Teaching Mathematics in Multilingual Classrooms

TEACHING MATHEMATICS IN MULTILINGUAL CLASSROOMS

Mathematics Education Library VOLUME 26

Managing Editor A.J. Bishop, Monash University, Melbourne, Australia

Editorial Board H. Bauersfeld, Bielefeld, Germany J.P. Becker, Illinois, U.S.A. G. Leder, Melbourne, Australia A. Sfard, Haifa, Israel O. Skovsmose, Aalborg, Denmark S. Turnau, Krakow, Poland

The titles published in this series are listed at the end of this volume.

TEACHING MATHEMATICS IN MULTILINGUAL CLASSROOMS

by

JILL ADLER Department of Mathematics, University of the Witwatersrand, Johannesburg, South Africa

KLUWER ACADEMIC PUBLISHERS NEW YORK /BOSTON /DORDRECHT/LONDON/MOSCOW

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To Taffy

TABLE OF CONTENTS FOREWORD BY DAVID PIMM

xi

ACKNOWLEDGEMENTS

xv

A NOTE ON TERMINOLOGY

xvi

CHAPTER 1 / The Elusive Dynamics of TeachingMathematics in Multilingual Classrooms 1. Introduction 2. Sketching the Dilemmas 3. A Journey into Theory, Research and Practice 4. Overview of the Book 5. Concluding Reflections

1 1 2 5 14 15

CHAPTER 2 / Complexity andDiversity: The Language andMathematics Education Terrain in South Africa 1. Introduction 2. Dealing with Apartheid’s Legacy 3. Changing Language-in-Education Policy and Practice 4. English Language Infrastructure across South African Schools 5. Language-in-Use in Day-to-Day Activity 6. Curriculum Policy, Mathematics and Language 7. In Conclusion CHAPTER 3 / Accessing Teachers’ Tacit and Articulated Knowledge 1. Teachers’ Knowledge - The Departure Point 2. A Sociocultural Perspective on Teachers’ Knowledgeability 3. Design: Issues in Teacher Selection, Data Sources and Analysis 4. The Noisy, Yet Often Silenced Issues 5. In Conclusion CHAPTER 4 / Dilemmas in Teaching: A Prelude and Frame 1. Introduction 2. Managing Dilemmas: A Practical Account of Teachers’ Actions 3. Dilemmas in Teaching: A Dialectic Account of Teachers’ Actions 4. Dilemmas in Teaching: A Context for Knowledge Growth in Teaching 5. A Language of Dilemmas: A Framework with which to Proceed

17 17 18 23 27 31 31 34 35 35 36 37 41 48 49 49 49 51 54 55

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CHAPTER 5 / Teachers Talking about Teaching: The Emergence of Dilemmas 1. Introduction 2. A First Level of Analysis 3. Commonalities 4. Divergences and the Emergence of Dilemmas 5. Producing the Dilemmas 6. In Conclusion

57 57 57 59 61 67 70

CHAPTER 6 / Language(s) as Resource and the Dilemma of Code-Switching 1. Introduction 2. Code-switching in Bi-/Multilingual Mathematics Classrooms 4. The Dilemma of Code-switching: Action in and Reflection on Thandi’s Classroom Practice 4. Dilemmas of Code-switching across Contexts 5. The Dilemma of Code-switching and Mathematics Education Reform 6. Concluding Comments

91 93

CHAPTER 7 / Dilemmas ofMediation in a Multilingual Classroom: Spotlighting Mathematical Communicative Competence 1. Introduction 2. Mediation and a Social Theory of Mind 3. The Context 4. A Vignette: An Incident and Observations 5. Dilemmas of Mediation 6. In Conclusion

94 94 95 99 102 107 112

CHAPTER 8 / The Dilemma of Transparency: Language Visibility in the Multilingual Classroom 1. Introduction 2. Talk as a Transparent Resource 3. Helen and her Focus on Explicit Language Teaching 4. The Context 5. A Vignette - Classroom Episodes 6. Dilemmas in Explicit Mathematics Language Teaching 7. In Conclusion

115 115 116 117 119 122 131 133

CHAPTER 9 / Central Dilemmas as Curriculum and Research Agenda 1. Dilemmas as Learning Curriculum for Language and Mathematics Teacher Education 2. What then of a Research Agenda?

72 72 73 76 85

135 139 140

TABLE OF CONTENTS

IX

ENDNOTES

144

REFERENCES

148

SUBJECT INDEX

155

INDEX OF NAMES

159

APPENDICES Appendix 1: Glossary of Terms Appendix 2: Table 1 and Table 2 Appendix 3: Methods of Data Collection

161 161 168 170

FOREWORD BY DAVID PIMM ‘Language is a public phenomenon.’ (Jan Zwicky, Wittgenstein Elegies) Questions surroundingthe interrelationship between language and mathematics have become an increasing focus of study over the past twenty-five years. Partly because of the places in which this work was predominantly carried out (as well as other dominances which contribute to what is seen as ‘normal’), the normalised setting was taken to be the monolingual mathematics classroom (simply referred to as ‘the mathematics classroom’). Against this was contrasted the multilingual mathematics classroom (on those occasions from the 1980s onwards when it did come more into focus), taken as either one in which the teacher and pupils did not all share a common first language or one in which there was an imposed language for education which may not have been a first language for either teacher or pupils. Another development in mathematics education over this same period was a move somewhat away from seeing ‘problems’ which had ‘solutions’ (which could make them go away) to a more plural, sophisticated perspective identifying at times mutually-conflicting aims, goals and intentions that required teachers to resolve them as best they could to their own (or others’) satisfaction, to manage them, to manage. In this book, Jill Adler has written an extensive account of her work exploring key dilemmas of secondary mathematics teachers’ practice as they teach. She speaks with a strong ‘I’ voice, grounded in her practice as a researcher and much more, while at the same time lowering the ladder on the activity of research itself by careful and thoughtful commentary, with both her argument and rationale intentionally and explicitly presented. Adler focuses on three main dilemmas of multilingual mathematics teaching and learning and comments on page 15 that “Teachers manage their dilemmas”, adding also in Chapter 3 that “Every day, teachers in South Africa (and in many other countries), manage their mathematics teaching in multilingual settings” (p. 35). David Wheeler (1984), in a different context, wrote about this dual sense of the verb‘manage’: Investigations have to be managed. [...] The idiom [‘I managed to’] is helpful, I think, just because it doesn’t have to meet very stringent conditions. ‘I managed to’ tells us just enough, I might have tackled whatever I was doing clumsily, inefficiently, longwindedly, unimaginatively, etc. but at least I ‘managed to’ so I must have ‘managed it’. (p. 25)

THE LANGUAGE OF DILEMMAS What is a dilemma? The word dilemma has an interesting etymology, originally meaning (according to the Oxford English Dictionary) “a form of argument involving an adversary in the choice of two alternatives either one of which is (or appears) equally unfavourable to them”. While not used strictly in this technical,

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logical sense, Adler’s mathematics teachers, in discussing issues of concern related to their teaching in multilingual classrooms, do find themselves faced with alternatives neither ofwhich allow them to function as fully as they would like. (The Oxford English Dictionary cites the words of one Mr Wilson in his book on logic from 1580, complaining that these dilemmas are such “so that what so ever you graunt, you fall into the snare”.) Adler’s dilemmas are at one and the same time much larger than Berlak and Berlak’s (1981) original sixteen(she draws inspiration from their work for her own) and yet at the same time considerably more specific. All are intimately connected withboth language use in classrooms and the fact that it is mathematics that is being taught. Mathematics places its own stresses and strains — it demands particular things of it. She identifies three key broad dilemma clusters: — code-switching (whether or not and when to switch languages in class, how to grant or assist in gaining access to the particular resources of particular languages, issues that arise when teachers and pupils value and use more than one language in class, switching between the language of instruction and pupils’ main/spoken languages); — mediation (dilemmas that arise when teachers shift towards learner-centred practices that involve more mathematical talk by learners, reforms that are clearly not context-neutral where “communicative competence was taken for granted ... a function of the underlying cognitive approach that assumed talk to be a benign cognitive tool, and that all learners were equally disposed to talking to learn.” p. 10); — transparency (of language as object and as means of communication, dilemmas that arise when teachers, in the interests of clarity and access to mathematical discourse, attempt to work explicitly on mathematical language in the classroom); She claims these dilemmas “lie at the heart of teaching and learning secondary level mathematics in multilingual classrooms in a changing South Africa” (p. 1) and I firmly agree with her. But a key point that she is at pains to make is that while these practices and dilemmas are often heightened in a multilingual arena, they are mostly not specific solely to such settings (code-switching would seem to be so, but see Zazkis, 2000). As a small instance of this, in the context of the third dilemma of transparency, Dave Hewitt (2001, p. 46) writes ofa secondary mathematics teacher in the U.K. : Emma Leaman has talked to me of spending a while with students asking them to chant a new word, so that they become practised in the physical movements of their mouths, tongues, etc. required to say such a word. This helps with the practice of saying the word, but there is still the job of helping students link the word with certain properties.

In much the same way, it can at times be easier to hear the potential for plays on words in a foreign language, because the materiality of language, its physical

FOREWORD BY DAVID PIMM

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features (the noises, the marks), is less invisible than for a native speaker simply throughgreater familiarity. But meaninghas tobe worked at much more. Adler, in Chapter 7, gives a telling example ofa student ‘slip’ between speaking of the ‘side’ and its near homonym the ‘size’ of an angle (the topology of sounds in any language is not the same as the topology of meaning). This is not just a ‘school howler’ — both terms make mathematical sense in the context of utterance. But statements using one word in place of the other flip in terms of their mathematical rightness. Rightly or wrongly, teachers always want to be able to infer from the correct use of language a correct understanding. But the fact that it is possible to say the wrong thing without a misunderstanding being present leaves them unsure of what to do when faced with such a slip. Do they focus on the language which requires disengaging with the content and interrupting the communication to what seems like a language lesson? Or do they let it pass, and risk a misunderstanding going unchecked? In some sense this is the core of the Adler’s dilemma of transparency. It feels reminiscent ofthe Copenhagen interpretation of light as endeavouring to resolve the dilemma of light being both a wave and a particle. It is not possible to attend to both of these perceptions at the same time, but it is possible to switch from one to another. But the question then become which do I deploy in any given novel situation. Likewise, the teacher can attend to language as object or language as means of communication. Neither one by itself is sufficient to teach mathematics successfully, yet both cannot be attended to at the same time. So the teacher needs to manage this, contend with it, live with it. Because each could be attended to at any given time, the teacher needs to manage their class’s attention. Underlying this, I would claim, is the fact that mathematics education has always had twin aims of fluency and understanding, even if at different historical periods and geographic locations one or other has been uppermost, either in practice or rhetoric. Understanding in mathematics is closely connected with language being invisible, while fluency is tied to it being visible (symbols taken as objects themselves rather than pointing to, representing, something else). which, for me, accounts for why it is so difficult (I am tempted to say impossible) to work on fluency and understanding at the same time. The dilemma in this duality for the teacher then lies in controlling the costs of the disruption in shifting attention from one to the other, managing the confusion it can bring, and remaining aware the fact that neither alone is sufficient to ensure both goals are achieved. Jill Adler has written a strong and insightful account of the reality and practicality of these dilemmas in mathematics classrooms while also providing a keen analysis and theoretical grounding for her work. I am confident it will be very widely read. Edmonton,Canada, March 2001

DAVID PIMM

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REFERENCES Berlak, A. and Berlak, H.: 1981, Dilemmasof Schooling: Teaching and Social Change, London, Methuen. Hewitt, D.: 2001 Arbitrary and Necessary: Part 2 Assisting Memory, For the Learning of Mathematics, 21, 1,44-51. Wheeler, D.: 1984, Gatherings, Mathematics Teaching, 106, 24-25. Zazkis, R.: 2000, Using Code-switching as a Tool for Learning Mathematical Language, For the Learning ofMathematics, 20, 3, 38-43. Zwicky, J. : 1986, Wittgenstein Elegies, Edmonton: AB, Brick Books/Academic Printing and Publishing.

ACKNOWLEDGEMENTS First and foremost is my deep gratitude and respect for the mathematics teachers who are so central to this book. They welcomed me into their classrooms, gave generously of their time and participated in the research process openly, selfcritically and with enthusiasm. I can only hope they benefited and were enriched by a process that undoubtedly enriched me. This book is the culmination of many years’ work. As such, it has been shaped and influenced by many colleagues in the field as well as students in my graduate classes. I thank them all. In particular, my thanks go to Shirley Pendlebury, Kathryn Crawford and Lyn Slonimsky for their generous, critical support in the early work. I am indebted to Mamokgethi Setati and Yvonne Reed for many discussions on schools’ English language infrastructure, and language journeys in multilingual classrooms, and to Mellony Graven for perceptive, detailed comments on the final text. I want to thank David Pimm, a friend and colleague in the field, for his critical comments and encouragement on the draft manuscript and for agreeing to write the foreword to the book. I also wish to thank the two reviewers of the manuscript, both of whose comments and insights contributed to its final form. Finally I thank Meg Dickson and Lesley Hudson who were both pivotal to the production of the final manuscript, assisting with macro and micro editing, proof reading and layout. And, of course, my family, Taffy, Joshua and Michelle, for unwavering love and support. TEXT ACKNOWLEDGEMENTS Parts of Chapters 5, 6, 7 and 8 have been the substance ofjournal papers. ‘Dilemmas and a paradox: secondary mathematics teachers’ knowledge of their teaching in multilingual classrooms’, reflected in some of Chapter 5 appeared in Teaching and Teacher Education, 11 (3), 263-274, 1995. Part of Chapter 6 appeared in ‘A language of teaching dilemmas: Unlocking the complex multilingual secondary mathematics classroom’ in For the Learning of Mathematics. 18, 24-33. ‘A participatory-inquiry approach and the mediation of mathematical knowledge in a multilingual classroom’ is the substance of Chapter 7 and appeared in similar form in Educational Studies in Mathematics, 33, 235-258, 1997. ‘The dilemma of transparency: Seeing and seeing through talk in the mathematics classroom’ is the substance of Chapter 8 and appeared in the Journal ofResearch in Mathematics Education, 30, 47-64, 1999.

A NOTE ON TERMINOLOGY An extensive and detailed glossary of terms used in relation to geographic context, schools and language-in-education policy in South Africa is provided in Appendix 1 (p. 161). Terms used to refer to geographic areas in South Africa like ‘non-urban’, ‘township’, ‘suburban’ and to schools in different communities all with their inevitable apartheid past, like ‘African township schools’ and ‘historically white schools’ require explanation. So does the use in this text of apartheid racial descriptors like ‘black’, ‘African’, ‘coloured’. The glossary explains the meanings of these various terms and why I have chosen to use them as I do. In addition, in language-in-education policy in South Africa, terms like ‘primary language’ or ‘main language’ are used in preference to concepts like ‘first language’, ‘home language’ and ‘mother tongue’. These latter are controversial in South Africa and thus require explanation, as does ‘English Second Language (ESL)’ and more general terms like ‘multilingual’ and ‘multilingualism’. In this section of the glossary, I have drawn liberally from the language-in-education glossary provided by Granville et al (1998). Like them, and as with the terminology referred to above, I use the explanations provided to position myself in relation to these terms and how they are being used in this book. The glossary is organised alphabetically, but within these two separated sections. Readers are referred to the glossary, particularly to assist their reading of the first three chapters of the book.

CHAPTER 1

THE ELUSIVE DYNAMICS OF TEACHING MATHEMATICS IN MULTILINGUAL CLASSROOMS 1. INTRODUCTION The inspiration for this book lies in the extraordinary diversity and challenge of post-apartheid South Africa. Its roots are also deeply personal, emerging out of my many years of experience in secondary mathematics classrooms, first as a teacher and then more indirectly, and for a much longer period, as a mathematics teacher educatorandresearcher. The story told is of three key inter-related teaching dilemmas that lie at the heart of teaching and learning secondary level mathematics in multilingual classrooms in a changing South Africa. Called the dilemmas of code-switching, mediation and transparency their simplicity masks the complexity of the classroom and research practices out ofwhich they have emerged — a complexity produced in the interaction between a changing socio-political and educational context, and the dynamics inherent in any mathematics classroom. The substance of this book lies in the stories behind the story, in peeling away layers of interaction between theory, research and practice that took place over a number of years within a research project in South Africa on teachers’ knowledge of their practices in multilingual secondary mathematicsclassrooms. The notion of a ‘teaching dilemma’ was key to capturing and opening up teachers’ knowledge of the elusive, complex and dialectical nature of teaching and learning mathematics in multilingual classrooms. Teaching dilemmas are at once explanatory tools and analytic devices for teaching. They make explicit the tensions inherent in teaching, tensions that are at once personal and contextual. At the same time a language of dilemmas can function as a source ofpraxis — of bridging abstract ideas (theory) and on-the-ground realities (practice). Teachers can use a language of dilemmas to reflect on and transform their practices so as to meet the mathematical needs of their linguistically diverse learners. The contribution of this book lies in the identification and elaboration of those teaching dilemmas mutually constitutive of and constituted by teaching and learning practices in multilingual secondary mathematicsclassrooms. In this first chapter, and by way of introduction to the book, I sketch each of the dilemmas of code-switching, mediation and transparency. These sketches are followed by a discussion of my own practice, its history and how I came to focus a study on the dynamics of teaching and learning in multilingual mathematics classrooms. My motivation was tied then, as it is now, to my interpretation of developments in the wider field of mathematics education. I thus include here an overview of influential research in the broad field of language and mathematics education and how the study relates to more recent developments in the field. I then briefly introduce the study and provide an overview of the chapters that follow. The

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chapter concludes with some reflection on my own goals in producing the book, and what I hope to accomplish through relaying its messages. 2. SKETCHING THE DILEMMAS

2.1 The dilemma of code-switching Jabu and Thandi1 teach secondary mathematics in African township schools in the south of Greater Johannesburg. Both are experienced teachers. Both are multilingual — they speak two or more African languages, as well as English. Their primary language is Setswana. They teach in schools where none of the students2 have English as their primary language, though English is the Language of Learning and Teaching (LoLT). Of particular importance, however, is that the school-leaving external examination in Grade 12 is in English. Like many mathematics teachers, they call on everyday life situations in an attempt to explain mathematics, and struggle with the complex journey their students need to travel between their informal mathematical talk — which is often in their students’ primary language, and formal written mathematics in English. This journey is difficult in any mathematics classroom, and doubly so in a multilingual class. Jabu: Eh, okay you see maybe some work is not done, I scold them in English, and if I feel that it has not got the message home, then I do it either in Tswana or Zulu, these are the languages in the school. Then ... I feel that I reach them. JA3:

And ifyou are trying to explain something new, some maths ...

Jabu: Okay ... In that case I think ofa problem in their day-to-day situation. Then, I feel, ifit will impact to use the language they use at home then I resort to it. ... what did I have recently? Oh yes! 2a - a. To me they said: [pointing to the a] ’‘There is nothing here, so the answer is 2a”. They see 2a and 0a and so get 2a. ... at times [teaching in English] is a problem. You feel bad that you don’t succeed to reach them ... I say “My girl, bring me that book”. Then I say “I have two books, and she brought one book”. And that, problems like that, I go on in English. But there are similar cases, like say half plus half and they have serious difficulties, and then I say, in Tswana: “I have a half a loaf, and a half a loaf, how many ...”?

Thandi: Sometimes you find that you get stuck because students cannot communicate — then, though not much, you resort to Tswana. You are careful because if you do that then they want you to do it all the time, and they turn the situation to a Tswana class. Then they will never improve.

Jabu and Thandi are clear about their dual task. Their first responsibility is to help their students understand and pass mathematics. It is also their responsibility to enable their students to proceed to further education and employment. For this students need to be competent in English, and in mathematical English. As a result of their dual task, they face a continual dilemma of whether or not to switch languages in their day-to-day teaching. If they stick to English, students often don’t

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understand. Yet if they “resort” to Setswana (i.e. switch between English and Setswana) they must be “careful”, as students will be denied access to English and to being able to “improve”. 2.2 The dilemma of mediation Sue teaches mathematics in a well-resourced ‘progressive’ private secondary school in Greater Johannesburg. Students are 99% black, her classes multilingual. The school draws on considerable external funding and is able to provide many students with bursaries. Sue’s main language is English. She learnt Afrikaans for 12 years at school, but no indigenous African language. The LoLT in the school is, unambiguously, English. Sue’s major concern is that her students, all of them, ‘understand’ their mathematics, that mathematics has meaning for them. She believes that mathematical ideas are formed through learner engagement, including discussion and argumentation and sees learner-centered practice as a critical route to meaning-making. She provides her classes mathematically engaging tasks and faces difficulties as she works with increased levels of learner-learner discussion. Sue: Because they are not sure of their own ideas they lose track in their arguments. One kid starts talking about one thing and the other points to something else and then they lose their argument. This happens in class a lot ...

A dilemma for Sue is that while it is important for learners to explore, explain and argue their interpretations and ideas, they easily “lose track”. Coming up against important conventional mathematical meanings requires her intervention. Yet, if she intervenes prematurely, she could, however unintentionally, discourage learners from expressing and exploring their own ideas. This dilemma is profound for Sue and it highlights a key challenge in our contemporary period where we strive at the same time for inclusion and voice and for greater mathematical access. Of course, appropriate and timeous mediation is a challenge for all mathematics teachers, particularly in the light of reform movements in mathematics. It is not specific to a multilingual classroom. However, Sue expresses what other teachers have said, that difficulties of learners communicating their mathematical ideas to the teacher and each other in English are exaggerated in a multilingual class setting. The dilemma of mediation involves the tension between validating diverse learner meanings and at the same time intervening so as to work with learners to develop their mathematical communicative competence: Sue: ... like there are some kids who are really not good at explaining themselves, and I don’t do anything to address that except to try to get them to explain it again because the class hasn’t understood. And they still do it badly, and then I say can someone help him? And by listening maybe he will get the chance to develop. I haven’t spoken to the English teachers about that but I want to because I am sure they have got strategies of actually developing a more exact way ofcommunicating. JA: And that would be an important thing to do? S:

I am wondering. I don’t know — these questions are starting to arise....

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2.3 The dilemma of transparency Clive, Sam and Helen all teach in historically ‘white’, suburban state schools that have deracialised rapidly. Their classes are multiracial and multilingual. All three teachers have English as their primary language, and like Sue, learned Afrikaans at school. Helen is multilingual. She speaks French and isiZulu in addition to English and Afrikaans. Clive, Sara and Helen all found that explicit mathematical language teaching was beneficial in their classes. Moreover, the benefits seemed to extend to the whole class, not just learners with main languages other than English. Helen, Clive and Sara are convinced that being explicit, particularly about mathematical discourse, is essential in a multilingual mathematics classroom. Clive: ... one is that they [learners] just don’t understand the use of words. They don’t understand the links between certain words, words which have similar meanings. ... and certain words which have rigorous meaning, like ‘at least’, ‘at most’, things like that, which are used in everyday language. I think black learners use them differently. They use ‘at least’ in particular ways ... Logical things and negations are difficultfor second language learners, and we have those things in mathematics. Sara: As I have said it has made one more aware of being careful about how you present things because you know there will be kids who don’t understand everything you say. Whereas beforeyoujust assumed that because kids spoke English at home they could understand everythingyou said, but they don’t ... Helen: ... I can’t think of an example, but it has happened to me several times. Where I would have assumed a few years ago in an all white class I would have just gone ahead and talked aways and now because there were black children in my class and I was writing up in a conscious effort to explain the English that I suddenly realised it was benefitting the English-speakers as well.

Helen was particularly alert to any confusions her learners had related to mathematical discourse. In one of her trigonometry lessons in her Grade 10 class (ages 15-16), for example, students were naming the ratios incorrectly, in ways that did not make sense to Helen. Some probing uncovered that students were confused by the words “opposite” and “adjacent” now referring to angles in relation to sides of a triangle, and not only to angles (as had been their previous experience in geometry). She then spent some time drawing students attention quite explicitly to these different uses of terms within mathematics. However, explicit teaching of mathematical language becomes more complex when students are involved in task-based activity, and generate informal ways of speaking mathematically. Inevitably, mathematical descriptions are partial or quasi-mathematical (Pirie, 1998), or, as Helen found, sometimes ‘they do it right but say it wrong’ ... in retrospect, when I look at that lesson, I went on but much too long [laughter] on and on and on and I keep saying the same thing and I repeat myself, on and on .. and I watch the video and I think I wonder why they are still sitting in their seats and I amfalling on thefloorfalling asleep. But the thing is, then, ifyou have sense that there is a shared meaning amongst the group can you go with it? um ... when the sentence is completely wrong? ... Can you let it go? Can a teacher use a sense of shared meaning to move on? I think this is a central question in [mathematical] discussion.

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Helen’s dilemma between implicit and explicit language practices pervades classroom practices, not only mathematics, and not only multilingual classrooms. There is always the problem in explicit language teaching of ‘going on too long’, of focusing too much on what is said and how it is said. Yet explicit mathematics language teaching appears to be a primary condition for access to mathematics, particularly for those students with main languages other than English or for those learners less familiar with school discourses. The three sketches above capture dilemmas of access, voice and meaning in teaching secondary mathematics in multilingual classrooms. The dilemmas are at once personal and contextual, a function of the teachers themselves, their mathematical and pedagogical goals, and the diverse multilingual contexts in which they work. Under any circumstances, teaching is a complex and sophisticated activity, imbued with tensions and dilemmas arising from the continual need to communicate for pedagogic ends. When a range of main languages are present in a classroom, the challenges of effective communication are highlighted. Increasingly, teachers all over the world are grappling on a daily basis with the fact of multilingual classrooms. They too will face dilemmas. The stories behind these three key teaching dilemmas in multilingual mathematics classrooms are developed theoretically and empirically as the book unfolds. They emerged through a qualitative study rooted in South Africa and the specificity of its urban, multilingual secondary classroom contexts. The study was undertaken between 1992 and 1996. Like any research, its roots go w ay further back and its branches have extended outwards since. 3. A JOURNEY INTO THEORY, RESEARCH AND PRACTICE

3.1 My practice My work in mathematics education in South Africa spans twenty-five years, and a range of institutions and learner levels. This time-span is telling: my overall involvement has been dominantly influenced by the apartheid years, and the struggle for democracy in South Africa. My particular interest was (and remains) educational quality and equity for all South Africa’s learners, as well as learner access to a meaningful education that produces mathematical competence and acknowledges and works with diversity. In the 1980s and early 1990s, concern with quality, equity and social justice in education in general, and in mathematics education in particular, was not confined to obviously unequal contexts like apartheid South Africa. Wider international movements in mathematics education named and explored race, class and gender inequities in mathematical performance. They sought reform of the narrow procedural nature of the mathematics offered in school and its related pedagogy. They also advocated more interactive and communicative classrooms and hence, meaningful learning. All learners were to be offered more open and authentic mathematical tasks that would engage them in conceptually rich mathematical ideas.

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Learners were to be given more of a knowledge voice in the classroom - a voice where diverse learner conceptions and ways of knowing were to be taken seriously, and brought into interaction with the set curriculum. The diversity of South African society made it obvious to me that a pedagogy built on increased and substantive communication between learners and teachers would place particular and additional demands on teachers and learners in multilingual settings. Yet nowhere in the research that accompanied and produced these moves in mathematics education reform, was the multilingual classroom problematised. Classroom communicative competence was taken for granted. In a multilingual class, the considerable challenge of changing classroom cultures to enable opportunities for mathematical conversations (talking to learn and learning to talk mathematics) would inevitably be further complicated by learners simultaneously working to acquire the language of learning and teaching - English I found myself constantly concerned about the possible consequences of more open and conversation-rich pedagogical practices in multilingual settings. Could we be sure that the nature of the practice sought, and its construction as ‘good’ and ‘high quality’ mathematics education, would indeed benefit all learners irrespective of context and culture? I was challenged by the demands of what I interpreted as a three-dimensional dynamic at play in a multilingual mathematics classroom: teachers and learners would need to work simultaneously for access to English, to specific mathematical discourses, and to ways of talking in school classrooms (classroom discourses). I shared my interest in communicating mathematics with practising mathematics teachers, both black and white, in the graduate courses that I offered at the University of the Witwatersrand in Johannesburg. Working from the assumption that knowledge is situated, made and not given, my seminars included critical engagement with the mathematics education literature related to the communicative and cognitive function of ‘talk’ in mathematical meaning-making, and changing classroom cultures to encourage talking to learn mathematics. We also dealt with the specificity of the language of mathematics, and studies of bilingualism and mathematics learning. That is, I drew from research and development in the wider field of language and mathematics education to provoke discussion of each of these three dimensions that I saw as constitutive of the dynamics of teaching and learning in multilingual mathematics classrooms. Each year, with each new group of students, the most interesting and heated session would be the one which grappled with the challenge and effects of having to communicate mathematics in English when English was not the main language of most learners and teachers. Each teacher invariably had a story to tell — either from teaching in multilingual classrooms or from his or her own learning of mathematics. The teachers did not (perhaps could not) prise apart the difficulties of access to mathematics as a specific discourse, from access to mathematics in English. Their stories revealed contradictory assumptions, including: - mathematics is difficult for everyone, irrespective of the learner’s main language since problems of understanding have more to do with the mathematics itself than with English as the LoLT;

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-

-

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learning mathematics in school in a language that is neither the teacher’s nor the learners’ main language places additional and complex demands on teachers and learners; learning mathematics in English is necessary, especially at a secondary level; language is learnt through use, thus learners need to be compelled to use English in the mathematics class; learners need to be able to use their main language in mathematics lessons — they can’t understand some concepts if they are only explained in English.

