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Mathematical understanding and Tongan bilingual students’ language switching : is there a relationship? Manu, Sitaniselao Stan 2005

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M A T H E M A T I C A L UNDERSTANDING A N D T O N G A N B I L I N G U A L STUDENTS' L A N G U A G E SWITCHING - IS THERE A RELATIONSHIP? by Sitaniselao Stan Manu BSc. (Mathematics/Physics) 1993 The University of the South Pacific, Suva, Fiji. MSc. (Mathematics) 1997 The University of Idaho, Moscow, ID, U.S.A. A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Curriculum Studies) UNIVERSITY OF BRITISH C O L U M B I A April 2005 © Sitaniselao Stan Manu, 2005 A B S T R A C T This study explores the relationship between Tongan bilingual students' language switching and their growth of mathematical understanding. The importance of this study lies not only in its ability to use the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding as a theoretical tool for examining the relationship between language switching and growth of mathematical understanding, but also in its ability to demonstrate the theory's applicability and validity in a bilingual context. Video case study was chosen as the most appropriate means of recording, collecting and examining the described relationship in a small-group setting. Two strands of data were collected between 2001 and 2002 from a selected number of bilingual students from five secondary schools in Tonga. Analysis of the students' language switching through the Constant Comparative Method resulted in the categorization of four main "forms" of language switching. These forms were identified, categorized, and developed from the data to provide a language for describing and accounting for the particular way Tongan students switch languages. The evidence from the data clearly demonstrates how language switching both did and did not influence and was and was not influenced by the students' growth of understanding through the construction of mathematical meanings. At the same time, language switching was found to definitely enable the expression of growth of mathematical understanding. This study proposes that the effect of bilingual students' learning and development of understanding in mathematics is largely dependent on the kinds of mathematical images each bilingual student associates with his or her language. Therefore this study introduces the notion of "evocative" language switching, used for identifying, retrieving, and guiding one's existing understanding and ability to work with images. The evidence from this study is certainly applicable to other Tongan-type bilingual situations that involve individuals using words with no direct or precise translation between a dominant Western language and an indigenous language. Ultimately, the findings of this study challenge the assumption that Tongan-type bilingual students have enormous problems in the classroom. Allowed the flexibility of language switching and thus access to appropriate terms and images in either language, they do not seem to be mathematically disadvantaged. T A B L E OF CONTENTS A B S T R A C T T A B L E OF CONTENTS LIST OF MAPPINGS LIST OF T A B L E S LIST OF GRAPHS LIST OF FIGURES LIST OF EXCERPTS LIST OF APPENDICES LIST OF ABBREVIATIONS A C K N O W L E D G E M E N T S CHAPTER 1: PREVIEW and RESEARCH QUESTION 1.1 Introduction 1.2 Formulating the Question 1.3 Responding to the Question CHAPTER 2: L ITERATURE REVIEW 2.1 Introduction 2.2 The Roots of the Study's Theoretical Framework 2.3 The Notion of "Understanding" in Mathematics Education 2.4 The Interaction between Language, Mathematics, and Understanding 2.5 Bilingualism in Mathematics Education 2.6 The Notion of Language Switching 2.7 Language Switching in Mathematics Education CHAPTER 3: C O N T E X T and THEORETICAL F R A M E W O R K 3.1 Introduction 3.2 The Tongan Bilingual Context 3.2.1 The First European Impact and its Effect on the Tongan Language 3.2.2 Language in Tongan Education 3.2.3 Mathematics Language in Tongan Mathematics Education 3.3 The Theoretical Framework 3.3.1 Introduction 3.3.2 The Development of the Pirie-Kieren Theory 3.3.3 The Nature of the Pirie-Kieren Theory The Theory as "Levelled but Non-linear" The Theory as "Transcendently Recursive" The Theory as a "Whole" The Theory as a "Dynamical Process" 3.3.4 The Modes or Layers of Understanding 3.3.5 The Two-Dimensional Model - A Mapping Technique 3.3.6 The Special Features of the Pirie-Kieren Theory Folding Back Acting and Expressing Complementarity "Don't-Need" Boundaries 3.4 The Appropriateness and Purpose of the Pirie-Kieren Theory CHAPTER 4: M E T H O D O L O G Y , R E S E A R C H DESIGN, and M E T H O D 4.1 Introduction V 4.2 Methodology: Video Case Study Research 92 4.2.1 Introduction 92 4.2.2 Case Study Research and the Notion of "Generalization" 95 4.3 Research Design 98 4.3.1 Introduction 98 4.3.2 Location 100 4.3.3 Participants 100 Selected Secondary Schools 101 Selected Forms (or Grades) 102 Selected Bilingual Students 104 4.3.4 Setting 104 Group Collaboration and Peer Discussion 105 4.3.5 Tasks 106 Topic of Investigation 107 Pictorial Sequence 108 The Set of Questions 110 The "Tongan" Task with Translated Set of Questions in Tongan 111 4.4 Method of Observation 112 4.4.1 Introduction 112 4.4.2 Video Analysis and the Notion of "Trustworthiness " 113 4.4.3 Advantages and Disadvantages of Video Study 115 4.5 Data Collection 120 4.5.1 Introduction 120 4.5.2 The Data Collection Technique(s) 121 VI 4.5.3 The Ethical Issues and the Role of a Non-Participant Observer 123 4.5.4 The Supplementary Data 124 4.5.5 The Collected Video Data 124 4.6 Summary 125 CHAPTER 5: D A T A A N A L Y S I S - L A N G U A G E SWITCHING 126 5.1 Introduction 126 5.2 Data Preparation for the Analysis 126 5.2.1 Familiarization with the Data 126 5.2.2 The Role of Vprism in the Data Analysis 127 5.2.3 Transcribing Video Data 128 5.2.4 "Flagging" incidents of Language Switching 131 5.3 Data Analysis: Categorization and Coding 131 5.3.1 Preliminary Categorization 133 5.3.2 Coding the Themes using Vprism 136 5.3.3 Clustering the Initial Themes 137 5.4 Definitions of General Themes 140 5.4.1 Types of Language Mixing 142 5.4.2 Types of Language Grouping 151 5.5 Evocative Nature of Language Switching 158 5.6 Summary 162 CHAPTER 6: A N A L Y Z I N G GROWTH OF M A T H E M A T I C A L UNDERSTANDING 6.1 Introduction 6.2 Data Preparation for the Analysis 163 163 163 vii 6.2.1 Familiarization with the Data and the Role of Vprism 163 6.2.2 Timed-Activity Tracing and Flagging 164 6.3 Data Analysis: Categorization and Coding 165 6.4 Mapping using the Pirie-Kieren Model 168 6.5 The Data Analysis - Selected Groups and Individuals 169 6.5.1 Selai's Growing Understanding of Patterns and Relations in Task 2 170 6.5.2 Malia's Growing Understanding of Patterns and Relations in Task 3 189 6.5.3 Alaki and Malakai's growing understanding of patterns and relations in Task 3 204 6.5.4 Christie, Ipeni, and Semi's growing understanding of the topic in Task 3 217 6.5.5 Meki, Nanasi, and Rosina's growing understanding of the topic in Task 3 225 6.5.6 Malia and Tupu's growing understanding of the topic in the Tongan Task 229 6.5.7 Niko, Pola, and Seini's growing understanding the topic in Task 3 232 6.5.8 Siona, Naati, and Samu's growing understanding of the topic in Task 3 235 6.5.9 Malia and Tupu 's growing understanding of patterns and relations in Task 4 236 6.5.10 Hehea, Lani, and Leisi's growing understanding of the topic in Task 1 239 6.6 Summary 243 CHAPTER 7: DISCUSSION and COMPARISONS 244 7.1 Introduction 244 7.2 Discussion: Relating Language Switching with Growth of Understanding 244 7.3 Responding to the Research Question 245 7.3.1 The Relationship between Language Switching and Image Making 246 7.3.2 The Relationship between Language Switching and the Move to Image Having 247 7.3.3 The Relationship between Language Switching and Image Having 249 viii 7.3.4 The Relationship between Language Switching and Move to Property Noticing 151 7.3.5 The Relationship between Language Switching and Property Noticing 253 7.3.6 The Relationship between Language Switching and the Move to Formalising 255 7.3.7 The Relationship between Language Switching and Formalising 257 7.4 The Relationship between Language Switching and Folding Back 258 7.5 The Relationship between Language Switching and Acting and Expressing 263 7.6 Relating Growth of Understanding and Evocative Language Switching 266 7.7 The Tongan Task, Language Switching, and Growth of Understanding 272 CHAPTER 8: CONCLUSION A N D IMPLICATIONS 273 8.1 Introduction 273 8.2 The Main Findings 273 8.3 Implications and Contributions of the Study 279 8.3.1 Contributions to the Pirie-Kieren Theory 279 8.3.2 Contributions to the Notion of "Bilingualism " and Bilingual Education 280 8.3.3 The Tongan Mathematical Language, Curriculum, and Bilingual Program 283 8.3.4 Implications for Teaching and Learning 285 8.4 Personal Reflections 288 8.5 Concluding Remarks 289 BIBLIOGRAPHY 291 LIST OF MAPPINGS MAP 1: Selai's growing understanding of patterns and relations in Task 2 174 MAP 2: Malia's growing understanding of patterns and relations in Task 3 196 MAP 3: Alaki's and Malakai's growing understanding of the topic in Task 3 209 MAP 4: Semi's and Ipeni's growing understanding of the topic in Task 3 223 LIST OF T A B L E S Table A Tongan counting system for coconuts (niu) 56 Table B Tongan discrete measurements for the length of "tapa" cloth 57 Table C Number of selected participants from each of the participating schools 103 Table D A typical transcribed video scene including "flags" of language switching 129 Table E The categorization of word-mixing 134 Table F Sample list of the kinds of language switching involved for one group 135 Table G An initial list of general themes of language switching 138 Table H Example of a timed-activity trace and "flagging" aspects of growth 164 LIST OF GRAPHS Graph A Percentage of language time used for Tongan and English (Grades 1-6) 61 LIST OF FIGURES x Figure 1: Contrasting the SUP and CUP models of bilingualism (Baker, 2001) 48 Figure 2: The Pirie-Kieren model for the growth of mathematical understanding 70 Figure 3: Complementarities of Acting and Expressing in each of the layers 85 Figure 4: The pictorial sequences for all of the tasks using square blocks and cubes 110 Figure 5: The role of the image as a mediator in the translation between languages 139 Figure 6: A model for the bilingual individual's shared underlying knowledge 145 Figure 7: The Tongan bilingual students' types and forms of language switching 157 Figure 8: Relating bilingual students' verbal expressions to their work with images 159 Figure 9: The continuing pictorial sequence in Task 2 170 Figure 10: Selai's work indicates a common numerical difference of four blocks 172 Figure 11: Pictorial image of the pattern using added square blocks for each diagram 175 Figure 12: Pattern and relation between the numerical totals and diagram numbers 179 Figure 13: Continuing pictorial sequence of square blocks in Task 3 190 Figure 14: Tupu's pictorial image - repeated addition of odd-number base layers 194 Figure 15: Malia's pictorial image - addition of two extra square blocks at the base 194 Figure 16: Continuing pictorial sequence of square blocks in Task 3 205 Figure 17: Malakai's pictorial image as a stack of vertical columns of square blocks 206 Figure 18: Alaki's pictorial image of odd-numbered horizontal layers 207 Figure 19: Repeated stacking of odd-numbered base layers 208 Figure 20: Alaki's pictorial images of the "extras" and relation to the "totals" 210 Figure 21: Ipeni recognizes the staircase pattern of each diagram as "sitepu" (steps) 219 Figure 22: Christie's initial description of the 4 t h diagram as descending columns 220 Figure 23: Meki's pictorial image as a stack of vertical columns of square blocks 226 Figure 24: Continuing pictorial sequence of square blocks in the Tongan Task 229 Figure 25: Image -addition of two extra square blocks at the ends of the base layers 233 Figure 26: The continuing pictorial sequence in Task 4 237 Figure 27: The continuing pictorial sequence in Task 1 239 Figure 28: Hehea's structural rule for constructing each diagram 242 Figure 29: The schematic overview of the data analysis steps 245 xi LIST OF EXCERPTS Excerpt 1 Substitution with equivalent words 145 Excerpt 2 Borrowing of non-equivalent words 147 Excerpt 3 Substitution with Tonganised words 149 Excerpt 4 Translation - Repetition (direct) and Reformulation (indirect) 152 Excerpt 5 Shifting between two languages 154 Excerpt 6 Transcript of Selai's image having in Task 2 171 Excerpt 7 Transcript of Selai's reformulation in Task 2 176 Excerpt 8 Transcript of Selai's property noticing in Task 2 180 Excerpt 9 Transcript of Selai's generalizing pattern in Task 2 181 Excerpt 10Transcript of Malia's borrowing of "triangular" in Task 3 190 Excerpt 11 Transcript of Malia constructing image in Task 3 193 Excerpt 12 Transcript of Malia noticing "square numbers" in Task 3 199 Excerpt 13 Transcript of Malia generalizing a pattern in Task 3 201 Excerpt 14 Transcript of Malia calculating the 17th diagram in Task 3 203 Excerpt 15 Transcript of Alaki associating equivalent words in Task 3 210 Excerpt 16 Transcript of Malakai noticing property "square numbers" in Task 3 213 Excerpt 17 Transcript of Malakai generalizing "square numbers" in Task 3 214 Excerpt 18 Transcript of Malakai generalizing "add prime numbers" in Task 3 216 Excerpt 19 Transcript of Ipeni seeing pictorial image and Semi recognizing "square" 218 Excerpt 20 Transcript of Meki, Nanasi, and Rosina discussing meaning of prediction 227 Excerpt 21 Transcript of Malia and Tupu discussing "multiple" in Tongan Task 231 Excerpt 22 Transcript of Seini and Pola borrowing "prime" and "composite" numbers 233 Excerpt 23 Transcript of Naati and Siona arguing on translation in Task 3 235 Excerpt 24 Transcript of Malia and Tupu discussing "triangular" in Task 4 237 Excerpt 25 Transcript of Hehea and Lani discussing keywords in Task 1 240 xii LIST OF APPENDICES Appendix 1: The Set of Questions (in English) for Task 1 to Task 4 308 Appendix 2: Tongan Translation of the Task Set of Questions 309 Appendix 3: Summary List of Notes 310 Appendix 4: Selai's LHS Form 3 Group 1 Answer Sheet for Task 2 320 Appendix 5: Malia's QSC Form 3 Group Answer Sheet for Task 3 321 Appendix 6: Alaki and Malakai's T C A Form 3 Group Answer Sheet for Task 3 322 Appendix 7: Christie, Ipeni, and Semi's LHS Form 3 Answer Sheet for Task 3 323 Appendix 8: Meki, Nanasi and Rosina's A F C Form 2 Answer Sheet for Task 3 324 Appendix 9: Malia's QSC Form 3 Group Answer Sheet for the Tongan Task 325 Appendix 10: Niko, Pola, and Seini's A F C Form 2 Group 2 Answer Sheet for Task3 326 Appendix 11: Siona, Naati, and Samu's Form 2 Group Answer Sheet for Task 3 327 Appendix 12: Malia and Tupu's Form 3 Group Answer Sheet for Task 4 328 Appendix 13: Hehea and Lani QSC Form 2 Group 2 in Task 1 329 Appendix 14: Map of the Tongan-Type Bilingual Societies in Oceania 330 LIST OF ABBREVIATIONS (i) Pirie-Kieren Layers of Understanding PK Primitive Knowing IM Image Making IH Image Having PN Property Noticing F Formalising O Observing S Structuring I Inventising (ii) Acting and Expressing Complementarity ido image doing ire image reviewing ise image seeing isa image saying ppr property predicting pre property recording map method applying mju method justifying A C K N O W L E D G E M E N T S xiv The completion of this project comes with a great sense of relief after taking on what appeared, at times, to be an unattainable task. Yet, with all the challenges I faced, or whatever obstacles lay ahead, I managed to propel myself beyond my own perceived physical, mental, or emotional limits. I attribute this realisation of my inner potential to my strongest personal virtue: faith. Faith allowed me to continue on, despite life's adversities, simply because I believed in myself, and more so, in God. With faith, came trust. Such a trust I placed wholeheartedly in my research supervisor, Professor Susan Pirie, who, in spite of her health difficulties, continued to believe in my work and in my abilities, particularly at times when confidence in myself began to falter. Susan's unwavering support, patience and understanding have been invaluable in realising the completion of this dissertation. I am doubly proud of this thesis because it also reflects Susan's willingness to share her intellectual gifts. As Isaac Newton once said, "If I have seen farther than others, it is because I was standing on the shoulders of giants." I would also like to extend my most heartfelt gratitude to the rest of my supervisory committee. Thank you to Dr. Lyndon Martin, for his continual support and advice, whether it related directly to my work, or guided me through the complex arena of financial support; and to Dr. Cynthia Nicol, for stepping in when I desperately needed the extra help. In addition to Susan's supervision, an "editorial angel" by the name of Dana Tye happened to come my way, and she was able to teach me a great deal about writing. I would not have been able to beat all the deadlines without Dana's extra help. I would also like to extend my appreciation toward the rest of the "MU-Group" at U B C for their friendship, unyielding support, and encouragement in relation to my work. The fulfillment of a dream to do a doctoral degree would have been unimaginable without the financial scholarship provided by the Canadian Commonwealth Scholarship and Fellowship Program under the administrative work of the International Council for Canadian Studies. A big "Malo 'Aupito" goes out to Ms. Diane Cyr, the Scholarship Program Officer, for her work, patience and understanding. X V I also would like to thank the Government of Tonga for allowing me to do my study in Tonga, along with the principals of the five participating secondary schools for granting me permission to carry out the study in their schools. I must also thank the parents, the teachers, and particularly, the participating students, for their co-operation, enthusiasm, and "classroom charisma" while engaging in the data collection phase. The work of these students has taught me the importance of being a good "listener", whether I am fulfilling my role as a teacher, or as an educator. Throughout this academic and personal journey, I knew I could always rely on the support, encouragement, and prayers of all my friends and relatives, especially the Tavo and Manu families, who were a source of ever-renewable energy and strength. I thank you all, wherever you are on this globe, including those in Tonga, Fiji, New Zealand, Australia, United States, and here in Canada (plus a few significant others in other parts of the world). A special mention goes out to all the people in my village, Matahau (Tonga), particularly those who grew up and breathed the same air with me, whether they are still at home, or residing overseas. Their constant support reminded me of how true it is that it takes a village to raise a child. I also remember to extend my gratefulness to all of my teachers, ever since I was first introduced to formal education, after stepping on the shore of 'Ata'ata Island in 1978. While the last two years of my program has proved the most challenging, my beautiful wife, Jennifer (aka "Jay") and her Canadian-based family has bravely assumed the role of my immediate family, and all cared for me so much, during this course of my study and during the period I was living abroad. Thank you to Jennifer: for your infinite patience, kind words of encouragement, and unshakeable belief in my ability to finish this thesis. I must also thank my sister and all of my brothers for the encouragement and support they have expressed to me in so many different ways, tracing back to my childhood years; not to mention of course, their willingness to fondly overlook the more trying aspects of their brother's character. A special love goes out to all my lovely nieces and nephew. Lastly, to my parents: I am grateful for their vision, patience, and faith in each of their children and in our strength as a family. I thank my Mom, Lotiola, in particular, for her continued xvi support, wisdom, strength and inspiration, especially when life tested our family in so many ways. Likewise, my Dad, Tu'a, who passed away in Tonga in 2002, would have been proud of me, his faith reaffirmed, in seeing what may have appeared to be just a dream turn out to be a reality. Thank you, Dad, for teaching me early on to be strong spiritually, emotionally, mentally, and physically. I would therefore like to dedicate this dissertation to him, and to my special brother Sepasetiano in Tonga, and to all of my grandparents, for whom I still carry so many fond memories. "Sepa" would have so easily filled my shoes, had he not been so unfortunate in his life and in his ability to further his education. I wish that all of my grandparents were still around to witness how far I have come in life, and to witness the fulfillment of a dream that was borne out of their unconditional love, the sharing of their many gifts, and their passion for a better life. CHAPTER 1: PREVIEW and R E S E A R C H QUESTION 1 1.1 Introduction Since the 1970s, linguists have hotly debated the concept of "bilingualism", a concept characterized by an individual's use of two languages. Subsequent discussions have led toward the field of education, provoking ongoing dialogue among mathematics educators about the role and effect of bilingualism on students' mathematical performance and understanding. Such discussions have given rise to a variety of claims and "myths" about the nature of bilingualism and its effects - good or bad - on mathematics education. In the past, educators have contended or simply assumed that bilingual students will find mathematics harder if the language of instruction is in their second language, and that such students are therefore naturally disadvantaged in mathematics in comparison to monolingual students (Gorgorio & Planas, 2001). Another naive position, described by Clarkson and Dawe (1997), asserts that bilingual students' first language is irrelevant to their understanding of mathematics. As a result, educators have adhered to the belief that, in countries such as Tonga, students that are more competent in the dominant or second language are actually better educated and more intelligent than their indigenous peers (Fasi, 1999). This false belief has spurred a growing opposition among educators toward the use of indigenous languages in formal education, particularly at the secondary-school level; furthered by critics' claims that indigenous language learning is actually detrimental to a student's education (Fasi, 1999), particularly the learning of English (Cummins, 1981). Clarkson (1992) has raised questions about such longstanding educational misconceptions, by proposing further exploration of the potential 2 advantages of students' use of their native vernacular in academic situations, a move also supported by Setati (2004). Research presented in this thesis challenges all of the preceding claims concerning the effect of bilingualism in mathematics education, in an effort to dispel existing cultural "myths" about the relationship between bilingualism and mathematics education in a Tongan-type bilingual context. The phrase, "Tongan-type bilingual context", refers to bilingual education systems similar to the one used in the South Pacific country of Tonga, and it reflects one of the many features of bilingual programs existing around the world. Indigenous groups such as the Tongans, which have emerged from a colonial past and are currently pursuing a cultural and linguistic renaissance, typically use a dominant second language, such as English, in their formal education. A l l of the island countries in the South Pacific (refer to map in Appendix 14) -currently associated with a dearth of published studies - offer clear examples of colonial languages, such as English (Tonga, Samoa, Fiji, Papua New Guinea, the Micronesian Islands, etc.), French (Tahiti, New Caledonia, and Wallis and Futuna), and Spanish (Easter Island [Rapanui] territories)1 being used primarily in secondary education. Such dominant languages are considered to have "superior" mathematics vocabulary than the Tongan-type bilingual students' first or indigenous language. Students who are taught mathematics through a dominant language like French or English, and who are learning mathematics within their own cultural environment, characterize the Tongan-type bilingual students. These students prefer to use their native language in mathematical activities, especially when they are engaged in discussions with their peers (Fasi, 1999). The inadequacy of indigenous languages in the language of mathematics and the students' lack of proficiency in the language of instruction are two of the main reasons why these students, as well as teachers, switch languages during mathematical discourse 1 Lotherington (1997) offers a brief overview of the state of bilingual education in the South Pacific. 3 (Celedon, 1998). Thus, learning mathematics in and through a second-language context presents a double challenge for both teachers and students: the difficulty in learning the mathematics (and its vocabularies), and the prior need to understand the language of instruction (Adler, 1998). Ultimately, this dissertation attempts to question long-held misconceptions about bilingual learning contexts that have overlooked the fact that bilingual students' growth of mathematical understanding may be similar to monolingual students, and that bilingual students can voluntarily "swap" or switch languages in the process of talking about, or doing, mathematics. In spite of these educational drawbacks and misconceptions about bilingual mathematics education, the National Council of Teachers of Mathematics (NCTM) has recognized that "students whose primary language is not the language of instruction have unique needs" (1989, p. 142). Edwards (1999) noted that N C T M (1993) has further acknowledged and addressed the influences of language and culture of "minority" students by publishing a series of manuscripts to help "all readers develop a deeper understanding of, become more sensitive to, and stimulate a desire to learn more about Asian and Pacific Island students and their unique characteristics" (p. vi). The current study offers a new perspective about the unique characteristics of Tongan bilingual students. Nevertheless, little attention has been focused on the ongoing problems faced by second-language learners in the field of mathematics education, demonstrated by the paucity of studies on this topic, particularly at the secondary level (Celedon, 1998). Yet the secondary-school level is a critical period for most Tongan-type bilingual students, especially for those with limited English proficiency, as they make the transition from instruction in one language at the primary-school level to another at the secondary-school level (Celedon, 1998; Fasi, 1999). It is at this transitional stage that research presented in this thesis investigates, within the Tongan-type 4 bilingual context, the nature of "growth of mathematical understanding", and how it relates to the notion of "language switching". 1.2 Formulating the Question Language switching, or "code switching" as some linguists refer to it, is described by Baker (1993) as the way bilingual individuals alternate between two languages, whether in words, phrases or sentences, and is a term widely used worldwide (Clarkson & Dawe, 1997; Celedon, 1998; Qi, 1998; Fasi, 1999; Setati et al., 2002). Growth of mathematical understanding, characterized by Pirie and Kieren (1991a) as a "whole, dynamic, leveled but non-linear, transcendently recursive process" (p. 1), is currently receiving much attention in the education community (Martin, Towers & Pirie, 2000; Powell, Francisco, Maher, 2001; Pirie, et al., 2001; Clark, 2001; Borgen & Manu, 2002). The prior mathematics and bilingual background of the author of this thesis allowed him to bring the two described phenomena together into a single thesis by asking: What is the relationship between Tongan bilingual students' language switching and their growth of mathematical understanding? This study therefore appears to recall an earlier challenge by Cuevas (1984), who envisioned future research efforts being directed toward exploring the relationships among selected aspects of mathematical performance and understanding and various language skills, because up to the 1980s, the dearth of research on this kind of relationship "becomes almost a void when one restricts one's attention to students from a minority language group" (Cuevas, 1984, p. 140). The employment of language switching among Tongan-type bilingual students remains a challenging phenomenon for mathematics educators. What such switching entails, why it occurs, and how it 5 relates to aspects of the students' growth of mathematical understanding, are all issues at the heart of the current study. The majority of the existing research in this area can be found in behavioral studies that assess and compare bilingual students' levels of competence in both language and mathematics, without examining the qualitative nature of language switching and its effect on (growth of) mathematical understanding. Chapter 2 reviews the three main bodies of literature specifically relevant to the chosen research area: mathematical understanding, language switching, and the connection between the two phenomena. Over the past 30 years or so, mathematics educators have been engaged in an ongoing discussion about what constitutes "mathematical understanding". Some educators have been able to identify and label various kinds of understanding. Yet, these classifications are limited and insufficient to account for the author's own mathematical understanding and recollections of valuable learning experiences in Tongan mathematics classrooms. As well, in accordance with Pirie's (1988) analysis, such classifications fail to explain a learner's observed levels of mathematical performance, or the complexity involved in individual mathematical understanding. Pirie's perspective on this problem led her, along with Thomas Kieren, to develop the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding - a theoretical tool for observing and describing how mathematical understanding in a specific topic grows over time for a particular learner or group of learners. Chapter 3 looks at the details of this theoretical framework and how that choice, in turn, guided the methodology, and subsequently, the analytical process involved in the current study. During the past 15 years, the emergence of the Pirie-Kieren theory has inspired a number of studies focusing on monolingual education, but no study has been done using the Pirie-Kieren framework in a bilingual situation. The research presented in this thesis redresses this omission. 6 A unique feature of this study therefore lies in the application of the Pirie-Kieren theory as a language for, and a way of observing and accounting for, the growth of mathematical understanding in a bilingual situation. Three years ago, an ongoing interest in the Pirie-Kieren theory initially led the author to publish, along with a colleague, a case study that demonstrated incompatibilities between students' written answers and observed evidence of their mathematical understanding (Borgen & Manu, 2002). But while that case study highlighted monolingual students' evidence of mathematical (mis) understanding, it raised the possibility and challenge of finding an explanation for the relationship between (growth of) mathematical understanding and the students' means of externalization - the language they used, coupled with the actions involved in the way they expressed their mathematical thoughts. Such curiosity guided the current research interest to pursue issues related to bilingual students' language use and their mathematical understanding. Because of the author's own cultural, linguistic, and educational background, Tonga was considered the most appropriate and effective setting for this study. In order to understand the Tongan bilingual context, a brief historical account of the Tongan bilingual and mathematics education program is also offered in Chapter 3. 1.3 Responding to the Question The complex and subtle connection between language switching and growth of mathematical understanding required an in-depth qualitative approach to research and data collection. In opting for a qualitative approach, "video case study" was chosen as the most appropriate means of 7 recording, collecting, and examining the described relationship in a small-group setting, a methodology chosen by previous researchers who have worked with the Pirie-Kieren theory (Towers, 1998; Martin, 1999; Thorn, 2004). Chapter 4 focuses on the study's methodology, research design, and the method of data collection used to address the research question. For the current study, two strands of data were collected between 2001 and 2002 from five secondary schools in Tonga, where teachers and students use both Tongan and English in mathematics classrooms in various combinations. Although the use of both languages is common in the Tongan mathematics classroom, only one major research project has taken place in Tonga concerning the effects of bilingualism in mathematics education. Fasi's (1999) comprehensive study on the relationship between language competence and mathematics achievement detected a significant correlation between the two variables, but failed to conclude whether one factor was a direct result of the other. The current study attempts to clarify this relationship through its response to the posed research question. Subsequent chapters explore the results and findings of an analysis of the relationship between language switching and growth of mathematical understanding in detail. Chapter 5 summarizes a list of the identified categories (or "forms") of language switching. Chapter 6 presents a detailed mapping and description of selected student groups and observed instances of their growth of mathematical understanding. Chapter 7 links the results and analyses of the previous two chapters in order to specifically address the research question. Finally, Chapter 8 summarizes the study's findings based on the previous three chapters, responds to the research question, and considers the implications of the study for research, teaching, and learning. CHAPTER 2: L I T E R A T U R E REVIEW 8 2.1 Introduction This chapter addresses the three main bodies of literature specifically relevant to the chosen research area: mathematical understanding, language switching, and the connection between these two. The link between mathematical understanding and language switching touches on the wider domains of bilingualism and mathematics education, as well as the role of language - in the sense of speech, or Halliday's (1978) notion of "natural language" in its spoken form - in mathematical understanding. The first main body of literature considers the notion of mathematical understanding, with a summary of the various philosophical beliefs, contemporary models and theories about, and of, mathematical understanding. The second main body of literature looks at the connection between mathematics and the natural language, but examines that connection on a smaller scale. This review begins with a brief discussion of the role of language in mathematics education, and then it extends to the use of two languages in bilingual situations. The focus then shifts to examining the nature of mathematical understanding in bilingual situations and the existence of bilingualism within a mathematical context. The third main body of literature explores the notion of language switching. This bilingual phenomenon has also emerged as a popular topic among mathematics educators, particular those involved with bilingual situations. However, language switching rests in the domain of language (Halliday, 1978); it is regarded as a natural language process and not a mathematical switch of any kind, and therefore only plays a secondary role in this study. 