: Five and one eighth, five and one half, five and one sixteenth, or five and one quarter? [G-points again at each fraction name written on the page with his pen as he says them] 116. C: It would be five and a half. 117. L: Okay. Five and a half would be the longest. [Writes this mixed number down on the paper using digits on the R side of the page to start a list of the numbers in ascending order from L to R.] Which would be the smallest? [G-points with 4th finger of LH to the fraction denominator words written on the page very briefly.] 118. C: Five sixteent (word left incomplete 30:00), five and one sixteenth. 119. L: [Puts 4th finger of LH onto edge of sheet to hold it steady for writing.] Okay. [Writes that at the L side of the page.] And if you were to put the other ones in order, here and here [Lionel has written 5 1/16, (left space) followed by 5 1/2 in the sheet G- points at the spaces for two numbers left between 5 1/16 and 5 1/2 on the paper with his pen] where would they go? 120. C: Five eight, five and one eighth would be next 121. L: Yeah. [writes 5 1/8 in the proper place on the page.] 122. C: and then five and one quarter. 123. L: [Writes 5 1/4 it its proper place to finish the row of numbers on the page. ] Okay. Yeah! Okay! (indicating the affirmative) [30:20] 124. L: (continues) Um. [reaches for rule transparencies with RH.] What’s really hard about this ruler [G-pointing to the tape measure still extended on the table top.] is that it’s five rulers in one. [sorts a number of transparency rulers now in a pile in front of him on the table top] 125. C: ooh. 126. L: Okay. 127. C: Okay. 128. L: Like it’s a, it’s a ones ruler [places the units transparency to top of the paper already in front of C on the table top.] Okay. 129. C: [G-starts nodding his head up and down] Okay. 180 130. L: It’s a half ruler at the same time [superimposes the halves transparency ruler on top of the units transparency ruler already on table top] Okay. Now the half ruler, it’s called half or one over two [G-sweeps pen tip through the first half inch interval at the zero end of the transparency ruler, pauses briefly, and then through the second half inch interval marked on the transparency in front of C. Then writes one half using conventional fraction notation on the paper and circles the denominator of this fraction.], because there’s two, the whole, each inch [G-sweeps pen through the first whole inch interval on the transparency ruler.] is split into two equal parts. [G-sweeps the tip of his pen again through each of the two adjacent half-inch intervals within the whole inch previously indicated on the half transparency.] Okay. Now it’s kind of complicated because they’re on top of each other. Okay. Now, next, next comes splitting it in, splitting each of the halves in half [G-sweeps his pen tip through an interval of a half-of-an-inch on the halves transparency and then G-moves his pen in a downward motion at the midpoint of this half inch interval as if inscribing a one quarter inch division], which would be quarters, ... [pulls out quarters transparency and places it on the table.] One, two, three, four [G-points to successive quarter inch intervals within a single inch on the transparency as he counts them] equal parts, 131. C: Okay, okay, got that, 132. L: and, 133. C: four equal parts (says aloud but to himself). 134. L: that overlays as well [positions and aligns the quarters transparency over the units and half transparencies already of the table]. Lets look at what we were measuring, (referring to the original 5/8” measuring tasks on the tape measure above and below 12” that led to the recognition of C’s problem locating fractions on the measuring tape). We were measuring five eighths. [Removes the stack of whole inch, half inch, and quarter inch transparencies from table top.] Now on this one (referring to the eights transparency that he is putting out on the table in front of C), let’s just double check that there are eight intervals, [G-points to divisions on the eighths-of-an-inch transparency ruler]. There are eight eighths in an inch [“eighth eighths” spoken with increased volume and more emphatically] [G-extends first finger of RH in air for emphasis while speaking]. Okay, so there’s, there’s sixteen sixteenths in an inch, [“sixteen sixteenths” spoken more emphatically for emphasis, circling the fraction 1/16 written previously as part of 5 1/16 on sheet 040519e in front of C on the desktop] 135. C: Okay 136. L: There’s eight eighths in one inch. [Circles the denominator 8 in the number 5 1/8 written previously on the paper in front of C.] There’s four quarters in one inch [circles the denominator 4 in the fraction 1/4 written previously], there’s two halves in an inch [circles the denominator 2 in the fraction 1/2 written previously]. [In this utterance each of the fractions per inch relationships, e.g. “eight eighths” are spoken emphatically with increased volume.] 137. C: an inch, Okay. 138. L: Okay. So, and it’s the spaces, [G- pointing at each successive eighth-of-an-inch intervals on the eighths transparency] 1, 2, 3, 4, 5, 6, 7, 8 [says eight louder than the numbers before it] [31:45] 139. C: Okay. 140. L: So that’s eighths [spoken at an increased volume.] [G-sweeps pen over the interval- of-an-inch just counted], and if we were to look at, um, five-eighths it would be, [G- 181 points with pen to the markings on the eighths transparency and then takes pen away] show me where five-eights would be then. […] 141. C: [G-points to individual increments on the transparency ruler as he counts eighths] Okay, so then 1, 2, 3, 4, 5 right to there. 142. L/R: [places he pen where C has indicated at 5/8”] It would be to there. Okay, so now lets compare that with our seven and five eighths, 143. C: Yeah, ... [laughs]. 144. L: [Lionel positions transparency ruler at 7 inches on the tape measure with C’s previously drawn line for 5/8” past 13 5/8” (top left of sheet PTO-040519e) also aligned with the transparency and tape measure] That white paper underneath. 7 and 5 eighths [G-holds his pen to mark the correct position of 7 5/8” where the 8ths transparency and tape measure are touching together.] 145. C: […] Yeah, its different, totally. 32:11 L aligns the eighths transparency beside the tape measure for C to compare with his tape measure. [32:13] 146. L: Yeah. How would you explain the difference, if you were to try to put your thinking into words? They are different [said emphatically] [G-still pointing at tm with pen].