There was a universality to the teachers’ stories. Most teachers asserted that learning mathematics in English while learning to speak English was a double challenge for learners as well as teachers. Yet, the dynamic of this challenge was elusive. It was either too deeply embedded in teachers’ tacit knowledge or less of a problem than they articulated. This elusiveness was reflected not only in teachers being unable to specify the challenges, but also in the absence of a multilingual focus in the actionresearch tasks they chose to carry out as part of their course requirements. Yet, dayto-day, these teachers faced, and found ways to manage, their double challenge. Herein lay the motivation for prising open teachers’ knowledge of their practices in their multilingual mathematics classrooms, and hence the starting point of my study. 3.2 The field 3.2.1 Research in bi-/multilingual mathematics education In 1994, summarising a conference on researching African classrooms, Rhubaguyma stated that “ ... more attention needs to be given to the ways in which teachers and learners actually get things done when there are two or more languages in day-to-day classroom practices” (Rhubaguyma, 1994, p.1). While the complex relationship between bi-/multilingualism and mathematics learning has long been recognised, research in bi-/multilingual mathematics education reported in the 1980s and early 1990s did not focus on classroom practices. For example, Dawe (1983), CockingandMestre (1988), Zepp (1989), Clarkson (1991), Durkin and Shire (1991) and Stephens et al. (1993) have all argued that bilingualism per se does not impede mathematics learning. Their focus was on cognitive functioning of learners in bilingual settings, and particularly learners whose “mother tongue” was different from the “medium of instruction”. Some of this research explored particular aspects of the mathematics register, like word problems, or logical connectives. Most of the research, however, explored the relationship between levels of bilingualism and mathematics performance, drawing extensively from, and building on, Cummins’ (1981) theory of the relationship between language and cognition, and his notion of the “threshold hypothesis”. Cummins distinguished different levels and kinds of bilingualism, and showed a relationship between learning, levels of proficiency in both languages, and the additive or substractive model of bilingual education used in a school. Secada (1992) has provided an extensive overview of research on bilingual

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education and mathematics achievement, and pointed to findings of a significant relationship between the development of language and achievement in mathematics. In particular oral proficiency in English in the absence of “mother tongue” instruction was negatively related to achievement in mathematics. This field of research has, however, drawn much criticism, largely because of its cognitive orientation and its inevitable deficit model of the bilingual learner (Martin-Jones and Romaine, 1986; Frederickson and Cline, 1990 in Baker, 1993, p. 144), criticism with which I agree. The argument is that school performance (and by implication, mathematics achievement) is determined by a complex of inter-related factors. Poor performance of bilingual learners thus cannot be attributed to the learner’s language proficiencies in isolation of classroom processes shaped as these are by wider social, cultural and political factors that infuse schooling. Notwithstanding the significant contribution of the pioneering work in the field of bilingualism and mathematics education, Rubagamya’s observation in the early 1990s that we knew little then about how things got done in multilingual classrooms was apt. It reflected my own experiences and concerns as I worked with this earlier research on bilingual mathematics education in my courses. My concern with this research went even further. In addition to not exploring how teachers and learners got things done on a day-to-day basis, none of the research drew directly on practice-based knowledge (Lampert and Ball, 1998), on what teachers themselves knew about the demands of their practices in these settings. More recently, research has shifted to classroom practices. Analyses have focused on interactions in the classroom, and how learners’ main languages interface with the language of learning and teaching and the ‘subject’ (mathematics) being learnt (e.g. Secada, 1992; Arthur, 1994; Ndayipfukamiye, 1994; Khisty, 1995; Moschkowich, 1996, 1999; Setati, 1998a, 1998b; Adler, Lelliott and Reed et al., 1998; Adler et al., 1999). The shift brings into focus a different unit of study. The cognising individual is replaced by discursive practices. Classroom discourse becomes the focus, and utterances between teacher and learners, and between learners themselves, the unit of study. This research has shifted the problematic from a deficit in the learner and what treatments might therefore be required for learners to overcome their disadvantage, to a location of the problem in the wider social order. The project instead has become understanding, describing and explaining a set of complex social interactions and relations in the classroom. There is a concomitant shift in theoretical assumptions from a relatively narrow focus on learning as a function of individual cognition, to a wider conception of learning as constituted in and through social, and particularly discursive, practices. Simply, classroom communication and learners’ communicative competence are not taken for granted. Moschkovich (19%) and Khisty (1 995) in the USA, and Ndayipfukamiye (1 994) in Burundi and Setati (1998a) in South Africa have all shown ways in which switching between the learners’ main language and English (or French) by learners and the teacher has enhanced the quality of mathematical interactions in the classroom. Setati and Moschkovich have since explored strategies teachers use to support learners’ participation in mathematical discussions in their multilingual settings. Setati’s current study of Grade 4 classroom interaction provides an

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interesting contrast between teachers who rarely switch out of English in the public domain, and one who does. Drawing on Cobb’s (1998) distinction between calculational mathematical discourse (talking about steps in calculations) and conceptual mathematical discourse (talking about concepts) Setati demonstrates that calculational discourse dominates in classrooms where switching is restricted. In contrast, when the teacher herself switches into the learners primary language in the public domain, this correlates with conceptual discourse becoming the focus of discussion Furthermore, in interviews where learners were asked to talk about the mathematics they had le arnt in this teacher's classroom, they were able to shift between talking about s teps in calculations and talking about concepts (Setati, 1998b). Moschkovich (1999) has provided a fascinating analysis of excerpts from a lesson where young Spanish-speaking 3rd graders in the USA were supported by the leacher in their mathematical discussions in English in the public domain. The strategies the teacher used included “modelling consistent norms for discussions revoicing student contributions, building on what students say and probing what students mean” (p. 18). Together these studies reveal that the relationship between language and mathematics education in bi-/multilingual classrooms is particularly complex. Understanding how teachers and learners get things done in these complex contexts requires much more than probing learners’ access to and proficiency in the LoLT, e.g. English in the USA or the South African context. It requires working with language-in-use in the classroom, and thus simultaneously with access to English, to mathematical discourse and to classroom discourses. As I have noted earlier, a threedimensional dynamic is at play. Moschkovich works with this dynamic to develop her argument that strategies used by the teacher in her study for focussing explicit attention onto mathematical discussion might well be of significance in all mathematics classrooms. Her assumption is that subject-specific discussion is not spontaneous for any mathematics student. It is learnt in a context of participation with others (teachers) who translate, model, revoice and probe the contributions of ‘newcomers’ (learners) to school mathematical practice. These more recent studies have focussed on code-switching as a teaching and learning resource, and on the teacher’s role as a language guide in the mathematics class. As key actors on the classroom stage, teachers are more present in these studies than in earlier bilingual mathematics education research. Yet, their knowledge of what they do and why as they manage the three dimensional dynamic in their day-to-day practice has remained backgrounded. I hope that through prising open some South African teachers’ knowledge of their practices in their multilingual classrooms, the study reported in this book will not only make a contribution to closing this gap, but in so doing, it will add depth to our increasing knowledge in this field. 3.2.2 General research in the field of language and mathematics education Austin and Howson’s article on Mathematics and Language published in 1979 placed language in all its complexity on the research and development agenda in mathematics education. So too it is known that extensive work on mathematics and

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language has been done by Pimm (1981, 1987, 1992, 1995). Pimm’s seminal book: Speaking Mathematically: Communication in the Mathematics Classroom published in 1987 provided the first theoretical analysis of mathematical language as it has come to be spoken and written in school mathematics practice. He explored Halliday’s notion of the mathematics register and learner access to mathematically valued written and symbolic form through day-to-day classroom communication. The field of mathematics and language has expanded considerably since then. Pirie, for example, did extensive research into and description of what constituted mathematical (as opposed to other) discussion (1988, 1989). Cobb and his coresearchers (e.g. Cobb, Wood and Yackel, 1992), together with numerous other studies driven by constructivist ideology (e.g. Murray. Olivier and Human, 1993) explored how mathematical meaning came to be co-constructed through learnerlearner interaction on mathematical problems. These latter studies point to the second issue that I raised as I described my practice in the previous section. A classroom culture that had learner-learner interaction at its centre was being promoted. Yet, communicative competence was taken for granted. This was a function of the underlying cognitive approach that assumed talk to be a benign cognitive tool, and that all learners were equally disposed to talking to learn. Taking communicative competence for granted was further symptomatic of the fact that bi-/multilingual sites had not been included in such studies. More recently, a result no doubt that experience has shown that communicative competence cannot be taken-for-granted a diverse literature linked to language and mathematics has emerged where the starting assumption is that language, and so too, forms of communication, need to be problematised. Studies range from a sociology of mathematical texts (Dowling, 1998), to a critical discourse analysis of mathematical Writing expected in investigative tasks (Morgan, 1998) and analyses of word problems and journal writing as genre (Gerofsky, 1996; Waywood, 1994). Together this research contributes to our understanding that communication in mathematics is never benign. Unfortunately, research and practice in bi/multilingual mathematics classrooms remain absent from these deliberations. In 1997, a panel discussion was held on “Learning mathematics through conversation: Is it as good as they say?“, and has since been fully reported in Sfard et al. (1998). The panelists interrogated the taken-for-granted ‘truism’ that has emerged in mathematics education that mathematics can (and should be!) learnt through conversation. The point of the debate was to question this seemingly obvious phenomenon, and make it an object of reflection (p.41). The paper is interesting and informative. It provides an analysis of different theoretical orientations to the centrality of conversation in knowing (cognitivist, interactionist and neo-pragmatist) as well as arguments from the panelists who work from these different perspectives. Sfard identified a common core to these diverse arguments: mathematical conversation has potential as a mode of learning. “[T]he question is not whether to teach through conversation, but rather how” (p.50). She acknowledged that orchestrating a productive mathematical discussion or initiating a genuine exchange were extremely “demanding and intricate”, thus pointing to a “decisive role” for the teacher. She concluded:

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... Communication skills cannot be taken for granted ... if conversation is to be effective and conducive to learning, the art of communicating has to be taught. How this is to be done, and what exactly should be learned by the students remains a question to which the mathematics education community has yet to give much thought (in Sfard et al., 1998, p. 51).

Sfard has thrown out a welcome challenge to the mathematics education community. But this challenge is not new. Hicks argued a similar point: The work of educational reformers such as those embracing the NCTM standards for mathematics teaching has led to important changes in the form of classroom communication .. [However] educational researchers and teacher-researchers may need to move one step further than altering the participant frameworks that constitute more traditional instructional practices. They may also need to begin serious enquiry into how the heterogeneous voices of students and teachers situationally constitute classroom discourses and what counts as academic knowledge (1995, p. 86).

For the teachers in focus in this book, what counts as mathematical conversation and how this is facilitated and developed in their classrooms is central to their mathematics classroom practice, constituted as it is by multilingualism. Indeed, they give communication a great deal of thought. Moreover, they share an acute awareness of linguistic differences in their classrooms and that they need to consider how their language practices enable or constrain not only the class as a whole, but the diverse learners within it. The point here is that it is precisely the challenge of establishing effective mathematical communication — of understanding the significance of the teacher’s voice in this, and learning how this is done in classrooms where there is diverse communicative competence in the LoLT — that has driven classroom-based research in bi-/multilingual classrooms. Here communication skills simply cannot be taken for granted. In mathematics classrooms in South Africa, and so too the North American research discussed above, there has been progress in dealing with the challenge Sfard poses. For example, Moschkovich’s (1999) research identifies strategies of the teacher as discourse guide in the mathematics classroom — strategies that, in her view, and mine, are important in any mathematics classroom. Yet this research is ignored, perhaps in ignorance of its existence, or, more likely, because it is viewed as siginificant only in and for explicit bi-/multilingual settings. The same can be said of the 1998 National Council of Teachers of Mathematics (NCTM) publication: Language and communication in the mathematics classroom (Steinbring, Bartolini Bussi and Sierpinska, 1998). This book had its origins in the working group on language and mathematics at ICME in Spain in 1992. In the years following and prior to publication, participants worked on their various papers, and additional contributions were invited. While the authors do not claim that the book is a comprehensive coverage of the field, the 20 chapters written by authors from a range of countries, provide a spread of current approaches to the field of language and communication in the mathematics classroom. There are numerous significant messages in the book, empirical and theoretical. In particular, across all of the chapters and their varying theoretical approaches is a rejection of a transmission view of knowledge and learning, and so of language as an unproblematic conduit of mathematical knowledge which learners then fail to grasp, or teachers fail to deliver

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appropriately (or both). This orientation is a significant move away from a deficit model of learners and teachers. In addition, there is an increasing shift away from a technical view of mathematical language towards engagement with the complexities of what counts as mathematical language, and the view that mathematics, like language, is a culture, formed and forming in use. In the introductory section of the book Sierpinska (1998) provides a similar analysis of three distinct theoretical approaches in the field to those identified by Sfard, above, reinforcing the shift towards a focus on discourses and their production of mathematics through classroom interaction. In addtion key messages in the book particularly the chapters by Sierpinska and Bartolini Bussi) reinforce the point Sfard made about the significant role for the teacher in mediating mathematical conversations. Sierpinska discussed Steinbring’s notion of the “epistemological dilemma in every mediation of mathematical knowledge” (in Sierpinska, 1998, p.55). Steinbring argues that mathematics is difficult not because transmission is impossible, but because the specificity of mathematics itself imposes stringent demands on communication. Mathematics is about relations, not about things, and relations cannot be experienced directly. Mathematical communication is therefore dependent on linguistic means. New topics mean new terms, symbols and definitions, all of which require mediation. In a similar vein, Bartolini Bussi presents conmunication in the mathematics classroom as a paradox: that meaning can neither be transmitted nor simply negotiated. Scientific concepts cannot be created anew but need to be assimilated as products of centuries of culture (Bartolii Bussi, 1998, p. 83). Meaning of these concepts is only found when students come to share mathematical discourse with others. Tension is inevitable as teachers move between supporting students’ personal senses and meanings, and established mathematical cultures. Throughout the NCTM book, aside from a brief mention of bilingualism as an issue in a chapter that deals with “access” (Clark, 1998), there is no reference to language and mathematics education in multilingual classrooms, let alone a chapter specifically focussed on the issues involved. As Morgan states in her review of the book, ‘‘issues of bilingualism are completely missing” (2000, p.94). Why is this so? Perhaps the issues of bilingualism were discussed in a different working group at ICME-6? Why then was no chapter commissioned, as were other selected chapters in the book. Is the ‘otherness’ of the multilingual classroom such that it is not conceived as having any potential bearing on the wider field? Or, more mischievously, do the kinds of practices observed to support learners in bi/multilingual mathematics classrooms appear to be too teacher-controlled? The kinds of practices supportive of mathematical discussion in multilingual classrooms discussed above, for example, by Moschkovich and Setati, include strategies where the teacher explicitly molds and shapes learners’ mathematical voices, as well as moments where the teacher is clearly in control of the discussion Perhaps, as I have argued elsewhere, it is because these explicit and directed mediational moves by the teacher run counter to dominant beliefs as to what counts as a participative mathematics classroom culture (Adler, 2000a), and that “too much teacher control is a ‘bad thing”’ (Morgan, 2000, p.95).

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And this would be despite increasing acknowledgement of the teacher’s role in supporting and promoting mathematical conversation. Speculation aside, the question remains as to why there is a continuing disjuncture between research on communication in bi-/multilgual mathematics classrooms on the one hand, and what could be described as more mainstream research on communication in the mathematics classroom on the other. For the teacher in a multilingual setting, all the issues raised above are simultaneously present and important. I hope that the study I report, focused as it is on teachers’ knowledge of their practices, goes some way to bringing these overlapping yet separate research areas under the same spotlight. 3.3 The Study The study that forms the substance of this book took place between 1992 and 1996 and during a time of massive political change in South Africa. A particular feature of changing landscape was the racial desegregation of schools. While the apartheid state formally collapsed in 1994 with the first democratic general election and the adoption of a new constitution, the deracialisation of South African state schools was already underway by 1991. Since the late 1970s, many private schools (particularly Catholic ones) had opened their doors to all South African learners. Deracialisation processes produced at least three different kinds of urban multilingual secondary schools. Firstly, there were, and still are, historically ‘white’ state schools where the learners are now racially diverse, and bring a range of main languages into the classroom. A good percentage of each class would be main language English speakers. The LoLT is unambiguously English, and the vast majority of the teaching staff are main language English speakers. Secondly, there are also private schools, and now some inner-city state schools where teachers are predominantly main language speakers of English, but the vast majority of learners are not main language English speakers. Here too, the LoLT is again unambiguously English. Thirdly, and in contrast, there are black urban township schools where neither the teacher nor the learners are main language English speakers, yet English is still the target language in the school. The study focused on six teachers, two from each of these three multilingual contexts, and their accumulated knowledge of their practices in their diverse multilingual settings. That teachers are knowledgeable about their practice was the starting assumption of the study. Teachers manage, and have managed, such complex teaching contexts. There is a great deal that mathematics education as a field can learn from teachers’ tacit and articulated knowledge of their practices in multilingual mathematics classrooms. While the three multilingual contexts noted above are still prevalent some three years after the conclusion of the study, there has been dramatic change at the level of policy both for language-in-education, and curriculum. In particular, pre-1994 only English and Afrikaans had official language status and these were the only official LoLTs at secondary level. Now there are eleven official languages - a recognition of the nine African languages spoken and used in and across South Africa, and the new

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Schools Act enables schools to choose their LoLT. In practice, however, English remains the language of power, dominant in government and finance, and consequently the adopted LoLT across secondary schools. Thus, although the deracialisation of historically white schools has intensified since 1996, the multilingual contexts in which the study was carried out are likely to constitute a substantial part of the landscape of urban secondary schools for some considerable time to come. The dominance of one language over others in multilingual classroom contexts is not exclusive to South Africa. In such classrooms, there will be unequal distributions of linguistic capital, in Bourdieu’s (1990) terms, that is diverse linguistic competence across learners in terms of access to the LoLT. It is obvious that those for whom the LoLT is their primary language bring with them to the classroom, a decided advantage. While the messages contained in this book derive from the South African context, its substance will appeal to and resonate with mathematics teachers and teacher educators working in multilingual settings elsewhere. Furthermore, schooling itself, and mathematics within it, invoke particular ways of speaking and meaning, discourses in Gee’s (1992) terms, or language games in Wittgenstein’s (1969) terms. Such linguistic competence will also be distributed unevenly across learners, and thus poses a challenge for mathematics teachers wherever they are. As I have already suggested, the messages in the book therefore extend to teachers whose classrooms are not multilingual in the obvious sense. 4. OVERVIEW OF THE BOOK In Chapter 2, I describe the complexity and diversity of the mathematics teaching and learning terrain in South Africa. I focus on language-in-education as well as mathematics curriculum policy and practice and look at how these have changed in the past decade. The central feature of the chapter is that despite policy pronouncements that value diversity and encourage multilingual teaching practices, the practical implications of working with more than one language in a mathematics class, particularly in the light of curriculum reform, remain complex and elusive. Chapter 3 focuses is on a range of methodological issues that were confronted as the research unfolded. The purpose here is to open up empirical, theoretical and political issues that inevitably shape any research endeavour and so too the story that can come to be told. In so doing, the approach to the study is described. Brief mention is made of the broad theoretical approach that informed the design and the study as a whole. Theoretical detail is provided in each of the chapters that follow. Chapter 4 discusses the notion of a teaching dilemma and its history in educational research as a prelude to the description in Chapters 5 – 8 of the three key dilemmas of teaching in multilingual mathematics classrooms. I have named these the dilemmas of code-switching, mediation and transparency. They capture key challenges that emerged through conversations with and observations of teachers in multilingual mathematics classrooms in South Africa and refer, respectively, to the dilemmas described previously:

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— dilemmas that arise when teachers and learners value and use more than one language in class, switching between the language of learning and teaching and learners’ main/spoken languages; — dilemmas that arise when teachers shift towards learner-centred practices that involve more mathematical talk by learners; — dilemmas that arise when teachers, in the interests of clarity and access to mathematical discourse, attempt to work explicitly on mathematical lan guage in the classroom. Chapter 5 describes the emergence of the dilemmas through the initial interviews with the teachers. The discussion that follows in chapters 6, 7 and 8 shows how the three dilemmas, and the language of dilemmas more generally, became the key with which to prise open and capture the complexity of teaching in multilingual mathematics classrooms. Where appropriate, the discussion extends to reflect subsequent research and development in relation to language practices in multilingual mathematics classrooms in South Africa. Moreover, as each of the dilemmas is described and explained, it will become clear that while dilemmas are expressed as binary oppositions, they are never ‘either/ors’ in the complex flow of classroom life. Instead, they are sources of praxis, of transcending inherent tensions in the dialectial teaching-learning process. In short, teachers manage their dilemmas. Sometimes teachers are fully aware of the choices they make, choices that are at once personal, practical, social and political and specific to mathematics teaching. At other times, in managing the complex three dimensional dynamic of access to the language of instruction (English), access to mathematical discourse and access to classroom discourse elements of their practice are obscured. Chapter 9 concludes the book with a discussion of implications for mathematics education, mathematics teacher education and research. The discussion highlights how the specific dilemmas identifed provide a language of description, by which I mean a way of talking about, reflecting on and questioning mathematical practices in complex linguistics contexts. Drawing on current debates in mathematics teacher education, and arguments for professional learning situated in practice, the book closes with the offer of the potential of cases built around central dilemmas as critical resources for professional learning and hence for mathematics teacher education. 5. CONCLUDING REFLECTIONS The structure and tenor of the book is a reflection of the particular way I have come to impose meaning on phenomena as a mathematics teacher educator and rese archer located in the academy. The first four chapters of the book discuss field, context and theory/method. This provides the reader with access to the particular setting that has given rise to its messages, to my selections from related fields in mathematics education and how I made connections between the particularity of my study and these wider fields, as well as to the assumptions about knowledge, world and being that infuse my work.

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While context, field and theory are never static in any practice, they provide an important backdrop for the empirical substance contained in Chapters 5-8, and thus offer significant means for readers to engage critically with the text. Of course, the inevitable linear presentation belies the actual scholarly process. There is always a dialectical interaction between the empirical and the theoretical, and this is reflected in the discussion in Chapters 5-8. Moreover, I am most certainly a different and more informed author at this point than five years ago. The presentation of my work in this chapter, and those that follow, is a product of both the completed research and my subsequent research and development experience in the field. I have written this book with a scholarly audience in mind, an audience of teachers and teacher educators engaged with mathematics education as a field of study, possessed of research interests in the field. While the specific classrooms are junior and senior secondary, the examples and reflections are such that the book can appeal to teachers at both primary and secondary levels. Moreover, language educators who have an interest in the language demands of different classrooms (i.e. classrooms that are not teaching language(s) per se) will hopefully find the book of interest to their work. This book, and the study on which it is based, is dedicated to all mathematics teachers who persevere with commitment and belief in assisting their learners’ access to mathematical know-how, across wide-ranging contexts of acquisition. I hope that in identifying and naming dilemmas of mathematical practice in multilingual settings, and in providing illustrations of dilemmas in practice, this book offers mathematics teachers and teacher educators possibilities for talking about and acting on the daily challenges posed by multilingual mathematics classrooms. The three cases described in the book emerge from practice, and so constitute the raw material from which cases or scenarios can be constructed to engage teachers in on-going professional learning where a focus of attention is managing dilemmas in multilingual mathematics classrooms. I offer the dilemmas of teaching mathematics in multilingual classrooms not as some constructed unassailable truth, but rather as possible objects and means — resources — in the practice of mathematics education, particularly teacher education. Whether they become such is a function firstly of the coherence and rigour of the story told. In addition, there needs to be a wider acceptance that the dynamics of multilingual classrooms are indeed worthy of specific engagement and focus. Ultimately, however, the value of the study lies not in whether and how the dilemmas are generalisable beyond the specific cases illustrated here, but rather on whether they are generative — taken up and used in educational research and practice elsewhere. It is my hope that the ideas explored here do indeed prove generative.

CHAPTER 2

COMPLEXITY AND DIVERSITY: THE LANGUAGE AND MATHEMATICS EDUCATION TERRAIN IN

SOUTH AFRICA 1. INTRODUCTION The breath-taking extent and pace of social, political and economic transformation in this early post-apartheid period is a defining feature of contemporary South Africa. Rapid change produces extreme and contradictory conditions that in turn throw a spotlight onto complex social practices. What might otherwise remain hidden or taken for granted in ‘normalised’ day-to-day practices can be seen in a new light and in new ways. Teaching and learning mathematics — curriculum-in-use — in multilingual settings is one such practice exposed by the spotlight of postapartheid South Africa. Long an abused fe ature of most South African schools, the multilingual classroom has now been accorded appropriate recognition as a fundamental mainstay of language-in-education policy, theory and practice. Such recognition is probably required of many educational systems throughout the world in our new millennium. Accordingly, an analysis of curriculum-in-use in the South African context offers an internationally valid and dynamic site for exploring complexity and opportunity in multilingual mathematics classrooms. What are the dynamics of teaching and learning mathematics in multilingual secondary classrooms in South Africa? What can we learn from South African teachers’ accumulated experience of such practice? These major concerns of this book beg additional questions that combine to form the main purposes of this chapter. What is implied by the notion of ‘a multilingual mathematics classroom’ in South Africa? What do such classrooms look like? How are they shaped by wider socio-political and economic conditions? In this chapter I will describe changing curriculum and language-in-education policy and practice, and locate these within the wider context of diversity, inequality and poverty within the country. By locating a study of the dynamics of teaching and learning mathematics in multilingual classrooms firmly within the South African context, my intention is at once to localise the research, and so enrich its story, and at the same time provide the means for its take-up and recontextualisation elsewhere. Placing educational research done in South Africa clearly within its socio-economic and political context is what enables others elsewhere to make sense of the conclusions drawn for their own practices in their different contexts. For it is always in the interaction between theory and practice, and between policy and on-the-ground realities that the issues embedded in the teaching and learning of mathematics in multilingual classrooms need to be understood. The extent of diversity, inequality and poverty in the country presents South Africans with enormous challenges — challenges that easily overwhelm. At the same time, each day, the many steps forward reinforce realistic possibilities of a more humane and democratic future, and within this, a qualitatively better education for

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all South Africa’s children. The South African landscape that I paint, therefore, is complex, difficult and contradictory. However, I paint it with a brush tinged with optimism, so bringing to life the different multilingual mathematics classrooms that form the focal points of the research that informs this book. 2. DEALING WITH APARTHEID’S LEGACY: POVERTY, INEQUALITY AND DIVERSITY It is necessary to stress at the outset that a full analysis of the problems and promise of the ‘new’ South Africa is beyond the scope of this book. The description provided here is brief. It does not and cannot do justice to the complexity of the socio-political economy of South Africa.1 However, such a full description will conclude that the ‘new’ South Africa is better than the old, not only in political terms but also in terms of quantitative data related to GDP (Gross Domestic Product), economic growth, balance of payments, overall distribution of income and social services, and in terms of what the majority of South Africans think.2 This said, the problems and challenges facing South Africa and its citizens remain enormous, requiring huge backlogs to be overcome in every field, a task made more difficult by political, social, financial, physical and human resource (what in South Africa is often referred to as ‘capacity’) restraints. Dealing with apartheid’s legacy requires an honest assessment of these problems and a realistic programme to deal with them. The following pages of this chapter provide this assessment in the areas of relevance to this book. Ways in which teachers deal with what seem at times overwhelming odds are found in the chapters that follow. 2.1 Income and employment While six years have passed since the first democratic elections in South Africa in 1994, and much has changed, apartheid’s legacy of structural racial inequality and widespread poverty remain defining features of the South African landscape. There are high levels of unemployment and huge disparities in annual income. According to the 1996 census (STATS SA, 1998), the poorest 40% of the population receive only 11% of the total income, while the richest 10% who constitute only 7% of the population, receive 40% of the total income. This said, there has, however, been a redistribution of income from the richest to middle income households and an emerging black middle class is blurring race boundaries (Reality Check, Survey Published in Independent Newspapers on April 28, 1999). A recent Statistics South Africa survey found that the annual incomes of African, coloured and Indian households rose substantially over the period 1990-1995 but that, over the same period, the income of the poorest, predominantly African, families had declined (The Star Newspaper, 16/06/2000). While income, GDP and other economic growth indicators have improved, employment in the formal sector has, in common with international trends, declined. Unemployment in 1996 (and there have been significant job losses since then) was 33,9% overall, with 52% of women

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unemployed. Moreover, 62% of those employed earn less than R1500 per month (roughly US$200). Other socio-economic indicators related to HIV-Aids, crime levels and domestic abuse confirm the high levels of economic and social inequality. Constituted as it is by this diversity, and deeply affected by excessive and increasing disparities between rich and poor, South Africa is indeed a microcosm of the growing economic divide between rich and poor countries in the new global economy. Within the economic growth, income and employment figures are complex racial inequalities which provide both challenge and opportunity. The very emergence of an entrepreneurial black middle class provides economic and development opportunities in industry, commerce, manufacturing and social services. At the same time the challenge is to ensure that the widening gap does not create a permanent class of the dispossessed. 2.2 Urbanisation The rich-poor divide is also an urban - non-urban divide. The term ‘non-urban’ is used in preference to ‘rural’. The latter, as it typically refers to agrarian communities, is too narrow a concept to capture the range of living and working conditions outside the urban areas of South Africa. The two richest provinces (Gauteng- where the cities of Johannesburg and Pretoria are found, and the Western Cape which contains Cape Town) are largely urban. The two poorest provinces (Northern Province and Eastern Cape) are largely non-urban. According to the 1996 census, of the 40.5 million South Africans, 18.8 million or 46.3% live in non-urban areas. Of the 21.7 million (53.7%) in the urban areas, over 7 million, that is one third of all urbanised South Africans, live in Gauteng. Gauteng, the industrialised hub of the country, accounts for 38% of the national GDP, and is 97% urban. Once again the changing demography of the country provides a powerful impetus for change and growth. At the same time changes like urban drift create development problems in both urban and non-urban areas arising from lack of investment, a lack of human and institutional capacity, and a range of social issues in education, housing and welfare. 2.3 Languages-in-use The 1996 Census also provides an analysis of “home language by province”. Only 8.6% of the entire population gave English as their primary or main language. This is close to the number for whom Setswana is primary language (8.2%) and in contrast to isiZulu (22.9%), isiXhosa (17.9%), Afrikaans (14.4%) and Sesotho (9.2%). The remaining five recognised and spoken African languages are constituted by smaller numbers who claim these as their main language. The point of these figures is that they reflect similar inequality profiles to that of income. Only a small percentage of the population has English as their main language. Yet English remains the language of power, commerce and government, and thus an advantage experienced by a privileged minority.