2.2 The Roots of the Study's Theoretical Framework Previous studies by Towers (1998) and Martin (1999) have put forward similarly detailed discussions regarding the nature of growth of mathematical understanding, and their findings have been discussed in conference papers and various publications (Martin & Pirie, 1998; Martin, Towers & Pirie, 2000; Towers, 2001). But in order to understand the nature of growth of mathematical understanding, much of the attention in this section is directed toward the notion of "understanding". This study's theoretical framework has its roots in constructivism, a theory of knowing and learning that has gained wide recognition for explaining the everyday reality of mathematics classrooms. The basic tenets of constructivist philosophy centre on a subjective construction of reality; constructivism therefore focuses on how ideas are created in the mind of the individual (Bauersfeld, 1988; Tall, 1991; Cobb, Yackel & Wood, 1992; Greeno, Collins & Resnick, 1996). Westrom (2001) describes constructivism plainly as follows: It is based upon the observation and assumption that every learner must construct his or her own knowledge. A teacher cannot 'put' information into a learner's brain; rather each learner makes his or her own mental constructions to build knowledge in his or her own mind. Of course these constructions are not the same for every student, nor do they always match what the teacher intended. The teacher is limited to placing information 10 and experiences in the student's environment. The student then combines this information and experience with previously learned information and experience to create new knowledge (p. 1). Constructivist philosophy lies contrary to the empiricist's and behaviorist's perspectives, whose ideas and beliefs are built exclusively on external observations of a stimulus and a response, so such a perspective "refuses to speculate about the internal workings of the mind" (Tall, 1991, p. 7). Yet, constructivists over the last two decades have gone on to suggest different perceptions of the individual's "reality". This expanded perception has led to the development of newer and more pragmatic ways of thinking, including "radical constructivism" and "social constructivism", and later, the emergence of "enactivism". von Glasersfeld (1987) is a major proponent of the principles of radical constructivism, in which an individual can only know for certain that which he or she has created in his or her mind. This philosophy, which is a re-interpretation of Piaget's notion of genetic epistemology (Martin, 1999), maintains that each individual's experience can only be understood within a particular context; that is, it is an experience unique to that individual, and therefore inaccessible to others, von Glasersfeld (1995) outlines two basic principles of his "radical" model of constructivism: one, "knowledge is not passively received but built up by the cognizing subject", and two, "the function of cognition is adaptive and serves the organization of the experiential world, not the discovery of ontological reality" (p. 18). The effect of this radical view involves reconstruction of basic concepts such as "knowledge", "truth", and "understanding" (von Glasersfeld, 1995). For instance, radical constructivism does not claim "ontological truth" (von Glasersfeld, 1989) -a perspective that has come under criticism, not only in denying knowledge as a "true" representation of the individual's world, but also in ignoring the social influences involved in the individual's construction of his or her notion of the world (Ernest, 1991). 11 Social constructivists, on the other hand, embrace the role of social interaction in the learning process, and thus recognize the vital relationship between individual and collective knowledge (B. Davis, 1996); a factor social constructivists (e.g. Lerman, 1992; Ernest, 1991; 1994) claim is downplayed in the radical constructivist's view. Cobb (1995) asserts that social interaction constitutes a crucial source of opportunities to learn mathematics. Furthermore, Cobb (1995) argues that radical constructivism and social constructivism simply evolved to address different aspects of any learning situation. The emergence of enactivism from constructivism offers an alternative, yet more holistic, way of explaining how individuals learn. Enactivism views understanding as an interactive process, as well as a continuously unfolding phenomenon, rather than a state to be achieved (B. Davis, 1996). Enactivist theory is based on the belief that, although understanding is still the individual's creation, the individual's mathematical understanding is linked with the environment in which that understanding was created (Varela, Thompson, & Rosch, 1991). Therefore, this understanding is manifested not by itself, or solely within the individual, but in relation to the mathematical space in which the individual is acting. Varela, Thompson, and Rosch (1991) first postulated the enactivist approach by drawing on modern and ancient philosophies, such as those found in biology, neuroscience, and Buddhism. These theorists went on to define and "situate cognition not as problem solving on the basis of representations, but as embodied action" (Towers, 1998, p. 11). Enactivism challenges radical constructivism by contending that i f one focuses entirely on the individual cognizing agent, then the "fluidity of the context", and the individual's role in the environment and larger community, are often unaccounted for (B. Davis, 1996, p. 8). This 12 perspective describes how "context is not merely a place which contains the student: the student literally is part of the context" (Davis, Sumatra & Kieren, 1996, p. 157). In addition, enactivism differs from social constructivism by focusing upon "who you are (a notion that subsumes the individual and the context), not just where you are (which considers the subject as separate from the environment)" (Towers, 1998, p. 12, original emphasis). This philosophical view recognizes how an individual's knowledge depends on being in the world, inseparable from his or her body, language, and social environment (Martin, 1999). Hence, enactivism "allows a space for discussing understanding and cognition which recognizes the inter-dependence of all participants in an environment" (Towers, 1998, p. 12). 2.3 The Notion of "Understanding" in Mathematics Education Since mathematics education emerged in the 1970s as a popular field for research, researchers have developed a growing interest in the notion of understanding in the context of mathematics. Skemp's (1976) article on the existence of various types of mathematical understanding is widely credited with sparking subsequent discussions about this subject in the academic field. In 1976, Skemp considered "understanding" as one of two terms in mathematics (the other being "mathematics" itself) that often leads to serious confusion and misinterpretation among mathematics educators. Skemp explains the need to distinguish between two alternative meanings for the term "understanding" in mathematics, since it is the lack of such a distinction that lies at the root of the many misconceptions and difficulties students, teachers, and educators encounter in mathematics education. In explaining the notion of understanding, Skemp (1976) differentiates "relational" from "instrumental" understanding. Relational understanding describes an individual knowing both what to do and why, while instrumental understanding involves the 13 individual mentally applying specific mathematical rules without using his or her capacity to reason. Furthermore, Skemp's (1976) distinction between the two types of mathematical understanding leads to his detection (Skemp, 1978) of another subtle distinction between what he considers as two different subjects: "relational mathematics" and "instrumental mathematics". In Skemp's view, the word "mathematics" (as a subject) is often confused with the word "understanding" (as a way of thinking). To clarify the difference between these two terms, Skemp (1978) notes that the content of mathematics in relational and instrumental learning may be the same, but the knowledge involved in each process is so different, that they might as well be regarded as two different forms of mathematics. In addition, Skemp (1978) describes the nature of instrumental mathematics as a subject that is easily understood, and, because less knowledge is involved in the learning process, the paths to the "right" answer are relatively quick. Relational mathematics, however, is more adaptable to new tasks and can be easily remembered as parts of a whole connected body of knowledge that is effective as a goal in itself. In addition, relational mathematics suggests that the quality of any relational schema is "organic". In this instance, a schema refers to a conceptual structure an individual can use to execute an unlimited number of plans. The organic quality of such a structure acts as an agent for growth. Towers, Martin, and Pirie (2000) note that a significant factor in Skemp's (1978) descriptions is the fact that the learner is seen as a cognizing individual, rather than a passive recipient of knowledge - a distinction welcomed by constructivists. The concept of "individuality" is demonstrated by Skemp's (1978) formulation of four main characteristics of relational learning, involving the gradual building up of a conceptual structure, as it can be distinguished from instrumental learning (p. 34): 14 1. The means toward understanding thereby become independent of particular ends. 2. Building up a schema within a given area of knowledge becomes an intrinsically satisfying goal in itself. 3. The more complete one's schema, the greater the feeling of confidence in one's own ability to find new ways of "getting there" without outside help. 4. As the schemas enlarge (since a schema is never complete), the process often becomes self-continuing. Using Skemp's (1976) model, and combining it with Bruner's (1960) distinction between analytical versus intuitive thinking, Byers and Herscovics (1977) proposed a tetrahedral model of understanding. Their model identified four complementary modes of understanding: instrumental, relational, intuitive, and formal. The "intuitive" mode involves the visual ability to solve a problem, while the "formal" mode represents the ability to connect mathematical notation and symbolism with relevant mathematical ideas. The nature of Byers and Herscovics' (1977) tetrahedral model, and the interplay between its modes, reveals understanding as a complex, dynamic process, unlike Skemp's multi-valued, static distinction (Martin, 1999). Tall (1978) also emphasized the "dynamics" of mathematical understanding, critically responding to earlier models he claimed would create an unnecessary, and seemingly endless, list of categories. Tall suggested a "dynamic interpretation, which sees the various kinds of understanding as different facets of a single development" (1978, p. 2). From Tail's perspective, mathematical understanding, therefore, arises from constantly changing mental patterns that characterize mathematical thinking. These changing mental patterns are called schemas, similar to Skemp's (1978) earlier classification. While Tall (1978) insists on de-classifying unnecessary categories, Skemp (1979) extends upon his own existing model by adding a third mode of understanding, called "formal" (or logical) understanding, defined similarly to one of Byers and Herscovics' (1977) tetrahedral categories. Skemp (1981) then goes on to identify within each 15 mode two complementary levels of thinking, which he calls "intuitive" and "reflective", and then later adds a fourth mode, that of "symbolic" understanding, to include understanding of the mathematical symbols and notations (Skemp, 1982). Bergeron and Herscovics (1981) critique the early models of understanding as heavily oriented towards problem solving, and declare them inadequate in describing the knowledge involved in concept formation. Bergeron and Herscovics' (1981) initial model consists of four levels of understanding in the construction of mathematical concepts: intuitive understanding, initial conceptualization, abstraction, and formalization. In 1982, they go on to clarify each different level of understanding by characterizing initial conceptualization as "procedural understanding" and then re-defining this level of understanding to encompass both the acquisition of mathematical procedures as well as the ability to use these procedures appropriately. Further development of their initial model led Herscovics and Bergeron (1988) to eventually present a two-tiered, extended model of understanding, used to analyze the development of particular mathematical concepts. This extended model shows that "the understanding of a particular mathematical concept must rest on the understanding of the preliminary physical concepts" (Herscovics & Bergeron, 1988, p. 7). Nickerson (1985) did not offer a model or theory of mathematical understanding, but instead discussed various aspects of understanding. Among his contentions, Nickerson noted that understanding is a context-dependent concept, in which an idea can be understood best from the context in which it is situated. But, he added, an incorrect answer is a reasonably good indication of a lack of understanding (Nickerson, 1985). Furthermore, understanding must also depend on the existing knowledge one has about the concepts involved, and requires not only having such knowledge, but also doing something useful with it. Understanding, for Nickerson (1987), can 16 also be seen as an active process, one that varies in its degree of complexity or completeness,2 and which can be described or investigated in terms of the "breadth" and "depth" of a concept's connectedness (Nickerson, 1985). Such a proposal, in which "deep" understanding can be demonstrated in a variety of ways, is also the basis of the previously discussed Herscovics-Bergeron model. Following Skemp's initial definition (1976) of the dichotomy between instrumental and relational understanding, Nesher (1986) argued for the importance of the connection between learning algorithms (or algorithmic performance) and conceptual understanding, and the fact that the separation between the two is "impossible" at any stage of learning. Nesher (1986) concludes that there are different levels of understanding, and that there exists a relationship between understanding and knowing algorithmic procedures: one contributes to the other. Nesher's (1986) notion of understanding is consistent with Nickerson's (1985) proposal: that understanding is never going to be complete. Nesher also observed that the levels of understanding are not always ordered, and that no one can know precisely which stages lead to understanding. By contrast, von Glasersfeld's (1987) radical constructivist view considers conceptual structuring and re-organization as essential to the individual's construction of knowledge, von Glasersfeld sees algorithmic operations, such as using multiplication tables successfully, as an inadequate demonstration of mathematical knowledge. He associates such types of knowledge with those learned through strict training - a process von Glasersfeld (1987) claims is mainly associated with the way animals learn, von Glasersfeld (1987) further argues that what 2 Nickerson (1985) discusses the difficulty with the so-called 'Hegelian' theory that "in order to know all there is to know about anything, one must know all there is to know about everything, because everything is related to everything" (p. 231). This view, according to Nickerson, is not very helpful because it denies the reality of different degrees of complexity in various levels of understanding. 17 determines the value of the re-organized, conceptual structures is "their experiential adequacy, their goodness of fit with experience, their viability as a means for the solving of problems, among which is, of course, the never-ending problem of consistent organization that we call understanding" (p. 5, original emphasis). This view recalls Tail's (1978) emphasis on the dynamic nature of understanding, von Glasersfeld (1987) further concludes that the "process of understanding.. .is analogous to the process of coming to know in the context of experience.. .it is a matter of building up, out of available elements, conceptual structures that fit into space as is left unencumbered by constraints" (p. 10). Unlike von Glasersfeld's (1987) radical view, the role of algorithms in mathematical thinking led Sfard (1991) to conclude that abstract concepts could be considered in two fundamentally different but complementary ways: "structurally" (as objects) and "operationally" (as processes). Sfard observes that the process of learning involves the interplay between these two incompatible approaches. She goes on to label three stages in concept development that she calls interiorization, condensation, and reification. "Interiorization" acquaints a learner with the processes that give rise to new concepts; "condensation" involves the ability of the learner in managing lengthy sequences of operations; and "reification" refers to a state of "ontological shift" in which the learner sees a familiar piece of mathematics in a totally new way (Sfard, 1991, p. 19).3 Sierpinska (1990), however, views understanding in an "active" sense, conceiving it as an act of grasping the meaning rather than a process or way of knowing. Moreover, Sierpinska sees understanding as an action that involves overcoming mathematical obstacles. Yet, Sierpinska 3 The next chapter describes how "reification" appears to resemble Pirie and Kieren's notion of "don't-need boundary" (refer to Section 18 recognizes that the level of understanding changes with growth of knowledge as a process, rather than as an act. She goes on to distinguish two categorized levels of understanding, involving "intuition" and "logico-physical abstraction", based on the previous work by Herscovics and Bergeron (1989). But Sierpinska's (1990) work appears to overlook the question of how meanings are constructed. In her model, intuitive understanding is based essentially on visual perception, while logico-physical abstraction involves one's awareness of one's logico-physical invariants. Sierpinska (1990) then concludes that depth of understanding can be measured "by the number and quality of acts of understanding one has experienced, or by the number of epistemological obstacles one has overcome" (p. 35). Kieren, Pirie, and Reid (1994) view such a perspective of understanding as "a multileveled activity in which a person perceives and overcomes epistemological obstacles" in order to move on to a higher level (p. 49). R. B. Davis (1984) initially built his own theory from the previous findings of Minsky and Papert (1972), suggesting that understanding occurs when a new idea is fitted into a larger framework of previously assembled ideas. This representational perspective, endorsed by mathematics educators such as Hiebert and Carpenter (1992), sees growth of understanding as a structured, connected network that grows bigger (in depth) and more organized (in complexity). R. B. Davis (1991) later surmised that a theory for the growth of mathematical understanding offered an exciting observation i f one could speak about "the development of the students' mathematical thinking"4 (p. 232). From this observation, evaluation of one's level of understanding moves from the view of taking "snapshots" of understanding at specific points in time, to mapping the growth of understanding of one's mental representations over a period of time. To conduct such an evaluation, R. B. Davis (1991) prefers the direct observation of videotaped recordings in 4 R. B. Davis (1991) actually directed this comment toward the nature of the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding that is the theoretical framework for this study. 19 settings such as small group discussions. Furthermore, he contends that mathematical understanding, as Nickerson (1985) observed earlier, requires not only having connected knowledge, but also doing something useful with it (R. B. Davis, 1991). The notion of connectedness in one's knowledge suggests that the more one knows about a particular concept, the more complex one's knowledge is, and the better one understands the concept (Nickerson, 1985). In the same manner, Hiebert and Carpenter (1992) later stated that the depth and strength of one's understanding increases as the network of representations grows, and as the interconnections "become strengthened with reinforcing experiences and tighter network structuring" (p. 69). This representational view also sees growth (of understanding) in the manner in which internal and external representations (for a particular concept) constantly affect each other, and the network of representations becomes larger and better organized. Given that the connected network described in Hiebert and Carpenter's (1992) model is continually undergoing re-organization and re-structuring, growth of understanding, then, can be considered dynamic in nature. Duffin and Simpson (1995) initially defined understanding as the "awareness of internal mental structures" (p. 167). A "breakthrough" in their work, they claimed, came about through their interpretation of Sierpinska's (1990) idea that an act of understanding took place through the use of network connections to solve problems. From there, Duffin and Simpson (1995) went on to name three components of understanding: building, having, and enacting. "Building" refers to the formation of the internal mental structures and the connections involved at a particular time during the learning process. Borrowing the idea from Nickerson (1985), Duffin and Simpson (1995) explained that a chain of connections from a single concept constitutes the learner's 20 breadth and depth of understanding. "Having", however, is the structural state of the connections at any given time, while "enacting", means the use of the available connections at a particular moment to solve a problem or construct a response.5 Throughout the discussion in mathematics education about the notion of understanding, the term "meaning" is used interchangeably in a mathematical context. B. Davis (1996) argues from an enactivist perspective that these two terms - understanding and meaning - can be separated by "rigid definition". B. Davis then describes how meaning is closely linked to "the objectifying realm of definition-seeking discussions", while understanding is closely aligned with "the interpretational realm of ever-evolving conversation (auditory sensory modalities)" (1996, p. 199). However, B . Davis (1996) argues that the distinction between understanding and meaning is less important than the common characteristic involved in each process: the "space of interpretation" (p. 199). For the most part, B. Davis accords a particular status to the word "meaning" within the field of mathematics education. He explains how the interpretations of "meaning" are as much denied in mathematics education as the "formulated" understanding is privileged, while the bodily and "unformulated" understandings are often neglected. In B. Davis' enactivist perspective, the bulk of meanings are "neither formulated nor strictly linguistic. They are, rather, lived through or enacted" (p. 205). As such, "meanings reside in the domain of language, and language, in turn, is generally cast as a mental (in contrast with a physical) capacity" (B. Davis, 1996, p. 205). However, the enactivist framework claims that understanding and meaning can exist not just in the words being spoken but also in the participants' joint action. Thus, according to B. Davis (1996), "the space of collective action is not merely a device in promoting individual sense-making; it is a location for (shared) meaning and understanding" 5 These three components of understanding appear to resemble three of Pirie and Kieren's modes of understanding: Image Making (building), Image Having (having) and Property Noticing (enacting). See description in Chapter 3 (Section 3.3.4). It is important to note, however, that Duffin and Simpson's concept of enacting is not, in any way, related to the philosophy of enactivism. 21 (p. 197), a fundamental notion in the enactivist philosophy. B. Davis' (1996) description of understanding as a dynamic process may invoke a sense of social interaction, and is often considered to be "more immediate, more fluid, more negotiable" than meaning (p. 199). On the other hand, meaning is more authoritative than understanding, and has a sense of "intention, of directedness, of pointing, of referring to something in particular" (B. Davis, 1996, p. 198). B. Davis' enactivist view concludes that, outside the teaching context, a mathematical idea is more concerned with meaning than with understanding on a broader, socially sanctioned level. 2.4 The Interaction between Language, Mathematics, and Understanding Since the 1970s, extensive research and interest in mathematics education intensified, shifting the focus to the teaching and learning of mathematics. As a result, teachers and students have been in the spotlight in most discussions about mathematics education. As Austin and Howson (1979) observed, the purpose of these studies and discussions "centres upon attempts to understand how mathematics is created, taught, and learned most effectively" (p. 161). In addition, Austin and Howson identified from Streven's (1974) UNESCO report the helpful distinction between three different levels of analysis: the language of the learner, the language of the teacher, and the language of mathematics. This distinction, according to Pirie and Schwarzenberger (1988), is essential to any study of the role of language in the development of mathematical concepts in a classroom setting. Other related factors have also been recognized for their major role in the learning and understanding of mathematics, including the impact of the classroom setting, available resources, the school environment, as well as the effect of the individual's home environment. (See Ellerton and Clarkson (1996) for a comprehensive review of such interplay between language and mathematics learning in the mathematics classroom.) 22 Academic discussions about the language of mathematics or "mathematical language" often lead to confusion, because of the difficulty in distinguishing between mathematical understanding and language understanding. Mathematical language refers to the use of the natural or verbal language in mathematical discourse, rather than the mathematical contents involved. Mathematical language typically refers to the words and phrases as they are used specifically in mathematics, or as metaphors in a mathematical context. Central to the concept of mathematical language is the notion of a register, a term Halliday (1978) coined to describe "a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings" (p. 195). While Halliday's notion of register appears to vary over time (Pimm, 1987), a mathematical register is described in this study as the collection of all the mathematical terms, including everyday words that often hold specific meanings within mathematical contexts; that is, the form of verbal language used in mathematics rather than the mathematical symbolism itself. For instance, terms in the mathematics register such as "square", "prime", and "odd" are often used in mathematical discourse, and the everyday words such as "increase", "line", and "period" often hold specific meanings within mathematical contexts. Linguists and mathematics educators have initiated discussions about the relationship between students' natural language and their mathematical language whenever they are engaged in mathematical tasks. Language itself is the focus of diverse fields of study; some of which - for , instance, semantics, semiotics, or functional analysis - can be related or applied to mathematics (Durkin, 1991). Durkin (1991), a mathematics educator whose research interest lies in language, asked why one needs to focus on, or at least refuse to ignore, language. After all, mathematicians are often known to work in abstract and highly symbolic representations, where precision and formalism are significant. Durkin (1991) then pointed out Pimm's (1987) discussion on the role 23 of language in communicating mathematical ideas, because language reveals the significance of both people and communication in mathematics. In addition, Durkin (1991) said that to ignore the role of language: certainly discounts many of the processes and problems of attaining any level of competence in mathematics: mathematics education begins and proceeds in language, it advances and stumbles because of language, and its outcomes are often assessed in language - the interweaving of mathematics and language is particularly intricate and intriguing (Durkin, 1991, p. 3). The role of language in learning is vitally important, especially if the assumption holds true that students' mathematical education takes place in language, and that it is therefore desirable to explore how this learning occurs and what problems and benefits it entails for education (Durkin, 1991). In this respect, viewing mathematics in terms of language use would shed some light on the issues of learning, teaching, and understanding mathematics (Pimm, 1987). Moreover, because mathematics education makes particular linguistic demands, such as explanation and justification, the act of learning mathematics could, in turn, make a more substantial contribution to learners' general language development (Durkin, 1991). Language is critical to many of the processes of learning and instruction, such as cognitive and communicative functions (Hoyles, 1985, cited in Pirie & Schwarzenberger, 1988). Language enables one to articulate, discuss, construct, and re-present the ideas and problems within the field of mathematics. In semantic theory, some theorists describe the principal function of language as "transmitting" meaning, whether mathematical or non-mathematical (Durkin, 1991). Such a function is disputed by the constructivist ideology, in which the individual is seen as responsible to his or her own "subjective conceptualizations and re-presentations" (von Glasersfeld, 1995, p. 141). Others, 24 however, including notable linguistic theorists such as Halliday (1978), often refer to the role of language in "conveying" mathematical meaning - a way of reflecting the ambiguity, uniqueness, and complexity of any meaningful construction. Such language complexity involves words often used to convey not just one, but also a multitude, of mathematical meanings. Consequently, students with language difficulties sometimes fail to interpret the words and associated mathematical meanings their teachers intended them to learn. Furthermore, language is also said to "mediate" a mathematical experience between several individuals (Austin & Howson, 1979; Durkin, 1991), which appears to possess a social function (see also Gorgorio & Planas, 2001, Secada, 1992). Social constructivists would argue that the construction of knowledge and understanding takes place through language whenever ones takes into account the contribution of all participants in any social activity, the value of group discussion, and the social environment (Cobb, Yackel & Wood, 1992). This social constructivist view of language describes how language is used as a tool "to construct knowledge and regulate access to that knowledge" (Veel, 1999, p. 185). However, Brown (1999) in discussing the relationship between language and mathematics, views how language conditions all experiences of reality while, from other perspectives, language distorts the experience of reality. For Durkin (1991), language brings together a great many dimensions of learning, including the syntactic, semantic, pragmatic, discourse, socio-linguistic factors, along with many other means of benefiting and enriching one's view of mathematics education (Durkin, 1991). But language also brings its own rules and demands, which, according to Durkin (1991), may not always be in "perfect correspondence with the rules and demands of mathematics" (p. 14). In that sense, language can also present its semantic and lexical ambiguities and inconsistencies when it 25 involves mathematics, thereby misleading and confusing some students. The semantic hurdles develop because of shifts in the application of particular words, and because mathematical symbols and words can be considered polysemous (having multiple, systematically associated meanings). Similarly, potential lexical ambiguities include the notions of homonymy (words spelled and pronounced alike but different in meaning) and homophony (words pronounced alike but different in meaning or spelling), illustrating the potential for confusion and misunderstanding for language-deficient students trying to learn mathematics. A l l of these are consistent with Durkin's (1991) description of varying levels of ambiguity in language use involving the "interplay of syntactic, semantic, inferential, temporal and contextual" clues (p. 9). Durkin's conclusions are hardly surprising, because language is a natural human creation, and unlike mathematics, it is "not clear-cut or precise" and is "inherently messy" (Halliday, 1978, p. 203). However, it should also be clear, as Durkin (1991) pointed out, that language is seen as only one of a number of concerns in mathematics education. Beyond the broad impact of language, others have focused narrowly upon the use of metaphors in mathematics education as the most potent and creative tool of any natural language (Pimm, 1987; Durkin, 1991; Pirie, 1998b; Lakoff & Nunez, 2000). Apart from highlighting differences in everyday word usage, metaphors are vital for the construction of meaning in mathematics (Pimm, 1987). In addition, much of everyday language is metaphorical in origin (Halliday, 1978), and so is the nature of the natural language in mathematical language. Halliday (1978) notes how mathematical expressions using the natural language "represent essentially concrete modes of meaning that take on a metaphorical guise when used to express abstract, formal relations" (p. 202). For Halliday, the difficulties of mathematical language relate to its extensive use of metaphorical and figurative concepts, such as the equivalence of the statement "four from six leaves two" with the numerical expression "6 - 4 = 2". Nolder (1991), illustrates with an 26 example how metaphorical terms such as "staircases", "wings", and "triangles" are used to describe cubes arranged symmetrically around a centre column of a tower pyramid; and she goes on to observe that the use of some of these metaphorical terms in mathematics education can turn out to be less helpful than others. Yet, in spite of the way such metaphorical expression pervades many facets of mathematical teaching and learning, Nolder (1991) argues that the use of metaphors usually goes unnoticed. Although the relationship between language and mathematics has long been explored (Halliday, 1978; Pimm, 1987; Durkin & Shire, 1991), an apparent division between the two subjects remains, along with a division between the way linguists and mathematics educators view each other's roles. Bell and Woo (1998) claimed, for example, that a discussion of, or about, language was absent from all 29 chapters of the 1992 American publication Handbook of Research on Mathematics Teaching and Learning. Furthermore, throughout the book, "assertions of the fundamental importance of language to mathematics teaching and learning tended to be sporadic rather than thematic" (Bell & Woo, 1998, p. 51). Veels (1999) described the alleged division more specifically when he wrote: To the mathematician, the research on mathematical language in language education journals frequently appears to be woefully inadequate in its understanding of mathematical knowledge. To the linguist, the research in mathematics journals seems horribly simplistic in the role it assigns to language in learning. The result of all this is that language educators and mathematicians rarely talk to one another (p. 185). Austin and Howson (1979), however, once suggested a logical way of bridging the described division. In their extensive review of this topic, they proposed that mathematics educators should pay attention to linguistics to better understand the language of mathematics, and to help make 27 the creation, teaching, and learning of mathematics effective (Austin & Howson, 1979). After all, mathematics education and language do have so much in common, and as a result, "co-operative research would pay dividends", whereby each may have much to learn from, and contribute to, the other's work (Austin & Howson, 1979, p. 162). In addition, to think about language and mathematics as separate entities seems inappropriate; rather, one should look at how "language is modified as a result of attempting to communicate mathematical ideas and perceptions which is of far greater import" (Pimm, 1987, p. 196). In mathematics education, language comprehension within a mathematical context can be explored by asking the following question: "Is language understanding different from mathematical understanding?" The available literature suggests various answers to this question, while aiming to explain the relationship between language and mathematical understanding and the various theoretical and epistemological interpretations of language understanding. The answers provided by the current literature can be drawn from one of five areas: (i) the analysis of the various synonyms associated with