… How do you, how do you make sense of that? 147. C: Um, well the lines [G-points at the markings on the tape measure between seven and eight inches with the fourth finger of his right hand] indicate, there’s three 148. L: intervals 149. C: lines, three lines, like there’s four, it looks like it goes by four (referring here to four thirty-seconds to each eighth). [G-pointing to 1/32” divisions with the finger nail of his right fourth finger as he talks about them.] 150. L: Each, for each eighth 151. C: For each eighth 152. L: Yeah. 153. C: Four 154. L: Yeah. 155. C: [inaudible] 156. L: There it is there [G-pointing to the transparency with the tip of his pencil] [32:50] 157. C: … Okay. [staring intensely at the measuring tape] … 1, 2, 3, 4, 1, 2, 3, 4. [G- pointing at the 1/32” intervals as he counts them] Yeah, it goes by intervals of four. 182 158. L: Okay, so each eighth has four little spaces [G-points to four spaces within one eighth on the measuring tape below 12”] 159. C: Four [G-points very briefly to the measuring tape and transparency on the table with his right index finger] little spaces [becomes inaudible]. 160. L: Let’s look over, what was the other one? We had five and an eighth and [G-points to that general location on the tape measure] 161. C: [G- points to 13 5/8 written earlier on the top left of paper 040519e] thirteen and five eighths. 162. L: and thirteen and five eighths. [Lines up the eights transparency with 13” on the tape measure where the smallest divisions indicate sixteenths-of-an-inch.] There’s five eighths again. [G-moves pen back and forth repeatedly as if tracing/extending the 5/8” mark off of the measuring tape onto the transparency ruler.] 163. C: [G-C nods his head slightly four times] 164. L: So what do you notice about that? 165. C: Now it’s different. Now it’s just the two interval (referring here to 2 intervals on the tape measure per eighth-of-an-inch as indicated by the eighths acetate). [33:27] 166. L: Yeah. If this is eighths (referring here to the increments on the eighths transparency and emphasizing the “ths” at the end of the word eighths) [G-sweeping pencil tip over the edge of the transparency ruler lined up along the edge of the tape measure] and there’s two spaces for every eighth, [G-points to two of the 1/16” spaces on the tape measure making one eighth] what fractions are shown by each line here (on the tape measure) between 13 and 14 inches? [G-sweeps his pencil tip between 13” and 14” on the tape measure as these are mentioned] 167. C: It’d be by … there’s … [counting] be 18? 168. L: Actually 16, because there’s eight, 1, 2, 3, 4, 5, 6, 7, 8 [G-points to each interval with tip of pencil while counting eighths within one inch on the acetate] and each is split into two 169. C: Two, so yeah, it’d be 16 170. L: Two times eight 171. C: Two times eight is 16 172. L: And this one over here would be [repositioning the eighths transparency next to the region on the tm below 12 inches], be what, from between seven and eight inches, if there’s four spaces for every eight, every eighth, how many little intervals has the inch [G-sweeps pencil tip over the interval between 7 and 8 inches] been split up to? 173. C: It’d be 4, 4 times, it’d be eight 174. L: Okay. Every eighth has four, so between 7 and 8 inches [G-sweeps pencil tip over this interval] how many little spaces do we have? 175. C: [C is staring at the tape measure and transparency lined up together on the table top and is counting silently] … 32? 176. L: Yeah! Did, 177. C: Okay. 178. L/R: now, I watched you lips move. Were you counting them? [G-sweeps pen tip from 7 to 8 inches] 179. C: Yeah, I was, I was counting by twos. 180. L: Okay, yeah, 2, 4, 6, 8. 181. C: to 32 [34:47] 183 182. L: Okay, now what’s tricky about this darn ruler is, first of all (referring to C’s tape measure) so from here to here [G-pointing at 0” end and then 12”] it’s measuring in 32nds [emphasizes 32nds by enunciating it more clearly and slowly and G-leaning into towards the centre of the work space on the table top], because there’s 32 little spaces [G-makes a small interval between this thumb and index finger] per inch. That’s the little lines. [G-points with his thumbnail to markings on the tm below 12”] And from here, after a foot [G-marks 12” on tape measure with thumb of LH and then sweeps pencil tip upwards from 12” location ending at a non-specific point above 12”], it’s measuring in 16ths. But what’s a little difficult here is that you’ve got your one, your inches ruler [lays down units acetate], then you’ve got half, half inches are, let’s look at, well it makes, I think you can make sense of where inches are, its just where the numbers (digits) are. [Lionel positions units transparencies against the tape measure starting with the whole inches scale.] [34:57] L leans forward as he emphasizes 32 intervals per inch [34:59] L makes small interval with thumb and index finger [35:02] L marks 12” withthumb and sweeps pencil in the increasing direction along tape measure 183. C: Yeah, I know where inches are. 184. L: The half inches are [aligns 1/2” transparency with tape measure] 185. C: That’s what screws everything up. 186. L: Okay. 187. C: Okay, the half inches. [G-nods his head up and down slightly] 188. L: Now, look at the lines that they use for the half inches. Those, next to the (whole) number lines, the 1, 2, 3, 4 the, the ones showing the half inch are the next longest lines. [G-points to a half inch spot on the tape measure and acetate] 189. C: Okay, yeah. 190. L: They’re the next longest ones. 191. C: Yeah. [Said emphatically.] They’re the next longest 192. L: So they’re, they put two rulers on top of one another [positions whole and half inch transparencies on top of one another]. 193. C: Okay. [35:49] 194. L: How many half inches does it take to make two inches? [G-points to 2” mark on transparency in front of C] 195. C: Two, (corrects himself) one, two, three halves. 196. L: Can you show me where they’d be on that ruler. 184 197. C: [G-points to the markings 1/2”, 1” and 1 1/2” on the superimposed whole inch and 1/2”transparencies as he counts] 1, 2, 3. 198. L: Okay. So actually, actually its spaces. [G-points to the part of the transparencies rule with his pencil] 199. C: Wouldn’t that be, 1, 2, 3, 4 then? [G-points to each space with his finger as he counts them] 200. L: Yeah. 1 space, 2 space, 3 space, 4 space. [G-sweeps pencil tip through adjacent half inch intervals one after the other and pausing briefly at the end of each interval as he counts them] 201. C: 4 spaces [***spoken almost in unison with L as he speaks these words] 202. L: So the line [G-points to the line a the end of the four half-inch intervals just counted] shows where you’ve got a full [G-makes a small but fast shake motion with his right hand] half, half-inch. We’ve got the first full one, the second full one, third full one, fourth full one. [G-once again sweeps pencil tip through adjacent half inch intervals one after the other in succession] 203. C: fourth full one, okay. [36:19] 204. L: Now, lets look at quarter inches [brings quarter inch transparency forward and aligns it, by itself, with the tape measure], and look at their lines. Now they’re not quite as long as the half-inch lines (on both the tape measure and acetate), but they’re longer than everything else (on the tm). … 205. C: … [C stares at the transparency and tape measure] (grabs hold of one side of the transparency to align it with the tm] 206. L: Oh, its kind of hard. [repositions the transparency over a while piece of paper so that the marking show up more clearly] 207. C: Oh, these ones right here then right [G-pointing to the lines on the tape measure] Second longest ones, Okay. 208. L: Yeah, point with a pencil 209. C: Okay. 210. L: it might be, just to make sure that we’re on, (…) cause this is really tiny stuff. 