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Within these overall national figures lie important provincial differences and so too urban - non-urban differences. In the poor, predominantly non-urban Northern Province, for example, only 0.4% of people have English as primary language. 90% of people in the Northern Province have either Sepedi, Tshivenda or Xitsonga as their primary language. As is explained in more detail below, there is limited English seen, spoken or heard in the streets of much of this province. In contrast, in Gauteng, there is a spread of primary languages (13% English, 17% Afrikaans, 21.5% IsiZulu, 13% Sesotho and 8% Setswana). As a predominantly urban province and the industrial and commercial heartland of the country, English is widely seen, heard and spoken. 2.4 Schools and schooling Similar inequalities pervade schools across the country. Differential distribution of material and human resources remains highly visible across South African schools. The relative wealth of schools in historically white middle-class suburbs in contrast with impoverished schools in black townships, in non-urban areas and in the increasing spread of informal settlements is well known. The recent Schools Register of Needs (Bot, 1997) reveals that a staggering 17% of all schools in South Africa lack basic physical infrastructure. There is serious overcrowding in some of these schools, with classes of up to 100 students, and in 23% of all schools there is no running water nor any toilet facilities in or close by the school. In short, learning and teaching is not only hampered by shortages of learning materials in such schools, but physical conditions actively detract from possibilities for focussed attention on learning and teaching. These conditions are compounded by the situation that all final external national Grade 12 examinations are still only offered in English or Afrikaans with the obvious exception of languages as subject. In short, large numbers of learners in South Africa live in a community where there is a high level of unemployment, and shelter and health facilities are minimal. They leave a poor home (be it urban or non-urban) to arrive at a poor and overcrowded school and face having to demonstrate their competence in subjects like mathematics in English. Even though there are possibilities for drawing on learners’ primary language(s) for learning, ultimately they will be assessed in English (or Afrikaans). Relatively speaking, a smaller, albeit growing, proportion of South Africa’s children are fortunate to live in communities of material comfort, and go to well-resourced suburban schools where the language in and surrounding the school is English. As will become clear through the stories told in this book, these ranging conditions bear down heavily on the teaching and learning of mathematics in school. 2.5 Teachers and teaching Most white, and some black, secondary teachers have a three-year university degree with specialisation in one or two teaching subjects, followed by a one year post graduate teaching diploma. They are professionally trained and have a strong subject

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knowledge base in their teaching. In contrast, most African teachers, primary and secondary, were only able to acquire qualification through three-year diplomas in segregated, apartheid-constructed Colleges of Education. There are large numbers of teachers, predominantly African, who are teaching mathematics and might be officially qualified (they have a three year post matriculation teaching diploma), but they have had minimal opportunity to develop and strengthen their subject knowledge base for teaching. A National Audit of Teacher Education and a followup audit of Mathematics and Science teacher education in 1997 (Hofmeyr and Hall, 1996; Arnott, Kubeka, Rice and Hall, 1997) reported that: - over 50% of our current secondary level mathematics and science teachers have less than one year of post secondary study in these subjects - over 50% of our current secondary level mathematics and science teachers have less than three years of experience - the majority of black teachers received poor quality training in mathematics and science as was provided by many Colleges of Education. Over and above the deep inequality and neglect produced through teacher education under apartheid, African teachers across the country, irrespective of their primary language and particularly at the secondary level, are expected to be able to teach their respective subject in English. One of the effects here is that as some teachers struggle to communicate mathematics in English, errors are fossilised and conveyed as such to learners.3 2.6

Performance indicators

These compounding conditions are accurately reflected in success rates of schools across the country. All Grade 12 students sit an external matriculation examination, and the 1999 results provide an indicator not only of educational inequality, but also of the dysfunction of large parts of the education system. In a ‘league’ table of school results published in South Africa’s largest weekly newspaper, The Sunday Times (16/01/2000), there was a 0% pass rate in 80 secondary schools across the country. In each province there were many more whose pass rate was below 20%. However contentious such indicators might be, and however problematic and further counter productive such league tables and their underlying assumptions might be, the conclusion from the tables is clearly that too many institutions are failing our children, and these institutions are predominantly in historically black urban townships and/or non-urban areas. To focus more specifically on mathematics, the pass rate in the 1999 examination was 43.4%.4 That half of the students who sat the examination failed mathematics is not only economically disastrous, but devastating to individual lives. In South Africa, as is the case elsewhere, mathematics functions as a critical social and economic filter. It is a requirement for most avenues of tertiary study, and often used as an indicator of potential employment competence. As part of post-apartheid discourse and practice, these 1999 matriculation figures hide racial descriptions. There can be no doubt that the distribution of failure in mathematics is heavily skewed in racial terms. Recently released results of the 2000

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matriculation examination show some improvement. The overall pass rate increased from 48% to 57%, and the number of schools with a 0% pass rate dropped from 80 to 43 (The Sunday Times, 28/01/2001). These signify a system slowly emerging from a devastating past. Sadly, however, the overall improvements hide the fact of no significant improvement in mathematics. In this context, the South African results (last of all the participating countries) in the Third International Mathematics and Science Study (TIMSS) of 1997 and TIMSS-Repeat (released in December 2000) were not surprising. I agree with Keitel and Kilpatrick (1999) that such results must be treated with caution and interpreted within the wider politics of international comparative assessment. As Cooper and Dunne (2000) argue there is sufficient evidence to question whether the performance scores obtained through the kinds of assessment strategies used in large scale international tests are indeed reflective of what learners know. South African learners were considerably disadvantaged in the TIMSS testing situation: all were tested in English; and the multiple choice testing format is not a familiar testing practice in South African schools. Moreover, sampling problems have been acknowledged in TIMSS-SA (Howie, 1998). Nevertheless, the very poor performance by South African learners on basic mathematics items in TIMSS supports the view that our education system continues to fail too many of our children. As will become evident later in the chapter, language-in-education policy and practice is a key factor here. 2.7 Fiscal constraint The extent of poverty in South Africa weighs heavily on the national budget. There are critical needs in all social services, and thus limits on amounts available for each of housing, health and education. In recognition of the damage done by apartheid education, and the need for redress, repair and development of educational opportunity across the country, the education budget is an enormous 21% of the GDP, higher than most other countries of a similar economic level. Within education, staff salaries (educators and bureaucrats) consume around 90% of the education budget in all provinces, There is little remaining for building of new schools and classrooms and for providing basic physical infrastructure in many schools, let alone for the development and distribution of learning materials. The vision for a democratic and prosperous South Africa is thus undermined by the inevitable costs of the extensive human resource and infrastructural redress and development required. 2.8 The context of practice As mentioned earlier, the South African landscape can easily produce despair and loss of hope for a better future. That is not my purpose here. On the contrary, I am continually invigorated by the extent of human agency in the country that confounds analysts of doom. Instances of progress and transformation, small and large, are in evidence across the country. These are seldom the subject of media attention and

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hence less publicised. My purpose in reminding readers of the extent of apartheid’s legacy is that such a backdrop is important to the full meaning of the stories told in this book. New language-in-education and curriculum policy are being mapped onto an environment of inequality, diversity and significant fiscal constraint. It is in the interaction between a vision for a new future and the history on which it is being built, that the tensions and contradictions of practice that emerge must be located in order to be understood. 3. CHANGING LANGUAGE-IN-EDUCATION POLICY AND PRACTICE It is widely acknowledged that education policies, and so too language-in-education policies, are overdetermined by economic interests and political ideologies (Vinjevold, 1999). In the apartheid era official language-in-education policy was specifically and explicitly designed to serve the apartheid state. It met with fierce resistance, the primary example being the 1976 Soweto Revolt (KaneBerman,1978). It is beyond the scope of this book to provide a full history of language-in-education policy in South Africa. I will nevertheless sketch out key language-in-education developments during the Apartheid era. After taking over power in 1948, the Nationalist Government passed legislation and extended the resources necessary to establish Afrikaans alongside English as a fully fledged official language and fully functioning ‘medium of instruction’ (LoLT) in schools. At the same time, different language-in-education policies came to define the racially segregated Departments of Education. According to Hartshorne (1992) prior to Nationalist Rule in 1948, there was a relatively loose policy of ‘mother tongue instruction’, certainly in primary schools. The duration of ‘mother tongue instruction’ varied from province to province. When the Nationalists came to power, African schools, specfically, were removed from provincial administrations and placed under the National Department of Bantu Education. The Bantu Education Act passed in 1953 stipulated that ‘mother tongue instruction’ be phased in across all primary school grades in African primary schools, with English and Afrikaans compulsory subjects from the first grade. In addition, English and Afrikaans were both to be used as language of instruction on a 50-50 basis in African secondary schools. African teachers were given five years to become competent in Afrikaans. Alongside these policies for African learners, policies for white, so-called coloured and Indian schools were also segregated along apartheid racial lines but came under different legislation. Learners in white, coloured and Indian schools were required to take both English and Afrikaans throughout the 12 years of school, one at a first language level, and the other at either first or second language level. Depending on department and location, the LoLT in these schools was either English or Afrikaans. A few schools were dual medium. English and/or Afrikaans were the primary languages of white, coloured and many Indian learners. In general, white, coloured and Indian learners were able to learn through the medium of their primary language in both primary and secondary schools. Many analysts trace the Soweto uprising in 1976 to rather belated attempts to enforce the controversial and highly contested “50-50” language policy for African

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learners first promulgated in 1953.5 Secondary students in Soweto, together with their teachers, came out in strong protest against the policy that attempted to bring Afrikaans in alongside English as a LoLT. The Soweto Students Representative Council rejected “... the whole system of Bantu Education whose aim is to reduce us, mentally and physically, into ‘hewers of wood and drawers of water”’ (in Simmonds, 2000). Some were killed, many detained and hurt. In the wake of the revolt and the turmoil that spread across the country, the government passed a new act in 1979 stating that at the primary school level, learners should be learning through the ‘medium’ of their ‘mother tongue’, but that the wishes of parents should be considered after Grade 4. Many African parents were vehemently opposed to ‘mother tongue’ as the ‘medium’ of instruction. This policy was viewed as part of the apartheid grand plan to deny Africans access to socio-economic advancement. In 1990, perhaps as a last gasp, the Nationalist government passed an amendment to the 1979 Act, giving parents the right to choose between: going directly for a ‘second language’ (e.g. English) as LoLT; a sudden transfer from the ‘mother tongue’ to a ‘second language’ as LoLT; or a more gradual transfer. While there is no systematic research evidence, it is widely held that many schools with an African student body adopted English as the language of learning and teaching (LoLT) from the first Grade (ages 6-7) i.e. a straight for English policy (Vinjevold, 1999). Nelson Mandela’s release and the unbanning of the African National Congress in February 1990 signalled the beginning of the end of the apartheid state. Multiple policy initiatives began across all social services. In terms of language policy a process was initiated to fully recognise the rich multilingual nature of South African society. The New Constitution adopted in 1996 for post-apartheid South Africa has given the country 11 official languages. Nine African languages - Sepedi, Sesotho, Setswana, siSwati, Tshivenda, Xitsonga, isiNdebele, isiXhosa and isiZulu - have been added to English and Afrikaans, the only two languages that enjoyed official status in the apartheid era. A critical feature of language policy initiatives has been the engagement with the hegemony of English i.e. the increasing naturalness of its symbolic power. In December 1995, in recognition of the historically diminished use and status of the nine official African languages of the people of South Africa, the Minister of Arts, Culture, Science and Technology announced the establishment of a Language Plan Task Group (LANGTAG). LANTAG’s task was to identify South Africa’s language related needs and priorities. The Minister noted that despite multilingualism being a “sociolinguistic reality’’ and being explicitly provided for in the Constitution, there was a “tendency to unilingualism” (i.e. English) in the country. LANGTAG articulated a multilingual policy for South Africa, and proposed the widespread use of the nine African languages as the means to countering the dominance of English (LANGTAG, 1996 in Granville, et al., 1998). Multilingualism has been given educational substance in the South African Schools’ Act, and through ongoing language-in-education policy initiatives. The most recent Language-in-Education Policy document (DoE, 1997b) advocates an additive model of multilingualism. Learners are to add languages to their repertoire of linguistic resources for learning. An essential feature of the additive. model is that the learners’ main language is maintained and developed, and used in the teaching

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and learning situation as a language of learning, alongside other language(s) of learning. Subject to any law dealing with language-in-education and the constitutional rights of learners, in determining the language policy of the school, the governing body must stipulate how the school will promote multilingualism through using more than one language of learning and teaching, and/or by offering additional languages as fullyfledged subjects, and/or applying special immersion of language maintenance programmes.(Department of Education (DoE), 1997b, p. 8).

Not only can South African schools now choose their language of learning and teaching (LoLT), but there is a policy environment supportive of the use of languages other than one favoured LoLT in school, and so too of language practices like code-switching — drawing on more than one language in the course of pedagogic action. Cultural diversity is to be preserved and valued through the maintenance and development of the range of spoken languages in the country, and learning and teaching is to be facilitated as learners draw on their languages as social thinking tools. Moreover, learners are to add new language(s) to their repertoires, and not subtract out their main language. Additive multilingualism is advocated ideologically and pedagogically, and is indeed ‘progressive’. It resonates with and reflects policy research elsewhere (JET, 1997). This widely acknowledged ‘good’ policy is already meeting significant on-theground constraints. Recent research suggests that most schools are not opting for main language as LoLT policy and practice (Taylor and Vinjevold, 1999; NCCRD, 2000), and that, ironically, there is a “decrease in ... primary language instruction in junior classes and consequent increase in English language instruction” (Taylor and Vinjevold. 1999, p. 216). This situation is not unexpected. Firstly, as the history described above reflects, main or primary language as language of learning policy has a bad image among speakers of African languages. It is associated with apartheid, and hence with inferior education: Parents’ memories of Bantu Education, combined with their perception of English as a gateway to better education, are making the majority of black parents favour English [as LOLT] from the beginning of school. even if their children do not know the language before they go to school. (NEPI, 1992, p. 13).

Secondly, urbanisation has created more complex and diverse multilingual classrooms. The presence of 5 or 6 different primary languages in one class gives further impetus to English as LoLT. Most learners in such classrooms. particularly at a secondary level, have English at least as one of their additional languages. English, it is held, then functions as the common language and as a potentially unifying language. In fact, English is becoming more and more dominant because the majority of parents want their children to learn in English. This point is forcefully made in the overall report of the 38 research projects undertaken across a range of schools during 1998 as part of the President’s Education Initiative (PEI) research project (Taylor and Vinjevold, 1999). The popular underlying assumption here is that a language is best and most easily acquired in use. Thus, although new language policy in South Africa is intended to address the overvaluing of English and Afrikaans in relation to

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African languages, in practice, English continues to dominate. Despite being the main language of the minority, English has become both the language of power and the language of educational and socio-economic advancement. English has both symbolic and material power. In Bourdieu’s (1990) terms, English is the dominant symbolic resource in the linguistic market in South Africa. Particular linguistic skills, like competence in English, are required of social actors for access to valuable social, educational and eventually material resources. Granville et al. (1998) draw on Bourdieu’s conception of linguistic capital to engage with LANTAG’s proposals for widespread use of the nine African languages. Starting from unequivocal support for the principles of mutlilingualism in the Constitution, they nevertheless challenge the assumption in the LANTAG proposals, that widespread use of African languages can in itself counter the status and increasing symbolic and material power of English They elaborate how Engllsh has come to have such power and point out that the dominance of English is, in fact, a global phenomenon. They quote Chrystal at length: English is used as an official or semi-official language in over 60 countries, and has a prominent place in a further 20. It is either dominant or well established in all six continents. It is the main language of books, newspapers, airports and air traffic control, international business and academic conferences, science, technology, medicine, diplomacy, sports, international competitions, pop music and advertising. Over twothirds of the world’s mail is written in English. Of all the information in the world’s electronic retrieval systems, 80% is stored in English. English radio programmes are received by over 150 million people in 120 countries. Over 50 million children study English at primary level, over 80 million study it at secondary level.(Chrystal in Granville et al., 1999, p. 261)

Bringing this home, daily local newspapers and weekly national newspapers are predominantly in English (there are also daily and weekly Afrikaans newspapers). For example, The Sowetan, a daily newspaper with major circulation in Soweto and neighbouring Johannesburg areas, is only published in English. There is only one Zulu newspaper, circulated largely in KwaZulu-Natal. Granville et al. conclude that all learners be provided access to English, on the grounds that without such access, the poor in particular will be further marginalised They include in their proposals specific recommendations for the study and use of primary and additional languages in school so as to reach the long term goal of primary language as LoLT for the majority. At this juncture, support for primary language as LoLT is growing amongst 6 practitioners and policy makers. Behind this is mounting evidence of the problems produced when subtractive multilingualism becomes practice (Macdonald, 1991; JET, 1997). A subtractive language environment is argued as a key factor in the production of high failure rates in schooling across the country (e.g. Alexander, 1997).7 Pedagogically the demands made of so many young children and their teachers here to learn in English makes no sense. There is further impetus for this argument when it is understood that all over the world, with the exception of some Asian and African countries, ‘mother tongue instruction’ is the norm (Heugh et al., 1995; Alexander, 1997). The benefits of ‘mother tongue instruction’ explain why Tanzania, post independence, dedicated resources to the development of the mathematics register in Swahili, and similarly in the Maori language in New

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Zealand. Barton’s report of the development of mathematical Maori suggests that there are good reasons for extending languages in this way (Barton, Fairhall and Trinick, 1998). Granville et al.’s dual proposals spotlight the contradictory conditions of language-in-education policy in South Africa. Pedagogical needs are undermined by on-the-ground politics of access. I agree with Granville et al. that there are two political necessities here: rights to English, and rights to primary language as LoLT. The immediate demand is for enabling access to English. Primary language as LoLT must nevertheless continue to be lobbied for and worked at as a long-term goal. As both practitioners and policy makers acknowledge, there are considerable challenges to the realisation of this goal. As long as English continues to function as the linguistic capital needed in the market place, and perceptions are that English is everywhere important when, for example, it is hardly used in the informal sector market place, there has to be an explicit programme to change parent and student resistance to African languages, and their favouring of English as LoLT. Less ideologically and more practically, the financial and human resources necessary for the development of all 9 official African languages as LoLT are enormous. There are two levels here. Firstly, there are enormous practical constraints on the development and delivery of learning support materials in all of the 11 official languages. Furthermore, in curriculum areas like mathematics, science and technology the situation is complicated by the under-development in the ap artheid era of these specific registers in any of the 9 official African languages. As pointed to above, the issue is not whether a mathematics register can be developed in each of these languages, but the resources required for such an undertaking. The double bind here is that if a language is not used internationally or for scientific purposes then without these kinds of exchanges, it can’t change. And if a language can’t change, then it will not selected (by society) for use for these kinds of purposes. The dominance of English in post-apartheid South Africa is not easy to resolve It will inevitably ramify in complex ways into classroom practice. As Baker (1993, p. 247) has argued: Decisions about how to teach [second language learners] ... do not just reflect curriculum decisions ... they are surrounded and underpinned by basic beliefs about ... [the learners’ main languages] and equality of opportunity.

Curriculum decisions are also a function of the wider context within which teachers find themselves. As already hinted at in the short stories told in Chapter I, within South Africa’s language-in-education context, teachers’ actions and reflections will be significantly shaped by what they believe to be in the interests of their learners, and hence their responsibilities. 4. ENGLISH LANGUAGE INFRASTRUCTURE ACROSS SOUTH AFRICAN SCHOOLS An aspect of the inequality and diversity across South African schools not yet discussed is what we have come to call their English Language Learning

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Environments (Setati, Adler, Reed and Bapoo, in press). The English Language Learning Environment refers to the extent of insfrastructual support in and around the school for English as LoLT. As noted, English has symbolic and material power across schools in South Africa. Yet, the conditions that support English as LoLT vary considerably. In particular, apartheid’s legacy is that with the exception of texts used for the teaching of language as subject (eg. isiZulu, seTswana, French, Portuguese), most teaching and learning materials currently available and used in South African schools are printed only in either Afrikaans or English. Ringbom distinguishes between ‘second’ (in South Africa now commonly referred to as ‘additional’) language acquisition / learning and ‘foreign’ language learning: In a second language acquisition context the language is spoken in the immediate environment of the learner, who has good opportunities to use the language for participation in natural communication situations. Second language acquisition may or may not be supplemented by classroom teaching. In a foreign language learning situation, on the other hand, the language is not spoken in the immediate environment of the learner, although mass media may provide opportunities for practising receptive skills. There is little or no opportunity for the learner to use the language in natural communication situations. (Ringborn, 1987, p. 27)

Ringbom’s distinction helps to describe the different conditions that pertain to schools in urban and rural/non-urban contexts in South Africa. 4.1 Non-urban schools In most of South Africa’s numerous non-urban schools, teachers and learners share the same main language. We can describe this as a bilingual setting. One primary African language is brought to the class by the teacher and learners, yet the adopted LoLT is English. All teachers will have been professionally trained in English, and are able to communicate in English but in the wider non-urban community context, English is rarely used outside of formal contexts. Learners in such schools typically only speak, read or write in English in the formal school context. Reading material (in any language) is limited to textbooks and in some schools learners have few opportunities to use these books, either because one class set has to be shared among several classes or because teachers wish to preserve a scarce resource. In general, together with an impoverished socio-economic context, the English language infrastructure of these schools remains extremely limited. Although English has become the de facto language of learning and teaching (LoLT) in all but the first three grades in these schools, it is more accurately described as a foreign language than as an additional language, because exposure to the language is almost entirely limited to the formal school context. This teaching and learning context is appropriately described as a Foreign Language Learning Environment (FLLE) (Setati, Adler, Reed and Bapoo, in press).

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4.2 Urban Schools The English language infrastructure of urban schools is quite different. Moreover, it is possible to distinguish at least two kinds of English Language Learning Environments in urban school contexts (See Figure 1 below). FLLE

ALLE

ALLE

Increasing support for English as LoLT across non-urban and urban schools

Figure 1. English language infrastructure across South African schools

4.2.1 Urban township schools ‘Township’ is the name used for residential dormitory areas close to major cities that were designated and separately developed for African, so-called coloured and Indian South Africans under apartheid. My focus here, however, is on the African townships. Such townships remain in post-apartheid South Africa, and schools within them are attended by African students. The vast majority of the teachers in African township schools are African, as are all the learners in these schools. Hence English is not the primary language for either teachers or learners, though it is the LoLT. Teachers in these schools are by and large multilingual. Many will speak two or more African languages in addition to English and/or Afrikaans. Learners are also likely to speak more than one African language, and will have ranging levels of English language proficiency. 4.2.2 Urban suburban schools These are historically white (either English or Afrikaans medium) schools, both private and State run. Their student bodies are now multiracial. Teachers in these

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schools remain predominantly white and either English or Afrikaans speaking, though the goal is that the teaching corps in these schools also reflect the demography of the wider society. Suburban schools are by and large well resourced, with parent bodies able financially to maintain and develop the material conditions of the school. Learners in these schools are immersed in a dominantly English language environment. From a language perspective, a feature common to township and suburban schools and classrooms is that they are multilingual. Learners bring to the class, a range of primary languages, either various African languages or a combination of African languages and English and/or Afrikaans. While this multilingual setting complicates teaching practices, the English language infrastructure of suburban and township schools is more supportive of English as LoLT than in non-urban schools. In township schools where English is not widely spoken in the street, there is, nevertheless, far more environmental print (for example, advertising billboards) in English (and in other languages) and teachers and learners have greater access to newspapers, magazines, television and to speakers of English. The degree of English in the environment in and around the suburban school is greater still. Multilingual learners whose primary language is not English, and who attend suburban schools, can be understood as immersed in an English language environment. In both township and suburban schools, it is appropriate to describe English as an additional language for learners whose primary language is not English. There are opportunities for learners to acquire the language informally outside the classroom, though the degree to which this is possible varies. As such, the urban school context is aptly described as an Additional Language Learning Environment (ALLE), even though in this one name, there is a masking of the differences between township and suburban schools. The classrooms in focus in this book were all in urban township or suburban schools. There were clear differences in their ALLEs, and these are described in more detail in the chapters that follow. The issue for education policy and practice is what it means to foster learning, teaching, independent thinking and respect for diverse language across this diversity in schools’ language infrastructures. How are fairness and parity to be established? In addition, miscommunication is not only a function of the language of communication. Many children are learning in what could be called educationally hostile environments. Wider or dominant cultural practices in the school function to alienate and exclude. Establishing fairness and parity was precisely the challenge thrown out by the National Minister of Education to the Department of Education’s National Colloquium on “Language in the Classroom”, June, 9-10, 2000. There are no simple ways forward in the complex linguistic context of South Africa. Diverse additional language learning environments are not unique to South Africa and post colonial contexts. It is useful to reflect on this situation for a moment by stepping outside of the South African context, and into immigrant communities that have grown in many European countries. As immigrant communities have grown in and across a number of European countries, and now extend to include a second generation, there are city schools where immigrant students are in the majority. They come from culturally ghettoised neighbourhoods around the school, neighbourhoods that tend to have strong cultural identities. The

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effect is similar to that of South African township schools where informal learning of the language of the country is available outside of school, but limited by a closing in of community life. The LoLT is available in texts, TV and so on, but not likely to be widely used in the street and home. In addition, inside the school, there are numerous primary languages making it difficult for the school to support the maintenance, development and use of all of these languages in the school.8 5. LANGUAGE-IN-USE IN DAY-TO-DAY ACTIVITY Just as there are different English language infrastructures across schools that are productive of advantage and disadvantage, and reflect the power of the English language, so there are different language-in-use possibilities across schools that intersect in interesting ways with the dominance of English. Very few teachers whose primary language is English at this juncture in South Africa’s history speak an African language. Most will have some level of Afrikaans proficiency as until recently this was a training requirement. However, while the new Norms and Standards for Educators (DoE, 2000) stipulate an African language requirement for all future educators, at present there is little pressure on English-speakers to acquire competence in another language. In the context of multilingual classroom settings this is a significant disadvantage. From the point of view of teaching and learning, teachers who are unable to switch are precluded from particular kinds of conversations with their learners. They cannot listen effectively to learner discussion on mathematical tasks when this discussion is held in a language besides English. From the point of view of learners, and relations of power that work in all directions in any social setting, learners easily exclude the teacher from their social talk in the classroom. In the suburban settings, monolingual English-speaking teachers are precluded from full social interaction with some learners. The interesting power dynamic here is that this can be controlled by learners. In contrast, multilingual teachers are strengthened by their facility across languages and so their ability to listen and talk to their learners in all circumstances and across a range of purposes. Their extensive linguistic resources extend significantly into the wider context of the classroom and school in ways that enable inclusivity. Overall, they are able to maximise participation for both their learners and themselves. 6. CURRICULUM POLICY, MATHEMATICS AND LANGUAGE The ‘new’ curriculum, Curriculum 2005 (C2005), which was launched with great fanfare by the National Department of Education in 1997, rests on three philosophical bases: an overall outcomes-based approach to education, an integrated approach to knowledge, and a learner-centred approach to pedagogy (Chisholm et al., 2000). As I write this chapter, C2005 is under review (DoE, 1997a). The review task team has reported to the National Department that the underlying principles are educationally progressive and sound, including the overarching Outcomes-based (OBE) approach. Based on extensive interviews with teachers, visits to schools and

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analysis of policy texts, the review team, nevertheless, concluded that the implementation of C2005 has been too rapid, and the interpretation of OBE far too jargonised and complex (Chisholm et al., 2000). The recommendations made to the Government are to slow down, and to simplify the vision. Government has accepted most of the review committee proposals. However, what these proposals mean in practice is not yet clear. Indeed, implementation of the government’s acceptance will require massive reorientation by the education officials in provincial and national departments who have invested in and promoted C2005 intensively over the past two years in particular It is beyond the scope and purpose of this book to provide a detailed account of curriculum policy development and change in South Africa in the past decade. While antiapartheid education has a long history, curriculum policy development for a post-apartheid South Africa can be said to have begun in earnest in 1990 with the setting up of the NationaI Educational Policy Investigation — NEPI (Jansen, 1999). How the range of policy proposals in the NEPI reports (NEPI, 1992; 1993) became C2005 is, however, a matter of controversy. In tracing an historiography of C2005, Jansen argues that the emergence of OBE as a curriculum approach, an approach not evident in the NEPI proposals, can be explained by the intersection of the competing demands of globalisation of education policy, the legacy of apartheid education, and the contestation of policy emphases within the education and training sectors in South Africa in the 1990s (Jansen, 1999, p. 16). Jansen and Christie (1999) provide extensive debate on changing curriculum policy, and OBE in particular, in South Africa. Controversy and shifting sands notwithstanding, some of the central tenets of new curriculum policy as developed in C2005 are likely to remain in some form in a revised Curriculum. I thus provide a sketch of its central tenets with a focus on mathematics. Curriculum 2005 promises a better education for all, one that at its outset was guided by principles of equity, success, flexibility and integration. This approach to education is distinct from apartheid education, driven as it was by knowledge fragmentation, racial segregation and inequality. All learners are to have access, and be able to learn successfully. They are to develop flexible and integrated knowledge, and learn through pedagogical orientations and processes that are learner-centred. The emphasis on integration in Curriculum 2005 is a response to what are still dominant curriculum practices. In South Africa, the knowledge transmitted and acquired in schools tends to be fragmented, abstract and inert. In contrast, the knowledge, skills and values encoded into the new mathematics curriculum has been significantly shaped by principles of integration: across areas of mathematics (pure, applied, statistics); and across formal and informal mathematics. Called the Mathematics Learning Area, and named Mathematical Literacy, Mathematics and the Mathematical Sciences (MLMMS), the new mathematics curriculum connects the various mathematical sciences on the one hand, and on the other, blurs the boundary around formalised mathematics. Mathematical literacy is expressed as a desired democratic competence. MLMMS is to induct learners into a broader field of mathematics than the current mathematics curriculum, enabling connections across various component of mathematics, and between mathematics and real world problem solving. MLMMS shifts from a strong focus on mathematical abstraction,

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to an emphasis on mathematical meaning — where learners can make sense of and use the knowledge, skills and values encoded into the curriculum. Mathematics is the construction of knowledge that deals with qualitative and quantitative relationships of space and time. It is a human activity that deals with patterns, problem-solving, logical thinking, etc., in an attempt to understand the world and make use of that understanding. This understanding is expressed, developed and contested through language, symbols and social interaction. (Department of Education, 1997a, MLMMS-2)

An emphasis on learner-centred practice complements the emphasis on integration and meaningful learning by advocating, on the one hand, learners’ active participation and involvement in the lesson. On the other, learner conceptions and meanings are to be elicited, encouraged, valued, brought into interaction with concepts and knowledge in the curriculum and so extended. Lemer-centred practice is widely advocated as co-operative, centrally driven by meaningful communication between learners and their teacher, and between learners themselves. In addition to the emphasis on conversational classroom practice where exploratory talk has a key function, MLMMS includes specific attention to mathematical language. In implementing C2005 and reaching specified outcomes for mathematics, teachers are also expected to initiate their learners into discourse-specific (i.e. mathematical) reading, writing and speaking (Department of Education, 1997a). What becomes evident in the sketch of new mathematics curriculum envisaged for South African learners is that it reflects developments and changes in mathematics curricula world-wide. Teachers are to provide learners with tasks that are mathematically rich and that relate to real world problem-solving. Teachers are to create a classroom environment where mathematical ideas are explored, discussed and developed, and learner meanings elicited, valued and worked with. As sketched in Chapter 1, such practices are challenging for mathematics teachers everywhere, but are undoubtedly complicated by multilingual classroom settings. I have argued in more detail elsewhere (Adler, 1998c; 2000a; 2000b), that an integrated, mathematically rich learner-centred curriculum requires considerable resources, material and human (i.e. cultural and social). In the epilogue to the NCTM book on Language and Communication in the Mathematics Classroom, Steinbring et al. point out that: The instances of mathematically rich and meaningful communication reported in this book all show at how high a cost they have been obtained, in terms of the teacher’s involvement, students’ intellectual and emotional investment, time and organisational skills. (Steinbring et al., 1998, p. 342).