the term "language understanding"; (ii).the understanding of the subjective meaning of words and symbols; (iii) the understanding of the objective meaning of words and symbols; (iv) the discussion about the depth of understanding within a particular language; and (v) the use of language in expression and external representation. Language understanding in a mathematical context refers to how language is used, talked about, and analyzed within mathematics education. Various conceptual terms are used synonymously with language understanding: most notably are the notions of "meaning", "sense-making", "comprehension", and "knowing". These concepts are profoundly intertwined, and as a result, they float around interchangeably within the students' (and teachers') everyday and mathematical discourse. Students (and teachers alike) are often heard to say, "I understand the 28 question", "I know what that word means", "I know (or understand) what you're saying", "How do (or can) you comprehend the written instruction?", "What do you mean?", and so forth. It is possible to distinguish the subtle difference between these remarks by remaining aware of the fact that language understanding, as is the case with mathematical understanding, is composed of particular characteristics. Similar to the process of mathematical understanding, understanding language - whether it is composed of a word, a phrase, or a sentence - is generally considered a context-dependent concept. This characteristic can be applied to the use of ordinary words, phrases, or algebraic symbols in mathematics. To understand a particular word being used in the natural language, it requires a context, which is often defined by the structure of the sentence or by the flow of the interaction, von Glasersfeld (1995) points out that almost every English word, including mathematical English words, contains more than one meaning while viewed in isolation. However, when such a word is spoken or written in a sentence, the context of communication usually eliminates all but one of the potential meanings. Consequently, the context or learning situation is central to determining the relationship between one's mathematical understanding and the use of language in any mathematical discourse. In the same sense as mathematical understanding, language understanding also varies in its degree of complexity or completeness. Nickerson's (1985) concept of "connectedness" in mathematical understanding can be applied to language understanding: one has to have a depth of connectedness within a particular language; hence, the more one knows about a particular concept, the better one understands it. The depth or "growth" of language understanding reflects the complexity in one's "language capacity" - the individual's accumulated knowledge of the language. Nickerson (1985) associates an individual's depth of understanding of a language with 29 how the individual uses a word, not only how he or she uses the word within a particular context, but also in his or her ability to use the word appropriately and meaningfully in a variety of ways. The verbal connectedness outlined previously is closely related with Paivio's (1971) notion of "verbal association", which describes the depth of language understanding by the complexity of the interconnection within the individual's language capacity. According to Paivio's "Dual Coding" approach - which attempts to describe how language and images interconnect -language understanding appears to literally mean a measure of how many words come to mind whenever a particular word or image is activated. Paivio (1986) therefore describes verbal association in such a way that its internal structure varies and that the smaller units are "organized into larger units in sequential or successive fashion" (p. 58). This sequential organization gives rise to the depth of understanding and the spread of connection among related words, a process Paivio (1971) called "associative". Studies have shown that connected language understanding must be built on an individual's prior or existing knowledge about the language, including the mathematical language (Baker, 1993; R. B. Davis, 1991). Hence, an individual can easily understand and make use of a mathematical concept in a particular language if the foundation of that language is sufficiently well developed (Baker, 1993), or if the process of learning language, including learning mathematics through language, is built upon a foundation of previously built-up understanding (R. B. Davis, 1991). Given the relationship between a natural language and the mathematical language contained within it, the process of understanding the language is considered similar to understanding the language of mathematics. This process does not refer to understanding the mathematical concepts, ideas, or images that are attached to the mathematics terms. The concepts, ideas or 30 images are like mental representations: they are "not built out of words" (R. B. Davis, 1991, p. 227). But each term, whether mathematical or non-mathematical, has a particular meaning or purpose.6 Sierpinska (1990) defines one possible explanation for the way an individual demonstrates understanding of a term by directing mental activities toward some "object"; the object is then called the "meaning" of the word. In an individual's mind, the objective meaning of words can transform into mental representations or images of the ideas denoted by words. The construction of mental representations, therefore, is one of the key features of the interpretation process. These mental representations are usually not about written words, but rather, about the ideas denoted by the words. In reference to the use of words, Hiebert and Carpenter (1992) claim that in order for words to acquire specific meaning, an individual must connect his or her mental representations of the written words with his or her mental representations of concrete materials. This connection allows the individual to create subjective meaning for the words. From a constructivist's point of view, the individual's own construction in using a word points only to the representational meaning he or she associates with that word (von Glasersfeld, 1995). Therefore, according to von Glasersfeld (1995), "the meanings of whatever words one chooses are one's own, and there is no way of [re-] presenting them" (p. 109). Moreover, for each individual, meaning and words are inexplicably intertwined, and in the mathematical context, an individual's subjective interpretation of a word reflects in some way the objective meaning of the word as defined by its creator (Lakoff & Nunez, 2000). Any difference or mismatch between an individual's construction of subjective and objective meanings can lead to misunderstanding. However, Pirie 6 Post-defense note: a substantive issue, not discussed in detail in this thesis, deals with how some philosophers, following an interpretation of Wittgenstein, would argue that mathematics is the use of language, and developing mathematical understanding is the same as learning ways of talking (Barton, 2005, personal correspondence). 31 and Schwarzenberger (1988) warn that a mere mismatch of a particular language, symbol, or notation should not instantly be considered evidence of a lack of mathematical understanding. The problem in language comprehension (whether it involves speaking, reading, or writing), or in difficulties understanding mathematical language, is often observed when students work with mathematical word problems. This kind of ambiguity is not new, especially with studies that cite difficulties when students work with simple relational English words such as "more" or "less" (Lean, Clements, & del Campo, 1990; MacGregor, 1993; Fasi, 1999). In fact, several studies have shown how students have difficulty in the transformation (or translation) of the relationships expressed in the natural language into corresponding mathematical relationships, and vice versa (Nickerson, 1985; Pimm, 1987; Fasi, 1999). In algebra, the advantage of the conciseness and precision in mathematical representation is often overshadowed by the abstractness and ambiguities of the mathematical terms and notations. Nickerson (1985) demonstrates this situation by describing how students struggle to express verbally the mathematical meaning of.the simple relationship denoted by the expression, " X = 2Y". For example: "One X equals two Y 's" , "For every X there are two Y ' s " or, "There are twice as many X ' s as there are Y ' s" . Moreover, Nickerson (1985) was also concerned with how certain algebraic tasks can be solved by purely syntactic methods that do not depend on comprehension of the meaning of the problem; thus alluding to the previously described separation between language and mathematics (Lakoff & Nunez, 2000). Lakoff and Nunez (2000) distinguish between the mathematical concepts, the written mathematical symbols for these concepts, and the words used for these concepts. For Lakoff and Nunez, words are simply part of the natural language, but words are not part of mathematics. 32 Therefore, in their "embodied" view of mathematics, the mathematical terms such as "square" and "odd" are only meaningful by virtue of the mathematical concepts attached to them; that is, to understand a mathematical word is to associate it with a concept. Hence, in their analysis of the mathematical symbols, including words, Lakoff and Nunez (2000) conclude: The meaning of mathematical symbols is not in the symbols alone and how they can be manipulated by rule. Nor is the meaning of symbols in the interpretation of the symbols in terms of set-theoretical models that are themselves uninterpreted. Ultimately, mathematical meaning is like everyday meaning. It is part of embodied cognition... From the perspective of embodied mathematics, ideas and understanding are what mathematics is centrally about (p. 49). However, an individual's understanding of a particular concept does not take place by reading a single word or text, but requires an active process of interpretation. A dialectic relationship between understanding and language exists through an explanation that is deeply reconciled to interpretation. In this regard, Sierpinska (1990) cites Ricoeur's (1989) didactical presentation of .such a dialectic relationship as "phases of a specific process": I propose to describe this dialectic as a passage, first, from understanding to explaining, and then from explaining to comprehending. At the beginning of this process understanding is a naive grasping of the meaning of the text as a whole. By the second stage, as comprehending, it is an elaborate way of understanding based on explanatory procedures.. .In this way, explaining appears as a mediator between two phrases of understanding (cited in Sierpinska, 1990, p. 26). Therefore, when an individual's interpretation involves language, his or her understanding takes into consideration more than just the subjective meaning of the word or phrase, or even the subjective meaning of the sentence. Moreover, the depth of an individual's interpretation often 33 includes more than the word(s) symbolizes and the concept(s) associated with it. Referring to words expressed through speech, von Glasersfeld (1995) quotes Vygotsky (1962), who stressed that "To understand another's speech, it is not sufficient to understand his [or her] words - we must understand his thought. But even that is not enough - we must also know its motivation" ,(p. 151). Nevertheless, von Glasersfeld (1995) adds that the "result of an interpretation survives and is taken as the meaning, if it makes sense in the conceptual environment which the interpreter derives from the given words and the situational context in which they are now encountered" (von Glasersfeld, 1995, p. 142). Thus, another individual, based on the words being spoken and the context for communication, might represent meanings differently than the individual who is actually deciphering the words. To broaden the discussion, Sierpinska (1990) gave an explicit explanation of the difference between the two statements relating sense and meaning: statement (a) refers to an objective description of meaning, while (b) reveals the subjective feeling or interpretation of the speaker. Sierpinska (1990) thus contends that the word "sense" has an objective meaning, demonstrated by the question, "In what sense are you using this word?" (p. 27) Since the question is answered with a sentence that contains the key word, mathematical meaning is therefore derived from context. The sentence itself, whose structure defines the mathematical function of the word, gives a mathematical sense to the word. Sierpinska adds that the sentence may exemplify, denote, state, or refer to something - its "reference" - that holds true (or false) in some mathematical reality. The meaning, then, of the mathematical word contains both the sense of the sentence and its reference. Moreover, Sierpinska (1990) notes that Ricoeur's (1989) "hermeneutic" philosophy - which is concerned with the interpretation of literary texts - defines the duality of sense and reference in 34 meaning. In other words, "sense" answers what the sentence says, and "reference" answers what the sentence is about. This "duality" can be observed, for instance, in the structure of the following statement: "Square the counting numbers to find the square numbers!" One method of writing this mathematical relationship in algebraic form is denoted by a2 = b , where a is an element of the counting numbers and b a square number. The sense of the statement refers to the equality between the two sets of objects: square numbers and the square-product of counting numbers. The reference denotes that the statement holds true in the field of real numbers, but not in the field of complex numbers. According to Sierpinska, a combination of the notion of sense and reference will provide the objective meaning of the statement. In addition, Sierpinska (1990), in her epistemological analysis of mathematical concepts, explains how language plays a role in mathematical meaning and understanding: Suppose we start from the informal language of mathematics. Let us find words and expressions used in defining, describing, working with the concept we are analyzing. Let us find sentences which are the senses in which these words and expressions are used. Then let us seek the references of these sentences. And then see relations among all these senses and references. This analysis can lead us to a description of the meaning of the concept in question.. .Understanding the concept will then be conceived as the act of grasping this meaning (p. 27). The social constructivist perspective goes further in relating language to learning. This perspective assumes learning is neither a static process, nor does it occur in isolation, but rather, learning is socially negotiated and expressed through language that focuses on explanation and clarification (Smith, 2000). It suggests that students in a social setting can interact through language during mathematical discourse. In addition, social constructivism views learning, and 35 hence, understanding mathematics through language, as a process of affecting one's meaning. Many experienced language teachers have long sought to avoid passing on superficial verbal learning skills. It is all too easy to get a student to say something - or to write it - without their necessarily understanding it (R. B. Davis, 1991). Perhaps, for these reasons, some mathematics educators and academics go on to suggest that learning ought to be made less dependent on language; and that teachers of mathematics, in particular, should emphasize the importance of learning through concrete operations on objects (Halliday, 1978). However, Halliday (1978) warns that there is no point in trying to eliminate language from the learning process altogether, a problem seen in mathematics textbooks deliberately designed for teaching mathematics without words. Rather than engage in any such vain attempt, one "should seek equally positive ways of advancing those aspects of the learning process which are, essentially, linguistics" (Halliday, 1978, p. 203). Verbalizing thought processes is a form of externalizing the internal dialogues or activities within the minds of individuals. It is then up to the individuals to construct their own conceptualizations and representations of whatever is being externalized. These external representations, through the use of natural language, may turn out to be the only way in which the individual's thought processes can be revealed and his or her potential realized (Halliday, 1978). In fact, Brown (2001) argues that an individual's framing of mathematical experience (including mathematical understanding) in words "should be seen as an integral part of the mathematics itself, inseparable from less visible cognitive activity" (p. 200). The question of individual expressiveness is therefore a central issue in mathematics education, as demonstrated by Pimm's (1987) question: "What is the connection between how things are described and how they are seen?" (p. 201). In fact, what an observer sees is a reflection of the 36 internal activity. Thus, a critical flaw in many structured mathematical learning situations lies in the gap between action and expression. On this point, Noss, Healy and Hoyles (1997) write: Indeed, students who are able to apply a correct method to any number of specific cases often cannot articulate a general pattern or relationship in natural language, and expression in algebraic symbolism is still more problematic.. .the evidence suggests that algebraic formulation is often disconnected from the activity which precedes it (p. 203). At the same time,,the role of language as a cultural tool reflects how "thought and language enhance each other's development" (Sierpinska, 1996, p. 42). Citing Piaget's (1959) observation that "language is molded on the habits of thought", Sierpinska (1996) argued that such a view expresses how language reflects thinking (p. 32). In addition, Vygotsky, according to Sierpinska (1996), was quite sensitive too, to the delicate relationship between language and thought, or as Sierpinska referred to it, the "subtle interplay between the natural language and spontaneous thought and scientific concepts" (1996, p. 41). While this section offers a comprehensive look at many existing models and theories of, or about, the notion of "mathematical understanding", Chapter 3 explains in detail an additional theory, known as the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding, and discusses why such theoretical framework is considered advantageous and more appropriate than others for this study. The next section of the current chapter, however, continues by looking at the notion of "bilingualism" in mathematics education. 37 2.5 Bilingualism in Mathematics Education During the past three decades, a growing awareness of bilingualism has triggered considerable interest in the field of mathematics education, particularly with regard to the question of how mathematics can be learned and understood in two languages. In general, bilingual issues are gaining widespread attention in a variety of arenas, not just in education, but also in political, cultural, and social circles (Cummins, 1981), perhaps because virtually every country in the world represents bilingual populations. Baker (2001) noted that numerous researchers in the past have extensively researched and publicly discussed the complexities associated with bilingualism and bilingual education, and yet, these subjects continue to challenge, and occasionally confound, linguists and mathematics educators alike. To add to the confusion, interested parties have yet to agree upon a precise definition of what it means to be "bilingual", mainly because of its multi-dimensional nature. In fact, the difficulty in defining bilingualism has been the source of major differences, engaging researchers, educators, and even politicians, in public debates (Cummins, 1981; Baker, 2001). The only generally accepted principle regarding the term, is that any issue regarding bilingualism is characterized by the involvement of two languages coming in contact with each other through communication or interaction. Yet, the degree to which this "contact" is defined, or how these two languages interact, remains a central source of disagreement and discord. In an effort to grapple with the broad nature of bilingualism, researchers have introduced varying typologies, largely to reflect an individual's grasp of two languages, such as the distinction between "non-proficient/proficient", "competent/incompetent", "partial/dual", "limited/full", "semi-balanced/balanced", "native/first/secondary", "minority/dominant", and so on (Lambert, 38 1978; Cummins, 1981; Larter & Cheng, 1984; Baker, 1993). Over the years, mathematics researchers basing their studies on bilingual situations have tended to focus on diverse populations, or they have conducted their research in widely different contexts, making it difficult to generalize from one bilingual group to another (Barton, Fairhall, & Trinick, 1998). In general, the effect of bilingualism on mathematics has been widely discussed (Dawe, 1983; Cuevas, 1984; Leans, Clements, & del Campo, 1990; Clarkson, 1992; Clarkson & Galbraith, 1992; Clarkson & Thomas, 1993; Moschkovich, 1996, 1999; Clarkson & Dawe, 1997) and evaluated through testing and assessment tasks in mathematics (Cummins and Corson (1997) offers a broad review of existing bilingual programs in the world). The results of these assessment tests have always been presented as the bilingual students' "levels of performance" in either language or mathematics. A mathematical result of this nature is hardly a clear indication of a student's mathematical understanding, since the student's difficulty has typically been attributed to a lack of language understanding rather than a lack of mathematical understanding (Whang, 1996; Fasi, 1999). Furthermore, some researchers, such as Fasi (1999) and Cuevas (1984), have argued that the use of assessment in a language that the students do not understand is inappropriate, especially when assessment is applied to bilingual students in a minority language environment. As a result, students' successful results on previous researchers' assessments cannot be interpreted as definitive proof of their understanding of the mathematics, a fact that lays the groundwork for a study that can more clearly and conclusively demonstrate the nature of the bilingual students' mathematical understanding. In the past, most bilingual studies in mathematics education have been largely influenced by Cummins' (1976, 1977, 1978, 1979, 1981) work, particularly in relation to his theoretical hypotheses on language acquisition and mathematical achievement. Previous studies used Cummins' hypotheses to demonstrate that bilingual students' competency in their first language 39 is significant for mastering both the second language, and the conceptual operations and functions in mathematics (Cummins, 1979; Dawe, 1983; Brodie, 1989; Clarkson, 1992; Clarkson & Dawe, 1997; Setati, 1998; Fasi, 1999). Various studies have also demonstrated the importance of bilingual students' level of competence in their second language, which is usually the language used for instruction (Cummins, 1979; Dawe, 1983; Cuevas, 1984; Leans, Clements, & del Campo, 1990; Clarkson & Galbraith, 1992; Clarkson & Thomas, 1993; Fasi, 1999). Other important research includes educational studies that report a sociological, linguistic, and psychological interconnection between one's bilingual status and one's ability to achieve mathematical understanding (Paivio & Desrochers, 1980). In any bilingual situation, each level of analysis in the language of the learner, the language of the teacher, and the language of mathematics (Austin & Howson, 1979) is concerned with at least two other layers - one for each of the natural languages involved. Likewise, in a bilingual situation, the language of mathematics can be examined by the way the mathematics is written, symbolized, expressed, taught, or learned in two different sets of vernaculars. As researchers to date have discovered, the complexities involved with bilingual learning illustrate the need to investigate the role of language in mathematical understanding (Brown, 2001). In fact, researchers and mathematics educators interested in studying bilingual situations have already been engaged in studying the relationship between natural language(s) and students' mathematical understanding (e.g. Dale & Cuevas, 1987; Ellerton, 1989; Ellerton & Clarkson, 1996); a sign of the attempt to bridge the division between linguists and mathematics educators, a division Veel (1999) described earlier. Nevertheless, depending on the researcher or educator's philosophical stance and his or her methodological approach, a range of perspectives abound between the two disciplines (Austin & Howson, 1979; Durkin, 1991), perspectives that 40 will ultimately affect the research findings and implications. At the same time, evidence of the crucial link between language and mathematics suggests the potential for further research on this topic, particularly in the way such link relates to how bilingual students can best learn, understand, or be taught mathematics with or without the use of language(s). Another meaningful challenge for mathematics educators and linguists relates to the development of new mathematical registers, tailored to be understood in a particular vernacular. Barton, Fairhall, and Trinick (1998), for instance, talk about how the development of a new Maori mathematical vocabulary during the past 20 years or so, a case that now demonstrates unforeseen semantic problems for new Maori speakers in mathematics education. In the Maori culture, and within other bilingual communities, both linguists and mathematics educators will inevitably face challenges in teaching, because "languages differ in their meanings, and in their structure and vocabulary, they may also differ in their paths towards mathematics, and in the ways in which mathematical concepts can most effectively be taught" (Halliday, 1978, p. 204). As a result, bilingual speakers, learners, and even teachers and educators, must pay attention to the unique characteristics of a particular language. Halliday's (1978) semiotic approach raises this important point: namely, the need to ask which mathematical idea or concept is most easily conveyed when the medium of teaching or learning mathematics is in this or that particular language. The significance of such a perspective is reflected in Jones' (1982) suggestion, that what one needs is knowledge of the aspects of that particular language that impinge directly on the learning or understanding of mathematics. Thus, both linguists and mathematics educators can learn and better understand each other if they jointly investigate the complexity of the process of learning and understanding mathematics through different languages (Austin & Howson, 1979). 41 The advantages of being bilingual have been confirmed in many studies conducted in various contexts and bilingual situations (Dawe, 1983; Clarkson, 1991, 1992; Secada, 1992; Stephens et al., 1993). At the same time, the findings of this group of researchers suggest that the language level of competence is vitally important to the bilingual students' ability to construct images, and consequently, their ability to ascribe meanings. For second-language learners, "meaningful understanding of the language, which is a direct result of the level of competence and proficiency of the learner in that language, is as important in learning mathematics as the mathematics content itself (Fasi, 1999, p. 60). The level of competence of the bilingual student in both languages has been the key to his or her advantage in learning mathematics. This concept is known as Cummins' (1979) "Threshold Hypothesis", which states that bilingual students must attain certain threshold levels of linguistic competence in both their languages in order to avoid cognitive disadvantages. In other words, the bilingual students' level of competence in both languages mediates the effects of cognition on their mathematical thinking. Moreover, Cummins (1981) proposes a second hypothesis called the "Developmental Interdependence Hypothesis", in which bilingual students' level of competence in the second language is described as dependent on their competence in their first language. The debate about the cognitive effect of bilingualism on bilingual students' mathematical thinking has involved not only theoretical issues, but sociological and political issues as well. For example, social and political issues are involved in the French Immersion Program pioneered, praised, and supported in Canada (Cummins, 2000), while most states in the United States do not support its philosophy and benefits (Collins, 1997; Baker, 2001). Baker (2001) reported that bilingual programs in California were outlawed, and that public opinion remained against such programs, owing to the perception that bilingual programs failed to foster social issues such as "integration". 42 In favour of bilingual education, the Maori education in New Zealand offers an example of another bilingual program that recognizes the benefits of bilingualism (Barton & Fairhall, 1995; Barton, Fairhall, & Trinick, 1995a, 1995b). A report by Ohia, Moloney, and Knight (1990) found "no evidence at all of Maori students being disadvantaged mathematically by being a bilingual unit" (cited in Fasi, 1999, p. 58), findings which, in fact, led to the development of the Maori's mathematical language. However, Barton, Fairhall, and Trinick (1998) presented a different scenario in which bilingual students' "thought patterns" were said to be dependent on language. In their view, i f new Maori speakers did not develop both languages equally, they were in danger of having only one conceptual pathway because "true bilinguals have two conceptual pathways available" (Barton, Fairhall, & Trinick, 1998, p.7). Based on this proposition, Fasi (1999) concluded that i f language changes, "distinctive thought patterns and conceptual pathways will also change" (p. 59). This line of thought resonated with the constructivists' view expressed by von Glasersfeld (1995), in which different languages determine different conceptualizations, in such a way that switching to another language may require "another way of seeing, feeling, and ultimately another way of conceptualizing experience" (p. 3). von Glasersfeld's view was reflected in Strevens' (1974) report, in which Strevens noted that there was a "major difference in mental preparation for mathematics learning" among different languages (cited in Fasi, 1999). Other useful observations can be drawn from Adler's (1995, 1997, 1998, 1999) extensive research on multilingual issues in South Africa. Her studies were based on "sociocultural theory of mind", and some of her findings support Cummins' hypotheses that bilingualism does not impede mathematical learning. Cummins (1976, 1986) and Ben-Zeev (1977) had pioneered the idea that mathematics education benefited from bilingualism because it required educators (and learners) to construct more meaningful terms. The benefits of bilingualism were also based on 43 the underlying assumption that knowing two languages opened one to more experiences, and as a result, contributed to one's mathematical knowledge. Fradd (1982) goes further to add that bilingualism has had positive effects on cognitive growth and divergent thinking; both of which are critical aspects in the individual's mathematical experience. Others, such as Coffman and Cuevas (1979), Lean, Clements, and del Campo (1990), and Clarkson (1992) reported a higher performance for bilingual over monolingual students in (test) assessment tasks. Pimm (1987) and Cuevas (1984) concluded that an understanding of how the second language is learned might contribute to how one understands mathematics, i f mathematics can indeed be considered a communicative language. However, those who concern themselves (e.g. Andersson, 1977; Ransdell & Fischler, 1987) with the negative effects of bilingualism often direct their concerns toward the bilingual students' deficiencies or incompetence with the language. They noted, in particular, that some of the linguistic disadvantages of bilingualism include the differences in lexicons and grammatical structures between two different languages (Austin & Howson, 1979). Skutnabb-Kangas (1986) discusses conflicting ideas about minority education. Secada's (1992) extensive overview of bilingual studies in mathematics pointed to reports in which the low level of proficiency in English as a second language related negatively to achievement in mathematics. Such findings, of course, coincide with Cummins' "Threshold Theory", in which limited competence in both languages resulted in negative cognitive outcomes (Baker, 2001). Although there are still debates about what level of bilingualism results in positive or negative effects for students, Grosjean (1982) proposed that bilingualism may have no major effect at all - either positive or negative -on the cognitive and intellectual development of children, a proposition that will be addressed further in this study. 44 From the perspective of researchers chronicling the downside of bilingual education, Adler (1995, 1998) notes a "paradox" for multilingual mathematics teachers, based on the amount of time they spend in the mathematics classroom teaching language, rather than teaching mathematics content. Bilingual students (of the Tongan-type) face an equivalent challenge by having to learn a second language while learning mathematics at the same time. As a result, these bilingual students misunderstand the mathematical ideas, or over-emphasize vocabulary learning at the expense of mathematics learning (Adler, 1998). This problem is not surprising, because almost half of the difficulties in solving mathematics word problems are language-based (Whang, 1996). However, Adler (1998) claims that the "dilemma of mediation" or the pedagogical tension between teaching mathematics, and teaching the language in which mathematics is delivered, cannot be resolved. As social constructivists claim, bilingual educators must also be cognizant of the cultural aspects of the languages involved (Smith, 2000). Crawford (1990), in a study of Australian aboriginal learners, acknowledged that many learning difficulties may be attributed not so much to a lack of language proficiency, but more to the cultural processes associated with the language. Brodie (1989) previously noted that the desirability of second-language learning in mathematics depended upon many factors, including: the emphasis on maintenance of first-language skills, the acknowledgement of students' natural modes of thought as determined by their first language and home culture, and the interaction of these factors with concept formation and mathematical symbolism. However, Adler (1995) pointed out that most studies are psycho-linguistic in nature, and that they do not examine the dynamics of bilingualism and mathematics in classroom settings. She further contends that the dynamics of teaching and learning mathematics in bilingual classrooms is about the "constant interplay in the cultural processes that constitute school mathematics learning between three analytically separable processes: proficiency in the 45 language of learning, access to the mathematics registers, and the social diversity and relations" of students (Adler, 1995, p. 265). 