211. C: This is one quarter [G-pointing at 1/4” point on tape measure with tip of pencil] (…) 212. L: Actually, lets line, line the inch up. Start at an inch, look at where the first quarter is. Start at one 213. C: Right here. [G-points at 1 1/4” with pencil tip on tape measure and acetate] 214. L: Yeah one, there’s one. 215. C: Two [spontaneously starts counting] [G-pointing at the markings at the ends of successive quarter inch intervals on his tm] 216. L: yeah, 217. C: three, 218. L: yeah, 219. C: and four, 185 [37:03] C counts first quarter inch interval [37:06] C counts second quarter inch interval [37:08] C counts third quarter inch interval [37:09] C counts fourth quarter inch interval 220. L: yeah. So those are quarters-of-an-inch. 221. C: quarters-of-an-inch. (repeating what L has said) 222. L: How many quarters fit into two inches? 223. C: 1, 2, 3, 4, 5, 6, 7, 8 [G-points to the endpoints of each quarter inch interval on the tape measure and transparency as he counts them] 224. L: eight 225. C: eight 226. L: four every inch. 227. C: four an inch 228. L: And in seven inches, how many quarters? 229. C: twenty-eight.[Said immediately without counting, nor even moving his pen] [37:28] 230. L: Yeah. Now (said loudly for emphasis), so, in fact what happens is, because they’re overlaid, [lays down the whole inch, 1/2” and 1/4” transparencies superimposed on the table in front of C] the, the quarters, if you’re counting by quarters, you’re going to use the half lines [G-points to the half inch mark and then 1 1/2” mark on the transparency stack] because those mark quarters too. So an inch and a half is an inch and how many quarters? 231. C: Wait, an inch? 232. L: An inch and a half 233. C: Inch and a half [spontaneously writes down 1 1/2 firmly on paper] 234. L: Which is to here, [G-points to 1 1/2” mark on the transparency stack] look on the, looking on the ruler. 235. C: One inch and one half, okay. 236. L: Is one inch and how many quarters? 237. C: … [G- points to 1/4”intervals on transparency from 0” to 1 1/2” as he counts them] be … inch and a half is how many quarters? Its well, six quarters, right? [writes 6/4 lightly ro the right of 1 1/2 on the paper] 238. L: Okay. (affirmative) 239. C: Alright. 240. L: Or one inch [G-makes a downbeat gesture in the air with first finger of RH to “one”] and how many quarters after the inch? [G-makes a sweeping/hopping gesture through the air with this same finger has he says “and how many quarters?”] 241. C: One inch and two [writes one whole to the right of 6/4] … [inaudible] [Then writes a two beside the numerator (6) and a two beside the denominator (4). Then writes a two in the right most numerator, and the overwrites it with at 3 finishing with an answer of 1 3/2 written on sheet 0519e] 186 38:42 What C has written on the sheet. [38:42] 242. L: Now six over four is three over two. [G-makes a small hopping gesture with the first finger of his RH as he says “six over four” and again as he says “three over two.”] But if we’re just going, we’re going to keep the one inch, [G-points at one inch mark on the acetate] we’re just going to change the one half part [G-sweeps finger tip between 1 and 1 1/2”]. Look at the ruler here [G-points to 1 1/2 on transparency with his index finger] 243. C: Okay. 244. L: one and, one and a half, one and a half is to there. [G-still pointing to 1 1/2”] 245. C: Here [G-placed his finger alongside L’s pointing at 1 1/2” on acetate], Okay. [38:55] C places his finger along side L’s pointing at 1 1/2” on the acetate 246. L: That’s the same as one [G-sweeps finger from zero to 1”and stops] and how many quarters? [G-then sweeps finger from 1” to 1 1/2”] [38:57] 247. C: One and one half [Said as a statement]. 248. L: [G-replaces index finger pointing at 1 1/2” with tip of his pencil] One and one half. 249. C: Okay 250. L: which is to here [G- sweeps pencil tip from 0 to 1 1/2” on the transparency ruler] 251. C: yeah 252. L: is one and how many quarter inches? 253. C: … Six. 254. L: Okay 255. C: Okay 256. L: It’s six quarters [G-sweeps pencil tip from 0 to 1 1/2”] 257. C: Six quarters, Okay. 258. L: or, one [G-sweeps pencil tip from 0 to 1” and then makes circle motion around this same interval] and how many more quarters [G-sweeps through each of the two quarter inch intervals between 1 and 1 1/2” on the transparency pausing briefly after 187 each one] past one, [G-sweeps again between 1” and 1 1/2” with pencil tip] to make, to here [G-points to 1 1/2”] 259. C: Two 260. L: Yeah. One and two quarters. 261. C: [writes 1 2/4 on paper[ [39:23] 262. L: The point is that the quarters use the half lines. [G-still pointing at 1 1/2” with pen] 263. C: . [nodding] Okay. [and barely audible--] quarters use the half lines. 264. L: Okay, so if you’re, if you’re looking on your ruler here [reaches for tape measure] at an inch and a half [G-points to the 1 1/2” point on the tape measure with tip of pen] that’s the same as one [G-points to 1” on tape measure with pen tip] and one quarter and two quarters [G- sweeps through each quarter inch interval between 1 and 1 1/2” as these are counted with a slight pause after each]. 265. C: Okay 266. L: Okay. The point is that the, the different kinds of fractions [G-points to an transparency on the table top] borrow the lines [G-tracing ruler lines the air with index finger] of other kinds of fractions. Now eighths [exaggerated enunciation of the “ths”] are these ones here [pulling our eighths transparency and aligns this with the tape measure], and we’ve already dealt with eighths. When we did the five eighths, we noticed there was four sixteenths, (corrects himself) four thirty-seconds in an eighth and higher up on the ruler [G-turns head to look at the part of the tape measure above 12”] there’s two sixteenths in an eighth. But look at those eighth lines. Okay, we’re looking at the lines here [G-points close to one inch on tape measure with eighth transparency aligned beside it], like from here to here is an eighth, [G-sweeps pen tip through an interval of 1/8”] but inside there’s even tinier lines, shorter lines. [G-makes small motions with pen pointing at the 1/16 and 1/32 lines within an interval of an eighth on the tape measure] 267. C: Yeah, Okay. 268. L: Okay, so there’s the first eighth [G-points at 1/8’ marking it with tip of pen]. Now this is a quarter line [G-points at 1/4” marking with tip of pen], but that would be our second eighth if we were counting by eighths. 269. C: Okay. 270. L: Third eighth [G-points with pen], half is, [G-points with pen and holds pen there until C answers] how many eighths? 271. C: Four eighths. 