If the costs of obtaining meaningful mathematical communication in school classrooms are so high, can they possibly be made widely available in the broader context of multilingual classrooms in South Africa? What does this mean for policy and practice in mathematics education in South Africa? As South Africa’s goals for a qualitatively better mathematical experience in school are mapped onto the diversity, inequality and poverty in and across her schools, we need perhaps to be cautioned by overarching curriculum and language-in-education policies and explore critically whether and how they can indeed be to the benefit of all.

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I have elaborated aspects of changing curriculum policy and practice in South Africa to provide readers with further insight into the current context. The seeds of the vision for a new curriculum were sown long before the launch of C2005, and in evidence in some progressive schools as well as curriculum policy development and debate in the early 1990s. Of particular significance to this book is that C2005 emphasises the kinds of learner-centred practice valued by Sue (see Chapter 1) and hence increasing possibilities for dilemmas of mediation across South Africa’s multilingual mathematics classrooms. 7. IN CONCLUSION In this chapter I have described the South African curriculum and language-ineducation policy context framed as they are by apartheid’s legacy of diversity, poverty and inequality. The description reveals that from a language-in-education perspective, a secondary mathematics classroom in a non-urban area is substantially different from one in an urban township which in turn is different from one in an urban suburb. In particular, the English language environments of these respective schools and classrooms are more or less supportive of learners for whom English as LoLT is not their primary language. The significance here is that all South African teachers are positioned by the symbolic and material power of English, and hence by “rights” to English as an access issue for their learners. They also live and work in a context where there are convincing political, cultural and pedagogical arguments for valuing learners’ primary languages and drawing on these as resources in learning and teaching. However, there are considerable constraints, in the medium term at least, on the development of all 9 official African languages as LoLT. In mathematics, the issue goes beyond the provision of learning materials to the development of a mathematics register in each language. These competing pressures on teachers and learners in multilingual mathematics classrooms are compounded by the vision of a more democratic education as encoded in new curriculum policy. Learner-centred practice and knowledge integration themselves imply language-rich mathematics classroom practices. Such are the complex conditions that ramify into all South African mathematics teachers’ practices. The purpose of the research that informs this book is to uncover the experiences and knowledge of secondary mathematics teachers as they go about their work in their diverse multilingual classrooms.

CHAPTER 3

ACCESSING TEACHERS' TACIT AND ARTICULATED KNOWLEDGE The empirical is always read from a perspective. The departure point of any research, and the way in which data is accessed, organised and interpreted, critically shape what come to be described as research outcomes. In this chapter I reflect on the approach taken to the research reported in this book, how it has been shaped in the broadest sense by a social theory of mind, and so too a social theory of knowinging and learning about teaching. and knowing and le arning mathematics in school. I focus on the range of methodological issues that arose as I worked to access teachers’ tacit and articulated knowledge of their practice across complex multilingual sites. In so doing, I describe critical elements of the design of the research. Brief mention is made of the broad theoretical approach that informed the study. Theoretical detail follows in appropriate parts in each of the chapters that follow. 1. TEACHERS’ KNOWLEDGE - THE DEPARTURE POINT My overarching purpose in the research was to come to understand, from mathematics teachers themselves, how and why they do what they do, how they manage the three-dimensional dynamic at play in their multilingual settings. I was thus not concerned with whether and how teachers were adopting multilingual perspectives and practices. Nor was I concerned whether or not teachers were attempting to change the mathematical and pedagogical practices in their classrooms. Every day, teachers in South Africa (and in many other countries), manage their mathematics teaching in multilingual settings. Starting an investigation of the dynamics of teaching and learning in multilingual mathematics classrooms with what teachers have learned and what they prioritised made good sense, practically and theoretically. In other words, teachers’ knowledgeability — their practice-based knowledge — was my departure point. I have come to use the notion of knowledgeability because it combines knowing with being. Knowledgeability indicates, firstly, that what teachers know is never static, and secondly, such knowledge is always related to what is being taught, to whom and where (Lave and Wenger, 1991; Wenger, 1998). Since 1990, all teachers in South Africa have been coping with changes in a democratising and deracialising society, and particularly changes in language-ineducation policy and in curriculum policy. More directly, some teachers’ classrooms changed, almost overnight, from being all white and English-speaking to being racially mixed and multilingual. In a situation of change, taken-for-granted assumptions are often fundamentally challenged. Thus, in addition to ‘ making-good sense’, I saw teachers’ experiences and their knowledgedeability in a context ofchange as fertile ground for coming to know more about teaching and learning mathematics in multilingual classrooms.

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This is not to suggest that there are no horizons to practical knowledge (Richardson, 1994), nor that teachers have some generalised decontextualised beliefsystem (Hoyles, 1992; Argyris and Schon, 1974). Teachers build theories through their practice, just as much as theories build teachers’ practices (Levine, 1993). Nor do I wish to suggest that teachers intentions are synonymous with classroom processes, or that teaching somehow unproblematically equals learning (Lave and Wenger, 1991). Teachers’ knowledgeability shapes and is shaped by their own histories, the context, culture and activity of classroom practice, as well as the wider educational practices and knowledge domains in which they participate. It is precisely from this sociocultural perspective that what teachers have come to know in and of their practice is a potentially productive and interesting route into an informed account of the dynamics of teaching and learning mathematics in multilingual classrooms. Furthermore, teachers’ knowledgeability is also a crucial dimension of the wider field of knowledge about teaching as a practice. As mentioned in Chapter 1, most of the research on bilingualism or multilingualism in mathematics education has, in fact, bypassed the teacher, a gap that needs to be filled. 2. A SOCIOCULTURAL PERSPECTIVE ON TEACHERS’ KNOWLEDGEABILITY Social practice theory (Lave, 1990; 1993; 1996; Lave and Wenger, 1991) and sociocultural theory (Vygotsky, 1978; 1986) are particularly useful interpretive frameworks for understanding issues related to the teaching and learning of mathematics in multilingual classrooms. This is partly because the assumptions about teaching and learning that are inherent in sociocultural theoretical approaches involve a merging together of what might be called the individual-social divide in psychological accounts of teaching and learning. Social practice theory and sociocultural theory are deeply infused into the study, its processes and outcomes. It thus forms part of what Lerman (2000) described as the ‘social turn’ in mathematics education research, and the need to provide accounts of teaching and learning mathematics that not only merged the individual-social divide, but that also understood the macro socio-political order as reflected in, and productive of, micro classroom processes and vice versa. My theoretical intentions were to find a way to bring the micro and macro, the personal and social, in their interaction, to bear on my interpretations of the complex practice of teaching mathematics in multilingual classrooms. The particular ways in which this interpretation has taken shape are detailed in the chapters that follow. As the research progressed, situated theory, that is, a theory of knowing as tied to being in context, has come to reflect the approach to knowledge and knowing that carved out the study reported here. Indeed, situated theory has offered new directions for research into how students experience school mathematics (Boaler, 1997), and for how teachers relearn their practice (Adler, 2000c; Stein and Brown, 1997). From this perspective, a teacher’s knowledge of her practice is at once a reflection of the teacher herself, her educational and personal history, her dynamic

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identity as a teacher, and of where she works, the situations (communities and wider socio-political order) in which she participates. Teachers’ thoughts and actions in their classrooms, tied up as they are with ‘becoming a teacher’, are a window into the dynamics of their teaching situations. 3. DESIGN: ISSUES IN TEACHER SELECTION, DATA SOURCES AND ANALYSIS During 1992 and 1993 I conducted field research (Rose, 1982), using qualitative methods, to find out how mathematics teachers’ in multilingual classrooms in South Africa manage their complex practices. As field research, it is in an interpretive paradigm, working in-depth and concerned with the meanings in, and complexity of, teaching in context with a critical stance towards human meaning (Erickson, 1986, p.122). The overall design, as will become clear below, drew from both ethnographic (Erickson, 1986) and case study methodology (Yin, 1994), becoming something in-between. 3.1 Selecting in and selecting out teachers and their multilingual contexts As with all qualitative methodologies, the number of teachers participating in this study was small. I selected six secondary mathematics teachers, all in urban schools in what is now called Greater Johannesburg. Greater Johannesburg is in Gauteng, the most multilingual of all the provinces in the country. Schools across the province are diverse, socio-economically as well as linguistically. The six teachers selected were specifically drawn from three different multilingual contexts. At that time, South Africa was in the process of negotiating its way out of its apartheid past. There was a great deal of turbulence that regularly spilled over into schools. As discussed in the previous chapter, schooling in some areas had all but become dysfunctional. In order to conduct a qualitative study at that time, it was important that I had an already established relationship of some trust, at least with the participating teachers. Access to schools and classrooms otherwise could have been blocked, even dangerous, or so unstable as to not enable significant data gathering. The six teachers introduced in the opening section of this book, Jabu, Thandi, Sue, Clive, Sara and Helen were all known to me. We had met and interacted as participating members of the community of mathematics teachers and teacher educators in the area. Some had previously attended courses at the university where I teach. I was able to contact them directly and invite them into the study. As a ‘sample’ their selection was thus a function of ‘opportunity’. Their selection, however, was more than opportunity-based. I wanted teachers from different multilingual contexts. Their selection was also a function of where they were teaching, and thus “purposive” and “theoretical” (Cohen and Manion, 1989; Rose, 1982). Jabu and Thandi were from two similar urban African township secondary schools. In these schools, neither the teachers nor the students had English as their

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main language. Setswana was the most common main language for students and teachers in the school but there were also students whose main language was another African language. Most of the teachers, including Jabu and Thandi, were multilingual, able to communicate in two or more African languages as well as English. Setswana was both teachers’ main language. English was the explicit LoLT in both schools. though English was a third, sometimes fourth language for learners. As urban African township schools, and as discussed in Chapter 2, the English language environment of each school was an additional language learning environment. While English was not the language of the immediate environment outside the school, it was available on TV, billboards, newspapers and so on. The formal school setting was not the only context for learners to encounter the English language. Helen and Sara were from similar and separate urban suburban secondary schools. These schools were recently desegregated, well-resourced, historically white state schools. English was the dominant language in and around the school. The teaching staff, and so too Helen and Sara, were white with English as their primary language. As of 1990, there were increasing numbers of pupils with other main languages. Classes in these schools were multilingual Students for whom English was an additional language were immersed in a dominant English language learning environment. One might say there was maximum support in such schools for learners with different primary languages but who were learning in English. The remaining two teachers, Sue and Clive taught in urban private schools, Clive’s in the Johannesburg inner city, and Sue’s in an historically coloured township. Both schools were relatively well-resourced. The students in each of these schools were black (African, coloured and Indian), and many did not have English as their main language. Moreover, they brought a range of main languages to class. English was the explicit LoLT in both schools, and most teachers, including Clive and Sue, were white and English-speaking. The surrounding communities were increasingly racially mixed, and there was more English in use in the wider environment than around the urban township schools. English, nevertheless, was an additional language for many of the learners in both schools. 1 All six teachers were fully qualified and experienced secondary mathematics teachers, and as intimated above, professionally known to me. They were themselves interested in language and mathematical learning and so in participating in a related study. Tables 1 and 2 in Appendix 2 provide biographical and infrastructural summaries of the six teachers and their schools respectively. A key issue in this selection and foregrounding of language and the multilingual setting is the extent to which these teachers typified the mathematics teachers in the field. They did not. Firstly, many teachers do not volunteer to devote time and intellectual and emotional energy to research. The teachers thus constituted a particularly reflective selection. Secondly, all six teachers were qualified both professionally, and mathematically. All had studied mathematics for degrees beyond school. This is certainly not typical of the field of mathematics teachers described in Chapter 2. My objective, however, was to find out about reaching and learning mathematics in multilingual classrooms. I thus geared the selection of teachers to include interest in language issues, and excluded what could become confounding

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issues like a teacher’s inexperience, or poor mathematical knowledge base. Of course, this selection, together with the schools all being in urban areas, has its consequences. The insights provided by the teachers in this study cannot be assumed to extend unproblematically, for example, to mathematics teachers in non-urban and thus ‘foreign language learning environments’. Nor do they easily extend to the critical situation in many mathematics classrooms in South Africa where teachers do not only confront the challenges of working in a multilingual settings and FLLEs. They also confront limitations in their own mathematical knowledge-base. 3.2 Multiple sources of data: Triangulation or rich and diverse voices? As teachers’ interpretations were central, in-depth individual and group interviewing and participant observation were the key methods of data collection (Erickson, 1986; Silverman, 1993; Hitchcock and Hughes, 1995). In summary, data was collected through initial semi-structured, in-depth individual interviews: videotaped classroom observations, individual reflective interviews, and follow-up workshops with the teachers as a group. The workshops included reports on action research projects that some of the teachers undertook, on their own initiative, in order to explore in more detail issues that arose for them through their participation in the research. Each of these data collection methods is briefly described in Appendix 3, in the order in which they occurred. Data were thus gathered in different contexts and through different activities, related to both what teachers said and what they did. The distinction between, and importance of both what we say and what we do, relates to what Argyris and Schön (1974) describe as “espoused theory” (theory to which we give allegiance) and “theory-in-use” (theory which governs actions) (pp.6-10). Argyris and Schön were particularly interested in whether and how what we say (about what we do or wish to do) might differ from what we actually do. The distinction between espoused and enacted theories can be read as a disjuncture, for example, in repressive terms, we do not like to look at what we do not want to see. This reading would bring the disjuncture into focus. Alternatively, the relation between espoused and enacted theories could be viewed as a complete relation, that is, there is no disjuncture. Much research on teachers’ beliefs and actions works on this assumption and its implication that changing actions thus means changing beliefs. Polanyi’s (1967) sense, and the way in which I view the relation, is that there is some relation between our espoused theories and our theories-in-use. There is overlap, but we cannot reduce the one to the other. Our knowledge is both embodied and discursive, and these are intertwined. We cannot always say what it is we do. And vice versa. We cannot always do what we say. Some of what we know is implicit, tacit. This study does not dichotomise these two windows on teachers’ knowledge about teaching. What teachers say and what they do (or how they act) are both seen as crucial, if sometimes conflicting or contradictory, parts of teachers’ knowledge. Moreover, from a situated perspective, what we say and do will not necessarily be uniform or consistent across different sites of practices. The different forms of data collected, and the different sites of practice in which they were collected, combined

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to provide access into and understanding of crucial aspects of teachers’ knowledgeability. The objective here was not a triangulation of data towards some fixed truths about teachers’ knowledge. Action and reflection in and across different sites provided different windows into teachers’ knowledge and experience. They became the sources for rich description and explanation, within and across teachers, of diverse perspectives and voices on multilingual mathematics classroom practice and their inter-relation. 3.3 AnaIysis as progressive focussing Hidden in the list of methods used above, was the progressive focussing built into the design, and hence a groundedness to the methodology (Hutchinson, 1988). I say ‘a groundedness’ because I am clear that the perspective brought to bear on the study as a whole, as well as my own views on the relationship between language, mathematics and mathematics education interacted with the data to produce the analysis. As stated above, there is an inevitable dialectic, a moving back and forth between the theoretical and empirical, that shapes all research. The initial teacher interviews were content analysed and key language issues raised by the teachers were identified. This analysis was the focus of discussion with the teachers as a group. What emerged even at that early stage of the research was that teachers in different multilingual contexts and different conditions of change emiphasised and prioritised differently. It was. nevertheless, possible to reach consensus on the ‘sum’ of key issues. The broad categories that were agreed on during the discussion were: the dominant use of English, particularly in classes where many learners’ fluency in the language was limited; what counted as mathematical discourse-specific ways of talking and meaning in the mathematics classroom and how teachers approached this; and whether learning from talk was a good thing. These broad priority areas were recorded, and were then used (by the teachers and myself) to reflect on their classroom practice. Interestingly, they reflect the constitutive elements of three-dimensional dynamic described earlier, where teaching and learning mathematics in multilingual classrooms simultaneously involves access to the LoLT, access to mathematical discourse and access to ways of communicating in the classroom. The categories emerging from the initial interviews formed the basis of categories of description (Marton. 1988) that were developed through the analysis and over time. Of course. classroom practice is its own context, as is a reflective interview with videotape as object of discussion. These took on their own life, shaped nevertheless by discussion of the initial interviews. New and more elaborated issues come into focus. And again, the videotapes, and the issues these raised for discussion in the reflective interviews, interacted with particular interests of each teacher in their specific context to shape what and how they selected as excerpts and issues for discussion and follow up during the workshops. Analysis of video interview and workshop data revealed noticeable presences and silences across different teachers and their different multilingual contexts. Although teachers in different contexts emphasized different issues. a common

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thread was the expression of tensions and contradictions in their practices. Teachers in different multilingual contexts revealed different teaching dilemmas as they spoke about their teaching, thus supporting the notion of teaching and learning as a contextualized social practice. Each dilemma has been introduced in Chapter 1. Tensions concerning code-switching (using more than one language in class) were emphasized by the teachers in township schools. Tensions related to mediation were emphasized by teachers who had tried to create more participatory-inquiry approaches in their classrooms. Teachers in suburban schools whose classrooms rapidly became multilingual faced the inherent tensions in explicit and implicit language practices in their multilingual classrooms, and what I have interpreted as the dilemma of needing both to see and see through mathematical language in class. While presences and silences across the initial interviews already pointed to the key and different dilemmas that emerged from the study (and these are described in detail in Chapter 5), the videos, reflective interviews and workshops became the means to develop rich analytic narrative vignettes (Erickson, 1986) to illustrate the key dilemmas, validated through levels of analysis — descriptive, interpretive and theoretical (Maxwell, 1992). Each dilemma is the substance of the stories told in Chapters 6, 7 and 8. My reflections so far have been on the who and how of the study, on what could be described as critical components of any empirical study. There were, however, a number of other challenges faced, issues that are typically not made visible in accounts of research in a book such as this. Instead, debate and discussion of more contentious issues are placed in texts on research methodologies, as if somehow, their meaning is best understood in some disembedded way. In the remainder of this chapter I make visible issues related to the specificity of the multilingual context, to data disruption, to research relationships and to voice as they arose in the context of the study with teachers in multilingual mathematics classrooms. Each is inevitably a part of, and a frame for, the story and stories told in this book. 4. THE NOISY, YET OFTEN SILENCED ISSUES

4.1 The Slippery ’yet productive specificity of the multilingual setting Through the interviews, videos and workshops, a process of interaction over two years and across situations, we, the teachers and myself, became aware of outcomes of the research that were not intentionally built into its design. Firstly, the data collection process had been a participative and generative one for the teachers. Three of the six were sufficiently motivated by issues that arose from them through their participation in the research to undertake small action research projects in their own classrooms. These projects formed the substance of their inputs into discussion in the research workshops. During this final phase of the study they also enrolled for further study in the university, establishing a formal base for their own research and development. Secondly, the workshops and meetings had been a ‘safe’ place (outside of particular school and staffroom politics) to discuss issues of interest and

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concern that might otherwise have been difficult. In Lave and Wenger’s (1991) terms, the group, together with myself constituted a small community of practitioners who continued over some time to participate in activities where the object of engagement was language and mathematics teaching in multilingual settings. Ironically then, we also came to see how with our broader focus on language, mathematics and mathematics education, the specificity of the multilingual classroom often slipped out of view. As the teachers confronted episodes of their teaching, many of the issues they discussed were not unique to a multilingual setting. It was rather that the setting provoked interest in and brought forms of language practice into focus. It was the multilingual setting that rendered the familiar strange. The multilingual setting was thus productive of insights that pertain beyond the particularity of such settings. In the chapters that follow, the specificity of multilingualism will slip in and out of focus as the three-dimensional dynamic comes into view in different ways. 4.2 Data collection, disruption and production All research processes contain stories about what it means to access and be in particular schools. Elements of the turbulent, diverse and unequal schooling system I described in Chapter 2 were exemplified in the six schools in the study. The initial interviews with the teachers in the two African township schools were conducted in small, sparsely furnished offices for use by Heads of Departments in the school. There was no general staff room in one school. In the other, a small office functioned as the staff room. In neither school were there private spaces for a teacher, a reflection of limited resources in each school. During each interview, there were a number of interruptions as other teachers entered the room. At one school, the interview took place at the beginning of a break as the teacher was free in the periods following. Despite a bell signalling the end of break and a return to class, the teacher drew my attention to the continuing noise and the fact that the bell had by and large been ignored. At that school most pupils chose not to return after the second break or to come back in their own time. As I left, over an hour later, pupils were moving in and out the main gate; three classrooms I passed had no teachers in them. Here was first hand evidence of the breakdown of a learning culture in township schools that both teachers talked about in their interviews. This school was situated opposite a migrant worker hostel that was the scene of ongoing violent confrontation between African National Congress (ANC) and Inkatha Freedom Party (IFP) supporters. The teacher explained how this political conflict inevitably spilled over into the school. Some Mondays, at the start of a new week and the end of a violent weekend of clashes, approaching the school gate involved passing dead bodies. The vivid backdrop of each teacher’s practice was brought sharply to light for me as I entered their schools for classroom observation during September and October 1992. Stark differences across schooling contexts were apparent. The second day of observation and videotaping in one of the township schools saw only half the class. The rest of the class was at a funeral of a young male from another

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school killed in a political clash. In the other township school I saw a repeat of what I had observed during the initial interview: while the pupils in the class being videoed were present after the second break (they enjoyed being filmed, I was told), much of the rest of the school was dysfunctional. Classrooms on either side of us had no teachers. Moreover, many pupils had left for home, or wherever else they might go. The teacher told me that there had been discussion with the staff as to whether I should observe and video in the school at that time of day. They agreed that ‘the truth will out’, anyway, so why try to hide the reality of their context. It is important to note the level of dysfunction in these schools in the light of the way in which I had selected teachers to work with. I had excluded schools where known levels of instability were such that access would either be blocked or data collection (as intended) rendered too difficult. Still there was the unanticipated. In contrast, the private and suburban schools were distanced physically, though not entirely emotionally, from the conflict in the townships. These four schools were calm, ordered, adequately or even well-resourced Day-to-day functioning was ‘normal’ with clear timetables and attendance as one would expect (pupils and teachers arrived and left on time). Some pupils in these schools lived in conflictridden townships and bussed into school. Their lives outside of school were often traumatised by increasing levels of violence. Valero and Vithal (1999) use the metaphor of the “North-South” divide and concepts such as “stability” and “disruption” to argue that research and particularly mathematics education research, in politically volatile contexts demands different kinds of methodologies. They start from the position that methods of research created in developed countries have a strong underlying assumption of stability, an assumption that in a post-modem world, let alone the developing world, requires rethinking. They go on to argue that: ... in contrast to this stability, developing societies are characterised by instability, given by the constant and abrupt reorganisation of political, social and economic forces. ... When the research process is obstructed by uncontrollable disruptions emerging from the very same unstable nature of the social context and of the research objects that are considered, then the whole process of research has to be reconceived to allow the disruptions themselves to reveal key problems that should be addressed in order to understand, interpret or transform the real issues of the teaching and learning of mathematics in developing societies. (Valero and Vithal, 1999, p. 9)

Valero and Vithal make an important contribution to debate on research processes and outcomes in a diverse world. There is a real danger that in restricting research sites to that which is stable, where there is a continuity between questions asked and methods intended and followed, we might in fact miss critical int erpretations of that which enables and constrains the teaching and learning of mathematics in such diverse contexts. I have written elsewhere that the issues raised by Valero and Vithal are in many ways also ethical - they are about what research processes are ‘‘right” and how “valid’ accounts of practice are produced (Adler and Lerman, in press). The “North“ has “written back (Ruthven, 1999), arguing that educational development, rather than research, is what is required in countries in social ferment. This is precisely the point — that somehow what is required transnationally can all be reduced to what is appropriate in the dominant world, and moreover, that the kinds

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of insights available through changing and volatile contexts are somehow of no relevance in the ‘stable’, however post-modern, world. It is precisely my argument in Chapter 1 that research in bilingual and multilingual settings, and here I am talking about difference as opposed to disruption, is othered, conceded as important for educational development. but not for generative research outcomes that could speak to pedagogical practices in dominant contexts in new ways. Given the wider political turmoil in which this research took place, and not withstanding the exclusion of schools where instability and disruption would have been the starting point, the process of data collection for this study proceeded remarkably smoothly — interviews, observations, workshops all took place as planned a function, no doubt of the opportunity and purposive selection of teachers and their own interest in, and commitment to, the research. However, it is clear that the contexts of observation and interviewing were different for the different teachers. In this way, they constitute different settings, productive of different knowledgeability. The overall perspective brought to bear on the study could incorporate this difference as possibility for further insight into diverse practices in context, rather than as a detraction from the study. Of even greater consequence in qualitative research where the researcher herself is the key research instrument, are the relationships between researcher and researched. As social events, data collection processes are never outside of or free from social relations and their consequences. There are three dimensions to my relationship vis-a-vis the teachers: (i) the benefits and constraints of familiarity; (ii) our distinct voices and (iii) our different racialised and professional identities. Each of these issues — familiarity, voice and identity — enter and frame the research account, and so require critical discussion. 4.3 Familiarity and professional relationships Since 1986 I have been employed in mathematics teacher education, working with both primary and secondary teachers. I am also involved in mathematics education teaching and research associations and through this activity and my teaching, I have come to know a number of secondary mathematics teachers in the Johannesburg region. I was thus able to approach appropriate teachers to participate in the study, teachers whom I knew had some knowledge and interest in the area. This familiarity brought both strengths and limitations to the research. Strengths lay in the already established relationship of trust between the teachers and myself, as well as a shared and acknowledged interest in the project. Familiarity at the same time imposes limitations: because teachers hew me and hence hew some of my professional interests, and moreover, because in some instances I had been their lecturer, they could have wanted to please me in their interviews by expressing views they thought I approved. My participant observation could also have been experienced as evaluative. I attempted to minimise these effects by assuring them at the outset that they were in no way the objects of evaluation in this study, but rather sources of knowledge in a difficult area where it was important to confront the realities they

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faced in their complex classroom activity. The point here is that while a different researcher might have been more distanced from the teachers, this too would have produced a research situation with its own benefits and constraints. Such relationships need to be acknowledged and worked with. They cannot be wished away, nor designed out by some notion of objectivity. Finally, as will emerge in the stones that follow, the teachers’ identities, and so their knowledgeabilities, were a function of their personal histories and their contexts. Their own interests came to frame the kinds of illustrations of dilemmas produced. In a recent paper, Setati (2000) has problematised conceptions of one-way power relations between the researcher and researched, particularly when the researched are teachers. She argues cogently that teachers enter research projects with their own interests and agendas, and exercise a level of agency and thus influence over the direction of the research There is no necessary relation of dominance — there can be a relationship of reciprocity. This assertion is strengthened by Jaworski’s in-depth analysis of her work with secondary teachers in the United Kingdom on the investigatory mathematical practices in their classroom. She described the knowledge growth of all the teachers, and not only herself, as a function of participation in the research (Jarworski, 1994; 1999). None of the six teachers I worked with are today school mathematics teachers. In the six years since completing the data collection, Sara has retired The other five are all still involved in education, two in mathematics teacher education, one in teacher support services in the provincial education department, and two in adult education. They are indeed a loss to a profession short of qualified and interested mathematics classroom teachers. Their interests and expertise, however, enter the system in other ways. The point here is that participation in research activity is never a benign affair. For at least half the teachers, the research activity was a catalyst for change. They began a process of participation in a community of educational research practice and so outside of classroom teaching per se. This outcome gives further substance to Setati’s (2000) argument that research in mathematics education that involves teachers can be reciprocal, neither directly on teachers, nor purely with them in some kind of equal relationship between researcher and researched. While the researcher clearly brings an agenda and interests to the research process, so do participating teachers. As did the teachers in Jaworski’s (1994) study, the teachers in this study can be understood as benefiting from it professionally, though in unintended ways, and ways that are not always visible during the research process. 4.4 In whose voice? Rose (1982, p. 119) makes an interesting distinction between participant concepts grounded in the data analysis and expressed in the words of the participants and theoretical concepts grounded in the data but abstracted out through the theoretical language of the researcher. In practice, Rose contends, we usually find a mixture of theoretical and participant concepts in any field research study. It is tempting to assign authenticity to participant concepts, to view the words of the teachers themselves as more accurate and meaninful representations of practice.