2.6 The Notion of Language Switching The process of switching between two or more languages is commonly known in studies of linguistics as "code switching". Ever since the concept of code switching appeared in the academic spotlight, researchers have focused on creating a number of related definitions and sub-categories, including the development of a special volume dedicated to the art of code switching (Eastman, 1992), along with other related literature and phenomena on the topic (Jacobson, 1998). As Halliday (1978) implied, the notion of code switching resides in the domain of language, not mathematics; a view shared by Lakoff and Nunez (2000). For that reason, it is important to consider first how code (or language) switching is interpreted in the field of linguistics, before examining how it is currently perceived, defined, and used in mathematics education. While researchers have investigated code switching from a variety of perspectives, most notably for what Macswan (1999) identifies as its sociological and grammatical properties, its sociolinguistic dimension underscores the social nature of language use among code switchers. In the early 1970s, a few published linguistic studies on code switching already existed, but Myers-Scotton (1992) cites and credits the work of Blom and Grumperz (1972) for exposing and promoting the sociolinguistic dimensions of code switching. Blom and Grumperz' (1972) study investigates code switching in the dialects of Norwegian in Hemnesberget, a Norwegian fishing village. Dialects, which are not considered types of code switching, are just "different ways of 46 saying the same thing and tend to differ in phonetics, phonology, lexicogrammar but not semantics" (Halliday, 1978, p. 35). Later, in 1982, Gumperz revised his model of code switching to cover its "situational", "metaphorical", and "conversational" aspects. Gumperz's (1982) study sparked great interest among linguists, who then went on to focus on various important features of code switching, such as its functional nature. The semantics issue associated with the term "code switching" poses a similar problem for linguists, because the term "understanding" defies easy definition for mathematics educators. Linguistic scholars are rarely in mutual agreement over which term to use at any given time. Halliday (1978), for example, refers to code switching as a particular form of shifting between languages that is actualized as a process within the individual. In this type of "code shift", Halliday notes, a code switcher "moves from one code to another, and back, more or less rapidly, in the course of daily life, and often in the course of a single sentence" (1978, p. 65). According to Halliday's definition (1978), Gumperz' (1971) notion of a "code potential" refers to the individual's "verbal repertoire". Furthermore, Jacobson (1998) defines code switching as "a structured mechanism of selection of two or more languages in the construction of sentences" (p. 1). Myers-Scotton (1998), on the other hand, describes code switching slightly differently, as a product of how a bilingual individual produces monolingual utterances in either language, although the individual may speak fluently in only one language. In a more general way, Milroy and Muysken (1995) define code switching as "the alternative use by bilinguals of two or more languages in the same conversations" (p. 7). It appears the key debate for many linguists revolves around the difficulty in identifying the role each language plays in the act of switching. Most linguists would agree (e.g. Poplack, 1980; Joshi, 1985; Myers-Scotton, 1993) that in any particular bilingual utterance, one language 47 dominates over the other. Myers-Scotton (1993) identifies a measure for this "dominance" to be the language with more morphemes - the smallest linguistic unit that has meaning or grammatical function. However, Sankoff and Poplack (1981) describe dominance in terms of which language can be attributed with the majority of phonological and morphological features of discourse. In either case, a language switching involves a "base language" - the dominant language - while the other will now be called the "embedded language" (also known as "donor language"). The embedded language refers to the language judged to be used in code switching, while the matrix language is the language code switching is judged to be coming from, and is recognized as a concept that needs meaningful clarification (Boechoten, 1998). However, the derivations and various definitions of categories and sub-categories of code switching have provoked further confusion among educators. One of these sub-categories is the notion of "code-mixing", which Kachru (1978) defines, as "the relatively unconscious integration of elements from the donor language into the base language to form a single composite code" (cited in Gibbons, 1987). To add to the complexity of this topic, other researchers apply a variety of terminology in connection with the concept of code or language switching. Grosjean (1982, 1985), for instance, describes how a bilingual individual can have two language modes: "monolingual language mode", when only one language is used while the other is "deactivated", and "bilingual language mode", where both languages are "activated". Thus, for Grosjean, the word "mode" defines a type of mechanical, linguistic function when one or both languages are used. This traditional view recalls the nature of the "Balance Theory", although Cummins (1979) calls it the "Separate Underlying Proficiency" (SUP) model (in contrast to his "Common Underlying Proficiency" model (CUP)). As Baker (2001) argues, the traditional balance theory is inappropriate because languages operate, as Cummins' (1979) model suggests, from the same central operating system (see Figure 1). 48 Figure 1: Contrasting the SUP and CUP models of bilingualism (Baker, 2001) For the purposes of this study, a crucial difference among linguists' definition of code switching comes from the use of the words "code" and "switching" themselves, because the term "code" varies so widely in its use by linguists and mathematics educators, and often suggests a technical connotation. In mathematics education, for instance, the term "code switching" has been used in different connotations. Zazkis' (2000) mathematical study on "code switching as a tool for learning mathematical language" distinguishes two different codes: everyday and formal mathematical language. Another contextual confusion is suggested by Setati's (2004) description of a switch in a multilingual classroom comprising three different codes: languages, registers, and discourse. Thus, for methodological and theoretical reasons, mathematics educators such as Dawe (1983), Clarkson (1992), and Celedon (1998), prefer to use the phrase "language switching", while others, such as Adler (1998), Setati (1998, 2002), and Moschkovich (1996) prefer to employ the linguistic terminology by using the phrase "code switching". Dawe's (1983) work with bilingual Punjabi, Mirpuri, Italian, and Jamaican students growing up in England provides evidence for the role of "language switching" in these students' mathematical understanding. In his examination of these bilingual students' mathematical, deductive reasoning abilities, Dawe (1983) noticed two processes of switching. He observed an "apparent switching 49 from one mode to the other during the reasoning process - often accompanied by a language switch as well" (p. 349). Dawe added that the complex nature of such a phenomenon makes it therefore very difficult to unravel or interpret. Gibbons (1987) defines switching as "either situationally determined alternation between codes, or as a relatively more conscious exploitation of the alternative values represented by the two codes, usually for social or rhetorical effect" (p. 75). ' Aipolo and Holmes (1990), in their study of Tongans living in New Zealand, offer a definition of code switching as a general term to "cover the use of two codes in one situation without pre-judging whether such behavior is conscious or deliberate" (p. 519). As well, Grosjean (1985) describes a "language switch" as a shift from one base language to another. When the base language is established, he explains, "language-mixing" can occur, in which the bilingual individual brings in the other language (the donor). This process then divides into two types: "code-switching" and "borrowing". "Code-switching" is characterized by a complete shift from one base language to another, while "borrowing" deals with adapting words or phrases into the base language. Grosjean (1985) also describes "code-changing" as a switch with at least a phrase or a sentence, while "code-mixing" - slightly different from his notion of "language-mixing" - involves inserting a single item (word) from one language into another. In addition, Jacobson (1998) brings similar or additional phrases to his list of switching terms, such as "code mixing", "code alternation", and "language mixing". Some of these labels are broken down into sub-categories, such as "intersentential" and "intrasentential" code switching, which are used to distinguish structured from non-structured switching (Macswan, 1999). The concept of "borrowing", in particular, turns out to be a concern for some researchers who have argued that borrowing must be carefully distinguished from code switching (Pfaff, 1979; 50 Sankoff & Poplack, 1981), although others find it difficult to separate the two concepts (Hill & Hil l , 1986). The degree to which bilingual speakers are aware of this process may differ with each borrowed item (Macswan, 1999). For example, Macswan (1999) explains how a monolingual English speaker might use the term "pork" without the slightest awareness that it was borrowed from the French (yet spelled "pore") during the Norman Conquest. But the phrase "tour de force" may be used with full awareness that the expression is of French origin. This phenomenon also provokes a discussion about what it means to be "bilingual", because, in some instances involving borrowing, for example, an English speaker using terms such as "pork" and "tour de force", is not in fact knowledgeable about the French language or culture. To examine the key concept of borrowing further, the difference between the use of the terms "pork" and "tour de force" can be explained by the fact that borrowed words are considered either "marked" (as unconventional) or "unmarked" (as conventional) - a distinction made useful by Myers-Scotton (1992). In using both terms, borrowing is seen as filling "lexical gaps" in which Myers-Scotton (1992) describes as the conventional form of borrowing taking place without conscious awareness. This distinction is further discussed in the next chapter to explain how Tongans borrow English words through a process called "Tonganisation", and how that act, in turn, integrates English words into the Tongan language. Ultimately, in view of all the described definitions, defining code or language switching appears to be a matter of theoretical or personal preference, rather than a defining characteristic for the research itself. Nonetheless, researchers, such as Eastman, (1992) and Boztepe (2003), have been led to conclude that the efforts to distinguish these terms are "doomed", based on the resulting confusion from all the defined categories and subcategories of code switching or language switching. 51 2.7 Language Switching in Mathematics Education In mathematics education, the notion of language switching has become a major topic of investigation within any bilingual (or multilingual) setting. What language switching is, what it entails, what factors cause it, and how it affects or re-presents students' mathematical thinking, are among the many questions that provoke the interest of researchers and educators. In citing work by Bishop (1979) and Lean and Clements (1981) in developing countries such as Papua New Guinea, Dawe (1983) has recognized the importance of both linguistic and cultural factors in students' preferred modes of thought. Dawe expressed concern about the sociolinguistic view regarding "language distance"(the disparity and mismatch between two pairs of languages) as an important factor in any bilingual study. Berry (1985) picked up on the same concern, and he explained that it might be easier for bilingual students to function effectively in a second language semantically and culturally closer to their native language, than a language considered remote. One reason for this phenomenon may relate to Cummins' (1981) view that students' native language has a strong influence on their cognitive processes. In addition, some studies have offered important information about the contextual and situational nature of language switching in mathematical discourse and cognitive activities. Moschkovich's (1996) study of Latino students in the United States and the nature of their mathematical and bilingual conversations recognizes how bilingual students switch languages between Spanish and English. Her study proposes a "situated model", in which language use and its relationship to mathematical learning depends on the learning situation. Although Moschkovich (1996) did not explicitly analyze language switching, she emphasizes the importance of the educational context in investigating the relationship between mathematical activity and the bilingual students' language choice. 52 Furthermore, Clarkson and Dawe's (1997) study of Vietnamese grade 4 students in Sydney and Melbourne, Australia, suggested the influence of the mathematical context and schooling environment upon these students' choice of language. These authors interviewed a sample of their participants to determine the central factors affecting their language switching. Clarkson and Dawe's study with these bilingual students mainly drew upon Cummins' theoretical framework. Among Clarkson and Dawe's findings was the view that the bilingual students' language choices were affected by competence in languages, semantic issues, and translation problems, affective responses, and memory factors. Clarkson and Dawe (1997) concluded that investigating language switching is very difficult, not because these two researchers lacked adequate methodological approaches, but because the nature of language switching is complex and "messy". In other research, Qi (1998) investigated language switching for Chinese learners in Canada involving mathematical problem-solving tasks in a second language (L2), such as English. Noting the sociolinguistic view of code switching, Qi offered a psycholinguistic definition of language switching as "the act of switching from L2 to first language (LI) as the language of thinking in the cognitive process of a bilingual person in an L2 composing task" (p. 414). He used the word "composing" to refer to the thinking process involved in the task. Qi (1998) suggested that the demanded levels of knowledge might be the basic variable influencing language switching and the choice of language use in composing tasks. Adler (1995, 1998) and Setati (1998, 2002, 2004) have identified various cues associated with the relationship between language switching and mathematical activities in multilingual settings. Setati (1998) also categorized types of switching in the classroom, although she defined each in 53 terms of the teachers' purposeful use of language for "reformulation", for "content of activity", and for "translation". Adler (1998) described such classroom-teacher situations in terms of a "dilemma of code-switching": a dilemma of access to both mathematical and language meaning. To focus on the Tongan setting used in this study, Fasi's (1999) study with bilingual students in Tonga attempted to categorize language switching into four main groups. Each classification was defined based on the purpose of the switch. For example, "substitution" was used as a convenient way for Tongan students to benefit from the precision of the English words rather than using lengthy Tongan explanations. "Explanatory" involved phrases aimed at relating ideas, defining and clarifying tasks, and explaining mathematical work. "Reformulation" involved paraphrasing the given information, and "repetition" referred to direct translation between the two languages. This study intends to expand upon and clarify further Fasi's results on the types of language switching that take place in a learning environment, while at the same time, use the results of specific classroom tasks to answer the research question. Given the varying nature of research, both on bilingualism and the notion of language switching, it is important to clarify how each of these concepts are defined in the context of this study. In order to define "bilingualism" and "language switching", it must first be clear that throughout this thesis, the word "language" refers to the natural or verbal language in the sense of speech (Halliday, 1978). Any other use will be specifically defined where needed, as was the case with the term "mathematical language", defined in Section 2.4. In this thesis, the term "bilingualism" takes on Cummins' (1981) broad definition as "the production and/or comprehension of two languages by the same individual" (p. iii). While bilingual proficiency can be described at different levels, Cummins' definition allows one to talk 54 about any individual's use of two languages within the course of a verbal exchange without being concerned with how much or how little the individual comprehends in either language. Thus, in this thesis, a "bilingual" individual refers to any individual who is engaged in any production and/or comprehension of two languages. Such production can be expressed verbally through "language switching", which refers to any alternation between two languages, ranging from mixing one or more words to changing in mid-sentence, or changing within larger blocks of words or clauses (Hoffman, 1999, cited in Baker, 1993). The decision to employ the term "language switching", rather than "code switching", aims to preserve the contextual and technical clarity of this study. A look at the bilingual context of this study and the study's theoretical framework is offered in the next chapter. CHAPTER 3: CONTEXT and THEORETICAL F R A M E W O R K 55 3.1 Introduction This chapter introduces the bilingual situation in Tonga and explores the role of language in the Tongan mathematics education program. The account of the Tongan context is based largely on a comprehensive study conducted in Tonga by Fasi (1999) on the effect of bilingualism on Tongan students' mathematical achievement.7 A brief historical account of the development and involvement of the two official languages in Tonga provided a basis for understanding the roles of these languages within the Tongan bilingual context and its mathematics education program. 3.2 The Tongan Bilingual Context 3.2.1 The First European Impact and its Effect on the Tongan Language Tonga is one of the Polynesian islands in the South Pacific. Prior to European settlement, all Polynesian languages were considered only as oral languages; there were no written forms for any of these languages (Fasi, 1999). "Tongan", the native and homogeneous language of the Tongans, is one of the ancient Polynesian languages belonging to the Austronesian family of languages known as "Oceanic". While the first visits from the outside world were recorded by famous European navigators of the 17th and 18th centuries, it was Christian missionaries who first 7 Fiefia's (1981) article also provided a valuable source on "Education in Tonga". introduced Western education to Tonga, and later developed a written form of the Tongan language. 56 British missionaries were the first Christian missionaries to arrive in Tonga in 1814, followed by French Catholic priests about a decade later. Both of these missionary groups have had a profound effect on the development of the Tongan written language. Exposure to Christian religious reading - which enabled the native people to read the Bible in their own language -was the main purpose for teaching the Tongan written language prior to the introduction of Western education. The missionaries were less concerned with introducing an English mathematical language because the native people of Tonga had traditionally used their own mathematical system for counting, measurement, and categorization of objects, which they included in their everyday activities. Table A and Table B, below, illustrate examples of the counting and measurement systems the Tongans developed and still use today. Number of Coconuts Number in Tongan English Translation 1 Fo' i niu 'e taha 1 coconut 2 Taua'i niu 'e taha 1 pair of coconuts 4 Taua'i niu 'e ua 2 pairs of coconuts 16 Taua'i niu 'e valu 8 pairs of coconuts 20 Tekau (or kau niu) 1 score 22 Tekau mo e taua'i niu 'e taha 1 score and a pair 40 uangakau (or kau niu 'e ua) 2 scores 100 nimangakau (or kau niu 'e nima) 5 scores 200 Tefua 10 scores 2000 Teau (or niu 'e teau) 100 scores 4000 uangeua (or niu 'e uangeau) 200 scores Table A: Tongan counting system for coconuts (niu) 57 Name of piece of "tapa" Length in "langanga" Approx. metric length (m) Fola'osi 4 02.4 Fatuua 8 04.8 Hongofulu'i ngatu ("tenth" ngatu) 10 06.0 Uofulu'i ngatu ("twentieth" ngatu) 20 12.0 Lautolu ("multiple of three") 30 18.0 Laufa ("multiple of four") 40 24.0 Launima ("multiple of five") 50 30.0 Lautefuhi 100 60.0 Table B: Tongan discrete measurements for the length of "tapa" cloth Under the influence of Christian missionaries and Catholic priests, and beginning with the introduction of the first Tongan orthography, the Tongan language continued to evolve along with the development of many subject areas, such as mathematics and science, and through growing interaction with the outside world. It was during this period that the missionaries realized that for some English words, no equivalent Tongan words existed. As a result, "borrowing" and "Tonganising" were the most common ways of incorporating English words into the Tongan language, and, in particular, to fill the lexical gaps in the Tongan language. "Tonganisation" was and is still a common and convenient way of creating equivalent Tongan words from English. This process is also known as a form of "conventionalization", in which English words are phonetically translated into Tongan and are generally used and accepted by the public in talking about everyday activities (Taumoefolau, 2004).9 As Fasi (1999) noted, some of these "imported" English words have become part of the everyday Tongan language, and no attempt has ever been made to examine more closely their necessity or the existence of 8 Churchward (1953) discusses how special terms and special numerals in Tongan are used, with "more or less regularity" (because sometimes numerals are also used), when counting coconuts, yams, fish, and other objects. 9 Lotherington (1997) describes such forms of conventionalization as "pidginized English", although this type of languaging is far more evident with Papua New Guineans than with Tongans. 58 corresponding Tongan words, if any had been or are still in existence. This practice of Tonganising foreign words is an unsatisfactory alternative to coining new words in Tongan, because no official rule or mechanism has ever been developed to regulate the importation or integration of foreign words into the Tongan language (Fasi, 1999). Consequently, Tonganising has created some social difficulties, particularly in the formal and informal uses of the Tongan language. For example, official government publications were often written in Tonganised words that were not connected to the common Tonganised language (e.g. stamp: official use, "sitamipa", versus common use, "sitapa"); various local poets and musicians took pride in the "elegant" Tonganisation of English words in their compositions (e.g. Britain: common Tonganised word, "Pilitania", versus the more elegant Tonganised word, "Polata'ane"); and various educational programs (government versus church schools, for instance) had their own way of Tonganising English words (e.g. page: government translation, "peesi", versus Catholic translation, "pasina"). In mathematics education, there were also complications and confusion, and as Fasi (1999) noted, the teachers tended to believe that if the borrowed words were more English-sounding in the Tongan version, the students would find it easier to learn the equivalent English words. For instance, the Tonganising words, "sikuea" and "sifia" are said to evoke in many Tongan bilingual students the same images associated with their concepts of "square" and "sphere", respectively (Fasi, 1999). (Further discussion in Chapter 5 will consider such Tonganised English mathematical terms as "borrowed" terms, if there were no equivalent words in Tongan.) In an attempt to clarify the situation, bilingual Tongan-English dictionaries were developed in Tonga. Churchward (1953), under the commission of the local government, wrote the first official grammar guide and dictionary of the Tongan language, but published only the English-Tongan and Tongan-English format in translation form, rather than offering a dictionary with 59 definitions of Tongan words in Tongan. Decades later, as a result of educational system changes, as well as ongoing developments at the social, political, and economical level, the existing dictionary had to be expanded to account for the evolution of the Tongan language. A "functional" bilingual dictionary was therefore developed in 1977, based on the most commonly used Tongan words (Fasi, 1999). But, as Fasi (1999) noted, most Tongans acquired English only through the formal school system, and could not be considered functional in English until they were at the secondary level. Fasi then observed: "It is rather odd that in order to find the meaning of a word in Tongan, one has to learn first to read in a foreign language" (1999, p. 12). Finally, beginning in 2004, a new Tongan Dictionary Project was commissioned for the development of the first monolingual Tongan-Tongan dictionary, to be completed by 2007, with a projected goal of collecting at least 20,000 Tongan words (Taumoefolau, 2004). This project has already faced many challenges, including having to avoid inventing "pure" Tongan words. Melenaite Taumoefolau, the head of the current project, announced that the project would include technical words (such as mathematics terms) "that have been conventionalized... and now have a public meaning" (Taumoefolau, 2004). Everyday Tonganised words such as "peteni" (pattern), "sitepu" (step), and "poloka" (block) came to be used in the everyday Tongan language as a result of this conventionalization process. So far, only about 6,000 words in total have been accumulated toward this project (Taumoefolau, 2004). 3.2.2 Language in Tongan Education Today, the indigenous Tongan language remains the country's national and homogeneous language. Tongan and English are the country's two official languages. English, however, is the recommended (or expected) medium of instruction at the secondary and post-secondary school 60 level for all subjects except Tongan Studies, and Religious Studies in most church schools (which cover the majority of Tongan secondary schools). Fasi (1999) noted that less than three decades ago, the Ministry of Education launched an integrated language program aimed at developing equal bilingual skills in Tongan and English for all Tongan students. The Ministry's objective was to make all Tongan students "competent" in both languages, and to ensure that both languages were used effectively, in and out of school. This initiative reflected the Ministry's dual effort - a dilemma similar to the Maori situation discussed in the previous chapter - to maintain the native language amid the onslaught of globalization, while improving English proficiency at the same time. From the establishment of the Education Act10 in 1947, the Tongan educational system, therefore, has evolved into the current language policy, summarized as follows: i . Both Tongan and English are official languages, and used in public examinations. i i . At the primary school level, both languages are used as the media of instruction with more Tongan at the lower levels (90 per cent at Class l ) u . The English component gradually increases to 50 per cent at Classes 4 to 6 (see graph in Graph A below). i i i . Both Tongan and English are required (or "compulsory") subjects up to Form 5. iv. The medium of instruction at the secondary level is English, except in Tongan and Religious Studies lessons. 1 0 In 1947, an Educational Act was established in which the primary schools would provide a general education in Tongan, the middle schools, a general elementary education using English as a medium of instruction, and the high schools, a general secondary education, mainly in English (Fasi, 1999). '' Elementary school starts at Class 1 to 6 (the Canadian equivalents of grades 1 to 6), and then at the secondary school level, it starts at Form 1 to 6, (the Canadian equivalents of grades 7 to 12). The curriculum is not the same as in North America, but closely follows the New Zealand educational system, although the South Pacific Board of Education centred in Fiji now governs the curriculum. 61 100 Percentage of Language Time 50 1 2 3 4 5 6 CLASS (grade) Graph A: Percentage of language time used for Tongan and English from Grade 1 to Grade 6 In many secondary classrooms in Tonga, both Tongan and English are used in various switching combinations. It is widely recognized that the teacher and the students move back and forth freely between the two languages (Fasi, 1999). While language switching is a common arrangement, Fasi (1999) claims no research on language switching or bilingualism has taken place in Tonga prior to his study, nor has there been an investigation into the most effective classroom methods to be adopted in such a bilingual situation. Citing the Minister of Education's Annual Report for 1995, Fasi (1999) further reports that the Ministry was confident the program would succeed by achieving competence in both languages by the end of Class 6 (grade 6), and that a smooth transition might also be achieved as the students moved through the middle or junior high school levels. At the secondary level, the medium of instruction is English, and all mathematical texts are in English, particularly texts from other countries, such as New Zealand, that are frequently used at Spoken/Written Tongan Spoken/Written English 62 the senior levels. But the effects of the current education policy are most significant between Form 1 and Form 3 (grade 7 to grade 9), when the Tongan bilingual students begin an "intensive exposure" to their second language, English. This study focuses, in particular, on this junior high-school period of the Tongan students' education, a time in which different levels of bilingualism are said to influence students' cognitive growth and functioning (Cummins, 1976). Moreover, when the language of instruction changes from using primarily Tongan at the elementary school level, to using mostly English at the secondary school level, the students are said to encounter twice, or even three times, as many learning problems as those who grew up in an English-speaking environment (Fasi, 1999). The difficulties escalate when the learner's first language is not the language of instruction, which is virtually always the case in Tonga. 3.2.3 Mathematics Language in Tongan Mathematics Education The number system used in Tongan education today is very similar to the system used in Great Britain and other western countries, which evolved from traditional base-ten counting systems. In the late 1970s, Tongan educators developed and distributed a mathematics textbook using "invented" words for certain concepts and mathematical terms. As Fasi (1999) observed, "the work was so full of invented words that teachers and those who were interested in mathematics questioned whether they were teaching mathematics or a new language" (p. 23). For instance, Fasi explained how the word, '"ulutefua" was created out of the native word stock and used for the corresponding mathematical word, "prime". '"Ulutefua" in Tongan means an only child, with the prefix '"ulu" referring to a head, and "tefua" as the root Tongan word of the number "one". Hence, '"ulutefua" literally means "one head". "Nomipa" was then used as a direct phonetic assimilation of the word, "number". So "nomipa 'ulutefua" was and is rarely used in the classroom to mean "a number that stands alone", the translation used in Tongan for "prime 63 number". In other words, the heavy English influence on the Tongan language has created double meanings for many words, creating potential confusion for students trying to understand the terminology - well before they can master the mathematical concepts themselves. As a result of numerous linguistic misunderstandings, Fasi conceded that teachers and students were so confused, that mathematics was neither taught nor learned, and therefore, the mathematics-textbook project was soon abandoned. Consequently, Fasi argued that when learners enter school, the main problem they encounter is not the mathematical symbols and systems, but the language used in the classroom. The proponents of the currently commissioned dictionary project (mentioned earlier in Section 3.2.1) in Tonga recognized this problem by avoiding the creation of a "Tongan purist language dictionary", as some people would have preferred (Taumoefolau, 2004). As the Tongan language has evolved, other concessions have been made to incorporate the influence of English on the.native vocabulary. For instance, the difficulties in finding a "real" Tongan mathematical vocabulary were in most cases resolved by "transliteration" (another term for conventionalization) of the English terms. The transliteration approach is still widely used today both in classroom teaching and in curriculum materials. Because English is the medium of instruction at the secondary school level, the mismatch between the language of mathematics and ordinary English plays a central role in the Tongan bilingual students' (mis) understanding of written mathematics texts and other resources. Fasi (1999) discusses an example of this scenario using the Tongan word for circle, "fuapotopoto", which in Tongan, encompasses all round shapes such as oval, ellipse, and even sphere. As a result of this confusion, transliterations were then used, and hence the notions of "circle" and "sphere" are commonly taught nowadays in the 64 mathematics classroom using the phonemic Tonganised words, "siakale" and "sifia", respectively. Another unique feature of the Tongan bilingual situation is the students' everyday use of the mathematical terms in both Tongan and English. In this instance, the students' prior learning and everyday experiences become sources of confusion when words are found to have different meanings in mathematical contexts. For instance, the Tonganised word "sikuea" (square) takes on various mathematical connotations - an area, a number, or a numerical property. However, the issue with finding proper Tongan words for many mathematical terms and concepts is complicated, not only by the lack of specific terms in the current Tongan language, but also, as Fasi (1999) argued, by the absence of many Westernized concepts in the life of the Tongan people. Concepts such as "probability", "negative numbers", and "absolute value", for example, have no equivalent functions in the activities of the Tongan people. With regard to the concept of probability, for example, one teacher in this study recounted how difficult it is for him to teach and distinguish between the meanings of non-equivalent terms, such as "very likely", "probable", and "almost certain", when these concepts have to be expressed and explained in Tongan. In 1995, the Ministry of Education, in co-operation with the Australian government, launched a new unified mathematics program for Class 1 to 5 (grades 1 to 5) for phased implementation, beginning a year later, in 1996. This new program recognized the important role language played in the learning and teaching of mathematics. This step was significant because the emphasis on previous syllabuses was mainly on language and mathematics (Fasi, 1999). One of the aims of mathematics education, specified in this new syllabus, was that "students will develop appropriate vocabulary and language forms, first in Tongan, and later in English, for the 65 effective communication of mathematical ideas and experiences" (Tonga Ministry of Education, 1995a, p. 4). The early stages of secondary education, Form 1 to 3 (grades 7 to 9), are the most difficult for most students whose English proficiency is too underdeveloped for them to cope with the language demands of the classroom. To attend to this problem, the new program reviewed, revised, and recommended the following processes as essential for the teaching and learning of mathematics (Fasi 1999, p.28): (i) Appropriate language should be developed and used by teachers and students during mathematical activities (ii) Teachers must be familiar with accepted language patterns for different mathematical processes, and different age groups: a. In Classes 1 to 4 (grades 1 to 4), all teaching should be in Tongan. b. In Classes 5 to 6 (grades 5 to 6), teachers should teach in Tongan but should give the English words and pronunciation for important concepts, and students should write and say the words. c. In Forms lto 2 (grades 7 to 8), the teaching should be in English, but concepts and ideas should be explained in Tongan and in English. d. In Forms 3 to 5 (grades 9 to 11), all teaching should be in English e. From Form 1 (grade 7) onward, an explanation in English that is not understood by students should be repeated in Tongan, and then repeated again in English. (iii) Students should use both appropriate oral and written language to gain meaning from their mathematical learning experiences. (iv) Teachers and students in all learning situations should use exemplary language modeling (in both English and Tongan). The terms "appropriate language" and "accepted language patterns" indicate recognition of the existence and common use of language switching by teachers and students in the classrooms. 