272. L: Four eighths. Five eights. Three quarters is six eighths. [G-continuing to point at each mark on the acetate/ruler combination as they are mentioned] 273. C: [nods] [40:36] 274. L: Okay, so the lines they borrow from one another. Now lets put this on top here. [Positions eighths transparency on top of existing stack of three acetates.] So right, right now we have four rulers, we have a whole inch ruler, a half inch ruler, a quarter inch ruler, eighth inch ruler laid on top of each other [places the stack of transparencies in front of G], and then there’s the sixteenths inch ruler. [Positions this final transparency on the stack] 188 275. C: [laughs] Great! (said sarcastically) 276. L: Okay. Now my ruler actually had 32nds [G-points to interval on top transparency between 0 and 1”], one thirty-seconds up to an inch and then after that, and I don’t know if I can get them to light up perfectly. [works to align the stack of acetates] If they don’t it can be really difficult to understand, but there you go. So all of those rulers are on top of each other. So, all those rulers are on top of one another, so. 277. C: Oh, so then, little ones, the little ones, all the little ones, if I have three sixteenths I would just count. If I have one inch and three 16ths, [G-chopping motion three times in the air away from the tape measure, coordinated with each syllable of the words he is saying “three 16ths”) with hand as he says one and three sixteenths] I’d just count the little ones–one, two, three. [G-makes three more chops in the air now with his 4th finger in unison with each syllable of these numbers.] 278. L: Yeah. Well it depends. [41:24a, line 277] C begins one in a series of chopping gestures with his right hand as he says one and three sixteenths [41:24, line 277] C ends the chopping gesture begun in the previous video frame [41:25, line 277] C makes three chopping motions with the fourth finger of his right hand as he counts “the little ones–one, two, three” 279. C: Right? 280. L: Depends what part of your ruler you are. If you’re over here [G-sweeps thumb over a region of the tape measure less than 12”] the sixteenths aren’t every line. [G- points to markings on tape measure below 12” with tip of pen]. Right? 281. C: yeah 282. L: if it was three and a sixteenth, we’re going to be counting pairs. 1, 2, 3 [G-points at the endpoints of successive pairs of marked intervals in turn using his pen as a pointer and then G-sweeping pen back an forth across this 3/16 interval]. 283. C: [G-points to the 3 3/16” point on the tape measure with his 4th finger, the same point that L is discussing and pointing at with his pen.] 284. L: If we’re counting 13 and one, three sixteenths it’s easy, 1, 2, 3. [G-points to each 1/16” intervals in succession while counting them on tape measure above 12” and then sweeping pen back and forth over the 3/16 interval] [41:37] C points to the 3 3/16” point on the tape measure that L is pointing at with his pen and discussing with his 4th finger 189 285. C: If, if you’re counting 13 and 286. L: and three 16ths. 287. C: three 16ths, 288. L: 1, 2, 3. [G-points again to each 1/16th them while counting them, now at 13” on the tape measure, and then G-sweeps pen across this 3/16 interval after in has been counted] because after a foot [G-sweeps pen tip from 12” on tape measure in the increasing direction] this inch, this ruler measures in 16ths. 289. C: Okay, measures in 16ths [41:55] 290. L: But down below, below a foot [G- sweeps pen tip between end of ruler and 12” and back] it measures in 32nds [G-points at smallest markings on tape measure below 12”]. So it’s not going to be every little interval. It’s going to be from here to here, one, two, three sixteenths [G-pointing to individual 2/32” intervals at the 3” point on the tape measure as they are counted-coordination]. And the way to check is, you say, how many of those, those [G-points in succession to a number of 16th markings on tape measure] fit in an inch? Well, 1, 2, 3, 4, 5, 6, 7, 8 [G-pointing to successive intervals while counting and then pauses with pen on tape measure] fit in half an inch, so how many in a whole inch? [G-points to next whole inch value on tape measure with the tip of his pen.] 291. C: four 292. L: If, I get eight of these little spaces 1, 2, 3, 4, 5, 6, 7, 8 [G-pointing to these intervals on the tape measure while counting] from here to here it’s going to be another eight, from here to here. [G-sweeps pen through interval from half an inch to the next whole inch on the tape measure] 293. C: Okay. 294. L: So those are sixteenths [G-continues to point at tape measure in region below 12”] It takes sixteen for a whole inch. [G-sweeps pen tip through a whole inch on the tape measure] 295. C: Oh, Okay, Okay. [Said louder, with confidence] [nodding] 296. L: Okay, You don’t have to count the whole thing. [G-again sweeps pen tip through a whole inch on the tape measure] 297. C: Yeah, that makes sense. 298. L: For quarters you might go 1, 2, 3, 4, [G-pointing to quarter inch markings while counting] Yeah, those are quarters. Halves, [G-points at half inch marking on tape measure with pen tip.] yeah. 299. C: Okay, yeah that makes sense. 300. L: Little ones you might do a shortcut. How many sixteenths in a quarter? 1, 2, 3, [G-points to 1/16” markings on tape measure-coordination] I know that’s a quarter for sure [G-makes circling motion with tip of pen over the interval of 1/4” just counted]. There’s four in a quarter. 4, 4, 4, 4 [G-points to each interval of 1/4” on the tape measure as he says 4, 4, 4, 4] those are sixteenths. 301. C: Okay. So if it was 32nds then you’d count the little small ones. [G-makes six small rhythmic hopping motions (flexing from his wrist) with his LH away from the measuring tape in the air as if counting markings on a tm and then points as if to the imaginary tm in front of his LH. (This starts just before the word “count”] 190 [42:49a] C making one of six small hopping motions with his left hand, here at the apex of a hop. [42:49b] C’s hand is at the bottom of the small hopping motion begun in the previous video frame. [42:50] C finishes this sequence of gestures with a pointing gesture into the air 302. L: Yeah. 303. C: And then 304. L: Yeah 305. C: [inaudible] Okay 306. L: Now, in plumbing you don’t have to, you often don’t have to get that close [G- points to smallest markings on tape measure below 12”]. You’re going to, [G-points to smallest markings on the tape measure above 12”] as long as you’re to the nearest sixteenth. 307. C: Okay 308. L: That’s close enough. … [43:04] 309. C: Okay. I’m getting. So say 12 and one quarter, right [grasps the extended measuring tape with two hands while L is still holding the measuring tape case], so if I was looking for 12 and one quarter, just like on the (pipe-fitting) project, these are all quarters you said, right? [G-pointing at 12 1/4” 12 2/4” and 12 3/4” in turn on the tape measure above 12” with his right fourth finger.] These are all quarters. 