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However, in a theory of practice, words are always an abstraction out of the practice, no matter who expresses them. Furthermore, the assumption that participants speak a truth unfettered by their language and context does not hold sway in a social theory of mind. Both Rose (1982) and Silverman (1993) criticise notions of pure theory-building, pure grounded theory which assumes that there is a starting point where the research comes at the theory-building process empty and blind. This is impossible. The only way we can see is with some orientation. Nevertheless, in qualitative research it is important to attend carefully to the data, to remain focussed on finding out from participants, from their point of view, what it is they know, but declaring the perspective which is brought to this listening. The issue then becomes: how is teachers’ knowledge described? What is gained and lost in using participant or theoretical concepts or both in developing a language of description? This study reinforces Rose’s contention that a mixture is in fact what occurs. The research discussed in this book is grounded in the voices of teacher participants but in the end, is presented in my voice — a voice that values and respects teachers’ knowledge, but is still a voice of a researcher with her own sets of interests and values. Through the research I have produced a language of description where theoretical concepts are grounded in practice — a language that, at once, illuminates teachers’ knowledge and offers possibilities for assisting teachers’ knowing and thinking about their practice. Nevertheless, Lather’s cautionary note is pertinent: ... language is a delimitation, a strategic limitation of possible meanings. It frames; it brings into focus by that which goes unremarked.(Lather, 1991, p. xix)

4.5 Racial, cultural and linguistic identities and the research context and process The discussion on familiarity above brought out issues of professional identity in researcher and researched. A key issue in any study related to language and curriculum-in-use in South Africa is the inter-relation between race, culture and language. The teachers in this study have been described and in so doing racialised. Who am I? I am a white, English-speaking South African female academic working in the field of mathematics education. I am also an experienced and qualified secondary mathematics teacher. English is my primary language and I learnt Afrikaans at school for 12 years. I would hesitate to call myself bilingual, but I can function in a number of situations with the level of Afrikaans proficiency I have accumulated. My knowledge of African languages however, is minimal. I can do no more than greet people in Sesotho and isiZulu, and I am familiar with some expressions and terms in each language. I cannot, however, understand a conversation spoken in either language. As I announce my race I am conscious that while race is always in the picture in South Africa, I did not focus on how it played out in the research process. The story I tell is thus partial. However, partiality is a feature of all social research, an unescapable reality rather than a necessary weakness. Depending on how research into complex social practices like learning and teaching in school is framed, the theoretical orientations adopted and the kinds of data collected, there will

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inevitably be more or less attention to whether and how race, for example, and its related discourses are revealed. Undoubtedly, a study on teaching and learning mathematics in multilingual classrooms that interrogates the production and reproduction of racial difference will produce new and challenging insights. In a context where English is the language of power, and where English speakers are by and large ‘white’, race, language and power are inextricably intertwined. I am inevitably privileged having English as my primary language. In the context of the research, my relative monolingualism was, at the same time, a disadvantage. Firstly, while observing, I could not understand interactions in classrooms that took place in one of the African languages. While these could be translated for me, I was also unable to converse with the teachers in any language besides English. In this particular study, the African teachers were fluent in English, and thus able to communicate in English. This has not been the case in more recent studies, where not being able to switch in the context of an interview, let alone informal conversations with the teachers, did detract from the nature of the data collected. Of course, identity, and the construction of a social space or setting in an interview, is not delimited by the language spoken. I raise this because it is important to understand that the power of English works to privilege, and in some instances also to constrain. I have mentioned the constraints, constraints that loom larger for me as I continue research in this field and am only fluent in English and partially in Afrikaans. If we shift to the classroom briefly, the African township classroom is constructed as the least supportive of English as LoLT across the classrooms in this study. From that perspective this might well be an appropriate description. Communication in English is in teachers’ faces all the time as they themselves have to work on how to express their thinking in English as well as hear learners’ understandings expressed in developing English. However, the teachers and learners in these classrooms often share a primary language; moreover all would have enviable levels of multilingualism and thus the ability to communicate across languages. Teachers in these classrooms can communicate with their learners at all times. In much of the work on multilingual classrooms, teachers whose primary language is not the LoLT, yet who are multilingual, are nevertheless often constructed and come to view themselves as being ‘a problem’. Multilingual African teachers in particular are constructed as ‘not having enough English’ as opposed to being able to work across languages. One of the significant challenges in South Africa is that with the power of English comes a status to those whose primary language is English. Such teachers are rarely constructed as ‘lacking’ in any ways. Yet policy is becoming clearer how this is a classroom disadvantage — South African teachers (and so too educational researchers) should all have some versatility with the major languages spoken in their areas. As in the discussion on race, the discussion here of cultural and linguistic identities in the research process is not because they are explicit foci in the research. I foreground them to highlight how privilege and disadvantage indeed work in complex and contradictory ways in the lived-in world. Such is the inevitable case in any research process.

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5. IN CONCLUSION In this chapter I have highlighted both usual and often hidden issues in the research design and process, and so brought into view, the messy and often contradictory social relations constitutive of social research. I have explained why and how I viewed teachers’ knowledge of their practices — their knowledgeability — as an important point of departure and route into an informed account of the dynamics of teaching and learning mathematics in multilingual settings. I described further how the overall study has been influenced by a social theory of mind and a conception of the multilingual mathematics classroom as a three-dimensional dynamic. While analytically separable, access to and proficiency in the LoLT, access to and competence in mathematical discourses and familiarity with classroom discourses all work together in moments of practice in a multilingual mathematics classroom. I now turn to an elaboration of the notion of ‘teaching dilemmas’ as prelude to interpreting and so learning from what the teachers said and did in and about their classroom practice.

CHAPTER 4

DILEMMAS IN TEACHING: A PRELUDE AND FRAME 1. INTRODUCTION That teaching in multilingual mathematics classrooms is dilemma-filled is not surprising. Classrooms, after all, are complex sites of practice. The value in identifying key teaching dilemmas and naming them is that they can then become objects of reflection and action. Familiar, taken-for-granted practices can be made strange. The notion of teaching dilemmas forms a part of the existing literature on teaching (e.g. Berlak and Berlak, 1981; Lampert, 1985; Edwards and Mercer, 1987; Jaworski, 1999). ‘Dilemmas’ were not part of my original focus or thinking. Indeed, the teachers did not talk about ‘dilemmas’ per se in their interviews or workshops. As mentioned in Chapter 1, the notion of a ‘teaching dilemma’ became the key with which to prise open teachers’ knowledge of their complex practices. A language of dilemmas became the means with which to organise and explain my observations. It was thus from the empirical work in the study itself that I turned my attention to understanding and developing dilemma theory, and a language with which to describe and explain mathematics teachers’ knowledge of their practice in a range of multilingual mathematics classrooms in South Africa. In this short chapter I provide focused discussion of the notion of a ‘teaching dilemma’ and some of its history and location in educational (and mathematics education) research. This theoretical elaboration serves as a prelude to, and frame for, the description in Chapters 5 to 8 of key dilemmas of teaching mathematics in multilingual classrooms. 2. MANAGING DILEMMAS: A PRACTICAL ACCOUNT OF TEACHERS’ ACTIONS Lampert (1985) has argued that from the teacher’s point of view, trying to solve many common pedagogical problems leads to “practical dilemmas”. Working from an example from her own primary mathematics classroom, she described facing the pedagogical need to position herself near the rather difficult boys in the class so as to keep them under her eye (a control problem) but in so doing found herself distanced from the girls in the class (an equity problem). She argued that teaching involved managing this tension. There was no one “right” solution to this tension in her specific context of practice, whatever analysis or theorising might have suggested. If she stayed near the boys, thus satisfylng her pedagogical need to have sufficient control in the classroom for teaching and learning to occur, she would betray her desire not only for gender equity but for encouraging girls in a field like mathematics. If she moved close(r) to the girls to satisfy the importance of equity in pedagogical practice, she would undermine classroom order which she believed was a prerequisite for teaching and learning. Thus it was not a matter of making a right

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choice. Resolution did not lie in one (control) or the other (equity). Rather, what was entailed was finding strategies that did not ignore but addressed the tension by diminishing conflict. Lampert managed her conflict by carefully reorganising the pupils’ seating. She also pointed out that she had fortuitous assistance with this as two boys were absent from class on that day. Lampert augmented her case for managing dilemmas by describing the dilemma faced by another primary mathematics teacher. Two pupils produced two different but meaningful responses to a problem, one of which matched the response in the text book. The pupils wanted the teacher to say which was correct. She did not want to undermine the pupils’ confidence in the text book, nor did she want to mislead them into thinking there was only one meaningful and correct response. The dilemma for the teacher was that the text book was her tool, but in providing only one answer and one method, it had limitations in relation to her other mathematical goals. This teacher managed her dilemma by working with the pupils on why both responses were correct. For Lampert, the notion of teaching as managing dilemmas runs counter to many images of teachers, for example, as technicians, simply implementing prescribed curricula, or as members of opposing camps making choices among dichotomous alternatives. These images portray the conflict in teaching as resolvable through choosing one or the other competing alternative. Lampert illustrates from her own and the other teacher’s practice, how the management of the dilemmas they faced was a function of their personal biographies and their identities as teachers. A different teacher, for example, might not be concerned, as was Lampert, with gender equity and with levels of classroom control that enabled both freedom and order. Such a teacher would probably not experience the situation in the same way, and might not face this dilemma. Lampert described a pedagogical dilemma as an “argument with oneself” that ultimately involved “personal coping strategies”. My capacity to bring disparate aspects of myself together in the person that I am in the classroom is one of the tools that I used to construct an approach to manage my dilemma (Lampert, 1985, p. 184).

Lampert’s argument for the management and use of conflict as opposed to attempting to resolve conflict in the face of competing alternatives — dilemmas — is compelling. Working as she did with both her own and another teacher’s practice in school, she offered a serious insider/outsider engagement with some of the very real complexities of teaching and learning and the practical work involved. Her roles as both teacher and researcher enabled an interesting account of “inside” (practical, local) and “outside” (theoretical, distanced) knowledge of teaching where these were not related simply as opposites but as “voices that engage one another in dialogue” (Cochran-Smith and Lytle, 1993). Lampert’s project was to argue for the “practical work involved in managing dilemmas”, from the teacher’s point of view. In this way we can understand how teachers manage their complex professional activity. Her position developed out of Schwab’s language of the practical in relation to knowledge about teaching (Schwab, 1978): the theory-practice dichotomy is dissolved in the practical. Effective teacher action is neither ad hoc nor theoretically

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driven. It is practical — in the teaching situation. It is also personal — tied to biography. From this perspective knowledge about teaching is thus not adequately portrayed in theoretical accounts that are divorced from the practical knowledge of teachers from their point of view. Lampert’s account of teacher knowledge and action is thus practical and personal, a focus she argues is lacking in Berlak and Berlak’s (1981) account of dilemmas in school teaching. However, it was neither accidental nor idiosyncratic that in the early 1980s, issues of control, equity and creative classroom organisation were part of her personal goals. Nor is it surprising, given the ‘child-centred’ discourses in mathematics education at that time, that her colleague experienced a dilemma around the authority of the prescribed text book. All these issues were deeply enmeshed in educational debates in the United States of America at the time. The emergence and management of dilemmas were thus as much a function of the wider social context of her and her colleague’s work as primary mathematics teachers in the early 1980s, as it was of their personal biographies and practical situations. An account of teachers’ knowledge thus needs to be situated in time and place, and extended beyond the personal and practical. 3. DILEMMAS IN TEACHING: A DIALECTIC ACCOUNT OF TEACHERS’ ACTIONS In their earlier work, Berlak and Berlak (1981) made an explicit attempt to develop a language of dilemmas that captured “contradictions that are simultaneously in consciousness and in society” (p.124). A language of dilemmas, they argued, can simultaneously represent contradictions that reside in the situation, in the individual and in the larger society, as they are played out in the form of institutional life, and particularly schooling: Each dilemma captures not only the dialectic between alternative views, values, beliefs in persons in society, but also the dialectic of subject (the acting “I”) and object (the society and culture that are in us and upon us) ... both the forces which shape teachers’ actions (those forces that press towards particular resolutions to a dilemma) ... and the capacity of teachers not only to select from alternatives, but to create alternatives. (Berlak and Berlak, 1981, p. 125).

The Berlaks’ project was to understand possibilities for change in schools and teaching and hence is broader than Lampert’s. They nevertheless shared with Lampert, a concern to recognise and acknowledge the complexity and wisdom in teaching, and held that teachers were able both to enquire into their practices and to change them, Their project was to develop a language that accounted for teachers’ actions, and so their knowledgeability, a language that could then be used to enquire into and change teaching. Berlak and Berlak developed a dialectical account of teachers’ action that challenged a subject-object dichotomy. They distinguished between habitual and reflective activity in teachers (and humans in general). Through reflective activity, teachers did not simply adapt to a given world but shaped and changed that world. Language or communication enabled persons to shape the conditions of their own adaptations.

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Where Lampert worked against a theory-practice dichotomy, the Berlaks wanted to address the sociological structure-agency dichotomy. They developed their dilemma language through an ethnographic study of a selection of primary schools in the United Kingdom in the late 1970s, and particularly those primary schools that typified the English open classroom at that time. As they struggled to organise their observations, it became clearer and clearer that, both within and across particular teachers and schools, there were no simple dichotomous ways of capturing the teaching they observed nor how teachers themselves understood their work. Teachers behaved in complex and often contradictory ways, and their understandings of their actions and teaching exhibited similar complexities and contradictions. Possible descriptions and explanations did not lie within teachers, nor their classrooms, nor the wider context, but rather in their inter-relations. The language of dilemmas became a means for capturing the complex relationship between the context of schooling and the dilemmas of teaching. The Berlaks developed three distinct sets of dilemmas that facilitated a description and explanation of what they observed in and across the schools and the teachers studied. “Control dilemmas” captured tensions over the locus and extent of control over students e.g. “whole child vs. child as student”. “Societal dilemmas” captured contradictions related to equity and social relations e.g. “equal vs. differential allocation of resources”. These were distinguished from “curriculum dilemmas” which captured contradictions in how teachers, “through their schooling acts transmit howledge and ways of knowing and learning” (p. 144). Some of these curriculum dilemmas were described as dilemmas of “personal vs. public knowledge”; “knowledge as content vs. knowledge as process”; “knowledge as given vs. knowledge as problematical”; “whole child vs. child as student”; “each child unique vs. children have shared characteristics” (pp. 135-175). From their empirical base the Berlaks revealed that teaching acts can be viewed as simultaneous resolution to multiple dilemmas. Certainly, Lampert’s dilemma can be viewed through both control (freedom vs. order) and societal (gender) dilemmas. In the Berlaks’ language of dilemmas these are “whole child vs. child as student” i.e. concern with freedom for the boys to be themselves, vs. order so that all could learn, and “equal vs. differential allocation of resources” i.e. how to manage teacher allocation of attention through proximity. The Berlaks distinguished dominant, exceptional and transformational patterns of dilemma resolution. Through dominant or exceptional patterns of resolution to one or more dilemmas, they were able to represent similarities and differences within and across teachers, and use their dilemmas to describe the variations, regularities and apparent contradictions they observed in classrooms. Exceptional or dominant patterns of resolution were not exhaustive of the possibilities. There were also solutions or resolutions where the pulls of opposite alternatives were joined, “when contending presses of the culture at least for the moment are synthesised and thus overcome” (p. 133). They called these transformative solutions, which, they acknowledged, were quite rare in practice. In contrast to Lampert’s argument that effective practical action in classrooms was about the management of tensions, the Berlaks’ emphasis and orientation was on resolution, on overcoming contradictions, or competing alternatives. But surely this is not an either/or? Both can be correct.

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There must be instances where tensions can be resolved, just as there are many tensions in teaching that are ever present in dynamic form and need to be managed. The Berlaks’ study illustrated the potential power of their dilemma language for “ordering the flux of classroom life, and placing the mundane events of daily school life in the perspectives of culture and time” (p. 37). But whatever the power of language, there are also limits. As they themselves point out, just as language can illuminate, so it can distort and fragment social activity. Language categorises and separates consciousness from action and from social context, making it difficult to talk and think about schooling as a continual process wherein context and consciousness are joined in the acting moment. In addition, while a language of dilemmas related to schooling is an effort to present the thought and action of teachers as an ongoing dynamic of behaviour and consciousness within particular institutional contexts of schools, the Berlaks remind us that dilemmas are not entities that may be physically located in a person’s head or in society. Rather they are “linguistic constructions that, like lenses, may be used to focus upon the continuous process of persons acting in the social world” (p. 111). Thus no matter how effective a description might be, it can never fully capture the intensities of feelings, pain and joy, angers and frustrations, that are part of any school’s daily life. The Berlaks’ project was about school change. They asked: would teachers change their patterns if they were more aware of the trade-off in and between dilemmas? This is an important question and of particular relevance in transitional South Africa. What is suggested by a language of dilemmas is that with a language that assists teachers to understand, talk about and act in relation to the tensions in their teaching, they might be better positioned for deliberative and transformatory action. Berlak and Berlak (1981) and Lampert (1985) aimed to describe and explain teaching acts in such a way that not only the complexity of teaching is captured, but also the deliberative and multifaceted way in which teachers think and act. To do this, they both worked with a language of dilemmas, though from different perspectives. In other words, both found in dilemma language a means of analysing teaching in ways that captured its complexity and teachers’ knowledgeability. There was a great deal in how both the Berlaks and Lampert talked about tensions in teaching that resonated with me as I was working to organise and make sense of what teachers had said to me and to each other, and what I saw in their classrooms. However, the dichotomy in their contextual and practical accounts of dilemmas was not reflected in what I observed. A full description of the dilemmas of codeswitching, mediation and transparency as they emerged through the study require an understanding of teaching dilemmas as at once personal, practical and contextual. Moreover, the Berlaks’ and Lampert’s research and the arguments they develop for dilemmas in teaching were developed in contexts very different in time and place from multilingual classrooms in South Africa in the 1990s. Neither focussed specifically on language-in-use in the classroom, nor on the kinds of mathematical dilemmas that are likely to arise in secondary classrooms. Thus, while I found support for the notion of a teaching dilemma as key in my study, I needed to extend a language of dilemmas in teaching in such a way that the tensions and

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contradictions in multilingual secondary mathematics classrooms could be made visible. 4. DILEMMAS IN TEACHING: A CONTEXT FOR KNOWLEDGE GROWTH IN TEACHING Recently, Jaworski (1999) reinterpreted her study of investigative mathematics teaching in the United Kingdom explicitly in terms of dilemmas. Drawing on her research with six secondary mathematics teachers, all of whom were interpreting ‘investigative’ mathematics teaching at the time, she describes tensions present in investigative approaches to mathematics teaching in secondary schools in terms of two key dilemmas: the dilemma between “inculcating” or “eliciting” knowledge, and the dilemma between being “didactic” or “investigative”. She argues further that these are tensions in teaching that teachers, through conversation with others in the community, including researchers, can use reflectively to build their knowledge of teaching. Jaworski gives practical meaning to the dilemmas through an example of a lesson on tessellation, where there were two competing or conflicting goals for the teacher. As part of her investigative approach, the teacher set tasks so that students could explore whether or not different quadrilateral shapes tessellated. She also wanted learners to come to know the “fact” that “all quadrilateral shapes tessellate”. Jaworski provides a reflectively rich account of the teacher’s struggles: learners saw how particular quadrilaterals tessellated, but that they were not convinced that this was so for all quadrilaterals. Should she tell them this “fact”? For Jaworski the tension in this example was illustrative of tensions in investigative teaching where teachers wanted to “encourage students’ autonomy in knowledge construction, but the construction had to result in particular knowledge”. These are clearly curriculum dilemmas, in the Berlaks’ terms, between personal knowledge vs. public knowledge, and knowledge as given vs. knowledge as problematical. Jaworski draws on Edwards and Mercer’s (1987) study of primary classroom practices in the UK in the early 1980s and so uses a different language to interpret these dilemmas. Edwards and Mercer described a general dilemma in learnercentred pedagogy of having to “inculcate knowledge while apparently eliciting it”. They saw “the teachers’ dilemma” as lying in “the problem of reconciling experiential, pupil-centred learning with the requirement that pupils rediscover what they are supposed to”. Jaworski argues that it is possible to describe this central dilemma as a result of “conflicting paradigms” or “irreconcilable cultures’’. Teachers could be understood as situated simultaneously in an “objectivist paradigm in which the required curriculum and its examination structures are based, and a constructivist paradigm in which the teaching is situated. They were also situated simultaneously in a pedagogical culture that was to enable learners to explore and have confidence in their own ideas and a culture where learners “wanted to be told. Indeed, as some have argued, the demands of curriculum were such that it was intellectually more honest to tell learners what they were expected to know (Driver, 1983).

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Like Lampert, Jaworski’s overarching concern is to relate the theoretical and the practical. She is clear that the one can never be the other, that the theoretical is always an interpretation of practice, one of a number of stones that could be told about practice. In placing the dilemmas in paradigmatic and cultural conflicts, she attempts to bring the personal and contextual into a relationship with practice. Her explanatory framework, however, is philosophical and sociocultural and does not extend to include societal issues of politics and economics. This is not her project. She argues instead that while dilemmas might result from conflicting paradigms or philosophies, or conflicting classroom cultures, for each individual teacher, they involve coping with moral and emotional issues “whose resolution is far from clear”. In other words, like Lampert, Jaworski is clear that no amount of theory per se will solve the problem for the teacher. However, like Berlak and Berlak, she finds in the notion of dilemmas, and the language they provide to talk about practice, possibilities for knowledge growth about teaching. Jaworski suggests that the selected classroom situations may be seen as starting points for further exploration of the kinds of questions dilemmas pose for school mathematics practice, that is, for knowledge growth about teaching. 5. A LANGUAGE OF DILEMMAS: A FRAMEWORK WITH WHICH TO PROCEED At this transitional juncture of South Africa’s social and political history, where choices have to be made in new educational policy formulation and its implementation in the flux of classroom life, a language of dilemmas offers a great deal. Having an accessible language with which to understand the forces, internal and external, that prevail in sets of circumstances might well provide the diversity of teachers with resources and tools to think and act constructively in their complex classrooms, resources that draw from the practical, the personal and the contextual in teaching. At a theoretical level, conceptualising dilemmas as at once personal, practical and contextual removes the dichotomy between Lampert’s practical account of teaching dilemmas, and the dialectical and contextual account provided by Berlak and Berlak. Berlak and Berlak express the hope that ... dilemma language will be useful in clarifying for professionals and the public some of the debates of schooling practices and their relationship to the major political and economic questions of the day, and for helping to identify alternative possibilities for making schooling a richer, more engaging and challenging intellectual, cultural and social experience for all students. (Berlak and Berlak, 1981, p. 9)

The Berlaks did so in the context of the conflicts over educational reform at the end of the 1970s in the USA and the UK. But their words could have been taken from the mouth of a researcher in current South African education as he or she confronts the realities of trying to carve more equitable, more meaningful and higher quality intellectual experiences in schooling for all South African chldren. And this needs to be done in an even more conflictual and contradictory context.

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Of course, Lampert, Berlak and Berlak, Edwards and Mercer and Jaworski are not the only educational researchers who have talked about dilemmas. As I noted in opening this chapter, teaching is complex and so dilemma-filled. There are no doubt numerous texts on classroom practice that highlight dilemmas in teaching. I have selected these for focused discussion because they pay explicit attention to dilemmas, illuminating what they mean in and for practice, and so to a language with which to describe them. Their motivations and descriptions were in sufficient resonance with my motivations, observations and interpretations to provide a framework from which I was able to capture teachers’ knowledge of their complex practice. Theorising dilemmas as at once personal, practical and contextual provides a framework through which it is possible to understand the management of dilemmas in practice — the practical (Lampert) — and that effective classroom action is about managing complexity rather than reducing it. At the same time it is important to understand dilemmas as at once personal and contextual (Berlak and Berlak), a reflection of the teacher-in-context. The notion of teaching dilemmas as at once practical, personal and contextual becomes a useful tool through which to capture the complexities and contradictions in teaching. Moreover, as Berlak and Berlak have exemplified and illustrated, a language of dilemmas needs to describe and distinguish intersecting dilemmas of school practice. Teachers can use such a language to reflect on their actions and on possibilities for change in relation to the wider social, political and economic context, or in Jaworski’s terms, for growing their knowledge about teaching. In Chapter 1 I referred to this action and reflection on both theory and practice, and resultant knowledge growth about teaching, as praxis. Teaching dilemmas are a source for action and reflection on action, for concretising theory in practice, and for theorising practice. Teaching dilemmas are a source for praxis, for working in and with the dialectic of theory and practice, It is within an overarching situated approach to teachers’ knowledgeability, and with a framework of managing dilemmas, dilemmas as personal and contextual and dilemmas as a source of praxis, that I now proceed to an analysis of the teachers’ articulated and tacit knowledge of their practices in their diverse multilingual and mathematical contexts, and so too to a further development of a language of dilemmas in teaching.