66 3.3 The Theoretical Framework 3.3.1 Introduction In attempting to answer the research question first presented in Chapter 1, a suitable theoretical framework must be used to explain the nature of bilingual students' growth of mathematical understanding. While language switching may be identified through its explicit form of verbal expressions, the growth of mathematical understanding is not that explicit. The evidence of any growth of mathematical understanding is often observed as a mixture of both verbal and non-verbal actions. The Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding offers such a language as a way of examining understanding in action, of interpreting the observed cues, and of accounting for any evidence of growth of understanding. The Pirie-Kieren theory was conceptualized and developed within a monolingual context, and has not been previously employed to investigate growth of mathematical understanding in the presence of two (or more) languages. This theory is nonetheless appropriate because it is a theory for the growth of mathematical understanding of a specific topic, by a specific person (that is, a group or individual), over time. A "specific person" implies the independent nature of the theory as a theoretical lens for observing growth of mathematical understanding, irrespective of what language(s) an individual or group of individuals possess. The Pirie-Kieren theory looks at how a learner gets to a particular mode of understanding through observations of the actions and language the learner uses, and how he or she comes to understand particular concepts in mathematics. A look at the theory's history, philosophy, and practice may first be necessary to understand its roles in the interpretation process. Further discussion of the purpose and appropriateness of employing the Pirie-Kieren theory is offered in Section 3.4. 67 3.3.2 The Development of the Pirie-Kieren Theory Since 1987, Susan Pirie and Tom Kieren have been exploring, and attempting to define, the complex notion of "understanding". Initially, Kieren (1988) worked on the area of elementary fractions, while Pirie (1987, 1988) was researching language in mathematics classrooms. Although Pirie and Kieren were working on different research areas in mathematics education, they were both interested in the notion of how one comes to understand anything at all (Pirie & Kieren, 1992a), rather than the behavioural ways of assessing what learners can and cannot do. Thus, instead of just looking at the notion of "understanding", Pirie and Kieren became more interested in investigating the "process of understanding". As a result, the Pirie-Kieren theory was developed, drawing on Pirie and Kieren's interests and personal and academic experiences, to look at learners' growth of mathematical understanding (Pirie, 2001, personal correspondence). In her early work on "understanding", Pirie (1988) found that a single category was inadequate to describe the complexity of a learner's understanding. She observed mathematical understanding as a "whole dynamic process and not as a single or multi-valued acquisition" (Pirie & Kieren, 1994, p. 165). In order to describe such understanding as a complex phenomenon, Pirie needed "an incisive way of viewing the whole process of gaining understanding" (Pirie & Kieren, 1989, p. 7). This led to the development of the Pirie-Kieren theory, initially based on the constructivist view of understanding as a continuing process of reflecting and reorganizing one's conceptual structures (von Glasersfeld, 1987). 68 As the theory continued to evolve, and become increasingly complex in its language, Pirie and Kieren came to believe that although understanding was still the creation of the learner, other factors were viewed as inter-related within the space in which understanding was created (Varela, Thompson, & Rosch, 1991; Towers, Martin, & Pirie, 2000). The Pirie-Kieren theory therefore shares the enactivist view that observes learning and understanding as an interactive process (B. Davis, 1996; Kieren, Davis, Mason & Pirie, 1993). Citing B. Davis (1996), Martin (1999) describes this interactive process of understanding "not as a state to be achieved but as a dynamic and continuously unfolding phenomenon", a view that recognizes the inter-dependence of all the participants in any particular environment (p. 35). Because learning was seen as a dynamic process, Martin (1999) said, "the creation of the diagrammatic model for the growth of mathematical understanding was therefore intended to provide a way of depicting this process" (p. 34). By adding this enactivist perspective to their initial constructivist approach, Pirie and Kieren, through an ongoing development of their theory, characterized growth of understanding as a "whole, dynamic, levelled but non-linear, transcendently recursive process" (1991a, p. 1). The addition of the enactivist perspective does not change the nature of the Pirie-Kieren theory, but rather points out the evolving use of the language within the theory (Martin, 1999). Since the Pirie-Kieren theory was first developed, various other researchers and former students have worked to expand and explore it further (Kieren, Davis, Mason & Pirie, 1993; Kieren, Pirie, & Reid, 1994; Martin & Pirie, 1998; Towers, 1998; Martin, 1999; Towers, Martin, & Pirie, 2000; Thom, 2004). The Pirie-Kieren theory has indeed expanded its application in both the research and in the mathematics education community. This study provides such an application and offers a new dimension to the power of the theory through using it to observe bilingual students' growth of mathematical understanding. At the same time, some features of the Pirie-69 Kieren theory remain open for further elaboration - a discussion reserved for the final chapter of this study. 3.3J The Nature of the Pirie-Kieren Theory The Theory as "Levelled but Non-linear" Drawing on Vitale's (1988) work, Kieren and Pirie (1991a, 1991b) initially characterized mathematical understanding as "levelled" or "level-stepping", because each level or "mode" of understanding was not exactly the same as the previous one. However, the word "level" carries with it the connotation of being hierarchical in structure. For Kieren and Pirie, understanding does not grow in a linear, hierarchical fashion or through attaining a particular level of understanding. They therefore do not believe that teaching can be implemented to achieve understanding at a certain level before the teacher moves on to the next task. This issue gives rise to Pirie and Kieren's recent preference for using the more neutral and descriptive term, "layer" (Pirie & Kieren, 1994; Towers, Martin & Pirie, 2000; Pirie, 2001, personal correspondence), which not only refers to any modes of understanding, but also reflects the complexity of one's paths of thinking within the Pirie-Kieren framework. In the Pirie-Kieren theory, eight potential layers of understanding are possible, through either informal or formal actions, for a specific person and relating to a specific topic. Beginning with the innermost layer, these layers, described in detail in Section 3.3.4, are labeled: Primitive Knowing, Image Making, Image Having, Property Noticing, Formalising, Observing, Structuring, and Inventising (see diagrammatic model in Figure 2). These layers of "thinking sophistications" demonstrate the complexity of how learners come to understand mathematics, 70 rather than the sophistication or complexity of the mathematics involved. Each layer contains all previous layers (except for Primitive Knowing) and is embedded in all succeeding layers, thereby illustrating the fact that growth of understanding is "a non-linear, non-monotonic, process-in-action" (Kieren, Pirie, & Reid, 1994). Figure 2: The Pirie-Kieren model for the growth of mathematical understanding While the modes of understanding are described as layered, a path of the growth of understanding is observed to occur through a continuous back-and-forth movement through the modes of understanding, as illustrated later in the mapping process12. In following such a path, learners remember and construct new understanding, based on their current and previous knowledge (Kieren & Pirie, 1991a; Pirie & Kieren, 1991b; Pirie, Martin, & Kieren, 1996). Similarly, growth of understanding is observed to be "non-monotonic", because growth of 1 2 Pirie and Kieren use this mapping technique to trace the pathway of growth of understanding (see the illustration in Figure 2). By tracing this pathway, the non-linear nature of growth of understanding is visible in two-dimensional form (see the detailed discussion in Section 3.3.5). 71 understanding does not progress outward, in a linear fashion, from one layer to another; understanding does not grow in a strictly hierarchical fashion. The Theory as "Transcendently Recursive" Following the work of Maturana and Varela (1987), Kieren and Pirie (1991a, 1991b) see human knowledge of mathematics as "recursive" in nature since understanding at each layer is in some way defined in terms of itself ("self-similar"), yet each layer is not the same as the previous layer ("level-stepping"). Citing Maturana and Varela (1980), Kieren and Pirie (1991a) use the metaphor of "recursion" to mean that "the individual can distinguish between ideas seen as products of their own mental actions and those which appear to them to be driven by outside sources" (p. 81). Within this recursive structure, a learner is observed to always exhibit a different degree of understanding, in which each layer of understanding is contained within succeeding layers (Kieren & Pirie, 1991a). However, Kieren and Pirie (1991a) warned that mathematical understanding of a particular concept or topic cannot be fully attained at any particular layer; rather, it is an evolving process between layers of understanding. For instance, a learner who correctly applies his or her formal understanding of the quadratic formula does not necessary mean he or she fully understands the concept of quadratic functions or has attained a formal understanding of all quadratic properties. In addition, Kieren and Pirie (1991a) take Margenau's (1987) idea of growth of scientific constructs to describe new transcendent knowing, not as a simple extension of knowledge, but as learning compatible with the knowledge previously held. So the notion of "transcendently recursive" refers to the structure of the theory repeating itself. A learner can keep going back, but it is never the same experience, because every time he or she re-visits a layer (either inner or 72 outer), the learner is re-visiting it with all the new understanding that he or she built up at the "return-to" layer during the process of growth of understanding. In other words, suppose a learner is working at an outer layer, " X " . If the learner moves back to an inner layer, " Y " , his or her existing understanding at that inner layer, Y , somehow becomes "thicker", and not at the same understanding as the original activity at that layer. Even if there was no prior activity at the inner layer, Y , or if the learner goes on to create new understanding at Y , his or her current understanding at Y is now shaped by his or her outer-layer knowing at X . Similarly, if the learner moves out again to the outer layer X , where he or she started, the learner's existing understanding at that outer layer is now shaped by his or her thick inner-layer knowing at Y . Therefore the learner's understanding is no longer the same as when he or she started, because the new inner-layer action simply alters the outer-layer knowing, and vice versa. The term, "thicker" means the learner has extended his or her understanding of the concept - possibly incorrectly through the introduction of misconceptions - and that the new "transcendent knowing frees one from the actions of the prior knowing" (Pirie & Kieren, 1989, p. 8). The Theory as a "Whole" The actions at each layer of the Pirie-Kieren theory do not constitute a complete understanding, but are simply named parts of a whole complex phenomenon. However, the recursive nature of the Pirie-Kieren theory is a way of describing a complex phenomenon, in which the "whole" at any time is structurally similar to, but not reducible to its previous states (Kieren & Pirie, 1991a). The Pirie-Kieren model therefore uses mapping techniques (to be discussed in Section 3.3.5) to illustrate not just how the path of one's growth of understanding unfolds, but also one's thinking sophistication or understanding as a whole complex phenomenon, and how each of the layers are dependent on the other. 73 The notion of recursion described earlier (in Section is used as an attempt to understand a piece of mathematical knowledge or a problem-solving act as a "dynamic whole", which means that "one can understand a piece of mathematics many ways at once" (Pirie & Kieren, 1992b, p. 244). To see the nature of the theory as a dynamic whole refers to both a condition and consequence of recursion called "level-connectedness", which suggests that a learner can "drop" back to a previously developed layer of mathematical experience to re-construct the basis for a higher-layer experience. Because of this "flexibility", which respects movement out to any of the other layers, the Pirie-Kieren theory demonstrates that it is possible, therefore, for a learner to understand a piece of mathematics in many ways at once (Pirie & Kieren, 1992b). The Theory as a "Dynamical Process" In their early writings, Pirie and Kieren (1989, 1991b) used the word "dynamic" for their theory in the sense of a dynamic within a group relationship. The word "dynamical" is now preferred, because it encompasses and reflects the interactive nature of learning and understanding, which is continuously affected by the environment (Pirie, 2001, personal correspondence). Thus, one cannot simply declare that the Pirie-Kieren view of growth of mathematical understanding is "not static". The term, understanding, as it is portrayed in the Pirie-Kieren theory, refers to an ongoing process, and the process of re-organizing one's knowledge structures within a specific environment characterizes the Pirie-Kieren theory as dynamical. In developing the theory, Pirie and Kieren (1994) characterized understanding to be "a dynamic process, not an acquisition of categories of knowing" (p. 187). 74 Likewise, in using the Pirie-Kieren diagrammatic model, the important aspect in any mapping is the dynamical nature of what the path exemplifies in terms of one's growth of mathematical understanding. Although the Pirie-Kieren model contains embedded layers of understanding, Pirie and Kieren (1992b) claim that it is the dynamic within and between the layers that is critical to observe. "Acting and expressing complementarity" and "folding back" are two features that describe the back-and-forth movements within and between the layers of the Pirie-Kieren theory (see Sections 3.3.6 for a discussion of these two features). One feature of this dynamical aspect can be seen through the interactions within the Pirie-Kieren modes of understanding, a system that can be seen as dynamic in the sense of what Capra (1996) called "ecological community". Capra (1996) describes the nature of an ecological community in the following way: A l l members of an ecological community are inter-connected in a vast and intricate network of relationships, the web of life... The fact that the basic pattern of life is a network pattern means that the relationships among the members of an ecological community are non-linear, involving multiple feedback loops (p. 298). This "ecological" view reflects the dynamic, back-and-forth flow that characterizes the uniqueness of the Pirie-Kieren theory. Words such as "connected", "disjoint", "(re-) organization", "structure", "(re) construct", "pattern", "root", "point", "inner", "outer", "layer", "path", "(re) tracing", all indicate a similarity of the Pirie-Kieren theory to that of a structural network. In addition, Capra's use of the words "non-linear" and "multiple feedback loops" resembles the recursive nature of the Pirie-Kieren theory; the term, "multiple feedback loops", of course, is also analogous to the Pirie-Kieren's notion of "folding back", although it may not reveal as much about the thickening effect involved in growth of understanding. The network 75 itself, in the sense of Capra's (1996) ecological idea of "network pattern", constantly undergoes re-structuring as the learner builds upon his or her understanding through a series of back-and-forth movements, thereby attaining thicker understanding. The Pirie-Kieren theory demonstrates the nature of understanding to be a constant, consistent re-organization of one's knowledge structures (Pirie & Kieren, 1994). 3.3.4 The Modes or Layers of Understanding The Pirie-Kieren model is a diagrammatic representation consisting of a sequence of nested circles or layers, developed to represent the eight potential modes of understanding. Figure 2 (Section shows a diagrammatic representation of the Pirie-Kieren Model. Although this two-dimensional representation of the model is not the theory, the diagram itself is a useful tool for mapping the path of one's growth of understanding. Further discussion of this mapping technique is contained in Section 3.3.5. A brief analysis of the layers within the model will be illustrated by using a hypothetical example of the way in which a student, say in grade 12, comes to understand the concept of linear functions. The eight potential modes of understanding of the Pirie-Kieren theory are offered, either as informal or formal actions, for a specific person, and relating to a specific topic. Layer 1: Primitive Knowing The process of a learner coming to understand a concept or idea starts at the Primitive Knowing layer. The word "primitive" does not imply prehistoric or low-level mathematics, but rather a sense of the starting point or "base knowledge" for the growth of understanding, "observed in a particular person on a particular topic at a particular time" (Kieren & Pirie, 1991b, p. 9). It is what a learner has in his or her mind when he or she enters a new mathematical situation, with 76 the exception of any existing knowledge, such as images about the topic to which he or she is about to be introduced. A n observer (such as a teacher) can only assume what the learner can do initially, and in making such assumptions, the observer can then build on that supposition about the learner's growth of initial understanding of that particular topic. Therefore, the observer cannot know in full what this primitive knowledge is, or is not, but his or her assumptions can largely be based on what the individual learner was exposed to in the previous classes or years of schooling. For the concept of linear function, a grade 12 student may have certain "primitive" knowledge such as the language used to express this concept, and possibly mental images and existing knowledge of lines, points, graphs, or any concepts that were learned in previous lessons or at lower levels, say in grade 10 or 11. In this latter case, the student has some knowledge already in the other layers or modes of understanding; Primitive Knowing is everything a learner knows that is unrelated to the topic to be taught. Although this layer may seem uninteresting in relation to the other layers, Ausubel (1968) has noted how significant such primitive knowledge is (and any existing knowledge) in influencing how students learn and perform mathematical tasks. He said, "If I had to reduce all of educational psychology to just one principle, I would say this: The most important single factor influencing learning is what the learner already knows" (p. vi). Layer 2: Image Making The labels for the next two layers are named using the word, "image". A study of this nature focuses on identifying the mathematical images held, accessed, made, modified, and worked with by the students as they engage in the task. An "image" is defined as any physical or mental representation or idea that the students may have for the investigated topic. These images are often specific, mathematically limiting, and context dependent (Martin & LaCroix, 2004). Pirie _ 77 and Kieren (1994) suggest that the word "image" is less open to ambiguity and misunderstanding than, say, "idea". The participants' work with images, (for the specified topic) in large part determines for them what is mathematically meaningful (or not meaningful) while they are working on the task. The second layer is therefore called Image Making, and the learner acts on his or her Primitive Knowing to develop particular images for the given task. These images need not be visual or pictorial, and could be mental ideas about the specified topic. The learner has to engage in activities that will bring forth these particular images and reflect on those activities and on his or her previous abilities in order to use these images in new ways. Thus, this Image Making layer is dependent on the learner's primitive knowledge in order to bring about his or her growth of understanding. Using the example of the grade-12 topic of linear function, the learner may begin making an image by plotting linear graphs or relations. When the learner performs an action or activity, such as plotting a linear graph, he or she is working at the Image Making layer, building images of linear graphs based possibly on his or her Primitive Knowing. Layer 3: Image Having At the Image Having layer, a learner demonstrates the use of a mental construct about the topic and is able to use that particular image without doing the activity itself. Image Having implies that the learner is no longer tied to the particular action, but is now able to carry out the particular activity through internalization of the constructed images - a concept Martin (2001) called a "mental plan". As a result, the learner does not need to do whatever it was that he or she was doing before at the level of Image Making. Furthermore, the learner does not have to make every image possible on that particular topic in order to have a meaningful image. A statement such as, "It should look like a vertical line!" when asked about the shape of the line x = 2 demonstrates a 78 student constructing a mental picture of how the graph looks, without having to draw it. Such a statement reveals that the learner has now characterized, developed, and brought forth his or her sense of the meaning of the word "linear". However, the learner's meaning of the word "linear" may not be the correct one, or what is needed to accomplish the task at hand. Layer 4: Property Noticing The fourth layer is called Property Noticing, and it refers to a learner looking at his or her images and trying to identify context-specific, interconnected properties by manipulating and combining aspects of his or her constructed images. The word "interconnected" is used to represent the learner making distinctions, connections, and relationships concerning the particular example he or she is working on and is also able to articulate. These connections are based on knowledge from the previous three levels or knowledge that arose from manipulating his or her direct images (Pirie, 2001, personal correspondence). The learner consciously sees the relationships and makes the connections, answering questions such as, "How are the images connected?" For example, a student may compare the graphs of y = 2x and y = 6x , and, after plotting the two lines, say, "Wow, 6x is a lot steeper going left to right!", and then add, "I think lOOx will be a lot steeper." In this instance, the student speaks of specific lines (that is, y = 6x and y = lOOx) and a specific property (that is, "steeper going left to right!"). But then, someone else might say, "Well, but y = -2x goes the other way around! What would happen, then, to its relationship with y = -6x ?" Hence, based on such student responses, and by comparing and seeing features of each line, the observer is aware of a student working at the Property Noticing layer. Pirie has alleged that teachers quite often do not teach Property Noticing, but instead put more emphasis on constructing images without helping the students to connect the properties of their constructed images (2001, personal correspondence). As a result, students are not able to make connections 79 or to progress in their growth of understanding, not just within Property Noticing, but also with regard to that layer's connections to other layers of understanding. Layer 5: Formalising Formalising refers to the next mode of understanding that involves a learner being able to articulate a general observation, which does not have to be stated algebraically. The learner abstracts from his or her noticed properties a method or generalization by stating the patterns or the rules, which are disconnected from specific examples, actions, or particular images. The learner, at the stage of Formalising, has therefore moved away from noticing specific properties, to generalizing and working with the concept as a formal object. It is at this layer that the learner may be capable of stating or recognizing formal mathematical definitions or algorithms, making the Formalising layer of understanding the focus of much teaching. Hence, the student can give some form of the general slope-intercept form {y = ax + b) of & linear function, a learner who knows that the magnitude or absolute value of "a" (the coefficient of "x") determines the steepness of the graph demonstrates a generalization of such a property for all linear functions of that algebraic form. Quite possibly, the learner may also Formalise that the constant value "b" determines the vertical translation of the linear graph, even without knowing whether b = -2 means moving the linear graph up or down. Layer 6: Observing In the case of linear functions, suppose a student has been taught the general algebraic forms: y = ax + b, f + j = 1, or ax + by + c = 0 . If the student is able to justify that these three algebraic equations are equivalent, and true for all linear functions, then his or her understanding of linear functions is beyond Formalising. Similar to Property Noticing, where a learner 80 constantly works with his or her images, Observing involves looking at all the generalized statements and making connections among them. Such ability demonstrates that the learner is ready to reflect upon, organize, and express such a formal activity as a theorem. It is a mode of understanding in which the learner is in a position to reference his or her own formal thinking, and he or she is able to see its consequences. Layer 7: Structuring An awareness of one's assumptions, together with one's observations, requires logical thinking in establishing relationships within one's formal observations. A learner at this stage is said to be Structuring, a stage requiring formal mathematical proof. Structuring involves the learner being confident in justifying and verifying statements or proofs through logical or meta-mathematical arguments. Such a statement expresses mathematical structure, independent of physical or algorithmic actions. In the case of linear functions, when a student says that a line is only a special case of the polynomial functions with the highest degree of one, it is an indication of the logical argument present in Structuring. Layer 8: Inventising The outermost layer is called the Inventising layer. A learner working at this layer is capable of "breaking away" and developing a new creation, new concept, or even a new question that may also lead to the formulation of other new concepts. Within a given topic, the learner working at this layer is assumed to have a fully developed, Structured understanding that may therefore allow him or her to break away from Structuring, to Inventising new possibilities. Such a circumstance happens, therefore, when the learner knows all the rules and he or she wants to change something. So the learner would say, "What would happen if . . .?" 81 Using the example of linear functions, a grade 12 student who seeks new properties using complex numbers would be Inventising by pursuing possibilities other than the existing theories of, or about, linear functions. So, in a way, Inventising is not about a sophisticated piece of mathematics that is very hard to accomplish, but rather a mode of understanding a concept that may or may not make sense for the learner. Learners at any level, therefore, are capable of coming up with some amazing mathematical results or new discoveries i f given the opportunity to Inventise, although such an outcome rarely occurs because it is not covered by the standard curriculum (Pirie, 2001, personal correspondence). 3.3.5 The Two-Dimensional Model - A Mapping Technique Pirie and Kieren have also developed a technique they call "mapping" to allow an observer to trace the pathway of a learner's growing understanding of a particular mathematical topic. Using the diagrammatic representation of their model, an observer can produce a "map" by drawing the learner's growth of understanding, as it is observed (see Figure 2, Section To produce such a map, the observer can plot points on the diagrammatic model and then, using connected or disconnected lines, and depending on how each of the successive events (represented by points) are related, show how the learner's understanding of the specified topic grows. It is important to remember, however, that in using such a mapping process, the observer is limited to sometimes only being able to see with hindsight what goes on inside the learner's thought processes. In the Pirie-Kieren model, "connectedness" is associated with having a path between the layers or modes of understanding. Without such connected paths, learners who unsuccessfully build connections may develop what Pirie and Kieren (1994) call "disjointed" or "disconnected" knowing. While observing learners with no connections between their new understanding and 82 their existing knowledge, Pirie and Kieren (1994) hypothesize that these "students will be unable to successfully build further understanding based on this disjoint knowing until they have in fact constructed the connection for themselves" (p. 185, original emphasis). In this respect, the Pirie-Kieren theory, as illustrated pictorially by their model, is analogous to Capra's (1996) ecological model of a connected network - a complex, dynamic whole, continually undergoing realignment and reconfiguration, as layers of thinking sophistication become thicker through ongoing recursion and back-and-forth movement. 3.3.6 The Special Features of the Pirie-Kieren Theory As a theory for understanding, the uniqueness and strength of the Pirie-Kieren theory is characterized by its special features. These special features -folding back, acting and expressing complementarity, and "don't-need" boundaries - are essential elements that add to the complexity of the theory. These features are examined in the context of this research study. Folding Back The metaphor of folding back is a very important notion in the Pirie-Kieren theory because it distinguishes this theory from any existing theory on mathematical understanding. One reason for this metaphor's importance is that in order for understanding to grow and develop, folding back is "an intrinsic and necessary part of the process" (Towers, Martin, & Pirie, 2000, p. 226). Kieren and Pirie (1991b) define folding back in the following way: when a learner is faced with a challenge, one that is not immediately solvable, he or she is prompted to return to an inner mode of understanding in order to reconstruct, and to extend his or her currently inadequate inner-layer 83 understanding. The learner needs to fold back to an inner layer to extend his or her inadequate understanding in order to progress at an outer layer. Folding back is not just going back; it carries with it the notion of superimposing one's current understanding on an earlier understanding, resulting in a thicker understanding at the "returned-to" layer. This returned-to layer is now informed and shaped by outer-layer awareness, and the inner-layer action is part of the recursive process necessary to build further upon the outer-layer knowing (Kieren & Pirie, 1991a, 199b). The returned-to layer is no longer identical to the original-layer understanding, because of the recursive nature and interdependence between each of the layers that enfold it. Martin (1999), identified four "forms" of folding back: i. Working at an Inner Layer: This form of folding back involves a learner shifting to work in a less sophisticated, perhaps less formal, mathematical way and can involve the learner either extending his or her current understanding by changing his or her earlier constructs for the concept, or through the generation of new images. In either case, the less sophisticated, inner-layer understanding is thickened by the new constructed understanding and by the existing outer-layer knowing. ii. Collecting at an Inner Layer: Folding back to collect entails retrieving previous knowledge for a specific purpose and reviewing or "reading it anew" in light of the needs of current mathematical actions (Martin, 1999, p. 176). Collecting is not simply an act of recall - in which case, it would be called "remembering". Collecting has the thickening effect of folding back. Folding back to collect occurs when students "know that they know" what is needed, and yet their understanding is not sufficient for the automatic recall of usable knowledge. 84 Hi. Moving Out of the Topic and Working There: This form of folding back accounts for those cases involving a learner moving out of the current topic and working on a different mathematical area. This phenomenon takes place in instances involving the individual having an incomplete or inadequate Primitive Knowing from which to build or develop his or her understanding. Such "incompleteness" requires a stepping out of the current topic to permit the developing and thickening of concepts from a different mathematical area. The purpose of moving out of the topic and working there is to work on one's Primitive Knowing and then to be able to work with it at the outer or current-layer knowing of the topic being studied. iv. Causing Discontinuity: This form of folding back occurs when an external intervention (for example, a teacher or a peer) causes a learner to fold back to an inner layer of understanding and work there. But on the return to the inner layer, the learner essentially "begins again", making a new image unrelated to his or her existing understanding. Thus, the external intervention causes a break in the developing understanding of the learner. The key feature in this case concerns the learner's inability to perceive the relevance of the need to fold back to the problem he or she is working on. Acting and Expressing Complementarity Pirie and Kieren (1994) believe that each layer beyond Primitive Knowing is composed of a complementarity of acting and expressing, and that complementarity is necessary to complete a layer of understanding (see Figure 3). Furthermore, growth occurs through "first acting, then expressing, but more often through to-and-fro movement between these complementary aspects" (p. 5). Acting encompasses all previous understanding, either mental or physical activity, and expressing concerns making explicit and articulating to others or to oneself what was involved in that activity. Reflecting is a critical component of this complementarity. The process of reflecting incorporates looking at how previous understanding was generated. 