310. L: Because there’s four (slight pause) [G-points to the end points of each of four successive quarter inch intervals between 12” and 13” with the tip of his pen] 311. C: [G-finishes his counting of quarter inch intervals by pointing to 12 4/4” when L points to the fourth quarter inch interval and speaks the word four] four 312. L: spaces [slight pause], 313. C: spaces 314. L: per inch. 315. C: So it’d be, right, this would be 12 and one quarter. [G-points at 12 1/4” on tape measure with his fourth finger.] 316. L: Yep. 317. C: And then for, lets, 12 and one eighth. This is one eighth here right? [G-points without hesitation to 12 1/8” on tm with his 4th finger.] 318. L: Yeah. Show me 12 and seven eighths. 319. C: 12 and seven eighths. [G-Points at 12” on tm with his 4th finger] 320. L: Use a pen. [Hands C the pen to point more precisely at the tm] […] 321. C: 1, 2, 3, 4, 5, 6, 7 [G- pointing to each 8th interval on the tm with a pen after 12” as he counts them aloud] so, so right here. [G-holds pen at 12 7/8” when he has finished counting.] 322. L: Yeah. That would be 12 and 7 eighths. 191 323. C: Okay. 324. L: Now, how many eighths are in a whole inch? [G-points to 13” on the TM with the tip of his pencil and holds this position as C, also holding his pen to the tm, figures this out.] 325. C: … um, there’s 1, 2, 3, 4, [G-hops over each 8th” interval in a deliberate manner with the tip of his pen and then says the interval number as he counts eighths on tape measure up to and stopping at 12 7/8”.] 5, 6, 7? [44:01] L holds his pen at 13” as C counts eighths from 12” to determine how many 8ths there are in a whole inch 326. L: In a whole inch, right to 13. [G-continuing to point at this location on the tm with his pencil] 327. C: eight [G-sweeps pen tip through the interval between 12 7/8 and 13”] 328. L: Yeah. 329. C: eight [44:11] 330. L: Now what is it about eighths that … gives you a shortcut so you don’t have to count them? How many eighths in an inch? 331. C: aah. How many eights in one inch? [laughs and shakes his head from side to side (no)] … Four, right? Wait [G-points to the measuring tape as he counts) 1, 2, 3, 4, 332. L: Remember eighths [G-sweeps his pen through an 1/8” interval on tape measure] are from here to here. 333. C: Okay 334. L: You just counted them. 335. C: Yeah. 336. L: What is it about the way that we write that and say it, that [takes paper from in front of C and writes “1/8” in the middle of the page] that tells you how many of those are in one inch? 337. C: … [no response] 338. L: Not sure of that? 339. C: Not sure, no. 340. L: Okay. 341. C: Not sure. [44:57] 342. L: [Writes one half at the left side near the bottom of the page.] How many halves are in one inch? 343. C: How many halves in one inch? 192 344. L: Yeah, how many half inches? [G-waves RH in sync with each syllable in “half inches” for emphasis] Show me half, … show me half-an-inch on your ruler. [G-leans forward to look at tape measure in front of C] 345. C: Okay, so half an inch would be right here [G-points with pen to the 7 1/2” point on the measuring tape and holds it there] 346. L: Okay, that would be seven and a half [G-sweeps pencil through interval between 7 and 7 1/2” on tm] 347. C: seven and a half 348. L: From here to here’s half-an-inch. [G-sweeps back and forth through interval with the tip of his pencil between 7” and 7 1/2” on measuring tape] 349. C: Okay 350. L: From here to here’s half-an-inch [G-sweeps through interval between 7 1/2” to 8”], so how many half inches in a whole inch? 351. C: [G-sweeps pen tip between 7” and 8” and then points to and counts divisions on the tape measure in the decreasing direction—what appear to be evenly spaced quarters] … Four. 2, 3, 4, 5 … [counting off what appears to be 1/4” intervals on his tape measure in one direction and then the opposite direction] 1, 2, 3, 4 [45:34] 352. L: I think I might not have done a clear job for you. [Positions the half-inch transparency up against the measuring tape aligning the transparency divisions with the corresponding markings on the measuring tape.] Those are the half-inches there. 353. C: [G-points at transparency with his pen and then moves the pen to the right (as if counting and without hesitation he corrects his earlier response.)] Oh, two. Okay. 354. L: So there’s two in an inch. [Writes the word “two” below the fraction 1/2 written on the paper, and then writes the fraction 1/4 next to the fraction 1/2 written earlier.] How many quarters are there in an inch? 355. C: [G-holds out four fingers and waves his right hand a number of times just off of the table top and then puts his hand down. This gesture starts just before and continues while he speaks.] There’d be four then, [45:51] C holds out four fingers and waves his hand as he says “four” in response to L’s question, “How many quarters are there in an inch?” 356. L: Yeah. [Writes the digit four on sheet beneath the fraction 1/4 written previously on the page] 357. C: Four. [G-the previous gesture continues and then ends as C says the word “four”] 358. L: How many quarters in a dollar? 359. C: [G-as L mentions the word quarter C raises his right hand off the table] 193 360. L: [G-lifts his right hand up off of the table, turns his palm up—to imply it’s common sense to invite C’s response— and then puts his hand back down.] [45:55] L turns his palm upwards to invite C’s response. 361. C: Four [G-just after L begins his gesture C repeats his previous gesture raising his right hand off the table while speaking the word “four” in unison with this gesture.] 362. L: Yeah 363. C: Exactly! four. 364. L: That’s why they call. [writes 1/8 in the row of fractions started on the paper] How many eighths? 365. C: There’d be eight [G-raising hand one time off of table top and stretching out his fingers in unison while speaking.] 366. L: Eight 367. C: Eight 368. L: [writes 1/16 in the row of unit fractions on the page] How many sixteenths? 369. C: [G-raises hand off of the table just starting before speaking] Sixteen. 370. L: Yeah, the size, the name of the fraction [circles the denominator 16 in the last fraction written] tells you how many are in [G-motions with right hand in the rhythm of his speech to emphasize the point being made] 371. C: in (affirming what L is saying) [while glancing up at L, G-taps table top twice with the second finger of his right hand] [46:08] 372. L: one whole, [G-raises one finger as he says “one whole”] whether it be an inch, foot, or a dollar. [G-continues right hand waving gesture] So, the reason we bring that up, if we’re going to work out 12 and 7 eighths, [G-points to this on tape measure with tip of pen to 12 7/8”] rather than counting eighths, 1, 2, 3, 4, 5, 6, 7, [G-pointing to successive 8ths on the measuring tape from 12” to 12 7/8”] if you know there’s eight [G-sweeps pen between 12” and 13”] to get to the next inch, seven is just one away from eight, so you can just count back one, from thirteen inches. [G-sweeps pen from 13” back to 12 7/8”] 373. C: [nods slowly] Okay … 374. L: Show me five and a, no lets make it six and five eighths. [writes this on sheet 040519e near the bottom] 375. C: [barely audible] Let’s see if I understand this perfect, Okay. Six