CHAPTER 5

TEACHERS TALKING ABOUT TEACHING: THE EMERGENCE OF DILEMMAS 1. INTRODUCTION The focus of this chapter is the teachers’ consciously articulated knowledge — what they said — about their practice. It presents an interpretation of Ianguage-related commonalities, divergences, presences and silences in the six teachers’ initial interviews, and thus what they were able to talk about as they stood back and reflected on their teaching. From my perspective as researcher, these were illuminating of how different contexts and conditions give rise to different dilemmas for teachers as they go about their work. 2. A FIRST LEVEL OF ANALYSIS The initial semi-structured interview with each teacher probed the context and ethos of his or her school, what were regarded as general challenges in teaching secondary mathematics, as well as the specific language issues each teacher thought he or she faced. The intention behind a conversation that included school ethos and context as well as general issues of teaching mathematics was to provide a context for interpreting the teachers’ accounts of language issues. I began my analysis using the broad curriculum categories of mathematical knowledge, context, teacher, learner, pedagogy (the relationship between teaching and learning and language, as well as sub-categories within each of these (for example, Learning in an Additional Language LAL, mathematical discourse and communication). These categories of interpretation arose firstly from my perspective of curriculum as relational (an interaction between learner, teacher and knowledge in context), and then from the form of, and assumptions underlying, the interview, and the teacher responses and discussion. Careful ‘listening’ to the data in the light of the field of language and mathematics education assisted the construction of the sub-categories and the generation of a ‘map’ (as in Figure 2 below) of each teacher’s interview. The tallies in each map against the broad curriculum categories described above reflected the number of teacher utterances in relation to that category and are summarised in the frequency table below. For example, in Helen’s transcript there were 23 instances of her talking about pedagogy. The six maps and the talks in each map together produced a picture of presences and silences within and across the interviews. As the frequency table below shows, context features prominently in both African township teachers’ interviews, a reflection of their difficult working conditions. Pedagogy (which includes approaches to teaching and learning, and classroom interaction) was noticeably more dominant for teachers who are trying to shift their classrooms to be more communicative and learner-centred. All teachers talked about mathematical discourse. Although they did not necessarily use the term ‘register’ they all spoke of the specificity of mathematical language, and particularly

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its symbolic form. All the teachers also spoke of LAL issues, issues of learning mathematics in English. Of significance here was that far more was said in relation to LAL by Thandi, Jabu and Clive, all of whom had fully, or almost fully, African classes i.e. English was an additional language for their learners. The silence in the two township teachers’ interviews in relation to what I have termed communication and culture was also interesting. Jabu and Thandi did not discuss communication breakdowns as attributable to a relationship between communication and culture. Table 3. Utterance Frequency Across Selected Categories

MATH KNOWLEDGE

LEARNER PEDAGOGY Explicit language teaching Approaches Interaction etc

CONTEXT

TEACHER

LANGUAGE LAL -Learning in an Additional Language Communication Mathematical discourse etc

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3. COMMONALITIES

3.I Communicating in English, the language oflearning and teaching 3.1.1 Policy and practice As urban secondary schools in Gauteng, English was the official language of instruction in all six schools and each teacher spoke of learners and/or themselves, using their main language at times. For example: Jabu: Eh, okay you see maybe some work is not done, I scold them in English, and if Ifeel that it has not got the message home, then I do it either in Tswana or Zulu, these are the languages in the school. Then ... I feel that I reach them. JA:

And if you are trying to explain something new, some maths ...

Jabu: Okay ... In that case I think of a problem in their day-to-day situation. Then, I feel, if it will impact to use the language they use at home then I resort to it.

(and a little later) ... when they come to talk to me on anything, I insist on them speaking to me in English most ofthe time. ... Clive: ... there is a rule in the school that they are not allowed to communicate other than in English ... I don't really stick to that too strictly, unless I have reprimanded someone and they break into vernac 1... so quite often I think they do communicate in vernac. But very often, I mean, they are learning maths in English. So they will talk about it in English. Sue: [Talk in class is] mainly in English, but when it gets very difficult or when they startfighting or arguing they will go into the vernac ... and I have my own policy, not a classroom policy. But, when I hear that happening, I will kind of walk to wards the group, listen, and then hope that they don't stop. If they do stop I say continue and once they have finished and if they agree or establish something then I say OK now explain to me in English or if there is a child in the group who doesn't understand vernac, and there sometimes is, then I say "OK! We don't understand, so can you explain to us?" So they are getting the chance to explain in English after they have understood what they are trying to do in the vernac.

3. I.2 Communication in English was difficult at times Each teacher mentioned some occasion when either they had difficulty in explaining through English, or learners had difficulty in expressing themselves in English. Clive: I find if a child wants to ask a question, they very often can't ask the question. They might know in their heads what they want to ask, but what comes out is a jumble ... and then sometimes I think I know what they have said but then I answer the wrong question. I can see from their faces ... there is a lot of difficulty with black children communicating what they want to ask.

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CHAPTER 5 Thandi: I don’t think they have a problem with hearing. They do hear what you are saying. But maybe answering hack, asking you a question in English, its a problem ... when they have to say it - saying in English that is the problem. ... at times ... you feel that you don’t succeed to reach them ... At times they Jabu: struggle to explain in English. And I say: ”use any language to explain the concept’: ‘cause I realise that he is battling with two things. First is the language and then the second thing is the math concept. Let him rather struggle only with the concept and pick up with the language later.

It makes sense that where English is an additional language for learners, they will, at times, resort to their main language in their learning, and also struggle when to express themselves in English. Furthermore, multilingual teachers who can speak the main language of the learners will do so at times. As I point to the ‘obviousness’ of the teachers’ comments, I do not wish to trivialise what they said. These excerpts reveal, even if somewhat superficially, that the experience of LAL learners in school was such that at times they were blocked from using verbal speech. And in terms of a social theory of mind, this must have some effects. Thandi: ... even though they can hear like I said, it would be much better if they could also talk about it in English - they would understand more - they would raise their views on it and it would make it easier to understand. I think communication is very important in learning maths, and ifyou can’t communicate then it makes it difficult

Teachers in this study knew that in their multilingual settings, teaching and learning mathematics in English inevitably meant using languages other than English in the class; and that at times, they and their students struggled to communicate with each other in English. 3.2 Difficulty with the mathematics register and mathematics symbolic form Aside from well-known difficulties with ‘word-problems’, all six teachers described difficulties with mathematical English and with algebraic language. For example: ... what did I have recently? Oh yes! 2a - a. To me they said: (pointing to the Jabu: a) “There is nothing here, so the answer is 2a”. They see 2a and 0a and so get 2a. Sara: ... “simplify” for Std 6 is a real problem - sometimes it means add the terms and sometimes it means other things ...

Some grappled with how and whether the mathematics register, particularly the use of ordinary English in mathematics, presents greater difficulties for LAL learners: Clive: ... one is that they (pupils) just don’t understand the use of words. They don’t understand the links between certain words, words which have similar meanings. They don’t get the subtleties. Like they don’t understand the difference between factorisation and division, that kind of thing, products and multiplication, units, for example millilitre and millimetre ... and certain words which have rigorous meaning, like “at least”, “at most”, things like that, which are used in everyday language. I think black learners use them differently. They use “at least” in particular ways ... Logical things and negations are difficult for second language learners, and we have those things in mathematics.

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Sue: Sometimes it is ordinary English that I am using to explain things. The technical words are a problem but I think they are for all kids. The mathematical English is also a problem for all kids, but maybe, ... there is this one issue of “divide by” which comes first? and “subtract”. Now that may be a problem for English-speaking kids but it may also be cultural ... At our school, kids say “this subtracted with that” and I have always interpreted to mean “this subtracted from that“, but it is not clear because “this added with that” can be the other way ... so there is a confusion over commutivity... (they) also say “this subtracted with that”.

The teachers in this study were thus aware that the mathematics register (mathematical English and mathematical symbolic) presented difficulties in general, that it could clash with everyday language, and that there were specific instances of clashes and confusions for LAL speakers. The specificity and challenges in learning to use the mathematics register and particularly algebraic language are well known and extensively discussed by Pimm (1987). These continue to generate interesting research across contexts. Studies continue to explore the relationship between mathematical performance, linguistic proficiency in the language of learning and teaching and the specificity of aspects of the mathematics register (e.g. MacGregor and Price, 1999; Clements, 1999; Yoong and Ramlee, 1999). Nevertheless, as argued earlier, the issue here is not simply one of the mathematics register and symbolic form. It is also about how mathematical discourses are learned (in English), within school mathematics culture. 4. DIVERGENCES AND THE EMERGENCE OF DILEMMAS More interesting than the commonalities across what all six teachers said, were the dimensions of language that emerged for some of the teachers in their particular multilingual settings. These language dimensions are interesting because they were linked to the context of the teacher’s work, to situations of change, and to the teacher him- or herself. They revealed the dilemmas that the teachers were facing in their practice. 4.1 Developing English vs. developing meaning: or, to code-switch or not Both teachers in the African township schools were multilingual, and fluent in both English and Setswana. Jabu, when he needed to, also spoke Setswana and/or Zulu in class. This asset was not shared by the four English-speaking teachers, though Helen could understand and speak Zulu. With this bi-, tri- or multilingualism in teachers comes the possibility of code-switching, of communicating in both English and the learners’ main language. There are, however, tensions in code-switching as teachers work to enable their students’ understanding, and to follow language of instruction policy (Rubagumya, 1994), tensions that create dilemmas for teachers: Thandi: ... in Standard 7 [Grade 9], where I asked a question and one answered in Tswana. And I said: “Can you please try answer that in English - I don’t understand that? ” and he said, crossly “No, mam, but you are Tswana - you are not white! ” He was angry. But it is not like that in Standard 9 [Grade 11] ...they like to work in English.

And later in the interview:

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CHAPTER 5 Thandi: Sometimes you find that you get stuck because students cannot communicate then, though not much, you resort to Tswana. You are careful because if you do that then they want you to do it all the time, and they turn the situation to a Tswana class. Then they will never improve.

Earlier in her interview, Thandi was also quite adamant that she “never translate(s) ...”. This could be related to the strong policy enacted in her school by a grouping of teachers which she explained as follows: Thandi: We have sort of formed a group. We have said that if there is someone who doesn’t come to school we confront them, and ifyou don’t appear in class we confront them. Ifwe pass your class and you are teaching in vernac, we confront you. So that is the thing the group adopted.

In current language-in-education policy, i.e. since these interviews were conducted, multilingual teaching strategies like code-switching are explicitly valued, and supported by the country’s constitution. However, as discussed in Chapter 2, the LoLT especially in secondary schools remains English The conditions that gave rise to the kinds of tensions Jabu and Thandi experienced are still largely in place. Linked to the reality that code-switching is not straightforward, Thandi said: Thandi: ... there are Xhosa speakers in the class - so if I am speaking Tswana then they complain I am favouring them ...

If anything, these tensions are now even more acute in the urban areas where, as discussed in Chapter 2, migration patterns have increased the multilingual character of many classrooms. On the other hand, Jabu acknowledged difficulties of teaching in English and explained how some ideas (those easily linked to everyday life) are conveyed better if explained in Setswana. Jabu: ... at times (teaching in English) is a problem. You feel bad that you don’t succeed to reach them, and then this is bad. But it only happens in the lower classes ... with the tens [Grade 12] it is English all the time.

Talking about the 2a - a problem above: Jabu: I say, “my girl, bring me that book”. Then I say “I have two books, and she brought one book”. And that, problems like that I go on in English. But there are similar cases like say half plus half, and they have serious difficulties, and then I say, in Tswana: “I have a half a loaf and a half a loaf how many ... ”?

He acknowledged that “unfortunately” at a Grade 9 level learners interacted with each other “in their own language”, but that “they need to talk to each other in a language that they understand”, and further, that if he did need to explain something, he would do it in both isiZulu and Setswana as these were the languages in the school. It is clear from these two accounts that while these teachers held different views in relation to English use in class (it is possible to argue that Jabu treats home language more as a resource than does Thandi), both spoke about using multilingual teaching, that is, they engaged in forms of code-switching. Their strategy was to use English themselves most the time. But in the lower classes it was simply not possible to do so all of the time. The dilemma apparent here is how to develop

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spoken English without jeopardising mathematical understanding in the lower classes and this dilemma emerged because of the particular multilingual context: teachers and many learners shared a main language and this was not the LoLT. The dilemma of whether or not to switch between English and learners’ main language in the mathematics classroom was not only a function of educational policy of English as LoLT. There were both political and pedagogical dimensions to this dilemma. For Jabu and Thandi, learner facility with English was also about access: access to tertiary education, “to college or technikon and there is English on campus”; and access more generally where without English “then the child will land up the victim”. The dilemma also reflects the common sense assumption noted in Chapter 1, and one that is reinforced by the teachers’ own learning experiences, that language is acquired in use. If English was to be acquired it should be used as much as possible.2 From what Jabu and Thandi said, as multilingual teachers they could and did switch in and out of different languages. They viewed code-switching as inevitable and necessary though “unfortunate” thus reinforcing Rubagumya’s (1994) claim that in contexts similar to Burundi, code-switching is tolerated rather than encouraged. Current policy that now advocates multilingual teaching was in fact already in practice, and has been for some time. But multilingual teachers are faced with a constant dilemma, particularly in the junior secondary school, over how to develop both mathematical meaning and spoken English; over how, when and for what purposes, code-switching should be used. Code-switching and treating language as a resource (and not a problem) makes both political and pedagogical sense. Yet, from what the teachers in this study said, code-switching in secondary multilingual mathematics classrooms in practice is not straightforward. An obvious question that arises is: does what teachers say actually happen in class? This question is the focus of Chapter 6. The dilemma of whether the teacher should use a main language besides English in the mathematics class (whether the teacher should code-switch) was obviously absent in contexts where the teacher was English-speaking, and did not speak or understand any African language. While the dominant strategy for Jabu and Thandi was to use and have learners use English as much as possible, this practice was taken-for-granted in settings where the teacher was English-speaking. The following kinds of speaking strategies emerged as these teachers talked about their multilingual settings, that is, where the teacher is not multilingual,3 and some or all learners bring to class main languages Werent from the LoLT: (a) being alert to learner-learner discussion that occured in another main language and asking that this then be explained in English (for the teacher and other learners) (Sue, see above); (b) encouraging learners to use whatever language was comfortable as they discussed and developed mathematical ideas with each other (the private space), but that when learners reported their thinking to the rest of the class (the public space) this needed to be done in English (Helen); and (c) repeating ideas in English different in ways (Clive) and asking learners to repeat words and phrases in English (Sam).

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Strategies (a) and (b) are common in LAL teaching and acknowledged as ‘good practice’.4 They are bound into pedagogical situations where learners are seen as active meaning-makers, and hence given an opportunity to discuss their ideas. This also accords with Levine’s (1993) research on second language teaching developments (in LAL) that emerged as ‘good practice’ for all language teaching. The latter strategies discussed above are more linked to traditional teaching modes. Each strategy, of course, brings its own dilemmas and these are dealt with in the next section. At this point of the analysis, it is necessary to posit that surely all of the above hold for all teachers, not only mathematics teachers? In the words of one of the teachers: Thandi: It is complicated - the problem is the maths. I don’t think they have a problem with hearing, they do hear what you are saying ... Because as I said, when you are teaching they don’t say “I don’t understand” the English you are speaking. It is the maths you did that they don’t understand. But when they have to say it - saying in English, that is the problem.

4.2 Developing mathematical communicative competence vs. negotiating and developing meaning As argued earlier, and this was recognised by all six teachers, it was not simply the English that was the problem: Helen: The problem [in group work - with discussion in any language, report back in English] is ... if all your discussion is in Zulu you get to the concept then you can’t report back in English so you can’t talk about it in English ‘cause you never developed it in English - they don’t develop the English to speak with. This is a difficult question it can be dealing with the problem or making it worse - and now I want to be able to explore it further. I am not sure which is better. If you start to try to develop the English while they are reporting, then you can be putting words in their mouths instead of hearing what they have construed. Sue: ... like there are some kids who are really not good at explaining themselves, and I don’t do anything to address that except to try to get them to explain it again because the class hasn’t understood. And they still do it badly, and then I say can someone help him? And by listening maybe he will get the chance to develop. I haven’t spoken to the English teachers about that but I want to because I am sure they have got strategies of actually developing a more exact way of communicating. JA:

And that would be an important thing to do?

Sue:

I am wondering. I don’t know - these questions are starting to arise ....

The experience of both Helen and Sue was that their learners clearly had difficulties in expressing their thinking. But this was not any thinking. It was specifically their mathematical thinking. Indeed, mathematics is hard to speak (Pimm, 1987). These accounts were from contexts of shifting pedagogy, from teachers whose pedagogical practice placed greater emphasis on learners’ meaning-making and

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included learners exploring mathematical ideas and then sharing these with the whole class. It seemed that some learners needed help to express their thinking both in English and mathematically This opened up two questions for the teachers. One (the last quote above) was the recognition that specific language help was needed. As mathematics teachers they had not also been trained as language teachers. So, where and how were such teaching skills to develop? The second was when and how to help with ways of speaking mathematically so that as the teacher you still listened carefully to what learners were trying to convey (their exploratory mathematical talk), that you assisted the negotiation and development of meaning with learners without blocking their meanings by prematurely working on how these were expressed (their display of mathematical knowledge). 4.3 The dilemma of implicit or explicit practices: whether or not to be explicit about mathematical language The above two questions that emerged for the teachers led on to the issue of explicit mathematics language teaching or what could be phrased as the dilemma of when and when not to tell, to be explicit about how mathematics is spoken, or to model mathematical language. The need for explicit mathematical language teaching was discussed by Helen and Sara, teachers in suburban schools whose classrooms had deracialised rapidly: Helen: As you were talking, something struck me ...is the assumption that everyone understands English ... you’ll say ... I can’t think of an example ... at points when I give an instruction, I write up the word on the board so that no-one is unclear of what it is and I have realised how many kids from English-speaking homes never knew that word, the one I wrote up was the one I was saying and that was interfering with their maths a lot ... I can’t think of an example, but it has happened to me several times. Where I would have assumed a few years ago in an all white class I would have just gone ahead and talked aways and now because there were black children in my class and I was writing up in a conscious effort to explain the English that I suddenly realised it was benefiting the English-speakers as well. Sara: I think one notices it more with the black kids ‘cause it is just so obvious. But there are some of the other kids who I only realised afterwards weren’t quite sure what I was on about and they had been too scared to ask ... JA

That is interesting - they get hidden?

Sara: Yes, especially if it is a big class and everybody seems to be carrying on and working away and it is only when you come and look underneath that little hand to see what is going on that you realise nothing is going on.

and later Sara: As I have said it has made one more aware of being careful about how you present things because you know there will be kids who don’t understand everything you say. Whereas before you just assumed that because kids spoke English at home they could understand everything you said, but they don’t and having the other children

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CHAPTER 5 there makes you aware that they don’t understand the more adult words — more aware of language.

These teachers’ experience was that explicit language teaching did help, and it helped everybody. Pimm (1987) and Mouseley and Marks (1991) also advocate explicit teaching of aspects of the mathematics register, and different genres within mathematical discourse. But the value of explicit teaching can lead to a deep pedagogical tension, and hence acute dilemmas for teachers. Learner-centred approaches to teaching and learning mathematics call for far more opportunity in classrooms for learners to investigate and build mathematical meanings and for teachers to listen to, build on, and interact with the meanings learners bring to and make in class. This is in tension with explicit mathematical language and genre teaching and both, in different ways, are about access to mathematical processes and its products for all learners. There are also strong interpretations of constructivism that conflate or reduce all teaching to intervention in learner’s meaning-making. For teachers who hold this view, explicit teaching will present a paradox. 4.4 Culture and Communication: is the problem language? vs. is it classroom culture? As mentioned earlier, what was also interesting was the absence of both miscommunication and the need for more explicit teaching in the interviews with the township teachers. Jabu and Thandi ascribed communication difficulties in their classrooms to English as LoLT, or to mathematical language, and not to ‘culture and communication’ issues. Three of the four English-speaking teachers in private and suburban schools told stories of communication breakdown. Sue: Its language as well. We tried [when setting classes off on investigative tasks] to explain it to the Std 5s [grade 7] in terms of exploring, like what do you do when you are exploring? You go looking for something but you maybe find other things along the way, and you try to ... we couldn’t talk about explorers in history because they hadn’t done that we talked about exploring an area like town or a new place for the first time like [school name] when you first get there how you explore it. But it didn’t really help as an analogy ... I don’t think, it may later ... but a lot of them maybe just didn’t understand what we were saying ...I mean the actual English I think may have been a problem with the Std 5s. Helen: I am not sure if I can explain them but they happen ... for example, we did a test at the end of algebra ... we developed a test where they had to develop a pattern ... the little squares that form a T ... we set that as a group-work test so part of the assessment was working as a group etc, and then apart from solving and handing in something written as a group, they had to explain what they did as a presentation to the class. The other three classes managed. My bridging class didn’t begin to handle the task and we had done three weeks of investigation, whereas the others had only done two weeks, we had done 7 to 9 investigations and discussed, groups, explain on the board etc. nothing, nothing, nothing, nothing, handed in pages of nothing, nothing it was weird.

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Sara: There are three who really battled at the beginning of the year. I mean the first test I did, one didn’t realise this was supposed to be a test and she just stared at it not knowing what to do. She hadn’t realised this was a test and that she must write the answers down. This horrified me. I was totally astounded. Surely her understanding couldn’t have been that bad that she could write nothing? And when I asked her she said she didn’t know she was meant to write the answers “down here” - she handed this page with I to 20 written on and the penny dropped only at the end and then she was too scared to tell me about it. .. I felt terrible about that. And since then, I have been a lot more specific to make sure everybody is clear what they need to do.

These were instances of communication breakdown. Why did they happen? The teachers struggled to interpret and explain them, and this points to the complexities of culture and communication in mathematics classrooms. Sue elaborated at other points in her interview: Sue: I think the issue of second language learning is a real one. I mean the fact that kids are not understanding all the language that is in the classroom, that their language is not the medium of instruction ... I mean sometimes it may be previous educational experience. A child might look blank because they are not used to what you are doing in class and you interpret that as them not understanding [the maths]. Sue: Again it is past experience ... because they are used to filling in words or doing exercises ... with the investigations it is difficult because often they are structured, like in the points of departure ones where there are like five questions. Each question is a real question and they think each question is like a one word answer [laughs]. So five minutes later they are finished.

Whether it was about writing a test on paper, or working in groups, or doing a mathematical investigation, these pedagogical practices carried with them cultural assumptions about being in a particular school and in the mathematics class. They were clearly not simply about English proficiency nor mathematical skill, but about these within classroom cultural processes — or in Bourdieu’s (1990) terms: cultural capital. They were situations where all three dimensions of mathematics teaching and learning in multilingual classrooms were in interaction, and the most difficult to interpret. They created what seem to be dilemmas of explanation (in contrast to dilemmas of action): is it the language and/or the mathematics? vs. is it the classroom culture? 5. PRODUCING THE DILEMMAS The teachers’ initial interviews (what they said about their practices) revealed that to a greater or lesser extent, mathematics teachers in multilingual settings in South Africa were already multilingual teachers. By this I mean they had developed and were aware of some practices and strategies to deal with and enable the multilingual learners in their mathematics classrooms. In relation to literature in the field, some practices, for example explicit mathematics language teaching and encouraging public reporting in class in English, constituted ‘good practice’ for all learners. Moschkovich’s (1999) recent research that identified teachers’ revoicing of learners’ mathematical utterances is another case in point. It makes sense then that these strategies should be captured, shared and developed with other teachers.

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But practices always have multiple effects. It was possible to infer from these mathematics teachers’ accounts that in their multilingual contexts they confronted situations that would otherwise be taken-for-granted, and some of these situations produced dilemmas. Jabu and Thandi, in their township schools, faced the dilemma of developing English vs. developing meaning, particularly at the junior secondary level. I have called this the dilemma of code-switching. Because they were bi- or in some cases even multilingual they could switch in and out of English and other main languages in their teaching, and the manner and extent to which they did this seemed to relate to the context of their school and to their own theories. The importance of English as the language of assessment and access, together with the increasing demands of the (English) mathematics register in the secondary school, created a pressure to teach in English and hence the dilemma for teachers. This dilemma relates to, but is not sufficiently specified in, the Berlaks’ curriculum dilemma of personal knowledge vs. public knowledge. This is the dilemma over what knowledge is the most worthwhile: that which connects with the knower or that which is more public and deemed needed to be known. The expression and display of mathematical knowledge in a learner’s main language (but not English) is probably more connected to the knower than to mathematical English, but it is the latter that is the route to scholastic success and further political and economic access. according to Jabu and Thandi, the problem of code-switching in the mathematics classroom seemed to disappear in the senior secondary school. How does this happen? What is it that teachers do? The dilemma of code-switching (of developing English vs. developing meaning) is the focus of the vignette in Chapter 6. Obviously, code-switching and/or the extent of English spoken by the teacher was not an issue for English-speaking teachers who did not speak an African language. These teachers, however, faced other dilemmas as they confronted the taken-for-granted in their particular contexts. Teachers who had attempted to change their pedagogy faced the dilemma of developing mathematical communicative competence vs. negotiating and developing meaning. I have called this the dilemma of mediation. In multilingual classrooms where learners were expected to talk to each other about their mathematics and to report back publicly on their thinking, different expressive competence was noticed. While this would be the case in any mathematics class, in multilingual settings it was often difficult to work out how both proficiency in English and understanding of mathematics were implicated. Was it the language (i.e. English) that was the problem, or was it the mathematical concept that was difficult for the teacher to explain and for learners to understand? In more analytical terms it was difficult to distinguish whether difficulties were linguistic or epistemic and how these were related. There was nevertheless, less of a tendency to reduce differences in students’ expressions of mathematics to ‘innate’ mathematical abilities, and so more of an openness to see expressive competence as linked to language and thus possible and necessary to act on in a specific way. I have discussed in previous chapters that we communicate to learn, and learning is about becoming communicatively competent Teachers thus need to assist learners’ talking to learn mathematics, and their learning to talk mathematically. The

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dilemma is how and when to act! Premature actions could silence the learner and prevent the teacher from really listening to the meanings the learner is trying to convey. On the other hand, delayed mediation could be destructive for the learner and create confusion in the class. What are appropriate and timeous mediational actions aimed at learners improving their expression of mathematical thinking? In Mercer’s terms (1995) how does the teacher assist the learner in moving back and forth between the learners’ informal expression of mathematical ideas and the formal discourses of mathematics? Linked to this is the argument that in mathematics classes peer discussion and reporting are two different tasks, and that reporting skills should not be left to chance, but explicitly taught. Too much emphasis on process and learners’ meanings can inadvertently perpetuate social difference by not making explicit to all learners exactly what is expected of them (Mouseley and Marks, 1991). Pimm (1992, 1994) has also argued that there is a specificity to the communication skill of reporting mathematical thinlung. Pimm raised the inevitable tension that if such skills are made explicit, then there is the danger that reporting becomes focussed on the form of the report rather than its substance. The dilemmas of mediation are clearly not new, nor unique to multilingual mathematics classrooms. They relate directly to the dilemma of eliciting vs. inculcating described by Jaworski, and to some of the curriculum dilemmas described by the Berlaks such as personal knowledge vs. public knowledge and each child unique vs. children have shared characteristics These are all at play in mediating the school mathematics curriculum and particularly so in classrooms where teachers have tried to shift their pedagogy to value and work with personal meanings, social interaction and diversity (Bartolini Bussi, 1998; Sfard, 1998). Multilingual contexts bring dilemmas of mediation inescapably to light. The crucial question now is whether and how the dilemmas that have been inferred from what teachers said relate to what they did in class. What happens in day-to-day classroom practice as teachers’ confront such dilemmas? This question is taken up in general terms in the concluding section of this chapter and is the focus of Chapter 7. Linked to dilemmas of mediation is what I have termed explicit mathematics language teaching. I have called this the dilemma of transparency. Explicit mathematics language teaching was a strategy consciously developed by teachers whose classrooms had recently changed from being homogeneously Englishspeaking to being multilingual. They claimed that being explicit about mathematical language in fact benefited all learners. They were forced to confront what they had previously taken for granted — that the learners, irrespective of their main language, were not equally competent in English. This diversity was previously hidden in a seemingly homogeneous English-speaking class. Here again is ‘good practice’ that needed to be further developed and shared. Two interesting and difficult questions arose from this ‘good practice’. The first was whether the silences in Jabu and Thandi’s accounts suggested that aspects of diversity were hidden in the seemingly homogeneous all-African township classroom. It relates to the question the Berlaks ask about whether and how teachers treat learners as unique or with shared characteristics, that is, whether teachers act differently towards different groups of children

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This is an interesting question in South Africa as we move beyond apartheid. Perhaps with a past where difference was reduced to race, diversity within township classrooms could now become more visible. The second was the deep tension and paradox mentioned briefly above. Learnercentred curriculum initiatives With their more democratic ideals of listening to learners, of creating space for their creativity and meaning, of less authoritarian classroom practice, require teachers to stand back more. At the same time, such classroom processes rely on communicative competence in learners as a means of sharing their ideas with others. The tension is that if communicative competence is to develop in mathematics, and this is equally important from the perspective of democracy, access and equity, then it requires mediation in general and explicit teaching of mathematical discourse in particular The dilemmas inferred from the teachers’ interviews are in support of a sociocultural perspective where the teacher’s role is understood as enabling learners to travel across boundaries between knowledge areas. What arises here is the dilemma of implicit and explicit practices, of when to focus on mathematical language (make it visible) vs. when to background language and focus on mathematical meaning (render language invisible). This dilemma, and its theoretical antecedents, are the focus of Chapter 8. Here again is the personal knowledge vs. public knowledge dilemma in a form that focuses on both language and mathematics simultaneously. Finally, some of the teachers’ accounts categorised as ‘culture and communication’ revealed situations that point to the interaction of English proficiency, mathematical understanding and classroom culture. None was attributable on their own to any one of the three dimensions of the dynamics of mathematics learning and teaching in multilingual classrooms. These accounts were clearly incidents of communication breakdown. Within a social theory of mind, we can explain their occurrence: attempts to communicate are through language: and in the classroom, through a great deal of, but not only, verbal speech. Language, as a bearer of meaning and motivation, is imbued with culture and history and hence not easily interpreted by learners with different cultural and historical experiences to those that underpin classroom mathematical processes. Practically, however, these incidents were difficult for teachers to explain and understand. They point to a dilemma of explanation for teachers that I described earlier as: is the problem language? vs. is it classroom culture? Again incidents of ‘communication breakdown’ were absent in the township teachers’ interviews and this suggests that either they were somehow hidden, or that in such contexts, this kind of communication breakdown was simply absent. Video data, reflective interviews and workshops, unfortunately, did not provide sufficient scope for further exploration of this dilemma through a focussed vignette. It clearly is an important aspect of working in multilingual mathematics classrooms and requires further study. 6. IN CONCLUSION The complexity of teaching and the inherent tensions in the teaching-learning dialectic are such that all teachers face dilemmas. Detailed and systematic analysis

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of teacher accounts of their mathematics teaching in diverse multilingual settings has revealed that teachers in different multilingual contexts faced different dilemmas, thus supporting the notion of teaching as a social practice. Specifically teachers in African township schools faced the dilemma of code-switching. They needed to ensure that learners understood the mathematics they were trying to teach and that they developed their competence in English. Teachers who had tried to make their classes more participative, communicative and inquiry-based faced complex dilemmas of mediation — of moving effectively between learners’ more informal expression of their mathematical thinking and more formalised school mathematical discourse. And teachers in schools that had suddenly and rapidly become multilingual and multiracial argued the benefits of explicit language teaching for all their learners. In relation to learner-centredness, however, tensions reside as to when to be explicit about mathematical language and when mathematical language needs to be backgrounded. Of course, teachers’ knowledge of their practices is more than they can selfconsciously articulate. Clearly these accounts need to be related to actual classroom practices, to what the teachers did. How did teachers act in the face of the dilemmas described here? How did they manage the dilemmas they faced? In Lampert’s (1985) terms, the questions become: How are teachers’ actions shaped by the practical and the personal, by who the teachers are at particular moments in their practice, and how, in the intersection, they manage the dilemmas they face? The Berlaks asked: Will teachers change their patterns — co-ordinate and rationalise them more carefully if they become aware of the trade-offs implicit in their classroom behaviour (sic); both the trade-offs represented by each dilemma, and the trade-offs among dilemmas?(Berlak and Berlak, 1981, p.165)

Are teachers’ actions in the face of dilemmas defaults to dominant practice, resolutions or transformations? How are such actions understood by teachers? Do teachers use a range of strategies as they make choices in the face of dilemmas. Are these choices tacit or informed and conscious? Or are they fall-backs to the familiar? Jaworski asked what it means to acknowledge the sociocultural and socioemotional contexts of learning in practical terms: What are the issues and dilemmas which such a position imposes on teachers? What are the consequences for teachers of avoiding such dilemmas? (Jaworski, 1999, p. 168)

In the next three chapters: each of which focuses on an incident in a classroom, I will attempt to answer these questions and give substance to each of the dilemmas while simultaneously developing a language of dilemmas particular to and illuminating of teaching in multilingual mathematics classrooms.