85 The acting/expressing complementarity feature of the Pirie-Kieren theory is central to observing the students' verbal and nonverbal actions. Pirie and Kieren have yet to investigate this feature further (Pirie, personal correspondence). Currently, only the complementarity "pairs" in the first five layers have been identified and defined (see the definitions below). While this feature requires further investigation, this study attempts to investigate any observable relationship between acting and expressing through the act of language switching. (See Pirie & Kieren (1994) for more discussion and examples on these complementarity pairs.) Figure 3: Complementarities of Acting and Expressing in each of the layers The first pair of instances involving acting and expressing complementarity occurs at the Image Making layer and are called image doing and image reviewing respectively. In image doing, a learner is engaged in activities associated with constructing an image of the situation. In this situation, the learner sees his or her previous action(s) as completed, and rejects returning to it in 86 anything other than a rule-bound way. But in image reviewing, the learner reviews his or her previous work and attempts to make sense of the situation. Such a situation allows for a constructive alteration of previous behavior without yet seeing an image. The next pair of instances occurs at the Image Having layer and are called image seeing and image saying respectively. A learner able to demonstrate that he or she has now formed an image (though not necessarily a complete or correct one) is said to be image seeing. In this case, the learner has gone further in his or her understanding and, through reviewing, has constructed some image for the topic. He or she would be able to articulate that the new "point" does not fit with his or her image of the topic, but would not be able to explain why. In image saying, the learner is able to express aspects of the image. Such behaviour articulates on features of the image. At this level, the learner is in a position to talk about his or her actions and why "something" does not conform to his or her image. The third pair of instances occurs at the Property Noticing layer, and is called property predicting and property recording respectively. In property predicting, a learner is able to connect previous images and recognize a pattern. This process deals with distinguishing and connecting features of the learner's constructed image(s), leading to a new kind of understanding. A learner who demonstrates property recording is able to express that he or she has an "organized" scheme(s) for the connection or pattern. Such behaviour articulates features of the observed pattern. The use of the word "recording" is not intended to imply that a written record be produced, but rather it must involve verbal or non-verbal explicit expression of some form. The last identified pair of instances occurs at the Formalising layer, and is called method applying and method justifying respectively. In method applying, a learner is able to apply his or her understanding of a particular concept in appropriate generalized circumstances, while demonstrating in method justifying that he or she is capable of justifying the generalization. "Don't-Need" Boundaries 87 The "don't-need" boundaries are of importance to understanding the nature of the Pirie-Kieren theory. The need to articulate Formalised understanding explicitly reveals the important role of language in explaining the move from Property Noticing to Formalising. These boundaries are observable in the Pirie-Kieren model, and are illustrated by the bold rings in the two-dimensional representation (see Figure 2 and Figure 3). These bold rings are called "don't-need" boundaries to convey the idea that beyond each of these boundaries, a learner is capable of operating in a particular mode of understanding without having to access previous modes, although these previous modes of understanding are embedded in the new level of understanding. With an image of a straight line, y - x, a student does not need to draw y = 2x, y - 3x, etc., each time to check its linearity. Likewise, working with the Formalised expression y = ax + b, the student does not need to check the formula with say, y = 3x + 4, to be sure of its linearity or accuracy.. As mentioned in the previous chapter (Section 2.3), the notion of "don't-need" boundary resembles Sfard's (1991) idea of "reification", in which the learner suddenly sees the familiar use of mathematics in a new light. Thus, a learner who is working at each of these particular modes of understanding, namely Image Having, Formalising, and Structuring, is capable of working with concepts no longer tied to his or her previous modes of understanding (Pirie & Kieren, 1994). The bold rings also illustrate the transcendent, recursive nature of the Pirie-Kieren theory; that "understanding is a transcendently growing whole" (Kieren & Pirie, 1991b, p. 21). The knowledge of a learner with connected understanding at a particular point in time is integrated with previous layers of knowing, and because of this interconnection, the learner's previous 88 knowledge can be called into current-layer actions. In contrast to the bold-ring boundaries, a learner is constantly working between layers around a "do-need" boundary, one that is marked by a non-bold ring. For example, a learner who is observed to be working at the Observing layer is constantly working with his or her Formalised statements or understanding in the same manner as a learner who constantly accesses his or her images, while working at the Property Noticing layer. 3.4 The Appropriateness and Purpose of the Pirie-Kieren Theory This chapter introduced the bilingual context of the study, and summarized the nature and features of the chosen theoretical framework. Ultimately, the importance of the chosen theoretical framework lies in its ability to explain and express in its own "language" the nature of a learner's growth of understanding, irrespective of his or her cultural background. This usefulness of the Pirie-Kieren theory thus extends, in this study, to observing how bilingual students' language switching interacts with their growth of mathematical understanding. The main question this study must therefore consider is the following: Why is the Pirie-Kieren theory appropriate in studying Tongan bilingual students' growth of mathematical understanding? To answer the preceding question: First, the Pirie-Kieren Theory is a theory developed for, and not of, mathematical understanding. A "theory o f is a theory that usually claims to explain everything. A "theory for" is an explanation, but it does not claim to be the only explanation. The Pirie-Kieren theory therefore offers a language for explaining, and a way of observing and accounting for the dynamical growth of mathematical understanding, which is appropriate to this study, taking place within a bilingual context. The theory has been given wide recognition as 89 exemplified by its influence in various curriculum reforms (Godino, 1996), and its application in the State of Wisconsin, where the Pirie-Kieren model is used by the mathematics teachers in a state-wide professional development program (Martin, 1999). Second, the Pirie-Kieren theory is not just a theory for the growth of understanding in a general sense, but rather, a theory for the growth of understanding of a specific mathematical topic. This is an important distinction because one cannot just say, "Hey, the student understands!" The task of an observer is to focus on and specify exactly what it is a learner understands, and how. The Pirie-Kieren theory becomes a "lens", and useful theoretical tool, for observing growth of mathematical understanding in any learning situation, with a chosen focus on the specified topic. It is therefore important to keep track of one's growth of understanding in that specific topic; choosing another topic or focus would result in a different perspective or path of analysis. Third, the Pirie-Kieren theory is a theory for the growth of understanding of a specific topic by a specific person. The word "person" is placed in inverted commas to indicate that the theory can be used to observe not just a single person's growth of understanding, but also to analyze a certain group's collective growth of understanding of any specific topic. The theory therefore includes the enactivist view of the "co-emergent of understandings", to cite the role of others in the interactive process. It must be remembered the observer can only make inferences based on what is observed through the learner's verbalization of his or her work, and any non-verbal externalizations that might be available, such as the learner's worksheet, or body movements and gestures; the observer cannot make any inferences based on what is not observed, and therefore conclude what the learner does not know. More importantly, the dynamic of growth of understanding varies from one person to another for any specific topic at any given moment. Hence, different persons (or groups) will have different pathways of understanding. 90 Fourth, the Pirie-Kieren theory describes the growth of understanding of a specific topic, by a specific person, over time. Time, of course, does not necessarily refer to a long period of time, say, in years or months, but it can refer to growth of understanding over a period of hours, or even minutes. Most importantly, however, the Pirie-Kieren theory does not take a "snapshot" assessment or create a "still-photo" picture of someone's growth of understanding. Finally, Pirie and Kieren (1991a) recognize that language has an "orienting function" in the constructions and abstractions of mathematical knowledge, a view they cite as consistent with Maturana and Varela's (1980) ideas. Pirie and Kieren do not believe that, during a course of interaction, language carries information between two individuals; rather, language "allows a child to orient herself to her own actions, particularly mental ones, thus facilitating abstraction and the recursive use of her own knowledge in the building of new patterns of knowledge" (Kieren & Pirie, 1991a, p. 81-82). This "facilitation" aids the personal internalizing of existing knowledge, and is part of the individual's understanding (Kieren & Pirie, 1991a). Thus, language use can facilitate knowing and ready it for further, more elaborate knowledge building. The orienting function of language described here has a significant implication toward the current study because language can influence a learner's path of growth of mathematical understanding in three different ways: (i) a provocative effect describes how the learner moves outward in his or her growth of understanding; (ii) an invocative effect describes how the learner folds back to work with inner-layer knowing; and (iii) a validating effect describes how the learner confirms his or her working within a particular layer. An external intervention can also influence a learner's path of growth of mathematical understanding in one of the three described ways. 91 The Pirie-Kieren dynamical theory, therefore, can be used as a powerful theoretical tool in studying and understanding the notion of "mathematical understanding". However, it has never been a theory for mere assessment, because it does not look at understanding as only an outward progressing, nor as an acquisition of specific categories of knowing (Pirie & Kieren, 1994). It is not a theory for testing what a learner can or cannot do, because such a theory does not reveal how the learner got to his or her answer or a specific point in his or her work, or even how the learner got to that state of thinking. Rather, the Pirie-Kieren theory is a theory for looking at how the learner gets to a mode of understanding, and how he or she comes to understand the mathematics in question. CHAPTER 4: M E T H O D O L O G Y , R E S E A R C H DESIGN, and M E T H O D 92 4.1 Introduction This chapter focuses on the study's methodology, research design, and the method of data collection used to address the research question. 4.2 Methodology: Video Case Study Research 4.2.1 Introduction In an effort to propose an appropriate methodology for the study, the research question was divided into two main phenomena: first, language switching, and second, growth of mathematical understanding. These two phenomena, in turn, steered the manner and direction of the investigation, and thus shaped the qualitative approach chosen for data collection and analysis. Unlike most linguistic studies, this study was not intended to evaluate bilingual students' language switching, nor was it an attempt to assess students' mathematical understanding through standard tests (or any such quantitative method). Instead, this study was intended to focus precisely on the relationship between language switching and growth of understanding for a particular mathematical topic. In order to investigate the relationship between these two phenomena, a qualitative approach had to be employed to investigate, in depth, the nature of a student's growth of mathematical understanding. Case-study research was deemed the most appropriate form of qualitative study, one that would involve a small number of 93 bilingual students, and examine how their language use (including language switching) related to their growth of mathematical understanding. This tactic, as Towers (1998) has demonstrated, provides a "deep and comprehensive description and interpretation" of the processes within a particular learning situation (p. 47). Because the Pirie-Kieren theory was chosen as the theoretical framework, a case-study approach seemed most appropriate for investigating such a complex process as the growth of mathematical understanding. Previous researchers have also dealt with this methodological choice while studying growth of mathematical understanding using the Pirie-Kieren model (Towers, 1998; Martin, 1999; Thorn, 2004). The methodology then selected for this study, termed "video case study", refers to a case study based on video-recorded data.13 Because of the essential role video plays in this study, the label "video case study" was used to emphasize the primary role of video-recording technique. This emphasis has been documented elsewhere, in similar video case studies of this nature (Maher, Davis, & Alston, 1992; Towers, 1998; Martin, 1999; Pirie, 1996, 2001; Powell, Francisco, & Maher, 2001). Video case study is not a new paradigm in qualitative research, since it has been employed previously in various fields for various purposes (Wood, Cobb, & Yackel, 1991; Davis, Maher, & Martino, 1992; Erickson, 1992; Cobb & Whitenack, 1996; Goldman-Segall, 1998; Hall, 2000; Lesh & Lehrer, 2000). In addition, several case studies involving video recordings have been widely implemented and discussed in many different fields: in anthropology and ethnography (e.g. Clifford, 1986; Rollwagen, 1988; Atkinson & Hammersley, 1994; Goldman-Segall, 1998); in sociology and social sciences (e.g. Grimshaw, 1982; Chaplin, 1994); in health sciences (e.g. Gross, 1991; Bottorff, 1994); in the arts (e.g. Gernsey, 1997; Ohler, 2000); in humanities (e.g. Hall, 2000; Powell, Francisco, & Maher, 2001), and law and 1 3 But such a study should not be confused with a case study video, which is not a process, but a product material, such as an informative video production used for educational or professional workshops. 94 business (e.g. Westerfield, 1989; Hawkins, 2001), to name a few. As well, video case study methodology has been documented in various contexts, even within a single discipline, such as mathematics education. For instance, previous mathematics education studies have been based on groups' interactions (Davis & Maher, 1990; Pirie & Kieren, 1992b, 1994, 1999; Cobb & Whitenack, 1996), teacher interventions (Maher, Davis, & Alston, 1992; Towers, 1998; Towers, Martin, & Pirie, 2000), individuals' learning and understanding (Pirie & Kieren, 1992b; Martin, 1999; Hiebert & Carpenter, 1992). More specifically, video case study research has been used to observe students' growth of understanding (Maher, Davis, & Alston, 1991; Pirie & Kieren, 1992b, 1994), an area in which the research question for this thesis (on language switching and growth of mathematical understanding) is framed and situated. Like all case studies, the advantage of using video case study involves what Geertz (1973) has called the quality of its "thick description" as opposed to the "thinner", more scientific modes of description. Stake (2000) defines a case study to mean "both a process of inquiry about the case and the product of that inquiry" (p. 436). Similarly, a video case study is both the process of learning about the case - aspects of the research question - and the product of that learning through video recordings. While the case study gives a "thick description" of an event, video becomes the efficient vehicle for the collection, storage, and analysis of the evidence or data. What is more, video case study brings the case alive on recorded tape, enabling researchers to revisit and reflect on the recorded scene for clarification, an invaluable tool for anyone wishing to verify or extend their knowledge of existing research. Nonetheless, it is important to question the empirical nature of video case study, and whether or not it can be seen as a useful and appropriate tool for examining the relationship between bilingual students' language switching and their growth of mathematical understanding. To address this question, on one level, case study must be considered the most appropriate methodology, and on another level, the use of 95 video recording must also be deemed a trustworthy source of data collection. As the following arguments will demonstrate, the integration of a case-study methodology and data collection through videotape was the best possible choice for investigating the proposed research question. 4.2.2 Case Study Research and the Notion of "Generalization" Case study has long been a standard and popular methodological research practice for studying the "particular", a term often referring to a specific characteristic or feature, or the observation of an interesting phenomenon (Stake, 1994, 2000; Tellis, 1997). This notion is characteristic of any qualitative study, and is often mentioned in comparison with the quantitative notion of "generalization", a term usually used to describe "something for all" (Yin, 1989). Case study research is also characterized, by its very nature, as an in-depth investigation of that particular phenomenon (Yin, 1989). For Stake (2000), the underlying epistemological question in any case study draws one's attention to what can be observed and learned from a single case: a view that resembles Yin's empirical approach, and one that is adopted in this study. Stake (2000) then describes case study research as merely "not a methodological choice but a choice of what is to be studied" (p. 435). Other researchers, notably Adelman, Jenkins, and Kemmis (1976) and Vaughan (1992), have focused on the particularity of case-study research, and its narrow applicability to the phenomenon being investigated. Yet, a dichotomy often arises whereby case study research emphasizes, on the one hand, its nature as a methodological technique, and on the other, its "particularity" with respect to the phenomenon being investigated. Born out of this debate is the emergence of video case study, in which its very nature and role in qualitative research combines both what is being investigated (the case) and how it is being investigated (the video). However, case study research may not only concern itself with a specific phenomenon. 96 The nature of each phenomenon can change when analyzed, observed, or conducted using different theoretical frameworks, different situations, or different methods of inquiry. At the same time, educational researchers will continue to debate the nature and substance of a case study, or case study research (Tellis, 1997). Like others involved in image-based research14, researchers conducting case studies are well aware of these studies' limited status - not because of the quality or the focus of researchers' work, but because of the nature of their work (Prosser, 1998, p. 99). In the case of the current study, the limitations imposed by certain forms of qualitative study were unavoidable, so it was important to remain aware of such limitations while conducting the research, and to offer appropriate counteracting measures when necessary. One such limitation of the methodology, for instance, related to Towers' (1998) observation, that the main objection to case study research is the statistically insignificant portion of a particular population being investigated. However, it was not the intention of this study to generalize. What is more, in defending such a position, other case study researchers claim they pride their findings upon knowing their cases extremely well, and as Towers (1998) further pointed out, in "recognizing the distinction between generalizing and over-generalizing" (p. 47, citing Wolcott, 1985). Such an emphasis allows for the selection of case study methodology with only a few participants because "one wishes to understand the particular in depth, not because one wants to know what is generally true of the many" (Merriam, 1988, p. 173). After all, committing oneself to generalization may draw the focus away from the features important for understanding the case (Stake, 2000). So case studies, including this one in particular, can be considered generalizable in so far as they are used in a general way to ask a question about a particular phenomenon, rather than in their claim to generalization of a particular concept as a conclusive 14 Image-based research encompasses studies involving film, video, photography, cartoons, signs, symbols, and drawings (Prosser, 1998). 97 inquiry (Towers, 1998, citing Lincoln & Guba, 1985, p. 110). But the fact that such a study focuses on a statistically insignificant sample challenges all researchers to "openly acknowledge what they are attempting to do" (Towers, 1998, p. 47). By so doing, readers may then be less inclined to criticism and confusion, enabling the author to get his or her message across. Yet, the concern toward attaining generalizability, or defining what is or is not generalizable, remains for some a weakness within qualitative research. The typical reaction to case study research is, "How can one generalize from a single case?" A notable counterpoint to this argument arises in the question of what exactly one generalizes from, a point made by Towers (1998) when she observed that "tensions are currently surfacing concerning what can and cannot be generated from" (p. 49). As well, Yin (1989) suggests a certain type of generalization can be made from case studies. He argues: The short answer is that case studies, like experiments, are generalizable to theoretical propositions and not to populations or universes. In this sense, the case study, like the experiment, does not represent a "sample", and the investigators goal is to expand and generalize theories (analytical generalization) and not to enumerate frequencies (statistical generalization). (Yin, 1989, p. 21) In relation to the topic at hand, Stake (1994, 2000) also claims that while the search for particularity competes with the search for generalizability, the nature of some studies are unique and complex, which, statistically (and practically) speaking, cannot be compared generally but may lead into what one may call a generalized theory. Therefore, in an effort to seek such generalization in this study, Glaser and Strauss' (1967) "constant comparative method" was employed, because it allowed for a general categorization of language switching to emerge from the analysis, while the analysis remained firmly grounded in the data. For the purposes of this 98 study, and to address longstanding concerns over what Yin (1989) called "sloppy", "lack of rigor", and researcher bias sometimes associated with case study research, certain features of the study were considered necessary. For instance, the case study approach took into account its role as an empirical inquiry, involving the use of multiple sources of evidence, and also the employment of the technique of "triangulation".15 Such an approach recognized the complexity of investigating a contemporary phenomenon within a real-life context, and also improved the reliability of the study, in order to respond to traditional criticism about the trustworthiness of interpretations based solely on observational methods. 4.3 Research Design 4.3.1 Introduction After deciding to use video case study with a relatively small number of students, it was important to choose the process of data collection. Various aspects of the research design were considered: location, setting, schedule, task, contact, participants, and so on. The prior consent of all participating parties - the government, the Ministry of Education, the schools, the parents, the teachers, and the students - had to be obtained before the study could take place. However, other aspects of the research process, such as drafted tasks and schedules, had to be continually revisited and revised or rescheduled, as the study progressed. Two separate studies were documented: one in 2001 and another in 2002, and these formed the two strands of the research. 15 Triangulation is a multi-method approach involving multiple investigators, multiple data sources, or multiple methods to confirm the emerging findings and to enhance the trustworthiness of the research (Merriam, 1988; Towers, 1998). Although this technique is sometimes associated with the notion of having "one truth to be told", enactivism advocates such as B. Davis (1996) argued that the belief that reality is "out there" could be replaced "with an acknowledgement of the contingency of interpretation" (p. 103). 99 The first study was conducted in Tonga between September and October of 2001, to collect video recordings of secondary school students' work. The second Tongan study took place the following year, between September and October of 2002. Each study took five weeks to complete. Although the schedule was tight, it was easy to effectively organize video-recording schedules, given the small number of students used for the study group, and the strong support for the study from all participants, among them principals, teachers, and students in particular. The decision to do a second study was guided mainly by what Glaser and Strauss (1967) would call the "emerging gaps" in data collection and data analysis used in the first study. Following the first study, and the researcher's return to Canada, subsequent data analysis revealed the need for further investigation of the research question, thus necessitating a second trip to Tonga. The data from the second study helped justify the data already collected from the first study, and also clarified certain unanswered questions associated with the research. Hence, the second set of data brought more depth to the data analysis. In reviewing the original data, LHS Secondary School1 6, for instance, was identified as a major additional source of data for the overall study because of the school's unique characteristics. In the Tongan secondary school system, LHS Secondary School, arguably, had the most resources and was the best-equipped secondary school campus. In particular, LHS distinguished itself from other schools in Tonga through its promotion of the English language, based on the school's strict, year-round enforcement of a "Speak English Only" policy. In addition, although LHS follows the Tonga secondary schools' mathematics syllabus, it enjoys a broad exposure to Western culture, through its unique relationship with affiliated church schools in Hawaii; and the school possesses a large number of staff from overseas. In considering LHS's exclusive approach, it was possible in this study to examine its mathematics students' use of English and Tongan. 1 6 All school and participant names used in this study are pseudonyms. 100 4.3.2 Location As previous chapters have made clear, this study focuses on the research question within the Tongan bilingual context. The researcher's cultural upbringing and experience as a mathematics educator and former secondary school mathematics teacher in Tonga, together with a bilingual background in both English and his native Tongan language, made him an appropriate person to conduct the study. The required assistance and permission were easy to obtain, including the full support of both the Tongan government and the Ministry of Education. The Tongan bilingual context (described in Chapter 3) presented the opportunity to examine a unique population: an isolated ethnic group internally homogeneous in its culture, particularly with respect to its national language. Also, with the exception of a few foreign students, most Tongan secondary school students (about 12,000 in total each year) are only fluent in their native, and everyday, Tongan language. However, these students are expected to rely mainly on the use of their second language of English - the language of secondary-school instruction - inside the classroom during school hours. At the same time, according to Fasi (1999), these students prefer to use their native language for mathematical activities, especially when they are engaged in discussions with their peers. 4.3.3 Participants The deliberate selection of the study's participants (schools, grade-levels, and students) was based on what Glaser and Strauss (1967) called a matter of theoretical purpose and relevance. Determination of the "purpose" and "relevance" of each choice depended, in this study, on the nature of the targeted data and its application to the research question (Chapter 5 describes how 101 some of the collected data turned out to be irrelevant for the study). Although Glaser and Strauss used these two terms to define specific criteria for selecting "comparison groups", their significance has a deeper implication for this study. In fact, Glaser and Strauss' (1967) notion of the "constant comparative method" was employed throughout the data analysis, in the development of themes and emerging categories associated with the Tongan bilingual students' acts of language switching. Further applications of Glaser and Strauss' constant comparative method are discussed in the next chapter. The researcher's role, therefore, was to choose particular schools and grade levels, and more importantly, to select bilingual students, based on their teachers' recommendations, who would help "generate, to the fullest extent, as many properties of the categories as possible" (Glaser & Strauss, 1967, p. 49). Furthermore, it was important to determine the theoretical purpose of choosing the participating parties in order to maintain what Glaser and Strauss (1967) called the study's "scope of population" and "conceptual level". Theoretically, the study's scope of population and the participants' conceptual level are defined by the choice of bilingual context, schools, grade-levels, and individual students. Both conditions - scope of population and conceptual level - were determined by the purpose of the study. Glaser and Strauss (1967) emphasized in this situation that the "scope of a substantive theory can be carefully increased and controlled by such a conscious choice of groups" (p. 52). Selected Secondary Schools Considering that all the schools in Tonga followed the same mathematics syllabus set forth by the country's Ministry of Education, it was appropriate to conduct the two studies within the main island of Tongatapu, where more than 70 percent of the country's middle and secondary school students were located. There were two key reasons for selecting these participating 102 schools: accessibility and diversity. The first criteria, concerning access to the schools and its participants, played a major role in deciding whom to work with, as well as when, and where, to work. The five schools involved in the study were among the seven largest secondary schools in the country, all of which had Form 1 to Form 6 (grades 7 to 12) students. These participating schools represented a range of religious denominations, academic, and social influences, namely A F C (a Catholic school), LHS (a Latter Day Saints school), QSC (a Methodist school), T C A (a government school), and THS (a government school). Each of these schools supplied students for the study, but all of the schools conformed to the mathematics curriculum and syllabus. Since the selected secondary schools also represented the diverse social and demographic range of secondary schools in the country, the individual students presented some unique characteristics for the study. Administratively, the schools were run either by the government (TCA and THS) or by any of the established religious denominations (AFC, QSC, and LHS). Various gender-based secondary schools could also be found in Tonga, particularly in the main island, where AFC, LHS, and THS were secondary schools with mixed students (boys and girls), and T C A and QSC, both of which offered boarding schools, were an all-boys and all-girls school respectively. Also, within the main island of Tongatapu, some schools were located in or near the capital (AFC, QSC, and THS), while others were situated in rural areas (TCA and LHS). Selected Forms (or Grades) This study took note of the fact that the middle-school level is a critical period for most Tongan-type bilingual students, especially for those with limited English proficiency, as they make the transition from instruction in one language at the primary-school level to another at the secondary-school level (Celedon, 1998; Fasi, 1999). It was at this transitional stage that the nature of "growth of mathematical understanding", and how it related to the notion of "language 103 switching" was investigated. The students were chosen from Form 1 to Form 3, because teachers and second-language learners in these age groups (11 to 14 years old) typically face serious communication problems in the classroom (Fasi, 1999). Judging from the Tongan Ministry of Education's guidelines outlined in Chapter 3, mathematics concepts are taught in Tongan up to Form 2 (grade 8) and then switched to English in Form 3 (grade 9). This middle-school shift in concept instruction is therefore a critical period for most of the Tongan bilingual students, particularly in mastering the mathematical language. As a result, some students struggle with the mathematical concepts when they switch languages (Fasi, 1999). Another problem is that while the language of instruction in the classroom is English, the majority of the Tongan secondary school students rarely use English outside their classrooms. In this study, the head of each school's mathematics department was responsible for choosing the mathematics teachers and classes involved in the research. Table C below shows the number and grade levels of participants selected from each of the schools: Strand One: FIRST STUDY (2001) PARTICIPATING SCHOOLS SELECTED FORMS NUMBER OF STUDENTS A F C 1,2,3 16 QSC 2,3 6 T C A 2,3 8 THS 2,3 8 Strand Two: SECOND STUDY (2002) PARTICIPA TING SCHOOLS SELECTED FORMS NUMBER OF STUDENTS A F C 2,3 12 LHS 2,3 8 QSC 2,3 9 Table C: Number of selected participants from each of the participating schools 104 Selected Bilingual Students In each school, the mathematics teachers were responsible for identifying students in their classes to participate in the study. Participants in the study were chosen using a "purposeful sampling" method, basing the criteria for student selection on teachers' identification of the following preferences: (a) Currently in Forms 1 to 3(11 to 14 years old); (b) Considered native Tongans, fluent (since infancy) in their native Tongan language, and deemed to have started learning English in primary school; (c) Keen to participate in group discussion; (d) Capable of verbalizing their thought processes; (e) Capable of switching languages (otherwise there was no point in selecting them); (f) Confident about being video- and audio-taped; (g) Most likely to be available when needed for any of the recorded sessions, especially outside school hours. Based solely on the criteria cited above, students' competency in either one of the languages or in mathematics was not a requirement for their selection. 4.3.4 Setting The setting in both studies was designed not only to engage the bilingual students in peer discussion and group collaboration work, but also to create an environment in which they could comfortably switch languages as they wished. This group environment was deliberately created to focus on the Tongan bilingual students' natural style of language switching. The word 105 "natural" in this case was associated with the desire to create a situation in which the Tongan bilingual students could switch languages voluntarily rather than in response to directions to do so. To adhere to the "natural" environment, a group setting was created with no external intervention or participation from either the students' mathematics teacher or the researcher (see a further discussion on this topic in Section 4.5.3). Group Collaboration and Peer Discussion In order to record the students' work, a setting had to be created to allow observation of their verbal and non-verbal activities through the video camera lens. Group collaboration work provided an appealing way to observe students' interaction and their mathematical work in action (Pirie, 1991), and it emphasized the importance of meaningful discourse in the classroom (Fasi, 1999; N C T M , 1991). This idea is stated clearly in Standard 3 of the Professional Standards for Teaching School Mathematics (NCTM, 1991), which says: Whether working in small or large groups, students should speak to one another, aiming to convince or question their peers. Above all, the discourse should be focused on making sense of mathematical ideas, on using mathematical ideas sensibly in setting up and solving problems (p. 45). Many educational researchers have focused on classroom talk and discussion (e.g. Pimm, 1987; 1991; Wood, 1990; Pirie, 1991, 1998a), and a great deal of study has been directed toward student-to-student talk and the process of group learning (e.g. Davis & Maher, 1990; Gooding & Stacey, 1993; Cobb & Whitenack, 1996). Although the terms "talk" and "discussion" do not necessarily refer to the same idea, it is essential that both activities are task-focused and that the style and level of their explicitness are taken into account (Pimm, 1987). According to Pimm, 106 "pupil talk" must be "focused, explicit, disembodied, and message-oriented" (1987, p. 42), while Pirie (1991) defines "peer discussion" as purposeful, mathematical talk in which there are genuine student contributions and interactions. Coincidentally, and conveniently for the purpose of this study, group work in a typical classroom setting is also strongly encouraged at the Tongan secondary-school level. Tonga's Ministry of Education (1996) recognizes the benefits of placing students in small-group work to highlight the advantage of mathematics students sharing and working cooperatively, using language to refine and consolidate mathematical understanding, developing mathematical understanding through involvement; acquiring problem-solving strategies, and performing tasks more effectively. In this study, students were assigned to work in groups of twos and threes, in order to maximize group discussion and individual participation; each group was encouraged to think out loud and to verbalize their thought processes while solving tasks. In addition, each group was asked to express any related work by writing or drawing on their worksheets. Thereafter, the non-participant role of both the researcher and the students' teachers was observed in order to create a "free" discussion environment for the students. 