and five eighths … [G-points to what appears to be 6” with tip of pen to start] So there’s eight. [G- moves pen to 7” and then G-counts three intervals of 1/8” down from 7 inches to 6 5/8”] So it’d have to be here 194 376. L: Perfect. Perfect. Sure. And, if you’re ever in doubt, if you’re ever unsure about the size that you’re measuring [G-sweeps pen upwards over a nonspecific part of the tm], and this happens, I think, to everybody until you’ve done it enough times, just count. If you think, oh is that an eighth? [G-sweeps pen over a small interval on the tm] Check it. 1, 2, 3, 4, 5, 6, 7, 8. [G-pointing in turn to the 1/8” intervals between two whole inch values on the tm.] Yeah, that’s eighths! [Said in a raised pitch.] And then you can count them. So make sure there’s eight of them in an inch. Or if you’re dealing with sixteenths, make sure there’s sixteenth, sixteen in an inch, [G-partially visible hand gestures used for emphasis] or maybe eight in half an inch, to save a little time. 377. C: Okay 378. L: Ah, if you want, if you’re not sure what spaces are on your ruler [G-sweeps over a region of the tape measure], count them [G-points to each of the smallest intervals below 12” (32nds) on the tm as he count them starting from a whole inch] 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, Okay, you’ve got 16 to here. [G-points to half inch mark] Another sixteen [G-sweeps pen from 0/32 to 1/2] and sixteen, [G-sweeps pen from half inch mark to next whole inch]. Those are thirty-seconds. 379. C: Thirty-seconds 380. L: Yeah. 381. C: Then you’d know, Okay, sixteen. [G-: pointing to half inch point between two whole inches on his tape measure] [47:46] 382. L/R: Yeah. … You’d know what? Can you explain that for me, what you’re thinking? 383. C: Okay, so you said this is [G-moving pen around on tape measure], I don’t know. I’m all confused now. 384. L: Okay 385. C: [laughs, moves way from the table and sits back in his chair] 386. L: What did I say? Each, [G-points to a 1/32” interval on the tm with pencil] each little interval 387. C: Each little interval [G-brings the tip of his pen to the same point on the measuring tape as L, as soon as he beings pointing there] 388. L: is how much? 389. C: Its, you said it’s thirty-seconds. From, like if you count, if I were to count [G- sweeps pen over a one inch interval on tape measure then sweeps pen again over this interval touching down on a few successive markings in rapid succession during the sweep], every one, every one would be thirty-second. 390. L: Yeah. 391. C: And then you said the next one would be every sixteen. [G-pointing to successive 1/16” divisions on his measuring tape] 392. L: Yeah. 393. C: so every two, every second one, sixteen. 394. L: Yeah. 395. C: So, wouldn’t every third one, be eight? [G-lifts pen off of tm and makes an exaggerated hopping motion (iconic gesture) in the air] 396. L: Not every third one, every pair [G-sweeps pencil through an interval of 2/16”] of sixteenths. 397. C: Oh, every pair. 398. L: So it would be every four sixteenths, or (correcting himself) every four thirty- seconds would be a (and pauses) 195 399. C: A, one [G-makes the pattern of 1/8 with his pen on the table top(iconic gesture of writing on the desk. Tracing of denominator coincides with saying “eight”] eighth. 400. L: Well let’s check that. If there’s eight eighths in an inch [writes “8 – 8ths” on paper at bottom right 040591e], if you spit that into four parts, eight times four is (writes “8 x 4=” on paper) Figure X: [48:43] What L has written for C to refer to on the paper. 401. C: is 32 402. L: 32 [writes 32 down and underlines it on the sheet to finish the equation] 403. C: Okay 404. L: And, then its, so you’re looking to the next longest line on the ruler, so the littlest ones are sixteen-ah, are thirty-seconds [G-points to a 32nd marking with his pencil], the next ones are sixteenths, [G-points to a 16th marking] the next ones are eighths [G- points again], (and) the next longest line is? 405. C: [G-C immediately brings his pen tip to the tm as L stops pointing with his pencil] Quarters. 406. L: Quarters. Yeah. 407. C: Quarters, [G-points to and taps the 1/4” marking on the tm repeatedly with his pen] Oh! [quietly (and reflectively)]… [with his gaze fixed on the tape measure] one quarter, two quarters, three quarters, four quarters. (appears to be visualizing without the need for the pointing gesture) [G- then points to a location on his tape measure with pen]. Okay, yeah, I think I got it now. [G-nodding his head indicating the affirmative] [49:22] 408. L/R: What’s different about your thinking now than 20 minutes ago? 409. C: [grasps each end of the tm extended on the table top] You know what? 410. L: Is there some, is there sort of, is there a picture you have now? 411. C: I guess, I’m more of a visual person. Like when you were, when you were writing it. [G-moves RH onto the paper that L had been writing on] When you were telling me it, [G-shaking head from side to side(no), RH takes back hold of end of tape-measure] I did not, I did not clue in, right. I just assumed, okay, but once you started going, like, once you started showing me visual, [G-moves hand to point at transparencies on table top] okay, this is how [G-makes a broad sweep over the tm with his RH]. Like basically, it all came down to, I didn’t know, I didn’t know [picks up tape measure with his LH while holding on the end of the tape with his RH] this was three rulers [holds TM on angle off of table top] in one, to begin with. I had no idea. [emphasizes the word no] [G-moves both hands, including tape measure to emphasize his speaking] 196 49:48 C picks the tm case up off of the tabletop. 49:51 C holds tm on diagonal off of table top whiletalking about his new understanding of the tm with L 412. L: Three? It’s like five. 413. C: No clue. 414. L: It’s wholes, halves, [G-counts on fingers by raising them in succession] quarters, 415. C: halves, quarters [spoken in unison with L] 416. L: eighths and sixteenths. Down there [G-pointing to first foot on tm] it’s five rulers in one. 417. C: Oh, it’s five rulers in one. 418. L: On the top, [laughs] and then 419. C: Yeah 420. L: it’s a centimeter ruler [G-points to bottom edge of tm] on the bottom. Yeah. Here’s a way that might help you. We’re, its kind of cheating a bit, but not really cheating. [50:15] 421. C: [G-sweeps pen tip over a short interval on the tape measure while L is speaking] (interrupting L) Like, like now I understand it because every two would be sixteenths (referring here to the smallest divisions marked on the tape measure below twelve inches G-while pointing at these with his pen), every four would be eighths [G-while pointing now at eighths division lines on his tape measure with his pen], every space would be quarters [G-makes a distinct sweeping gesture with his pen through a 1/4” interval], um, every two spaces would be halves [G-while pointing, in turn, at each of two quarter-inch intervals within a half inch on the measuring tape]. 