CHAPTER 6

LANGUAGE(S) AS RESOURCE AND THE DILEMMA OF CODE-SWITCHING 1. INTRODUCTION The induction of learners into mathematical discourses, and to informal and formal spoken and written mathematics, is widely acknowledged as a complex affair. All learners come into the school with informal ways of talking which they can bring to bear on their mathematical learning. The valued goal in school mathematics cIassrooms is formal, written mathematical competence. Teachers thus need to encourage movement in their learners from informal spoken language to formal written mathematical language. Pimm (1991), suggests that there are two broad routes to facilitate movement from informal spoken language to formal written mathematical language. The first route is to encourage learners to write down their informal utterances and then to work on making the written language more ‘mathematical’. The second is to work on the mathemtical formality and selfsufficiency of the spoken language prior to its being written down. Pimm’s suggestion gives mathematical focus to the general distinction between two kinds of ‘learning talk’ in school. (i) Exploratory talk, which is informal, and a necessary part of talking to learn because learners need to feel at ease when they are exploring ideas (Barnes, 1992, p. 126); and (ii) discourse-specific talk which is part of learners’ apprenticeship into the discourse genres of subjects in the school curriculum (Wells, 1992, p. 291). These are different ways of speaking mathematics and, as many argue, there is no guarantee of moving from one to the other without enculturation into subject-specific discourse and ways of describing the world mathematically (Nesher, 1998; Bartolini Bussi, 1998). In a bi-/multilingual mathematics classroom the movement from informal spoken language to formal written mathematical language is complicated by the fact that the learners’ informal spoken language, their exploratory talk, is likely to be in their main language and so in a language that is different from the LoLT. There is thus a longer and more complex journey with diverse routes for teachers and learners as they move between informal spoken mathematical language (in the learners’ main language) to formal written mathematical language (in English). An underlying assumption in all of the above is that learning talk, and particularly exploratory talk is part and parcel of the approach to teaching and learning in school classrooms. A further, related assumption is that the approach to knowledge, too, is exploratory. As will become evident through this chapter, these assumptions cannot be made about many mathematics classrooms and particularly classrooms where the learners and teacher share a main language and it is different from the LoLT. Irrespective of approaches to mathematical knowledge and to teaching and learning, an obvious resource and strategy for teachers and learners

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when the discursive journey in class includes navigating different languages, is code-switching. This chapter focuses on teaching dilemmas of code-switching. In classrooms where the main language of the teacher and learners is different from the LoLT, there are the ongoing dilemmas for the teacher as to whether or not she should switch between the LoLT and learners’ main language, particularly in the public domain, and then whether or not she should encourage learners to use their main language(s) in group and/or whole class discussion. These dilemmas pivot on learners’ need to access the LoLT, as this is the language in which, ultimately, critical assessment will occur. I start the chapter with a brief introduction to the notion of code-switching and a discussion of the forms this takes across the range of South Africa’s multilingual mathematics classrooms. The dilemma of whether or not to switch languages in class was particularly strong for Thandi. I thus move on to tell a story about Thandi’s classroom practices, her actions and reflections, as these illuminate dilemmas of code-switching and illustrate how they can take shape in the flux of complex classroom practice. Thandi’s story is further powerful in that it illustrates how talking about, and acting on a dilemma can be a source of praxis for a mathematics teacher. In the latter part of the chapter I extend the illumination from Thandi’s classroom into a brief discussion of a teacher development research project in South Africa which included a focus on teachers’ changing language practices in primary and secondary schools in urban and non-urban mathematics classrooms. This research has shown that although multilingualism and code-switching practices now have official sanction and legitimacy in South Africa, and teachers are more comfortable with their use, dilemmas of code-switching remain strong for mathematics teachers. As argued in Chapter 4, teaching dilemmas are situated, related to context. Indeed, the teacher development research referred to above suggests that the dilemmas of code-switching are most acute for teachers in non-urban, primary classrooms. I conclude the chapter by raising critical questions for mathematics education reform, questions that arise when dilemmas of code-switching are explored and interpreted across contexts. 2. CODE-SWITCHING IN BI-/MULTILINGUAL MATHEMATICS CLASSROOMS Code-switching is when an individual (more or less deliberately) alternates between two or more languages ... code-switches have purposes ...[and there] are important social and power aspects of switching between languages, as there are between switching between dialects and registers. (Baker, 1993, p.77)

Code-switching refers to the use of more than one language in the same conversation. In language and learning literature, code-switching is distinguished from code-mixing, and code-borrowing where the latter refer to insertion of single words or short phrases within a sentence in another language. There is a great deal of code-mixing and code-borrowing in learner-learner conversations in urban

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township mathematics classrooms. As learners engage in exploratory talk, and this occurs largely in their main language, mathematical English is mixed into their speech. For example, words like ‘equals’, ‘fraction’, ‘less than’ become part of a conversation in Setswana In the discussion in this chapter, and indeed the book as a whole, I have classified code-mixing and code-borrowing as code-switching. In schools and classrooms, code-switching occurs most obviously in bilingual or multilingual settings where learners are learning in an additional language i.e. their main or primary language is not the language of learning and teaching (LoLT). At its most general. code-switching entails switching by the teachers and/or learners between the LoLT and the learners’ main language(s). In South Africa, codeswitching has been observed as a “main Iinguistic feature in classrooms where the teacher and the learners share a common language, but had to use an additional language for learning ... the learners’ language is used as a form of scaffolding” (NCCRD, 2000, p. 68). These classrooms are typically in non-urban schools where English is more like a foreign than an additional language, and have been likened to post-colonial settings in other parts of Africa. In such contexts, teachers tend to dominate classroom talk. As a result, investigations into switching practices tend to focus on the teacher. In a study of language-use in South African classrooms undertaken by the National Centre for Curriculum Research and Development (NCCRD), teachers across subjects were seen to rely on the use of learners’ primary language to explain difficult concepts. We teachers switched for a range of purposes (e.g. for social regulation, for rephrasing instructions etc.) code-switching took on a dominant form. The teacher would introduce the lesson and/orconcept(s) involved in English, and then re-explain in the learners’ primary language. The NCCRD report sites a range of research (Arthur, 1996 in Botswana; Lin, 1996 in Hong Kong; Ndayipfakamiye, 1996 in Burundi; Hornberger and Chick, in press, in South Africa and Peru) on language-in-use across similar kinds of post-colonial contexts. All these report a teaching and learning style that offers limited opportunities for learning talk, both exploratory and discourse-specific. Learning talk in the target language in particular tended to be restricted to single word answers to the teacher’s questions. A common argument across the research mentioned above, including the NCCRD research, is that as long as learners and teachers are working in a language that is not their mutual main language, and the wider environment is not supportive of English as LoLT, these kinds of teaching and learning styles and forms of code-switching by the teacher will persist. Coupled with this argument comes a strong advocacy for the development of ‘mother tongue instruction’ and additive bilingualism. Of course, as a feature of language-in-use in a classroom, code-switching is part and parcel of wider teaching and learning styles. Code-switching will thus take on different forms in different curriculum contexts. Many mathematics classrooms across the world remain dominated by teacher exposition and demonstration followed by learners doing exercises from the textbook. However, the reform movement towards more meaningful, communicative and investigative mathematics in school has had an impact particularly in some urban schools, in South Africa. At the same time, classrooms in urban contexts have become more and more multilingual. In many such classrooms there are opportunities for learners to talk

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amongst themselves while engaged with mathematical tasks and/or exercises. Here learners have more vaned interactional opportunities as well as opportunities to harness their main language as a resource for exploratory mathematical talk (Adler et al., 1998). In situations where the teacher is monolingual and English-speaking, there is obviously an absence of switching by the teacher. Even in urban settings where the teacher is multilingual, possibilities for explaining and working in more than one language are reduced as learners bring three or more different main languages to the classroom. Here too, code-switching tends to be harnessed more as a resource for and by learners. In these urban multilingual settings, irrespective of the teacher’s main language, English tends to dominate in the public speech channel — both the teacher and learners use English in the public domain. However, codeswitching is prominent in learner-learner interactions, and so in exploratory talk. As discussed in Chapter 1, code-switching has been a focus in recent mathematics education research in bi-/multilingual settings. Moschkovich (1996) and Khisty (1995) in the USA, and Ndayipfukamiye (1994) in Burundi and Setati (1996) in South Africa have all shown ways in which switching between the learners’ main language and English (or French) by the teacher enhances the quality of mathematical interactions in the classroom. I have referred before to Setati’s study of Grade 4 classroom interaction in African urban townships. In a number of the classrooms, the LoLT was English but the teachers and learners shared the same main language (Setswana). Setati described an interesting contrast between teachers who rarely switched out of English in the public domain, and one who did. Calculational mathematical discourse dominated in classrooms where switching was restricted. In contrast, when the teacher herself switched into the learners’ primary language in the public domain, this correlated with conceptual discourse becoming the focus of discussion. Moreover, when learners in this teacher’s classroom were asked to talk about the mathematics they had learnt in their interviews, they were able to shift between verbalising steps in calculations and talking about concepts (Setati, 1998b). As research and development in language and learning in bi-/multilingual settings has shifted from regarding the learner as in some way deficient to embracing the presence and use of more than one language in teaching and learning as a resource, so code-switching has become a taken-for-granted ‘good thing’ It makes immediate sense that learners whose main language is not the LoLT should draw on their main language(s) in the learning process, particularly in the learning of content subjects like mathematics. It is nevertheless incumbent on the mathematics education community to render all taken for granteds problematic, no matter how much sense they make. The story told from Thandi’s classroom practice, and the extension of the exploration of code-switching practices into other classroom contexts in South Africa, helps to do just that. As highlighted in Chapter 1, during data collection in Thandi and Jabu’s classrooms (1992 – 1993), they were both teaching secondary mathematics in urban township schools. Thandi and Jabu are both multilingual. Setswana was the primary language of most of the learners in both their schools. Jabu and Thandi themselves have Setswana as their main language. They thus both taught in schools where the teachers and many, but not all, learners shared a main language that was not the

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LoLT. As hinted at in Chapter 1, Jabu and Thandi both faced teaching decisions that invoked dilemmas of code-switching. In particular they faced the dilemma between developing learners’ proficiency in English (the LoLT) vs. ensuring that learners understood the mathematics. A related tension for teachers was between the need to model mathematical English on the one hand, and a concern then about the teacher “talking too much” on the other. In Thandi’s classroom, switching to Setswana (by learners in their group discussions), and modelling of mathematical English (by Thandi in the public domain) were practised but seen as problematic. 3. THE DILEMMA OF CODE-SWITCHING: ACTION IN AND REFLECTION ON THANDI’S CLASSROOM PRACTICE In her report on her own action research, and to which I will return later, Thandi wrote about the dilemma of switching out of English, the LoLT, and into learners’ main language’ : This is a dilemma because as a maths teacher I would like to have my students to understand the mathematical concepts and at the same time to have them master English as a language, especially that they learn mathematics in English. Grasping the concepts might mean allowing the students to use the language they understand better; in which case they will be free to communicate in their groups although the usage of English will not improve. On the other hand if they are forced to have their discussion in English they may either not do as required or they may withdraw and not communicate enough in their groups.

3.1 The context As a reminder, Thandi’s primary language is Setswana. She also speaks English, Sesotho and isiZulu. At the time of the study she was a mathematics teacher in Mohlakeng township, west of Johannesburg and neighbouring on, but not part of Soweto. Her school was a typical large state urban township secondary school. It was overcrowded, with limited resources. Since the mid-1980s a culture of learning had all but broken down in the school, a reflection of the serious political turbulence in South Africa at that time. On the day that the episode below was videotaped, nearly half the class of 55 students was away at the funeral of a student from a neighbouring school, yet another young victim of political violence. There was a constant noise from outside Thandi’s classroom — evidence that many other learners in the school were not in class or that there was no teacher in the class next door. Most of the mathematics and science teachers in the school belonged to the same teachers’ union. They formed a group within the teaching staff at the school and they had established a set of rules for their conduct as a way of dealing with the breakdown in the teaching-learning culture. In her initial interview, Thandi said: Thandi: We have sort offormed a group. We have said that if there is someone who doesn’t come to school we confront them. and if you don’t appear in class we confront them. If we pass your class and you are teaching in vernac, we confront you. So that is the thing the group adopted (my emphasis).

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Thandi was a member of this group. She firmly believed in its policy and only used English in the class I observed. In her interview she described what for her were the problems of using her main language, Setswana. She would “run out of words” if she were to try to explain mathematics in Setswana. Moreover, not all her learners were Tswana-speaking. Embedded here were social and political concerns of equity on the one hand and access to English, the language of power, on the other. Thandi was revising linear inequality graphs in preparation for linear programming. The lessons I observed were focussed on such graphs, and reflected Thandi’s belief that mathematics is “not rules but reasons”. She constantly asked learners to explain why they shaded graphs as they did. The lessons consisted of learners drawing inequality graphs in pairs or in groups, and then whole class interaction on their solutions to questions posed. The episode below occurred in the last quarter of the last lesson I observed, and shows Thandi explicitly focussing on the mathematical language (subject-specific talk and writing) related to the comparative concepts not less than, at least, not more than and at most. There was no actual code-switching in this particular episode. However, together with Thandi’s reflections, it illuminates how and why she worked with mathematical English in the way that she did, and the effects of her actions on learners’ interpretations of her mathematical messages. 3.2 Episode: Thandi’s year 11 linear inequalities class KEY: [brackets within a data extract — researcher commentary] T-Thandi S1, S2 - students whose names are not used Ss - all students () very short pause ... - longer pause bold - when speaker places particular emphasis Thandi: And note that inequalities can be given () sometimes inequalities are given () inequalities may not always be given in mathematical symbols They can be given in verbal symbols and you should be able to recognise if they say “not more than” what will it be? () OK? () And I want us to look at that because sometimes I can use the words “not more than ” () I can use the words “not more than ” () So you need to check as io whether if I use the words “not more than” do I mean greater than, or less than, or greater than and equals to or less than and equals to. () OK () So I made a table there and I am going to compare my verbal statement () whatever statements I make verbally and then the mathematical symbol we use for that statement. So (she draws a table)

verbal/word

mathematical symbol

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Thandi started with the statement “not more than”. She related it to the everyday use of “not more than 50 cents” and drew from the class that the mathematical symbol here was the “less than or equals to” symbol. And she filled this into the table. She then moved on to both “at least” and “not less than”. She used the everyday example of “there are at least 10 people at the meeting’’ and there was a quick response that the symbol here was “greater than or equals to” and she filled these into the table. We pick up the lesson where she continued from there with “at most”, now separated out for attention.

verbal/word

mathematical symbol

not more than

less than or equals to

at least/ not less than

greater than or equals to

at most Thandi: If I say you can spend “at most R50”, what do I mean? huh? S4:

[inaudible]

Thandi: Mogapi says “equals to”. You can spend at most R50. Does that mean I can spend R50 exactly? ... Who agrees with him? ... Let me write: [and writes “at most R50” on the board]. What do I mean “at most R50”? ... Peter disagrees. Peter:

You must get more than R50?

Thandi: You think more than R50? ... Sabie’s hand is also up. Sabie:

plus minus

Thandi: What do you mean plus minus? Sabie:

Not much more than R50 or less than R50.

Thandi: More than R50 or less than R50. Is that what you are saying? Sabie:

[trying, mumbling] ... greater ... [then puts head in hands and laughs shyly]

Thandi: How can we write that with a mathematical symbol? Sabie:

X greater than R50, less than R50

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Thandi: Huh? X greater then R50, less than R50. Huh? and what is that? If I say you go to the shops and you can spend at most R50 how much would I () would I be happy i f you spend seventy rands? [inaudible..]. Ss:

no, no [mumbling]

Thandi: So what do we say?More than or less than? .... Ss:

[mumbling]

Thandi: Peter says it must be more than R50 and Mogapi says more than R50 and Siza says equals to R50. huh? Ss:

[some mumbling]

S5:

Less than R50?

S6:

Equals to R50?

[some interchange that is inaudible] S7:

I think, um, the one must get more

Thandi: More? Which means I am saying it means the same as not less than? because (interrupted] S7:

No ...you spend more.

Thandi: How different is it from “not more than” or “not less than”? () How are you going to write it in symbols? S7:

You are going to use greater than

Thandi: Which means you are saying “at most“ is the same as “at least”? huh? S6:

No. It is just greater than.

Thandi: Oh. You mean this one is just greater than () not also equals to? Ss:

[some together] yes

Thandi: And others are saying no. OK. I want you to go home and check the meanings of “at most” and “at least” () and what is going to be your symbol for each. And I gave you examples to think about. You can spend at most R50. OK?

3.3 Teaching and learning style and restricted patterns of interaction Difficulties were apparent. Thandi struggled through a questioning process to scaffold the meaning of at most. Students were confused, first offering ‘‘equals to”,

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then “plus-minus” and then “one must get more”. The first guess “equals to” could have been a function of the table (we have had < and > , so we now need something different). The guessing that followed is not uncommon in mathematics classes where there is a culture of the well-known I-R-F (initiation-response-feedback) pattern of teacher-learner interaction. Thandi fed back on all student responses, attempting through this to lead them to the appropriate interpretation of at most. While observing the class, I wondered whether the use of Setswana in class would have helped and to what extent the task itself was the problem. I noted that Thandi’s tacit practice (not fully captured in the transcript extract here) included a great deal of repetition and reformulation of the (mathematical English) verbal statements she was trying to teach. She talked in a way that served to repeatedly model mathematical English. I wondered about the effects, both positive and negative, of the repetition and reformulation, whether this practice was more prevalent in multilingual classrooms like Thandi’s and whether this served to effectively model mathematical English. 3.4 Shifting discourses and the three-dimensional dynamic What the episode also reveals is the complex practice of changing discourses. To illuminate the mathematical meanings of not more than and at least Thandi shifted explicitly between everyday and mathematical discourses, and between verbal and symbolic forms, creating “chains of signification” (Walkerdine, 1988, pp. 121, 128). On the basis of extensive analysis of how children use relational terms like more and less in both their home lives and in school, Walkerdine challenges notions that children’s successful or unsuccessful use of these terms at school connotes ability in some decontextualised way. Rather, use of relational terms is tied to “regimes of meaning” (p.32) produced in cultural practices or sets of discourses in which the children are inserted, and which they bring with them into the classroom. Walkerdine showed empirically that for the children in her study, more as a relational term was in constant regulative use in their everyday lives (e.g. “I want more”, “Would you like more?’) and contrasted with no more rather than less. In fact, less did not occur in the everyday discourses she analysed. The point here is that more and less as contrastive relational terms are specific to pedagogic discourse. Children’s greater pedagogical facility with more in contrast to less is thus a function of familiarity with more, and not an ‘inability’ to cope with less, nor with less being an intrinsically more difficult concept. Walkerdine’s analysis challenges common sense notions, particularly in school mathematics, that unproblematically assume everyday contexts brought into the classroom will necessarily make mathematics more meaning-full. The example of more/ no more/ less points to difficulties that might well arise in the mathematics class if teachers assume that a familiar opposite of more is an unproblematic less. Even worse would be if teachers then attributed learners’ difficulties with the concept of less to their ‘ability’.2 In this context then, everyday notions can be used in school to make connections with mathematics. But as teachers move into everyday relations in the attempt to

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contextualise and make more sense of mathematics, these bring in other signifiers that could, in fact, cause confusion, perhaps even pain. Mathematical meanings thus have to be prised out of their everyday discursive practice and situated in a school mathematical discursive practice. As Walkerdine argues: ... non-mathematical practices become school mathematical practices, by a series of transformations, which retain links between the two practices. This is achieved, not by the same action on objects, hut rather by the formation of complex signifying chains, which facilitate the move into new relations of signification which operate with written symbols in which the referential content of the discourse is suppressed .. (Walkerdine, 1988, p. 128)

As the episode above reveals, Thandi brought in the everyday with success until she came to at most. Here we come to some of the limits to Walkerdine’s work. Walkerdine’s empirical base is elementary mathematics. The signifiers in the episode in Thandi’s class do not lie only in the everyday use of most. The table that Thandi harnessed as a pedagogical resource itself seemed to operate as a signifier. Students offered symbols that were not yet in the table reflecting the anticipation (acquired in previous classroom processes) that what comes next must be different. There were also signifiers at play here that derive from previous mathematical learning, signifiers tied to changes within the mathematics register. Registers have to do with the social usage of particular words and expressions, ways of talking but also ways of meaning...pupils at all levels must become aware that there are different registers and that the grammar, the meanings and the uses of the same terms and expressions all vary within them and across them ... (Pimm, 1987, pp. 108-9)

What Pimm is arguing here is that even within the mathematics register, meanings shift. Mathematical meanings are not forever fixed but shift in relation to mathematical use. Students need to become aware of such shifts. In language learning, most would be associated with more. Similarly in earlier mathematical learning, most would have been associated with more and hence with greater than. The issue for Thandi was not simply prising at most out of its everyday use, but also out of the previous mathematical association between more and greater than, and into the new meaning of at most which was now the negation of greater than. Thandi and her learners faced a double shift in meaning, and thus a much more complicated signifying chain. This, in fact, resonates with why Thandi separated out at most for focused attention. In the reflective interview on the video she discussed how when students see the word most, they write greater than. She thus believed it was important to focus explicitly on at most on its own. Furthermore, from a Vygotskian perspective, mathematical meaning is not simply a matter of awareness. In Vygotskian terms, at most is a “scientific concept” (1978, p. 130; 1986, pp. 172-173), linked with and emergent from other concepts. It is bound in with meanings of related concepts and their use. Shifting into the everyday might well not be sufficient to attach the appropriate new conceptual meaning.

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3.5 Absence — the teacher’s dilemma of switching out of English For Thandi, all this was complicated by the fact of her working solely in English, and what I came to call the dilemma of code-switching. In her reflective interview we discussed code-switching and her tacit use of repetition and reformulation. Thandi explained that switching to Setswana would not necessarily have helped since there is little in Setswana to distinguish greater than, from greater than and equal to, and so she would have “run out of words”. ... in Tswana it becomes a problem because () um () like if he explains in Thandi: Tswana, then when it comes to the [terms? — unclear] our language is unique; and when you come to “at most” and “at least” then what are you going to say? For in our language, “greater and equals to” and ‘‘greater” () there is a little difference. I have to use a long sentence for ‘greater than and equals to”. JA:

And for “at most”? and “at least”?

Thandi: That is going io be problem to say it in Tswana, “at most” and “at least”. That is why I talked of “not more than” and “not less than”. I feel if they resort to Tswana, then, when they come to those terms what are they going to do?

... even English speakers battle with those terms. () Would it help to explain JA: the idea in Tswana and then shift to English? Thandi: I think if I was to explain in Tswana I would run out of words. And for my mixed class it would also be a problem because not everyone speaks Tswana. So must I do it again in Xhosa and then Zulu? I would definitely run out of words and go back to English. For example, I can explain it in Tswana but if I am trying to say “at most” I would say something like “the limit is this” .... I would explain what it means but trying to find words to say “at least”? ...

Thandi’s use of English and her focus on at most, were intentional. In contrast, her repetition and reformulation were not intentional. In a follow-up conversation she said that it was not her explicit purpose to model mathematical English. However, as a result of observing herself teaching on the video, she had become aware that she repeated and reformulated when she herself felt less secure with what she was trying to explain and when she felt she had to show her students that she, the teacher, knew the mathematics. Thandi had since noticed that she was much less repetitive when she worked with primary students. The higher she moved up the levels in school, the more exaggerated was this action. Thandi’s actions and reflections here confirm a recent analysis of classroom discourses in classrooms in South Africa and Peru. Hornberger and Chick (in press) describe what they call “safetalk” as a dominant feature in classrooms in both countries. “Safetalk” is mainly the use of chorused, one-word student answers to teachers’ talk and questions. They see in these interactions, a practice where teachers and learners “preserve their dignity by hiding the fact that little or no learning is taking place” (Hornberger and Chick in NCCRD, 2000, p.11). In a later conversation, Thandi commented further that, at that time of the study, with the breakdown of a learning culture, the only thing she could ensure as teacher was that she had done her job. In all the chaos, she could at least make sure she had conveyed

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the content, over and again. But, on reflection, she regarded this as “talking too much” and so faced another dilemma. This episode clearly reveals and illuminates the three-dimensional dynamic at play in the teaching and learning of mathematics in multilingual classrooms. As discussed in Chapter 1, teaching and learning mathematics in multilingual classrooms is not only about access to the language of learning (in this case English). It is simultaneously about access to the language of mathematics or subject-specific discourse and to scientific concepts, and about access to classroom cultural processes. We see the criss-crossing of discourses that Thandi and her learners had to manage. Accessing at most and at least can, for example, be through the ordinary English use of these terms, that is, through contextualisation in the everyday. However, Thandi’s difficulty here was not only chaining across these different discourses (mathematical and everyday) but rather within mathematical discourse as well. She needed to try to delocate the meaning of most from more than and relocate it as at most and the negation of more than. For Thandi, however, all this was complicated by the fact of her working solely in English, and the dilemma of code-switching. In addition to her overarching concern that if she herself switched and explained in Tswana, then she would be denying her learners access to English, she was also concerned as to how she would manage switching given that there were Zulu and Xhosa-speaking learners in the class. Moreover, even if she did wish to switch she would have problems because the mathematical words she needed were not available in Setswana. Thandi’s insights and experiences illustrate that code-switching in a multilingual classroom is no straightforward matter, both in terms of which language is used if the teacher is to switch, and then, for example, how to find appropriate mathematical language in Setswana. 3.6 Presence - the dilemma of learners switching to their main language(s) The most astonishing revelation for Thandi in her video was her observation that, in fact, outside of whole-class, public domain speech, her students worked in their main language a great deal of the time. This was a complete surprise, providing a means for her to rethink her strong views that both she, as teacher, and her learners, should only use English in class. In her action research report she wrote: ... during the maths period, students are expected to work in English, this has been policy in my class since I started teaching them; I always thought that they practised it, or at least I should say they gave me the impression that they do. The video, however, revealed that to me during that particular period, ten groups out of twelve had their discussions in Tswana, Zulu or street language (Tsotsitaal). This raised a lot of questions in me. Can’t they understand English or does communication in English make it difficult to have a discussion?

She observed further that when her students worked in groups, they did so in varied ways. For example, some groups functioned like a small class with one taking on a teacher role. In others, one did the work and others copied. She wondered what benefits they derived from group discussion.