4.3.5 Tasks Considering the research question, and its chosen theoretical framework and methodology, the purpose of creating an appropriate task for the study was two-fold: to allow language switching among individual students by facilitating group discussion in both languages, and more importantly, to enable mathematical understanding of the specified topic to grow. It was therefore evident, based on the Pirie-Kieren theory as a theoretical framework, that a task or set of tasks required an investigative approach. Using this approach, the intention was not to present 107 the students with a task involving only one right answer, but to give them an open-ended problem in which they could explore various possibilities and argue for their own ideas and problem-solving methods. The set of questions cited below was used to verify the appropriateness of these tasks: (i) Wi l l the students' mathematical thinking be externalized in some observable form? (ii) Wi l l a variety of strategies and approaches to the tasks likely emerge? (iii) Wi l l each task challenge the students' growing understanding of the specified topic? (iv) Wil l each task provide a context for seeing patterns and relationships? (v) Wil l each student be able to make a connection with the previous mathematical knowledge and everyday experiences? (vi) Wi l l the students be likely to formulate conceptions and inquiries that set the stage for thinking about more complex ideas or sophisticated forms of thinking? (vii) Wi l l the tasks promote meaningful and rich mathematical discussion through debates, questions, conjectures, generalizations, and explorations? Topic of Investigation In Pirie and Kieren's terms, the Property Noticing layer is a crucial mode of understanding mathematics that classroom teachers often neglect (Pirie, 2001, personal correspondence). One of the observations characterizing students working at the Property Noticing layer concerns their ability to distinguish relationships between images: students' mental or physical representations in relation to their work on a particular topic. The topic, "patterns and relations", was considered for the purposes of a study set in the Tongan educational context, because this topic encouraged possible work at the Property Noticing layer. In addition, Tonga's Ministry of Education has clearly stated its assumption that Tongan bilingual students can learn mathematics by investigating "mathematical patterns, relationships, processes and problems" (1996, p. 2). 108 According to the ministry's guidelines, "Students should be given opportunities to discover and create patterns, and to describe, record, and explain relationships contained in those patterns". Thus, consistent with the Tongan mathematics syllabus for all grade levels, the topic, "patterns and relations" was chosen for investigating students' growth of understanding in this study. What is more, patterns and relations are central features of any Tongan student's everyday life; an attention to intricate patterns and designs being a uniquely rich element of the Tongan culture, as is displayed in various handicrafts, the most prominent of which are tapa-making and weaving. From the perspective of this study, the students' prior knowledge of such shapes, patterns, and designs illustrates the primitive knowledge these students bring to the task, and provides a background for their growth of mathematical understanding. The employment of the Pirie-Kieren theory allows for observation and analysis of the students' construction of mathematical images and a study of the way the students relate, create, and expand patterns and their existing knowledge. Pictorial Sequence The next important job was to construct a task that the students would be interested in exploring. After choosing patterns and relations as the topic for investigation, a simple pictorial sequence involving square blocks and cubes came to mind as a way of engaging students in their exploration. These kinds of pictorial representations have appeared previously in various studies (Pimm, 1987; Foreman, 1998; Thorn, 1999). Each pictorial representation was set up to play a major role in the students' image-construction processes. Although the tasks involved working with two-dimensional square blocks (first study) and solid cubes (second study), counting played a huge role in the students' work, and the mathematics that came out of these tasks was already established as rich from a teaching perspective. In each task, students were shown a different set 109 of pictorial sequences of square blocks (see Figure 4), or what Foreman (1998) called "tile arrangements" and Pimm (1987) named "bricks", to engage them initially in "visual thinking". Maier (1985) described this visual thought process as: thinking that draws upon the processes of perceiving, imaging, and portraying. Perceiving is becoming informed through the lenses, through sight, hearing, touch, taste, smell, and also through kinesthesia, the sensation of body movement and position. Imaging is experiencing a sense perception in our mind or body that, at the moment, is not physical reality. Portraying is depicting a perception by a sketch, diagram, model and some other representation (p. 3). Foreman (1998) argues that the major premise of such visual thinking is that carefully designed tasks based on sensory experiences would: enable students to develop meaningful mental and kinesthetic images of mathematical concepts and processes. These images help students understand, retain, and recall this information. Listening to others talk about their thought processes and mental images fascinates students and prompts new ideas. As students explore mathematics in this manner, ideas makes sense, math anxiety diminishes, and confidence grows (p. 142). In addition, the tasks were intended to provide a mathematical-oriented environment, whereby the students were able to create, manipulate, test, and explore their ideas. Thus, through students' actions, the gap between their physical world and their imaginary one would diminish, their experiences and the symbolic world of mathematics become connected, and hence, students' mathematical understanding would be enhanced. 110 1st diagram 2nd diagram 3rd diagram 4th diagram 5th diagram 1st sequence • B D 1 I 2nd sequence • I—| — 1 I 1 1 1 1 3rd sequence - m\ i 1 1 4th sequence B G 1 Tongan Task sequence !• 1 1 Figure 4: The pictorial sequences for all of the tasks using square blocks and cubes The last pictorial sequence was used for the Tongan Task, in which the accompanied set of questions was directly translated from English into Tongan (see Section The Set of Questions A set of questions was drafted in English for the two studies that would accompany the pictorial sequence. The role of this set of questions was mainly the following: (a) to guide and validate the work of the students with respect to the given pictorial sequence; (b) to allow analysis of the students' knowledge of the English language - the language used in their mathematics classes; (c) to engage them verbally in a group discussion. The word "prediction" was intentionally used to mean, "making an educated guess". An educated guess in this task implies the need for an I l l individual to have spotted some pattern in the given pictorial sequence in order to offer a reason for the predicted answer; (d) to allow language switching; and (e) to occasion students' growth of mathematical understanding in the specified topic. The English version of the set of questions which accompanied each pictorial sequence is shown below (refer also to Appendix 1): For each sequence: (1) Draw the 4 t h diagram in the sequence. (2) How many extra squares (in addition to the 3 r d diagram) did you draw? (3) How many squares are there in total in each diagram? (4) Predict first how many more square blocks will be needed for the 5 t h diagram. a. Draw it and check your prediction. (5) Discuss the 6 t h diagram. a. Do you need to draw it in order to know how many square blocks will be needed? (6) What can your group say about the 7 t h diagram? a. How many square blocks in total were needed? (7) Can your group see a pattern in the number of squares you add each time? (8) Can your group see a pattern in the total number of squares used in each diagram? (9) Can your group make any prediction(s) for the 17th diagram? (10) Can your group make any prediction(s) for the 60 t h diagram? The "Tongan" Task with Translated Set of Questions in Tongan The same set of questions was also translated into the students' native Tongan language (see Appendix 2), together with a different, yet similar, pictorial sequence (see Figure 4). The purpose of this task was to see whether the Tongan students would still switch languages, and to observe any significant variation or similarity of such verbal action to their working with the English version of the questions. At the same time, the task attempted to look at the possible reason(s) for these bilingual students' language switching. This objective stems from the claim 112 that Tongan students think in their native language while working with mathematics problems (Fasi, 1999), a claim that might have been significant in explaining why Tongan students would still switch languages. 4.4 Method of Observation 4.4.1 Introduction The search for an appropriate method of data collection was driven by the following questions: (i) How is it possible to capture, categorize, and analyze bilingual students' acts of language switching? (ii) How is it possible to engage and occasion bilingual students to grow in their mathematical understanding, and how can such processes be captured?17 (iii) How is it possible to capture, define, and analyze the relationship between the investigated phenomena in (i) and (ii)? From the outset, it was important to ask why video recording was appropriate in the context of this particular study and to probe the significance of using such a technology in a qualitative study of this nature. Researchers such as Cohen and Manion (1989) have noted that the heart of every case study lies not in the problem to be addressed or the researcher's approach, but in the researcher's method of observation. Because the method of research is so important, the employment of video as an observational tool in this study was central to the trustworthiness of the findings. This is because video, as Asch, Marshall, and Spier (1973) observed, could be to the researcher "what the telescope is to the astronomer or what the microscope is to the 1 7 Towers (1998) defined the notion of "occasioning" as "a situation in which a growth of understanding is allowed for, not caused" (p. 229). This definition, however, refers to how a "teacher intervention" influences understanding. 113 biologist" (cited in Henley, 1998, p. 48). Observation may therefore be seen as the primary function of video, yet video recordings are far more useful in many areas of scientific inquiries than simply as observational tools. Unlike a telescope or a microscope, video records data instantly, and new digital technologies these days are capable of transmitting data directly from all kinds of technological tools. 4.4.2 Video Analysis and the Notion of "Trustworthiness " Case studies, especially those involving video-recorded data, may not be replicable, a condition often associated with any experimental study. Yet the most important factor in any qualitative inquiry of this nature is a determination of the "trustworthiness" of the study - a term used, in contrast to the quantitative notions of "reliability" and "validity", to reflect whether or not the study's methodology, methods, data collection, analysis, or findings are demonstrably justified. Clearly, the choice of methodology, and the reason for employing case study research in this study, is important for any in-depth inquiry. Equally important is the method for collecting and analyzing data, in order to establish trustworthiness and even to determine the "broad applicability" of any qualitative study - a phrase used by Towers (1998). Yet, Bottorff (1994) suggests that in some case studies, certain circumstances might call for observation beyond conventional strategies, such as the use of direct observation. Bottorff described such circumstances, often characterized by situations such as the following: when behaviors of interest are of very short duration, the distinctive character of events change moment by moment, or more detailed and precise descriptions of behaviors and/or processes than is possible with ordinary participant observation are required (1994, p. 244). 114 Bottorff (1994) also argued that a participant observer could not simultaneously monitor all verbal and non-verbal behaviors, even failing to remember most of these instances when the "moment" had passed. It is for these kinds of reasons that the decision to employ video recording as a choice of observation strategy in this case-study research was based. Since the introduction of film, along with the advancement of digital video and computer technology, the role of video in qualitative and, likewise, quantitative research, has become increasingly specialized and sophisticated. The evolution of the technology obviously shapes methodological strategies, effectively opening up new avenues for research. As a result, the integration of video technology within case study research can be considered a significant step toward attaining trustworthiness of a qualitative study. But, as Mignot (2000) asserts, the use of video case study, like many other methodologies, is not just a question of what resources are being used, but how and why they are being used for a particular case or purpose. Hence, Bottorff (1994) attributes successful video-recordings to careful planning and the expertise or competence of the researcher. Ultimately, establishing the degree of trustworthiness is crucial to the success of all qualitative inquiries, particularly in how the notion of trustworthiness relates to the subjective interpretation of data. Adelman (1998) contends that arguments expressed against qualitative studies may be avoided by "honestly acknowledging our certainty and uncertainty in an inference, by being explicit about reasoning underlying our inference from low to a high level of generalization, and by studying the causal process" (p. 159). Towers (1998) goes further in this aspect to note Sanjek's (1990) notion of "theoretical candour" which involves the explicit admission of the "problem addressed, the methods used and the interpretations made" as a way in which the trustworthiness of case study research may be strengthened (p. 51). In this study, the use of 115 video-recorded data, along with think-aloud, video-simulated-recall, and non-participant observer techniques (will be discussed in Sections 4.5.2 and 4.5.3), allowed the researcher to overcome the inherent weaknesses present in this and other forms of case-study research. 4.4.3 Advantages and Disadvantages of Video Study Clearly, the primary function of video as a research tool is data collection and data analysis by scientists, researchers, and academics. Video has also been acknowledged in research as the primary data itself (Pirie, 1996b; Pirie, et al., 2001; Powell, Francisco, & Maher, 2001; Pirie, 2001, personal correspondence; Borgen and Manu, 2002). However, despite the prevalence of video data, little is known or written extensively about its nature in research (Hall, 2000). Nevertheless, the use of video data as a research tool does provide researchers with a number of distinct advantages over other methods of data collection. The first advantage of video data over other data sources, such as a transcribed script, comes from video's ability to capture moving pictures and sound, two valuable attributes in educational research (Bottorff, 1994; Towers, 1998), as opposed to the use of a still picture or photography. In any observational research, Eisner (1991) argues that sensitivity is required toward how something is said and done, in addition to what is said and done, in order to understand what is going on. The "what" may very well depend on the "how", and video provides one with the opportunity to make such a critical distinction. For example, the benefit of using video data in educational research was demonstrated in a recent study conducted by Borgen and Manu (2002) analyzing the mathematical understanding demonstrated by a particular grade 12 monolingual mathematics student. On paper, the student had presented a picture-perfect solution to a calculus problem. Using a video recording of the student's work, a detailed study of the process was 116 carried out to determine what this particular student understood in relation to her written work. In the final analysis, an examination solely of the video transcript, or the simple evidence from the student's written work, would have lacked the details significant to determining the nature of the student's understanding (or lack thereof) of the problem and solution. The recorded video data revealed clear evidence of the student's lack of understanding about key aspects of the mathematical problem. The second advantage of video-recordings concerns what Grimshaw (1982) refers to as the "density" of the video data. Even a two-minute videotape provides the viewer with a vast amount of information that can be deemed much richer than other kinds of recordings, such as a transcribed video or audio data. This characteristic of video data is very significant to this study, because evidence of growth of understanding (such as an act of folding back) can occur in a matter of seconds. However, a researcher must be aware that a video is there to capture and follow, but not to dictate the observed event. In addition, even a video camera cannot record everything there is to know, since any video-recorded material is a selective, rather than a complete record of a particular event, and because all recording devices are subject to inherent mechanical limitations (Bottorff, 1994). Thus, the researcher must make some effort to compensate for such limitations through other supported means of recordings or strategies. For example, researchers are encouraged to accompany video data with supplementary data such as ethnographic field notes (journal entries), copies of the students' written work, or records of open-ended meetings with their subjects, all of which contribute rich data to the case. The third advantage of video recordings is what Grimshaw (1982) refers to as the "permanence" of its data. In this aspect, video preserves what Towers (1998) describes as the "natural colouring" of an event, or what Prosser (1998) calls in film and photographic images "instant 117 appearances". From this perspective, video data is an unalterable record of the actions in time and space, together with the recorded sounds, because the video-tape freezes, but does not alter the observed action, as it happens. Thus, video data allows a time delay for any observer for re-viewing the stored recorded data in the analysis. At the same time, video data allows the researcher a degree of flexibility in the choice of what needs to be observed, recorded, analyzed, or reported. This advantage relates to what Prosser (1998) calls "reflexive data", often used to describe the opportunity for post-event reflections. Thus, video data allows a researcher the luxury of re-visiting and re-viewing selected events as often as necessary in a variety of ways, and the capability of attending to the desired feature(s) of the tape at leisure (Bottorff, 1994; Pirie, 1996b; Martin, 1999). Consequently, video data delays decision-making (Towers, 1998), which Yin (1989) argues to be a key feature of any video case study research, since it permits the researcher to reflect more deeply before interpreting the data. The time involved in reviewing video data may also present another of its major strengths. That is, any data presented on video, beyond mere verbal activity, may be considered relevant to the study (Pirie, 1996b). Pirie (1996b) reported in one of her studies that her classroom video recordings would seem to be the "least intrusive, yet most inclusive" way of studying a phenomenon, particularly when such data is coupled with a study of the students' written work and any field notes of the classroom in general. If it is taken as a primary data source, video data contains the continuous sequence of action, both verbal and non-verbal, as it occurs in real time (Powell, Francisco, & Maher, 2001). Erickson and Wilson (1982) further extol such benefits by maintaining, "such records permit systematic analysis of verbal and non-verbal behaviours in the event recorded" (cited in Adelman, 1998, p. 154). In short, video data achieves contextual validity - also known as "contextual data" (Prosser, 1998) - based on its potential to capture continuity, movement, and contextual speech. In addition, Towers (1998) says that video data's 118 subjectivity may offer its greatest advantage, through its ability to address concerns associated with "descriptive validity". She argues, "This category deals with what Maxwell calls the 'factual accuracy' of an account and, as such, concerns itself with errors of omission as well as commission" (Towers, 1998, p. 79). On this point, Pirie (1996b) noted that any data that is gathered or lost depends on the researcher, where he or she places the camera(s), or even the type of microphones he or she uses. However, the use of videotaped recordings in case studies presents a few disadvantages, largely related to the expenditure of time and resources. For instance, the delayed approach to research by reviewing and revisiting taped events can be costly, tiresome, and time-consuming, and may even turn out to have no relevance to the study (Towers, 1998). In order to remain effective, the researcher needs to maintain his or her focus on the context of the study, keeping in mind its framework and overall purpose. As Bottorff (1994) warns, the "merits of filming with a fixed or movable camera are still under debate and may depend on the focus and purpose of the study, the type of setting, and the characteristics of the participants" (p. 249). Such a focus, together with the video angle used, can freeze researchers to one particular setting of the observed event, and therefore the recorded video data is often considered to be selective, and lacking in contextual data. The contextual restrictions of video data are noted by Bottorff (1994), who assumed that video recordings are limited, but not "sensitive to the historical context of the observed behaviours" (p. 246). In short, researchers are apparently restricted in their view of the event to what is recorded through the video camera lens. The effect of video on the subjects is another major concern. A relaxed environment is therefore important, given the generally accepted fact that the presence of the video camera in any video 119 case study is not invisible to those being videotaped, and also does affect the participants' behaviour during the recorded session (Towers, 1998). This effect is known as "procedural reactivity" (Prosser, 1998). The subjective nature associated with video analysis presents another disadvantage in using video data as a primary source of data. According to this argument, the dynamic nature of any video image is widely open to subjective interpretation. An example of such potential misinterpretation is noted by Goldman-Segall (1993), who asks: "is closing of the eyelid.. .a twitch, a wink, or a conspiratorial communication?" (cited in Pirie, 1996b). In other words, the dynamic nature of video calls into question the trustworthiness of both the data and the analysis. Maxwell (1992) did raise a critical argument when he said that "validity" is not an inherent property of a particular method, "but pertains to the data, accounts, or conclusions reached by using that method in a particular context for a particular purpose" (cited in Towers, 1998). In this regard, Goos and Galbraith (1996) quote Ginsburg, pointing out that, "in any form of research the significance of the data must always be judged relative to the researcher's explicit or implicit theories and assumptions" (p. 236). Nonetheless, much of the criticism to date about qualitative research has been directed towards the observer's perception of what he or she sees on video, regardless of the observer's training in the interpretation process (Towers, 1998). Because the interpretation of observed actions is a serious concern in video analysis, the researcher must be sure to only make inferences based on participants' verbal or non-verbal actions, without making any assumptions about what the participants cannot do. 120 4.5 Data Collection 4.5.1 Introduction During the first week of data collection with the targeted schools, the researcher organized the study's daily and weekly recording schedules. Beginning in week two, the study focused on the chosen groups in one school, for one week, before the research moved on to another school for the following week. Most of the videotaping sessions took place outside normal classroom lessons, either during one of the school's one-hour breaks, or for two hours after school, with the exception of the Saturday sessions at A F C and TCA. A l l necessary videotaping appointments were scheduled through prior arrangement with the teachers and the students, along with the prior approvals of school principals and each student's guardian(s). The video-recording sessions involved the use of a static digital video camera, set up about 10 yards away, and placed directly in front of the students, with a separate microphone wired from the camera to the students' work-table, with the camera lens focused on the entire group. A l l of the students remained seated on one side of their work-table, facing the camera, without moving outside its frame. In addition, there was an interview prior to or after the recordings regarding the students' language use in both Tongan and English in order to gain an understanding of their knowledge of both languages: which language they used at home or at school, which language they preferred for their mathematical work, which language they used with their peers or mathematics teacher, and which language they thought they preferred to use in thinking, etc. 121 4.5.2 The Data Collection Technique(s) One way to gain trustworthiness in qualitative research is to conduct a collective case study or a joint study of a number of similar cases: in this instance, an examination of several individuals from five different secondary schools. Hence, as Towers (1998) suggested, a multiple case study offers an alternative approach to looking at a specific phenomenon. In addition, two techniques were employed to enhance the trustworthiness of the recorded data: (i) "think-aloud" and (ii) "video-stimulated recall". In the first technique, video data is usually gathered in the recording phase by encouraging participants to verbalize their thought processes. This instant verbalization enabled the researcher to match what the students were saying, doing, and writing. In spite of what others might say against this process (for example, Marland, 1984, cited in Pirie, 1996b), this verbal revelation, coupled with the students' non-verbal activities, was the only form of external expression that enabled the observer to get as close as possible to the mind of the protagonists (Pirie & Kieren, 1994). Furthermore, Pirie (1996b) suggested that this think-out-loud process could be used in essentially two ways: (a) as a means of improving students' thinking abilities, and (b) as a means to study students' existing cognitive abilities, which is particularly effective in videotaping a single student. Pimm (1987) supported this claim, adding that thinking aloud helps "the pupils to clarify and organize the thoughts themselves" (p. 23). Moreover, thinking aloud can be enhanced through group-work settings involving the participants interacting with each other. The idea of situating students in collaborative group work allows them contentment and space, by working with their own peers. 122 With regard to the video-simulated-recall technique, the researcher conducted an interview while both he and the participant(s) watched the videotape, in order to clarify, if necessary, the participants' initial thoughts and actions. This process had to be conducted immediately or shortly after the recorded session (Bloom, 1954; McConnell, 1985; both cited in Pirie, 1996b), lest participants find genuine recollection too difficult. Therefore, the video-stimulated sessions were never delayed longer than one day to ensure students clearly recalled what they had been thinking about at the time of the recording when they watched a replay of the recorded session. Thus, it was hoped that the participants were discouraged from explaining what they ought to have been thinking, which could have interfered with the interpretation process (Pirie, 1996b). For the weekday groups, if a recording session was done during the break, a video-stimulated recall session then followed right after school in the late afternoon hours. This offered opportunities for the researcher to view the recorded tape between sessions, and to prepare questions needed to clarify aspects of the students' work. If the recording was scheduled after school, a post-mortem on the recorded session had to be delayed until the next day. With the weekend groups, however, the video-stimulated recall sessions took place shortly after a lunch break during the Saturday sessions. Various other research protocols were in place to ensure the information was interpreted from all possible angles. During the video-stimulated sessions, a member of the group took charge of the video remote control and he or she (or any group member) could stop the video randomly to elaborate about a "special" moment that might offer some insight into their recorded mathematical activities. In these "post-mortem sessions", the researcher (or the students' teacher) also actively participated in the discussion and their questions either emerged during the video-stimulated session or the questions were prepared prior to the session. The researcher's role was to query any activities related to students'. > 123 language use or mathematical understanding: particularly during incidents that showed evidence of language switching and/or growth of understanding of the specified topic. 4.5.3 The Ethical Issues and the Role of a Non-Participant Observer In previous educational studies that focused on the process of the growth of understanding, either the researcher (e.g. Martin, 1999) or the teachers were actively involved (e.g. Towers, 1998) as "participant observers".18 Unlike these earlier studies, this study deliberately excluded outside interventions to allow students the freedom to discuss the tasks among themselves, in their own language, without being forced or coached to speak in a particular way, although the teachers were invited to participate, i f they wished, in the video-stimulated recall sessions and to thus allay concerns over the trustworthiness of the study. In this case, the researcher's decision to assume the role of "non-participant observer" did not preclude his impact on the integrity of the research. As Towers (1998) sensibly noted, "doing research involving human subjects is always a matter for ethical concern" (p. 61). One key ethical aspect to consider was the avoidance of "harm" to participants, whether such harm took place bodily, mentally, or emotionally: a concern previously raised by Towers (1998). To avoid such "harm", the interviews within video-stimulated sessions took place in an atmosphere of utmost respect, considered essential in a relatively isolated society that has yet to consider video technology a household item. 1 8 Atkinson and Hammersley (1994) refer this to a situation where "observation is carried out when the researcher (or teacher) is playing an established participant role in the scene studied" (p. 248). 124 4.5.4 The Supplementary Data While the collected video data was treated as the primary source of evidence, supplementary data provided significant information for video data analysis. The supplementary data included the accompanying field notes and students' work sheets. The field notes were recorded during the video-stimulated recall sessions, especially when the sessions were audio-recorded. The students' work sheets were significant pieces of evidence in the video analysis, and because a different colored pen was used for each student, each student's work was easily identified. After the task, a group meeting involving all participating students and their mathematics teachers was conducted and video-recorded to discuss issues related to the data, participants' experiences, and the research question itself. In one case, some of the teachers from one school (AFC) from the Mathematics, Tongan, and English departments were invited to discuss with participating students their use of language switching in mathematical work. This discussion session proved to be very useful, offering insights into students' learning experiences and teachers' perceptions of the sources and reasons behind the students' use of language switching in mathematical discourse. In addition, the Teachers' and Pupils' Guide Books in each of the grade levels were collected as references to previous work completed by each student and to gain a sense of the scope of the mathematics syllabus throughout the secondary school years. 4.5.5 The Collected Video Data The bulk of the video-recorded data was based on students' completion of the tasks. In the first study, a total of 24 hours of video-recorded tapes were collected from 16 different groups, together with an additional five videotaped hours and four audio-recorded hours of video-stimulated recall sessions. In the second study, a total of 26 hours of video-recorded tapes were 125 collected from 11 different groups, along with an additional five hours of video-recorded recall sessions. This video data was compressed, burned, and stored into mpeg-files. The analysis of this video data was then carried out using video-analytical software called Vprism. The following chapter discusses the role of Vprism in detail (see Section 5.2.2). 4.6 Summary This chapter outlined the methodological approach, design, and method of data collection used to address the research question. These methodological issues were critically reviewed with respect to the actual data collection process, which took place in Tonga between 2001 and 2002. The next chapter begins with the first phase of the data analysis - categorization of the types and "forms" of language switching. CHAPTER 5: D A T A A N A L Y S I S - L A N G U A G E SWITCHING 126 5.1 Introduction The data analysis in this study was approached in order to identify and examine separately both aspects of language switching and growth of mathematical understanding, outlined in Chapters 5 and 6 respectively, with the results of two distinct analyses integrated into a holistic analysis in Chapter 7. Chapter 5 begins, therefore, with a look at the way various categories of language switching were identified, categorized, and developed from the data to provide a language for describing and accounting for the Tongan bilingual students' language switching. Various descriptive examples are drawn from the data to illustrate the Tongan bilingual students' distinct types and forms of language switching. This chapter concludes with a discussion on the data of a particular nature of language switching and how it relates to the bilingual students' mathematical activities, particularly in working with mathematical images. 