422. L: Okay. Show me 16 and 3 eighths. 423. C: Okay. [extends the tape measure] 16 and 3 eighths. It would be right here then [G- points at 16 3/16”] … [L uses silence—lack of an immediate response to indicate that C is not correct] quarter … This is one quarter right? [G-sweeps pen back and forth over the interval between 16” and 16 1/4”]. 424. L: That would be a quarter yeah. 425. C: That’s a quarter. … That’d be, yeah, so its three eighths is right there. [G-points to 16 3/16” with pen and continues to hold this position while L counts 16ths in the next utterance.] 426. L: And lets count how many of those spaces are in an inch. 1 space, 2, 3, 4, 5, 6, 7, 8 [slight pause at this the half inch mark], 9, 10, 11, 12, 13, 14, 15, 16. [G-sweeps pencil tip through each 1/16” interval as it is counted] [paused briefly to let C speak] 427. C: Okay, there’s sixteen spaces [G-sweeps pen back and forth through the air] so it’s going to be by sixteen 428. L: So we wanted 16 and 3 eighths. 197 429. C: Oh! [G-moves pen to point at marks on the tm] So if it’s sixteen it’d be, this would be, that would be one eighth? [G-pointing to 1/8” interval on tm] 430. L: Yeah. 431. C: That would be two eighths, three eighths. [G-pointing to successive 1/8” intervals with his pen on the measuring tape as he counts them--coordination.] 432. L: There you go, yeah. So you always got to be careful where you’re measuring on the ruler, [G-points to ruler with pencil tip] and maybe, I think, for the time being, you’ve got to double check how many of those 433. C: yeah, yeah 434. L: spaces are in an inch [G-waves pen in the to emphasize what he is saying] 435. C: yeah 436. L: Until you can just eyeball it. [G-points index finger outwards and shakes hand to emphasize words being spoken] 437. C: Yeah. [51:51] 438. L: Okay. Now, my ruler is, [extends a second tape measure in front of C] I think this is just an old ruler but on the top it’s in sixteenths, [G-sweeps thumb over the top edge] and on the bottom its thirty-seconds [G-sweeps thumb over bottom edge]. So there’s no metric on it. What that means is, that if I always want to measure in 16ths I can just measure on top [G-sweeps index finger back and forth over the top edge of the tm] and avoid the hassle, but I don’t know that you’re going to find that [G-points at C’s original tm from the tool box] with a, with a. So here’s a, here’s a way you might cheat. [takes C’s tm case in his hand] 439. C: Okay. 440. L: If you know these are sixteenths [G-sweeps thumb over inches edge of tm beyond 12 inches] and you only want to measure in 16ths. [C glances at L at this point.] Lets say you wanted to measure ah, 12 and three sixteenths [writes 12 3/16 on bottom right margin of sheet 040519e], 441. C: yeah 442. L: or no, or no, lets go 10 and three sixteenths. [writes a zero over the digit two that he has just written to make 10 3/16] If you started measuring at ten, [G-points with his thumb at ten inches on the tape measure and repositions tm in front to C.], instead of measuring at zero [G-points to zero end of tm with LH and then returns back to point at the 10” mark] start measuring at ten. Where would 10 and three sixteenths be? [G- points with RH at interval above 10” on tm] If you put one end of your pipe here [repositions edge of paper at 10” to represent a pipe length, aligning paper bottom edge along length of tm and lines up left side of paper with 10” on tm) [G-makes a slicing gesture to signal the end point of the pipe at 10” and then immediately sweeps his hand up the tape measure from the ten inch point to show where the pipe would be positioned for measuring.] 198 [52:38a] L aligns the paper with the 10 inch mark on the measuring tape and with his left hand makes slicing motion as he says, “put one end of your pipe here.” [52:38b] L then sweeps his hand up the measuring tape to show where the pipe would be positioned for measuring. 443. C: Oh, you’d just count one, two, three, four, five, six, seven, eight, nine, ten, [G- pointing to each whole unit line with his index finger as he counts these on the tape measure.] and then, 444. L: Careful, you started [G-points in the general direction of where C started to count on the tm with index finger] counting this as one. [G-points to 10” mark on tm with pencil tip] 445. C: This is one 446. L: This is zero. [G-pointing a 10” with pencil] [spoken more emphatically] One! [spoken emphatically] [G-sweeps pencil tip from 10” to 11” on the tm] 447. C: Okay. 448. L: two [now speaking normally] [G-sweeps pencil tip through each inch interval touching down on each whole number mark as it is counted)[] three, four, five, six, seven, just look at these numbers (referring to the ones digits of each whole numbers inscribed on the tm) [G-points with a pencil at the ones digit of the 17 inch mark on the tape measure], eight, nine, ten. And now you’re into the sixteenth inches [G-brief sweep of pencil across tm markings and then points at these 16th markings with his index finger)[], you’re away [G-sweeps LH index finger back and forth over the lower part of the tm near the tip] from the thirty-second inches. 449. C: Okay 450. L: So you could just use the ruler here [G-sweeps RH index finger back and forth in the vicinity of 20”] for sixteenths 451. C: So, it’d be right here, right? [G-points with his index finger to 20 3/16” the correct location on the tm in answer to the question of finding 10 3/16” starting at 10”] 452. L: For three and three eighth, and ten, what was it, ten and? 453. C: [G-points to 10 3/16 written on paper on table top] Ten and three sixteenths 199 [53:09] C points to 10 3/16 written on paper on table top. 454. L: Yeah. 455. C: So, it’d be right there [G-continues to point to tape measure]. 456. L: [G-points to 20 3/16” point with his pencil tip] 457. C: Okay. 458. L: Exactly. So this is a way of avoiding having to deal with that [G-sweeps index finger back and forth over the section of the tape measure below one foot.] 459. C: Yeah. [laughs and nods head up and down] … 460. L: Often if you want to be really precise [extends tape measure] 461. C: good one. 462. L: you have, if you want to be really precise with your measuring you wouldn’t measure to the end [wiggles end stop piece on tm back and forth a number of times to show play], because this is, there’s a bit of play there. It might not be perfectly accurate. You might start at ten anyway. [adjusts length of measuring tape] And if you want ten and a bit, you’d go from ten to twenty. [puts down tm on table top] [53:34] 463. C: Yeah, that’d be better. [Picks the measuring tape up right off of the table top for the first time during the tutoring session, sits back in his chair, extends it in front of himself and stares at it intensely] Like in my case it’d be better because it’s, I’ll know automatically, this is sixteen [G-sweeps his left thumb back and forth over division markings on the measuring tape above 10 inches while staring intensely at it and raising his eyebrows in unison four times in sucession.] [53:41] C lifts tm right off of tabletop & sits back in his chair with it in his hands. 200 [53:46a] C’s facial expression immediately prior to raising his eyebrows. [53:46b] C raises his eyebrows while explaining that using the strategy of measuring exclusively on the region of the measuring tape above 12” would be better in line 463. 464. L: after a foot [G-points to the one foot mark on the tape measure with his fourth finger] 465. C: And you can see it, exactly, after a foot. [G-points to a series of one inch intervals on the tm with his thumb as he counts them on the interval past ten inches or one foot] so, one, two, three.... I’m probably going to end up doing that. It’s way easier to just, cause this [G-points to the interval below one foot on the measuring tape], this is going to confuse me like crazy [G-sweeps his thumb over the markings below 12 inches on the tape]. Cause I have to remember, okay thirty, thirty-seconds, [G-holds out his thumb as he says 32nds] and this way I know it’s sixteenths, [G-sweeps this thumb over a small interval above 12” on tm] and you can see it clearly, right, Like the spacing- wise. [G-Moves the measuring tape to place it directly into L’s view as he says this]. 466. L: Yeah, it’s not as [inaudible] 467. C: It’s a lot easier to understand. [G-waves the extended measuring tape in the air.] So maybe that’s why yesterday’s project, I was having such a problem with it. (Laughs and extends and looks at markings on tm) I think there’s, I think there’s some other guys who don’t understand that either. 468. L: Okay 469. C: And their projects are [G-rocks extended tape measure in a teeter-totter motion at 54:11] a little bit short. Maybe, maybe that may be the case, [looking at markings on tm again] I don’t know. 470. L: I can help them too. [54:16] 471. C: …[G-more counting off inches and moving his hand about on his tape measure—practicing/rehearsing what he has learned. Looks intensly and silently at his tape measure for 63 seconds] 472. L: […] 201 [55:20] 473. C: [G-points to 13 3/8” on tape measure] So would this be 3 and 3 eighths right here? This is one? [This is similar to the task he had problems with earlier, to find 16 3/8” at line 421 of the transcript] 474. L: Show me with a pen [passes pen to C]. 475. C: Okay 476. L: Your fingers are pretty fat there. 477. C: Yeah [laughs]. 1, 2, 3, [G-pointing to successive 1/8”markings with pen along the tape measure] 478. L: Perfect. 479. C: Okay. 480. L: And, 481. C: I just want to make sure that 482. L: 13 and five sixteenths, sixteenths would be? 483. C: What, 13 and five sixteenths? 484. L: [spoken in unison with C] five sixteenths. 485. C: So this would be one, [G-pointing a 13 1/16” in tm with pen and then counting off 1/16” intervals in succession] two. It’d be right here five 486. L: Okay. Okay. 487. C: And then one quarter, [G-pointing at the end points of successive quarter inch intervals with pen] two quarters, three quarters, four quarters. Right? 488. L: Yeah. 489. C: Okay. […] Yeah, I’m going to do it that way. [puts pen down] It’s, thanks L [chuckles] 490. L: You’re welcome 491. C: Perfect 492. L: I’m happy I could help you. 493. C: [Gets up from table while maintaining eye contact with L] Now, now I know why the [...] [C is moving away from the table off camera] 494. L: Sure 495. C: Okay, perfect. 496. L: Now you’re not the only person who’s had, like I made this [has transparencies in his hand] just because it’s a common problem. And, actually it was an ironworker who I needed to help explain it to, so I thought this might be a way to help. 497. C: Yeah, it is. It just, it’s really good. I’m just going to go back to the shop. Thanks Lionel [C walking out of the classroom]. 498. C: Sure, you’re welcome. [video tape ends 56:31] 202 APPENDIX II: UBC BEHAVIOURAL RESEARCH ETHICS BOARD CERTIFICATE
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Learning mathematics for the workplace : an activity theory study of pipe trades training LaCroix, Lionel N. 2010
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Title | Learning mathematics for the workplace : an activity theory study of pipe trades training |
Creator |
LaCroix, Lionel N. |
Publisher | University of British Columbia |
Date Issued | 2010 |
Description | This study examines a single pipe trades pre-apprentice within a one-on-one impromptu tutoring session making sense of fractions-of-an-inch on a measuring tape within the context of a pre-apprenticeship program for the pipe trades. The multi-semiotic analysis of this event is framed using cultural-historical activity theory and Radford’s theory of knowledge objectification. From these complementary perspectives, mathematics is considered a culturally situated purposeful activity. Specifically, mathematics learning involves a cultural-historical, socially, and semiotically mediated process of objectification, (i.e., a process in which one becomes progressively aware and conversant, through one’s actions and interpretations, of a cultural logic of mathematical objects). The analysis focuses on the pre-apprentice’s and tutor’s joint activity during this encounter, drawing on video data and various artifacts used. This entailed slow-motion and frame-by-frame analysis of the video to assess the role and coordination of various semiotic systems, actions, and artifacts. Particular attention is paid to: the semiotic system of cultural signification, norms of practice, contradictions or conflicts that serve to motivate this activity, specific objectives of or sub-goals in the learning process for this student, semiotic processes used both by the student and tutor in the objectification process, as well as changes to the subjectification of both the pre-apprentice and researcher-as-tutor in this process. This analysis informs Radford’s theory of knowledge objectification by showing, through fine-grained analysis, relevant aspects of its dynamics and by calling attention to a new form of iconicity and a process of semiotic extraction, both original contributions to research. It also shows various ways in which a learner’s subjectification is evident in the process of learning mathematics. The results have a number of practical implications for the teaching of mathematics generally, and mathematics for the workplace in particular, by drawing attention to the social, cultural, historical, and mediated dimensions and dynamics of mathematics learning activity. The findings also illustrate the complexity of learning to measure by identifying a number of processes and conflicts involved and practical ways these are negotiated or resolved. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-30 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0055148 |
URI | http://hdl.handle.net/2429/27022 |
Degree |
Doctor of Philosophy - PhD |
Program |
Curriculum Studies |
Affiliation |
Education, Faculty of Curriculum and Pedagogy (EDCP), Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2010-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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