5.2 Data Preparation for the Analysis 5.2.1 Familiarization with the Data Before focusing on elements related to the research question - either language switching or growth of mathematical understanding - it was essential to develop an overall feeling and familiarity for the data itself, particularly the video-recorded footage. As a result, the researcher gave no attention at this stage to the question, the mathematical topic, or the theoretical 127 framework. Instead, the data was subjected to a "familiarization phase", involving the viewing of each of the recorded digital videotapes at least twice,1 9 without pausing, to get a sense of each video-recorded session as a whole, and a sense of the context within which the students' verbal and nonverbal actions took place. Following each video viewing, the follow-up data was also checked, which included a thorough review of the recorded video-stimulated recall sessions, field notes of those sessions, and the students' worksheets. 5.2.2 The Role of Vprism in the Data Analysis Because of the time-consuming nature of video-data viewing and analysis (Towers, 1998), a large amount of data was first stored and managed through computer video software called "Vprism". Hence, all of the recorded digital tapes were first compressed into mpeg files. The use of the Vprism software allowed the researcher to time-code (using a built-in coding program), transcribe (using a separate, built-in transcribing program), and analyze all of the videotaped sessions within the same frame. This filing process allowed the researcher to link and code the video and its transcript for particular events or categories, thus synchronizing video playback with portions of the transcript by linking key portions of the video with the research text. As a result, the researcher used Vprism to build a history of each individual and group's recorded verbal and nonverbal activities, in which all video clips were annotated with notes and transcripts, and later used to query video collections to find specific items of interest, examples, spoken utterances, notes, subjects, or events. Vprism therefore allowed the researcher to keep track of time, individual speakers, and their observed verbal and non-verbal actions all at once. The Vprism coding program was also used to allow various categories of language switching to 1 9 Each video-recorded tape involved at least two viewings prior to the actual data analysis. This process included: (i) watching each recorded tape at least once prior to each video-stimulated session; and (ii) viewing each tape without pausing during the compression of the recorded digital tapes to mpeg files. 128 be entered simultaneously without having to code the same video segments repeatedly. The capacity of Vprism as video computer software thus offered the researcher a flexible way of breaking down or "de-constructing" a particular video into smaller video segments and re-constructing those segments to provide a holistic analysis of the research subject in question. For each video-recorded tape, data preparation for analyzing the bilingual students' language switching using Vprism involved two main processes: transcribing and flagging. 5.2.3 Transcribing Video Data The need to thoroughly investigate students' language switching called for a transcribed version of the recorded videotapes. Pirie (1991) described how the simplistic nature of a transcript data, "allows us to stop time, to wind back the clock and repeat transient utterances, and therefore enables us to form considerable hypotheses as to the thought processes of the participants at that moment" (p. 143). It was only through transcribing that an explicit and detailed analysis of students' verbal expressions was brought to the fore, making the transcription a more effective working tool, rather than working directly with the video-recorded sound. Although the transcripts provided a more manageable way to analyze the students' acts of language switching, a qualitative study of this nature recognized the importance of treating the original video data as the primary source (Pirie, 1996). Nonetheless, the transcribed video scripts provided an efficient and transparent way of identifying the bilingual students' language switching, along with a necessary means of expressing the data for presentation purposes. The researcher's fluency in Tongan and English allowed him to transcribe the majority of the recorded videos. The recorded video-stimulated recall sessions were used to help the researcher clarify what the students had done verbally and non-verbally. In addition, some of the students participating in the Tongan research would, on occasion, offer to watch themselves on video and offer their own 129 transcription of the events. The students' assistance was helpful, and appreciated, but all of the transcripts still had to be double-checked by the researcher during subsequent video viewings. Table D, below, features a two-minute transcribed video scene, as it appeared in Vprism for a particular group (LHS Form 3 Group 1) in Task 3. The Vprism scene (like Table D) was set up as follows: the "Unit" label showed the group and the tape used (if more than one tape was used for a particular group); the "Time Code" showed the exact time on the tape when a particular event occurred; the "Speaker" column was used to label each participating student using letter abbreviations; the "Transcript 1" column detailed each action (verbal and non-verbal); and "Transcript 2" was used as an alternate section, in which other features and analytical processes could be recorded, such as locating incidents of language switching. Unit: LHS Form 3 Task 4 (Tape 2) Time Code Speaker Transcript 1 Transcript 2 03:16:30 S: Can you see a pattern in the total number — — of square blocks used in each diagram? 03:17:00 Oh, ko e ha e total? Borrow the word "total" from question. (Pointing at the word "total" in the question.) 03:17:08 K: Twenty-eight — ua-valu. Directly translate meaning of a number. 03:17:12 S: Ko e ha e: taha, tolu, ono, hongofulu? (Points at the written totals for Question 3.) 03:17:23 K: Triangular numbers! Borrowing a non-equivalent English term. 03:17:30 S: Triangulars: tolu, ono, hongofulu, taha-nima. Pick up the word "triangulars" for a peer. 03:17:50 ALL: "Yes, they are all triangular numbers." ' 03:18:00 S: Predict the total for the 17lh diagram? Reads question in English. 03:18:09 T: Predict — Ko e ha "predict"? Extract keyword "predict" for translation. 03:18:16 S: Koeme'a — koehakoa? Shifting from English to Tongan. 03:18:25 K: Ko e guess! Translation with equivalent word. 03:18:30 S: Kesi — fakafuofua? Table D: A typical transcribed video scene including "flags" of language switching. 130 The analysis of language switching, at this stage, was based on the actual Tongan-English transcript, rather than the translated copy. The transcripts of the students' Tongan utterances were later translated into English solely for presentation purposes. Once the observed verbal and nonverbal actions from the video were transcribed (using Transcript 1), each occurrence of language switching then needed to be identified. The language-switching pattern in the student's language use was characterized by any alternation between Tongan and English, whether in words, phrases, or sentences. The video data collected from Tonga provided evidence comparable with Baker's (1996) finding, in which the bilingual students' verbal alternation ranged, "from one-word mixing, to switching in mid-sentence,.to switching in larger square blocks" (p. 86). Each of the students' verbal responses was considered an "utterance", described as a word, or group of words that made up a unit of sense or meaning, usually separated from the next utterance by a pause (Sorsby & Martley, 1991)20. The preceding description offers each utterance as the verbal unit of analysis and the basis for categorizing the bilingual students' language switching. In other words, a student who says, "Are those square numbers — one, four, nine, sixteen?"21 indicates two utterances, separated by a pause in between. Thus, from a research perspective, isolating each incident of Tongan bilingual students' language switching became relatively straightforward. The students' verbal expressions were assumed to be instantaneous and explicit externalizations of their thought processes while working on tasks, starting from reading the task, executing it, and reporting on it, such that all thoughts were being used at the same time (Graham, 1997). 2 0 There were instances involving single words or phrases as responses. Taken in context, and in relation to the preceding and following responses to the word, most of these instances were considered meaningful in and of themselves. This semantic consideration led to determining the types of language switching involved. 2 1 The underlined words in this example represent the Tongan utterances being translated into English, and the symbol"—" denotes a pause in the students' verbal expression. 131 5.2.4 "Flagging" incidents of Language Switching The identification of specific features and instances related to the students' language switching involved a process called "flagging". This process highlighted aspects of the bilingual students' language switching that could quickly be attended to or identified. Using Vprism, the Transcript 2 section was used to enter the "flags" to provide an instant account of each incident of language switching. This flagging included incidents involving a student switching entirely from speech in one language to another. For example, student "S", in Table D, above, showed such a shift from English [03:18:00] to Tongan [03:18:16], although there was no flagging when she used Tongan between the coded times [03:18:16] and [03:18:30]. However, flagging involved working with each piece of data in preparation for the next stage in the analysis: categorization. 5.3 Data Analysis: Categorization and Coding Once the transcribed video data was prepared, the next stage in the analysis moved to the development and categorization of emerging "themes" of language switching. These themes constituted the needed language for talking about the particular way the participating Tongan students switched languages. Categorizing each incident of language switching was a demanding task, involving an ongoing development of general themes using the constant comparative approach with all available data. This approach, employed as a way of "grounding" the emerging and developing themes in the data, recognized the role of the researcher in purposefully constructing categories for the current study. 132 It is important to note here, that in determining how to organize the data, the concepts of "purpose" and "relevance" (discussed earlier in Section 4.3.3) offered the main reasons to exclude most of the THS data from the study. These excluded data contained nothing relevant in connection with the research question, and therefore the purpose of the study. The initial analysis after the first study showed that the collected data lacked evidence for language switching - a basic requirement in the investigation. Since THS is arguably the top academic secondary school in Tonga, its students can converse fluently (and perhaps think) in English, and thus do so for most of their group discussions. Thus these students could use English without any language switching, making related data irrelevant to the study. However, one of the students from THS did demonstrate language switching and this student's activities are reported in this study. The initial categorization process began by applying to the data an existing list of language-switching themes Fasi (1999) had developed in his study with Tongan bilingual students (previously discussed in Chapter 2, Section 2.7). Fasi's data came from clinical interviews with students, and from observing their verbal responses from a psycholinguistic perspective. In this perspective, Fasi (1999) grouped instances of Tongan bilingual students' language switching into four main themes - "substitution", "explanatory", "reformulation", and "repetition" - each theme was defined based on the purpose of the switch. These general themes were soon found to be incomplete or inappropriate for the current study, acting instead to constrain the constant comparative process, rather than facilitate understanding (Towers, 1998). There were also difficulties in assigning certain incidents of language switching to one of Fasi's categories. In addition, Fasi's definitions of explanatory and substitution were found to be too vague, and too broad, for the purpose of the data analysis; thus allowing significant features of the words or phrases involved in switching to be easily overlooked. For instance, it became apparent that Fasi's definition of "substitution" did not differentiate between what is distinguished in this 133 study to be three different types of words: equivalent, non-equivalent, and Tonganised words - a distinction that became apparent in the course of the current study's data analysis. Because Fasi's (1999) labels and definitions were deemed inapplicable to the current study, one of his labels ("explanatory") was rejected and others ("substitution", "reformulation", and "repetition") were redefined for the current study. It was then necessary to return to the data to construct new themes, and to find commonalities among certain types of language switching exhibited by the participating students in the current study. 5.3.1 Preliminary Categorization At the beginning, it became clear that Tongan students frequently "mixed" English words in their Tongan mathematical discourses. The analysis, therefore, shifted at this stage to look at the nature of the specific words, phrases, or group of words used in any language-mixed situation. While most of these situations involved mixing English and Tongan words, it was possible to discern switches from Tongan to English, and from English to Tongan. In addition, it became evident that students were using mathematical and non-mathematical terms. So an initial categorization of the mixed words and phrases was divided into five classifications: Equivalent Words: Words that were interchangeable or "translatable" between the two languages. Borrowed Words: Words that were either extracted from the task (questions) or non-translatable. Concrete Words: Words that referred to objects or events that were available to the senses. Functional Words: Words that functioned as verbs, and were used to designate a particular process. Abstract Words: Words that meant ideas or concepts that had no physical referents. Table E shows an example of a list of words and phrases used in mixed sentences for one group of Form 3 students (QSC Form 3) in working with Task 3: 134 Equivalent Borrowed Concrete Functional Abstract Words Word(s) Wordfs) Wordls) Wordfs) plus/tanaki diagram triangular (shape) plus no seven/7 blocks/fitu sequence fa ("four" blocks) tanaki hake ("add-up") yes taha-fitu/seventeen in total five blocks lau (read) like sixty/60th/ononoa predict odd numbers 'ai ("do") imagine first/lst/fika 'uluaki sixth diagram question tanaki ("add") mo'oni ("true") seconaV2nd/fika ua discuss fitu ("seven" blocks) square the number something thiroV3rd/fika tolu seventh diagram seven (blocks) valivali'i ("colouring") know nine/hiva pattern seventeen squared 'ova taha ("over one") faikehekehe twenty-five/ua-nima extra picture 'ova-'ova mai ("over") ("difference") draw/ta extra blocks katoa e ("all that") change manatu'i eleven/taha-taha second diagram fika nima ("No. five") you multiply ("remember") answer/mo'oni fo'i fa ("Four" blocks) liliu ("change") 'uhinga thirteen/taha-tolu next odd number predict ("meaning") difference/faikehekehe number seven you explain numbers/fika fifth diagram tanaki fitu ("add seven") times/multiply/liunga sixty minutes tanaki fiha ("add how many") totals/katoa forty-nine blocks tohi leva ("write then") diagram/fakatata square numbers seventeen times seventeen seven square nima ki he ("five to that") total fitu 'e fitu ("seven by seven") lau sekoni ("count seconds") Table E : The categorization of word-mixing While this list above shows an example of the type of words the participating students used when working with Task 3, the categories were found to be incomplete or inadequate in explaining some of the observed incidents of language switching. In particular, it turned out that these categories were not mutually exclusive, and hence needed to be redefined. For instance, the words "diagram" and "total" were initially defined as "borrowed words", which included both non-equivalent words and words used in the questions. Most of these words were also equivalent words; some of which were also concrete words. In addition, the described categories above only 135 dealt with switching at the word-level, but did not account for switching at the sentence-level. It was therefore necessary to return again to the data to construct new themes, and to find commonalities among certain types of language switching exhibited by the students. This return involved reviewing all the incidents of language switching that were initially flagged in Vprism. As illustrated in Table D earlier, all of the flags in the Transcript 2 section set up the initial categorization process: what each particular switch entailed in terms of its content and language-switching pattern. Table F, below, shows a compiled list of the descriptive flags used in identifying the types of language switching involved for the same group, QSC Form 3, in Task 3: Substituting an English word to emphasize a point. Tonganising an English word in everyday discourse. Relating mathematical work to a keyword in question. Borrowing a word or phrase used by a peer. Labeling in English the mathematical concept. Integrating with a non-equivalent Tonganised word. A peer recalling a concrete English word. Recalling a translated-equivalent Tonganising word. Indirectly translating a concrete Tonganising word. Borrowing a non-translated equivalent English word. Substituting with a Tonganising word. Reformulating the Question in English. Indirectly interpreting question to construct image. Substituting with an everyday or common word. Mixing an English word into a Tongan sentence. Activating an image from a word or phrase. Directly translating from English to Tongan. Paraphrasing the English question in Tongan. Defining and clarifying tasks using translation. Changing language from English to Tongan. Translating an answer from Tongan to English. Determining the meaning of a keyword. Translating a statement using Tonganised word. Substituting with an equivalent English word. A peer explaining constructed image in English. Identifying a key word from the question. Rephrasing the constructed answer in English. Substituting with a key-word from the question. Directly associating two English words. Shifting in reading questions to work in Tongan. Extracting a keyword from the question. Associating a word or phrase with an image. Table F: Sample list of the kinds of language switching involved for one particular group (QSC Form 3) The type of compiled list shown in Table F, above, became the basis for using the constant comparative approach in generating the needed categories of language switching. 136 5.3.2 Coding the Themes using Vprism During the ongoing process of language-switching data analysis, a clear structure began to emerge for labeling the nature of students' language-switching behaviours, including the reason for the switch, the sentence structure involved in the switch, and the content observed within each switch. Vprism was then used to code the data - a technical process of categorizing each act of language switching and any specific incident of interest for further analysis. In using the Vprism coding program, various event categories led to the generation of a list of "Event Types", starting with the most frequently observed events of language switching. Each new category required the creation of a unique code, using up to five characters at a time. An example of the coding process used in this phase of the analysis is shown as follows: " P T K Q " : Peer Translates a Keyword from the Question. "SRSE": Self-Replacing within a sentence a Substitution of an Equivalent word. " S M L N " Self-Mixing within a sentence a Loaning of a Non-equivalent word. By allowing for the systematic creation of a list of codes, the use of Vprism made the categorization process a lot easier and less time-consuming. Categorization was an ongoing and recurring process that involved re-visiting the existing themes with each new piece of data. The Vprism system enabled the researcher to quickly reference the old codes, whenever an event was categorized, or a new code added to the existing codes. Each previous category was revisited and modified when new evidence was observed, leading to the emergence of a new category or the grouping of the incident with a category on the existing list. This re-categorization process continued for each new incident of language switching until all pieces of data were exhausted, 137 and no new category emerged, a stage Glaser and Strauss (1967) call the "saturation" point. Another advantage of working with Vprism was the case of "collapsing" of every identifiable act of language switching into a suitable set of general themes. In Table D, for example, the uses of the words "total" [03:17:00] and "predict" [03:18:09] were grouped under a general theme called "substitution" to indicate their use within each utterance. (The summary of the themes is discussed in Section 5.4.) 5.3.3 Clustering the Initial Themes The categorization process described in the previous section eventually led to the emergence of a new list of types of language switching. Such a categorization process resulted in the creation of various general themes of language switching, observed occurring in both strands of data, and among all the participants, irrespective of their form-level or school of origin. Table G, below, shows a list of the generated themes at this stage of the analysis, identifying an initial generalization of the students' language switching. The list offers a description of each general theme. These general themes were later refined (see next section), and further data analysis revealed a clearer picture of all the types of language switching that took place: Inducing - mainly involving a switch at the word- or phrase-level when non-verbal actions or particular images directly activate specific labels or verbal representations in a particular language. Translation - any language switching that deals with the accessing "routes" between two languages, resulting in either "direct" word-to-word encoding (if not involving images), or "indirect" encoding (if involving i direct activation of images). 138 Substitution a language switching that involves inserting an English word, for example, into a Tongan sentence or utterance. This type of language switching only occurs at the sentence level, as a word or phrase from one language is substituted into another language. Switching describes how bilingual individuals switch verbalization from one language to another in expressing their thought processes. Keywording describes the way students extract a particular word or phrase from an external source (e.g., from questions, or from peers' language use). Table G: An initial list of general themes of language switching for the Tongan bilingual students. During the theme-labeling process, the use of the term "inducing" became problematic, because this term solely defines the relationship from nonverbal to verbal language, rather than also including a description of the reverse relationship. This limited definition prompted the introduction of another term, "activation", considered to be a type of language switching, involving the interconnection between images and language. This new definition was later revised and incorporated into another type of language switching called "grouping", which is discussed in Section 5.4.2. Another problematic term was "translation", which was used in the literature in a wide variety of contexts. Two existing definitions of translation in bilingual mathematics education included: one, the direct and indirect relationships between two verbal languages (see Figure 5), and two, the correspondence between language and mathematics. The distinction between these two definitions rested with the "mediating" role of images (in the process of meaning construction), and whether or not images are considered in the translation process. It turned out that this basic distinction determined the significance of differentiating "translation" as a term for talking about language switching, and treating it as a process involving working with images. 139 (^^TCLISH)^^ • • • " ^ ( T O N G A N J ' % ^idiretf^* IMAGES (Existing Knowledge)] Figure 5: The role of the image as a mediator in the translation between two languages Eventually, the initial list of general themes, described above, developed into the identified forms of language switching (see the next section). Although the term is not explicitly defined in Martin's (1999) thesis, the word "form" is deliberately chosen to describe a pattern in the way the Tongan students switched languages, or as a term to indicate, within the Tongan context, a particular way the students switch in and out between two languages; thus, the preference of the word "form" over similar synonyms, such as "style", "type", or "kind". Form is not just about a particular language-switching pattern, but it is also meant to reflect its purpose and effect on the students' mathematical actions. Nonetheless, the validity of these categorized themes was established largely through "triangulation", whereby the developed themes were shared and discussed with other researchers, colleagues, and especially, committee members. In particular, the aforementioned people were given samples of the data and were asked to consider whether the themes were appropriate with respect to the transcribed excerpts. Initial findings from the study were also presented at two major conferences in North America2 2, and again, examples of transcribed excerpts were offered for the participants to differentiate the categorized themes for the types of 2 2 See the references to a poster presentation at the PME-NA, Athens, GA (Manu 2002), and a short oral paper presented at the AERA meeting, San Diego, CA (Manu, 2004). 140 language switching involved. A l l of these activities contributed to the refinement of the defined themes and made it possible to decide, in the early stages, which themes should be deemed distinct categories and which could be included within other categories. This kind of "independent verification" by the aforementioned people offered validity to the language-switching categories included in the final list. 5.4 Definitions of General Themes Three years of relentless engagement with the available data, trying to determine how the general themes of language switching were to be identified and categorized, prolonged the data analysis considerably. While using the constant comparative method, it became apparent that the Tongan bilingual students' alternation patterns could occur at any point in the conversation (or speech) in two main structural ways: STRUCTURE 1: A switch occurs within an utterance. In this situation, a switch occurs within a single sentence, a verbal behaviour also described as "flip-flopping", or "unstructured" (Macswan, 1999). It involves a single word or phrase from one language being inserted into an utterance in the other language. STRUCTURE 2: A switch occurs between utterances. In this situation, a switch occurs between sentences, a verbal behaviour also described as a "structured" alternation in utterances between Tongan and English. This situation occurred far less frequently and was very much dependent on the bilingual individual's degree of proficiency in both languages. 141 Each structural type cited above consisted of a unique type of language-switching activity, which could be distinguished at either the word- or sentence-level. The nature of each switch depended on the purpose and need for its cause. The following statements [LI-LI3] from the Tongan data illustrate STRUCTURE 1 - the insertions of English words and phrases, including Tonganised English words, within Tongan utterances: Line Transcript 1 [Translated] Transcript 2 [Transcribed] LI "The composite numbers." "Ko e composite numbers." L2 "What is that pattern there?" IToneanised word peteni.~\ "Ko e ha e peteni he?" L3 "Just use row." '"Ai pe ko e row." L4 "So the extra square is how many? Two. eh? '"A ia ko e fo'i extra square ko e fiha? Ua, he?" L5 "The pattern is prime numbers." "Ko e peteni ko e prime numbers." L6 "It's one and three — those two are prime." "Ko e taha mo e tolu ~ ko kinaua ia ko e prime." L7 "Just use block — make another block." "'Ai ko e poloka — make another block." L8 "The composite number — what is it?" "Ko e composite number — ko e ha ia?" L9 "Prime factor — a number that has onlv two factors." "Prime factor, ko e mata'ifika 'oku ua pe hono fakitoa.' L10 "Because so that we know — the order of the pattern." "Because, ke tau lilo 'a e hokohoko 'o e peteni." L l l "There is no such words as 'intruction'!" '"Oku 'ikai ke 'i ai ha lea ko e 'intruction'!" L12 "What is 'predict'?" Is it guess?" "Ko e ha "predict"? Ko e guess?" L13 "Oh. what is the total?" "Oh, ko e ha e total?" In contrast, the following statements [L14-L21] illustrate STRUCTURE 2 - the alternation between Tongan and English utterances: Line Transcript 1 [Translated] Transcript 2 [Transcribed] L14 L15 L16 L17 L18 "Triangular numbers — three, six, ten, fifteen." "One by one, one: two by two, four: three by three-nine; four by four, sixteen — square numbers." "Just use block — make another block." "Oh, I already knew it — add the prime numbers." "Triangular numbers — tolu, ono, hongofulu, taha-nima.' "Taha 'a e taha, taha; ua 'a e ua, fa; tolu 'a e tolu hiva; fa 'a e fa, taha-ono — square numbers." "'Ai ko e poloka — make another block." "Oh, 'osi 'ilo ia 'e au — add the prime numbers." "Prime factor — a number that has only two factors." "Prime factor, ko e mata'ifika 'oku ua pe hono fakitoa.' 142 L20 L19 "The difference is only the answer — — only the answer is change." "Draw the fourth diagram in the sequence. "Ko e faikehekehe pe ko e fo'i mo'oni — — only the answer is change." "Draw the fourth diagram in the sequence. L21 So it is one there [1" diag.], four there." [2nd diag.] 'A ia 'e taha he [1" diag.) — fa he." [2nd diag.] "Twenty-eight — twenty-eight." "Twenty-eight — ua-valu." To further understand the difference between the two described structural types of switching, a critical distinction - often at the centre of most debates about language-switching term definitions - must first be explained. Any act of language switching describes an interaction between two languages that come in contact within the course of a single conversation, whether at the word, or sentence, level. The Tongan data showed one language dominating the other within the course of a single conversation. This inter-language contact relationship is distinguished by one language being considered the primary or "base language", and the other language being the secondary or "embedded language". Although these two concepts continue to be debated, they are defined and used in this study specifically to identify language switching in particular situations. The terms "base language" and "embedded language" will be used throughout the thesis. Differentiating the base language from the embedded language led to labelling the two main structural types of language switching as "mixing" and "grouping". These types and forms of language switching were then ready to be defined. 5.4.1 Types of Language Mixing Language mixing is a type of language switching involving a language change within an utterance, which makes up a unit of sense or meaning. For example, the statement "Find the total number of square blocks needed — then draw the fifth diagram" illustrates a mixing in which the base language is Tongan (represented by the underlined translated English words), and the words "total number", "draw", and "fifth diagram" are embedded into a Tongan sentence. However, the 143 statement "Hanga pe 'e koe 'o square 'a e fika 'o e taiakalami —" ("You just square the diagram number —") shows integration of the Tonganised word "taiakalami" (diagram) within a Tongan sentence, and insertion of the non-equivalent word "square". Definition 1: A "language mixing" refers to a language change at the word-level, whereby a word or a phrase from the embedded language is inserted or integrated23 into an utterance in the base language to make up a single unit of sense or meaning. Since the structural change of this type of language switching occurred at the word-level, the analysis of this switching pattern (and how it related to the students' mathematical activities) focused at the "micro" or word-level. Further analysis of the Tongan data revealed incidents involving a change between the role of the base language and the embedded language, particularly among those students proficient in both Tongan and English. Tolini and Mapa, for example, were Form 2 (grade 8) students who worked on Task 2. At one point during their group's discussion of the task, Mapa used English as the base language while calculating the total for the 7 t h diagram to be "four sevens — twenty-eight." Then he turned to Tolini, and said, "Yeah — you can tohi the pattern. Write the pattern." In this instance, Mapa inserted the equivalent word, "tohi", in place of the English word, "write". Hence, Mapa's example showed why the role of the base language could change from Tongan to English with Tongan bilingual students, as long as these students were capable of using English as a second language. The insertion or integration of words and phrases in language mixing gives rise to two different types of language mixing, defined as forms of language switching: borrowing and substitution. 2 3 The difference between an act of insertion and an act of integration involves the types of words used in mixing. If Tongan is the base language, insertion involves English and Tonganized words that are not part of the Tongan language, and integration involves Tonganized (English) words that have become part of the Tongan language. (See Definition 4 to follow.) 144 Definition 2: Substitution is a type of language mixing in which an equivalent word (or phrase) from the embedded language is inserted or integrated into the base language to replace within an utterance the meaning associated with its equivalent word. Definition 3: Borrowing denotes the process in which a non-equivalent word from the embedded language is loaned and inserted to act as a "stand-in" term within an utterance in the base language. Equivalent words are directly interchangeable (or "translatable") between the two languages, and they are assumed to refer to or embody the same meaning (idea, concept, or image) for a particular individual. Substitution, therefore, refers to those words that have equivalent words in the base language (although they may not necessarily evoke the same meaning between two different individuals), and hence do not really meet any lexical need in the base language (Myers-Scotton, 1993). The substituted words are viewed as being built-in and stored in the bilingual individual's "underlying language capacity", which refers to the individual's available verbal repertoire. Therefore, these equivalent words are easily accessible whenever the individual chooses to use them in group discussion or mathematical discourse. During the study, bilingual students often used one language (English or Tongan) as the base language and then "threw-in" substituted words from the embedded language, without necessarily affecting their intended meaning. A l l of these equivalent words form part of the language foundation in which a bilingual individual verbalizes his or her cognitive processes, which is a component of the individual's existing knowledge and Primitive Knowing that he or she brings to the task. Figure 6 illustrates how, in a mathematical context, the bilingual individual's underlying mathematical knowledge - Primitive Knowing and existing knowledge of the topic - may be "shared" between his or her la