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Thandi followed up her questions in a small action research project as part of her contribution to the follow-up workshops in the wider research project. She was interested to understand from her students themselves, how they viewed their participation in their group discussions. She interviewed her students and asked about their Tswana and/or Zulu discussions. The students blamed each other (for example, “he started ...”), or suggested that their talk in Tswana was a “slip”. They said that English use was better, giving the usual access/power rationalisations: they “need it for work”; they are “examined in English”. They also said that it was better to have discussions in English because there were other languages in the class besides Setswana. Thandi and her students were thus mutually implicated, at least at the level of what they expressed, in the view that the use of main languages other than English in the mathematics classroom should be restricted. In the interviews she also probed the students’ views of the group work she set up, how their groups worked and how they felt they benefited from such activity. She emerged from her action research with a major reformulation of her practice. She argued now that she needed to embrace code-switching as a resource in her classroom, and she also saw that her “talking too much” was bound up with the way in which she had constructed the tasks in her classroom. She had to provide more opportunity for learners’ meanings and informal expressions of their mathematical ideas i.e. for more exploratory talk. As quoted earlier (see p. 76), she wrote about her dilemma in her action research report She wanted her learners “to understand the mathematical concepts and at the same time to have them master English as a language”. If she allowed learners to use their main language, “the usage of English will not improve”. On the other hand, they might not be able to communicate with each other “if they are forced to have their discussion in English”. 3.7 The dilemmas of code-switching as a source of praxis Thandi went on in her action research project to reflect on the kind of tasks that would be appropriate for group discussion, as well as how to organise groups for productive and constructive functioning. She talked with me about how, in changing her tasks, she could provide opportunity for learners to work on and talk about concepts like at least and at most. She would be able to hear how they used ordinary English as well as Setswana or isiZulu in their informal talk about these concepts. She would then be able to work on how to revoice these informal expressions in formal mathematical language. She was adamant that “reporting-back” by learners would need to be “in English”, as this would then provide learners the practice they needed with mathematical English. While Thandi’s action research was focused on switching by learners, through her action research, she found a way to manage the dilemma of code-switching that could be considered transformational. She was able to shift to seeing the languages in her classroom as a resource rather than only as a problem, and so move her ideas beyond the commonsense notion that mathematical English is best learnt if the main language of the learners is never used. As a result of her own action research and her reflections in her interviews, Thandi grappled with the potential benefits of code-switching, both by herself as

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teacher, and in relation to learners’ use of their main language(s) as a resource for learning. She also talked about the importance of appropriate tasks. Recognising and engaging with her dilemmas in the context of her work became a means for action and reflection on action. But not in any simple way. There are no straight forward answers in her real and very complex secondary classroom. It is not a matter of whether or not to code-switch, nor whether or not to model mathematical language, but rather when, how and for what purposes. 3.8 Dilemmas as personal and contextual, analytic and explanatory. Thandi’s story illuminates how the dilemma of code-switching, and of modelling mathematical language are at once personal, practical, contextual, and mathematical. Thandi’s actions, including reformulation and repetition, were not tied simply to her pedagogical beliefs, but also to her social and historical context and her positioning within it, including her own confidence of working mathematically in English. In particular, in the South African context, where English is dominant and powerful, Thandi’s decision-making and practices were constrained by the politics of access to mathematical English. Thandi might value using languages other than English in her mathematics classes to assist meaning-making. But this pedagogical understanding interacts with strong political goals for her learners, for their access, through mathematics and English, to further education and the workplace. In addition, her decision-making on code-switching inter-related in complex ways with the mathematics register on the one hand and its insertion in school mathematical discourses on the other. Language choices in day-to-day classroom interactions are simultaneously, and in contradictory ways, impacted on by “issues of pedagogy and effectiveness, politics and ethics” (NCCRD, 2000, p. 20) The story from Thandi’s classroom practice brings to life the dilemma of codeswitching, and further illustrates that teaching dilemmas are at once explanatory tools and analytic devices for teaching. They make explicit tensions in teaching specific to particular contexts. As we have seen, a language of dilemmas can, at the same time, function as a source of praxis. While dilemmas are expressed as binary opposites, they do not function as once and for all either-ors in the ebb and flow of classroom practice. Thandi used the notion of dilemmas to reflect on, and consider, how to transform her practice so as to more effectively meet the mathematical needs of her linguistically diverse learners in her township classroom. 4. DILEMMAS OF CODE-SWITCHING ACROSS CONTEXTS Earlier I described the journey from informal spoken mathematics to formal written mathematics as a demanding one for any mathematics teacher. It is particularly so in a context of curriculum reform and learner-centred practice where there is emphasis on eliciting and working with learner meanings and hence the informal ways in which these are often expressed. The journey is even more complex in a multilingual mathematics classroom where teachers and learners have to navigate between informal spoken mathematics in the learners’ main language to formal written

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mathematics in English. The dilemma of code-switching as illustrated by Thandi’s classroom practice does not extend to all levels of the journey as there was little opportunity for learners’ informal speech or exploratory talk. Additional questions need to be asked. What kinds of code-switching demands and practices arise in multilingual mathematics classrooms where exploratory talk is encouraged? What kinds of code-switching demands and practices arise in multilingual mathematics classrooms where there is less support for English as LoLT than was the case in an urban township secondary classroom Bike Thandi’s, in particular, in non-urban classrooms? Are these demands different yet again in primary classrooms, and nonurban primary classrooms? What also of the post 1996 context where the new constitution and language-in-education policy explicitly legitimates multilingual practices, yet, as explained in Chapter 2, this occurs in a wider national and global context of increasing dominance of English? In terms of the focus of this chapter, and with an understandmg of dilemmas as situated, teachers across different multilingual contexts are still likely to face dilemmas of code-switching, as LoLT issues are no less pressing across these contexts. But it is likely that these dilemmas will be experienced in different ways. It was possible to explore these questions as part of a larger teacher development research project between 1996 and 1999. 4.1 Recent research on code-switching across levels and regions In 1996, the University of the Witwatersrand introduced an in-service teacher development programme: the Further Diploma in Education (FDE) in Mathematics, Science and English Language Teaching. At the same time a research project was launched with the aim of investigating teachers’ ‘take-up’ from this programme (Adler et al., 1997; 1998; 1999). The study included 23 mathematics, science and English Language teachers, primary and secondary, urban and non-urban. Of the 9 mathematics teachers, 5 were primary teachers, and 4 were secondary teachers. Two of the secondary teachers and one primary teacher worked in urban townships schools. The remaining 6 teachers were all working in poor non-urban schools. English was an additional language for all the teachers and all their learners. As discussed in Chapter 2, in non-urban schools, although teachers and learners tend to share the same main language, they learn and teach in a context of very limited English language infrastructure. I referred to this as a Foreign Language Learning Environment. Typically, English is only heard, spoken, read and written in the formal school context. In contrast, in more urban environments, teachers have the added complexity of having to work with learners who bring a range of main languages to class, but at the same time there is a more substantial English language infrastructure in and around the school. As an additional language learning environment, the linguistic context is more supportive of English as LoLT. In these diverse context,3 mathematics teachers face the double challenge of teaching their subject in English while their students were still learning English, and in non-urban schools, English is more like a foreign than an additional language. In line with current curriculum and language-in-education policy, the FDE programme’s approach to the multilingual context in which teachers work, has been

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to encourage code-switching as a means for enabling learners to talk more freely in class, and so to use their main languages as a learning resource, for talking to learn. In addition, the programme worked with teachers to understand and use more learner-centred approaches, in particular eliciting and working with the mathematical conceptions and meanings learners bring to class. One of a number of foci in the research project was whether and how teachers and learners code4 switched in class. We were interested to see whether these practices shifted in any way over the three years of the research, and then with what possible consequences. 4.2 Occurrence of code-switching across contexts Elsewhere we have reported on the language practices across all the teachers i.e. in mathematics, science and English language (See Setati, Adler. Reed & Bapoo, in press). Table 4 summarises the code-switching practices of the primary and secondary mathematics teachers in the study, over the years 1996, 1997 and 1998. As the table reveals, most of the teachers code-switched, as did their learners. Codeswitching was observed during the base-line study in 1996, and thus was an already established practice of the teachers in the study when they entered the FDE programme. What can also be observed in the table is that, in general, the extent of switching increased over the three years of the study. The form that this took in most classrooms was as follows: Teachers used English predominantly in the public domain. They switched to learners’ main language(s) for reformulation in public whole class teaching, and for interaction with individual learners or small groups. Learners also mainly used English in the public domain. Their talk here was limited to short phrases, single words or recall of procedures. There were, however, opportunities in class for some exploratory talk in learners’ main languages. What is hidden in Table 4 is that the increased use of main languages by learners, and hence code-switching, was part of an organisational shift across most of the teachers to group work as a pedagogical strategy. In 1997 and 1998 teachers provided learners opportunities to discuss their work with a partner or group, and all learner-learner interaction that we observed took place in the learners’ shared main language. This organisational shift reflected teachers’ interpretation of learnercentred practice in their classrooms. Of significance here are the differences between the primary and secondary teachers on the one hand, and the urban and non-urban teachers on the other. Of particular interest is that code-switching occurred least in the non-urban primary classrooms, those described as foreign language learning environments (FLLEs). Elsewhere we have described primary mathematics teachers code-switching practices in more detail (Setati and Adler, in press). Teachers and learners in the secondary mathematics classrooms observed made greater use of code-switching in comparison with the switching practices observed in the primary mathematics classrooms. This was a complete surprise. We were similarly surprised to find teachers and learners in additional language environments making greater use of code-switching in comparison with those in foreign language learning environments.

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Table 4: Record of codings from observation schedules 1996, 1997, 1998

Code-switching by teacher - CST 0: 1:

teacher only uses English in all verbal interactions teacher occasionally switches from English to main language(s) for reformulation in public domain and in limited individual/group interactions 2: teacher switches from English to main language(s) for reformulation in public whole class teaching, and uses main language(s) as major language of interaction with individuals and small groups 2+ 2-: used to indicate that the teacher switches, as described in 2 above, but with more or less frequency 3: teacher switches between English and main language(s) as necessary for the flow, order and content of teaching in public whole class teaching and uses main language(s) as major language of interaction with individuals and small groups XX: teacher not teaching mathematics during observation period in that year

Code-switching by learners - CSL 0: 1:

2: 2+: 3: 4:

learners only use English in all verbal interactions learners use limited English in public domain (responding to teacher questions, typically short phrases or single words, procedures require); occasionally have opportunity in individual/ group interactions to use main language(s) for questions/ exploratory talk use English in public domain (still limited to short responses), with good opportunity for exploratory talk in main language(s) used to indicate learners switch, as described in 2 above, but with more frequency switch as needed in whole class interactions; use main language for exploratory talk switch as needed in whole class interactions; use main language for exploratory talk: there is explicit attention to reporting in English in the public domain on work done in groups.

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This reflects one of the most significant findings of the entire research project: how complex language issues are in non-urban schools where there is very limited English infrastructure in the surrounding community for teachers to build on in school. Exposure to English is via the teacher. This puts pressure on teachers to use English as much as possible. Mathematics teachers in non-urban schools, particularly in the senior primary levels (Grades 5 - 7, ages 10 - 13), argued strongly against frequent code-switching in class. We also found that both rural and urban primary mathematics teachers felt acute pressure to teach in English because their students are still, and need to be, learning English. That code-switching is an established practice in many South African mathematics classrooms, outside of current in-service initiatives, and irrespective of changing policies is not a surprise. In most the classrooms and schools we visited, learners were clearly not sufficiently proficient in English to have all mathematics and science teaching and learning take place through English (Madonald, 1991). In addition, that English was more ‘seen’ in non-urban foreign language learning classrooms, can be understood as teachers regarding it as their task to model and encourage English, as the classroom is the only context in which learners have this exposure. The increase in code-switching in most classrooms, though less so in non-urban primary classrooms, was further explained by the teachers themselves through their interviews. In 1997 and 1998 most of the teachers talked explicitly about how their involvement in the FDE programme gave them more confidence in using codeswitching. Their participation in the programme provided them with stronger pedagogical rationales for drawing on learners’ main language(s). In other words, an established practice was legitimated through teachers’ engagement with language practices advocated and used in the programme. In the words of two of the teachers, the FDE “liberated” them with regard to code-switching. Before I joined the FDE, I thought it a mistake to talk in Tsonga in a mathematics class and I said whether they understand or not I must talk English (STI, non-urban, 1998 interview) It is easier for them to ask questions if they use their mother tongue - they become more free. It is easier for them to explain exactly what they want (ST4, urban, 1998 interview)

4.3 Dilemmas of code-switching persist Code-switching nevertheless remained a difficult practice for all the teachers, both practically and ideologically. All the mathematics teachers in the study expressed some form of the dilemma of code-switching illuminated in Thandi’s story. In the words of one of the secondary teachers in the FDE research project: Code-switching is good only when it is used properly ... I mean if you just allow your students to use just Tsonga they just talk, talk, talk Tsonga too much ... but maybe if you ask a question and you see that a child is struggling to say something properly in English, but maybe he has got some ideas, if you allow your students to talk in Tsonga it helps. You find that he has got brilliant ideas or the answer you wanted or something like that or the misconception ... after you have code-switched to Tsonga, you can

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In the FDE study, the primary teachers experienced this dilemma more acutely than secondary teachers. For example, in PT5’s classroom (see Table 4), Tshivenda was only used by learners in their group discussions. All speech in the public channel in PT5’s classroom was in English. She was clear that “learners needed to use Venda to understand”, but was concerned that “they can get spoilt”. The dilemma of code-switching was most acute for the non-urban primary mathematics teachers where the school is the only place learners can hear English being spoken. The view of the teachers was that even if learners did not always understand what is being said in English, they needed to hear English being spoken, and the teacher was thus compelled to use English as much as possible. One of these teachers (See PT1 in Table 4) who shifted from no switching in 19% to limited switching in the public domain in 1998 expressed the dilemma and the related tension between the pedagogical and political through the contradictory views she gave in her interviews. In 1997 she said: “I use code-switching because learners do not understand English”, and in 1998 felt that “code-switching does not benefit learners” who in the end have to be able to do mathematics in English. In other words, not all teachers were ‘liberated’ by the engagement with code-switching practices in the FDE programme. As suggested above, the dilemma of code-switching persists across ranging multilingual classrooms in South Africa where the main language of teachers and learners is different from the LoLT. However, dilemmas of when and how to switch are experienced more or less acutely, depending on the level of English language infrastructure in and around the school. Moreover, the experience of the dilemma is counter-intuitive in terms of the English language proficiency levels of learners. The less the English infrastructure, and despite learners’ poor levels of English language proficiency, the more teachers feel compelled to only use English. As captured in the three-dimensional dynamic in multilingual classrooms, a language practice like code-switching is never separated in actual practice from approaches taken to mathematics, and teaching and learning styles in the classroom. Across levels and all three subjects, some of the teachers worked at enhancing learner participation, and at eliciting learner meanings. The particular form that this took across many classrooms was increased group work, though as we have argued elsewhere, form was rarely accompanied by substance. For example, where learner meanings were elicited, these were often not worked on in any way (Adler et al., 1999). Two of the 9 mathematics teachers (one primary, PT5, and one secondary, ST4) provided their learners with opportunities for more exploratory learner-learner talk. In ST4’s Grade 11 class, learners worked in groups on exercises from their prescribed text book. In these lessons, all interaction between learners and between ST4 and his learners while they were ‘on task’ was in either Sesotho, or isiZulu, with a fair splattering of mathematical English (mainly terms e.g. ‘exponent’, ‘base’) mixed into these main language conversations. English was only spoken by the

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learners and himself as focus shifted from the groups to whole class reporting of solutions, and ST4’s related mediation of these in the public domain. The journey here appeared to be a formalising of spoken mathematics in group work, followed by formal written productions that accompanied public reporting. The switching practices and related ‘language’ journey in PT5’s Grade 7 classroom were more complex, and more problematic. PT5 was the only mathematics teacher who attempted more open tasks and goals (see Adler, 2000b for a more detailed description of this teacher and her changing mathematical practices). In one of her sets of lessons in 1998 she had her learners categorise and then graph the variety of waste material they had collected in their school grounds. Learners worked on and completed their tasks, particularly the construction of their graphs in their groups. All group interaction was in Tshivenda. PT5 circulated among the groups, interacting with learners in Tshivenda. She realised fairly quickly that most groups were struggling to set up a scale for their graph, and moved to the chalkboard to mediate and so assist with this general difficulty. In so doing, she effected a double shift from learners’ informal talk in Tshivenda to her own formal mathematical talk in English, shortcutting the language journey. There was little to no opportunity for learners to explore their thinking about graphical representation and scale either informally in English, or more formally in spoken Thivenda before being driven into formal written graphical representation. The point here is not to castigate PT5. It is, instead, to raise a question as to the possible effects of short-cutting the language journey. The kind of abbreviated journey described in PT5’s classroom was also seen (though in a range of forms) in the English language and science classrooms where teachers had attempted to embrace exploratory talk on tasks. The teachers struggled to set up and support various stages in possible journeys from exploratory talk in a main language to subject-specific written forms in English. PT5 moved to formal witten mathematics but the journey for her learners was abbreviated and so, possibly incomplete. Some of the English language teachers who were able to elicit learner productions through more group work and learner-learner interactions, did not carry these through to written work at all. Working in two languages in more learner-centred post-colonial classrooms is clearly not easy. Code-switching by learners and the teacher, and the dilemmas this provokes, is only one part of the complex journey that needs to be travelled. Other aspects of this journey, though with a focus on the secondary urban classroom, are raised in Chapter 7 and 8 following. 5. THE DILEMMA OF CODE-SWITCHING AND MATHEMATICS EDUCATION REFORM The dilemma of code-switching, and of building mathematical English communicative competence takes on added significance in the context of curriculum reform everywhere, not only in South Africa. As the above research suggests, a valorising of learner-centred practice, and a less interventionist or transmission role for the teacher needs careful and critical consideration in the light of the discursive demands in multilingual mathematics classrooms. I pointed to the critical issue of

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the teacher’s role in initiating learners into mathematical discourse in English in Chapter 1 in my brief discussion of Moschkovich’s (1999) recent research in a bilingual primary classroom in the United States of America. In her study of a classroom where most learners were Spanish-speakers, Moschkovich documented the significant effects of practices like revoicing by the teacher. Here, in the whole class setting, the teacher listened to and worked with learners’ mathematical language productions. She was then able to revoice these in line with appropriate mathematical discourses The teacher understood her role as including the modelling of mathematical talk for learners who were struggling simultaneously with concepts and their appropriate naming in English — the language of learning and teaching. A similar argument has been made by Fradd and Lee (1999) with regard to science teaching in bilingual settings in the USA. They too found that induction into scientific discourse is not best facilitated by ever-present ‘group work’ organisation in the classroom. In their report on a study of science classrooms, Fradd and Lee posed the question: does the total move from whole class to small group work benefit English Second Language (ESL) learners? They argued that learning science is dependent on the learners’ ability to comprehend and communicate concepts and understandings. Learners need to develop the language to question, inquire and explore, i.e. they need to acquire the discourse of school science. They argued further that the indirect nature of exploratory talk (in groups) made it difficult for learners to acquire these specific participation rules on their own, and as a result a fully exploratory science classroom learning environment may limit, rather than enhance ESL learners’ opportunities to learn. Fradd and Lee argued that learners could benefit from both explicit teacher-led activities and from exploratory teacherfacilitated activities. They advocated a research agenda to effectively implement science inquiry in ways that would enable all students to succeed and where teachers need to link the nature of science with students’ experiences and interactional styles. Fradd and Lee express for science the concern I raised in Chapter 1 related to the disjuncture between mainstream research on communication and conversation in ‘the’ mathematics classroom and research in bi-/multilingual mathematics classrooms. Research in bi-/multilingual settings has a great deal to say about the implementation of enquiry-based learning in mathematics (and science). If this research is not heard, then there is the danger that curriculum policy across a range of countries which invariably include commitments to equity, might well undermine such goals through ignorance of what counts as ‘good’ teaching and learning across a range of linguistic classroom contexts. In Chapters 7 and 8 following, with their foci on dilemmas of mediation and transparency of language, I hope to deepen an understanding of communication and conversation in the teaching and learning of mathematics by bringing the concerns of mainstream and bi-/multilingual mathematics education and language research under the same spotlight.

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6. CONCLUDING COMMENTS In this chapter I have illuminated the dilemma of code-switching through a story of Thandi’s classroom practices. The dilemma was experienced in terms of both the teacher and the learners switching ‘out of English’ while engaging with teaching and learning mathematics. I also argued that the dilemma of code-switching at once names the tensions and contradictions in practice and provides a source of praxis. Thandi was able to talk about and act on her dilemma and so work towards transforming her practice. I have also described how dilemmas of code-switching persist across multilingual contexts in South Africa, though code-switching practices inevitably take on different and more complex forms across primary and non-urban contexts. In non-urban contexts, the dominance of English works to subtract out learners’ main language, and diminish use of learners’ main language as LoLT. Together these play havoc with teaching and learning styles. In particular, learners are provided little opportunity for learning talk in any language. The devastating consequences for mathematical learning were described in the overview of the South African context in Chapter 2. I also described how in a primary mathematics classroom where there were more ‘open’ tasks and goals, and with these, more opportunities for exploratory and informal mathematical talk in learners’ main language, the journey from this to formal written mathematics in English was abbreviated and possibly incomplete. The FDE research project, precisely because it worked across levels and regions, illuminates just how much context matters when it comes to language practices in the teaching and learning of mathematics. That dilemmas of code-switching persist across these multilingual contexts provides support for the notion of teaching dilemmas as analytic devices, and sources of praxis. What needs to be understood, however, is that the management and resolution of dilemmas across such ranging contexts will inevitably be different. It is thus interesting to reflect briefly on the de facto situation that language-ineducation policy, and policy for curriculum reform in South Africa and elsewhere is always in a generalised form. Teachers’ varying uses of code-switching across contexts suggests that language-in-education policy needs to engage more seriously and explicitly with what multilingual practices like code-switching can and do mean in the day-today realities of diverse classrooms contexts. In particular, in the context of mathematics education reform, policy research and development needs to embrace the specificity of demands on teachers who work in contexts with limited English language infrastructure. What, for example, does learner-centred mathematics practice mean in such contexts, and what role does code-switching by the teacher and the learners play? Disaggregating the multilingual mathematics classroom in policy, research and practice is a significant challenge for mathematics education in South Africa, and, I believe for the wider mathematics education community. At the same time, we need a great deal more research on classroom communication and conversation where the spotlight indeed is wide enough to encompass the diversity that constitutes ‘the’ mathematics classroom.

CHAPTER 7

DILEMMAS OF MEDIATION IN A MULTILINGUAL CLASSROOM: SPOTLIGHTING MATHEMATICAL COMMUNICATIVE COMPETENCE 1. INTRODUCTION A participatory-enquiry approach to teaching and learning school mathematics is often driven by the twin goals of (1) moving away from authoritarian, teachercentred approaches to learning and teaching and to mathematical knowledge itself, and (2) improving socially unequal distribution of access and success rates. The underlying assumption is that this kind of pedagogy provides a more meaningful and effective way for students to learn. As described in Chapter 2, these goals and assumptions underpin curriculum transformation in post-apartheid South Africa. In a participatory-enquiry approach,1 or, as it is described in current curriculum policy in South Africa, a learner-centred approach to school mathematics, creative thinking is encouraged so that learners can develop confidence in their own ideas. Typically, they are provided opportunity to engage with challenging mathematical tasks, either alone, but more likely in pairs or small groups. The knowledge learners bring to class is recognised and valued. Diverse and creative responses are encouraged, and justifications for mathematical ideas sought, often through having learners explain their ideas to the rest of the class. The task-based, interactive mathematical activity that is provided in such a class offers learners a qualitatively different mathematical experience, and hence possibilities for mathematical learning and knowledge development that extend beyond traditional ‘telling and drilling’ of procedures. A participatory-enquiry approach places particular communicative demands on both teachers and learners. In this chapter I argue that in a multilingual classroom, a participatory-enquiry approach to teaching and learning mathematics creates dilemmas of mediation for teachers, particularly the dilemma described in Chapter 5 as developing mathematical communicative competence (subject-specific discourse) vs. negotiating and developing meaning. As described in Chapter 5, those teachers whose pedagogical approaches were more interactive and participatory expressed concerns constitutive of this language-related dilemma of mediation. Indeed, the dilemma between developing mathematical communicative competence and developing meaning became the means with which to capture the complex demands of mediation of teachers who encourage enquiry and participation in their multilingual mathematic classrooms. Of the six teachers that I worked with, Sue had gone furthest in establishing a participatory-enquiry approach in her mathematics classroom. Sue’s mathematics lessons were noticeably different from most others, where learners all do the same thing in the same way. Despite curricula constraints, and in contrast to dominant school mathematical practices in South Africa, Sue had created a classroom culture that provided learners with confidence to enquire, interact and develop their

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mathematical intuitions. In this chapter I explore dilemmas of mediation through a vignette of Sue’s practice. As I argued in Chapter 5, teaching dilemmas, including language-specific dilemmas of mediation, are not exclusive to a multilingual context. Indeed, in any mathematics classroom there are diverse communicative competences. It is the multilingual context, however, that brings the range of dilemmas of mediation inescapably to light. Managing these dilemmas can entail trade-offs some of which are conscious and deliberate, others tacit and even unaware. Trade-offs are tied up with worthwhile goals that sometimes conflict in moments of practice. In particular, working to meet the dual goals of validating diverse student perspectives (which entails working with informal expressive language and learners’ conceptions) together with developing mathematical communicative competence (which in turn entails access to formal mathematical language and to specific mathematical concepts) is extraordinarily complex within the time-space relations in a school classroom. Trade-offs are inevitable. Moreover, teachers are often acutely aware of dilemmas in shaping informal, expressive and sometimes incomplete and confusing language, while aiming towards the abstract and formal language of mathematics. What can be obscured, however, is that a participatory-enquiry approach, and the possibilities it offers for learner activity and student-student interaction, can inadvertently constrain mediation of mathematical activity and so, access to mathematical discourse and related concepts. The suggestion from the analysis of a lesson in a participatory-enquiry multilingual classroom is that the multilingual classroom context exaggerates this possible constraint. In so doing, it throws a spotlight onto issues of mediating mathematical communicative competence, a competence that is important for all mathematics learners. 2. MEDIATION AND A SOCIAL THEORY OF MIND Dilemmas of mediation point to theoretical issues that I will discuss before I turn to explore dilemmas of mediation through an incident in Sue’s classroom. Firstly, embedded in a participatory-enquiry approach are assumptions about language and learning — talk is understood as a social thinking tool (Mercer, 1995). There are also assumptions about mathematics as a practice. Mathematics is something you think about, enquire into, and communicate with others. In this conception of mathematical practice, learning mathematics and becoming mathematical entails talking to learn mathematics and being able to talk mathematically. In Lave and Wenger’s (1 991) terms, becoming knowledgeable about a practice, like mathematics, is the fashioning of identity in, and as part of, a community of practice (pp. 50-51). Becoming knowledgeable means becoming a full participant in the practice, and this involves, in part, learning to talk in the manner of the practice, i.e. being able to talk mathematically. Within this notion of learning as situated, Lave and Wenger distinguish between “talking within” and “talking about” a practice (p. 109), where both are entailed in learning to talk. For example, in participatory-enquiry approaches to school mathematics, students often work on

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tasks together and then report on their working to others in the class and to the teacher. While engaged in tasks, learners could be said to have the opportunity for talking within their mathematical practice. Then, either to the teacher, or to other learners, or both, they talk about their mathematical ideas. In Lave and Wenger’s terms, a participatory-enquiry approach is a situation and practice that provides learners with both these opportunities for learning to talk mathematics and so to become knowledgeable about their school mathematics. Lave and Wenger’s concept of learning to talk is thus useful for exploring how learners in Sue’s class talked within and about mathematics. A theoretical question, however, is that Lave and Wenger developed their conception of situated learning in contexts of apprenticeships where there is a situated and continuous movement from peripheral to full participation in a practice (p. 53), and so a seamless movement between talking within and about a practice. The school, however, is a very different context from those of apprenticeships (Adler, 1998b). Indeed, previous discussion has pointed to the difficulties teachers face in assisting learners to move between informal and formal mathematical talk. Embedded in a participatory-enquiry approach, and thus a particular view of learning as participating through talk in mathematical practice is, perhaps, an absence of attention to the movement between informal talk on a task amongst learners, and expressive forms needed in the public mathematical domain. Here learners’ mathematical talk needs to be interpreted and understood by others — other learners, the teacher, and the wider mathematical community. Discontinuities in school mathematics practice extend beyond forms of talking and significantly into the world of concepts. Learners bring to and construct in the classroom, intuitive mathematical conceptions and meanings. These can be understood as products of their current activity, their everyday knowledge and their prior mathematical learning. Often, these intuitive and spontaneous ideas are different from, or less rigorous than, those required in the mathematics curriculum and accepted as mathematical convention and knowledge. Vygotsky (1978) recognised the school as a distinct context entailing distinct kinds of activities leading to qualitatively different kinds of knowledge from those acquired in everyday life, in play or in work. For Vygotsky, schooling and formalised instruction lead specifically to the development of metacognitive awareness on the one hand and to the development of what he called “scientific concepts” on the other (1986). The learning of new word meanings in school is not through direct experience with things or phenomena: rather, it is through a system of concepts. Vygotsky distinguished “scientific” concepts from “spontaneous” concepts, those concepts that are formed in our everyday activity. For Vygotsky, scientific concepts are part of a system of concepts, and they are deliberate and self conscious. In contrast, spontaneous concepts are unsystematised and saturated with experience. Nonetheless, scientific and spontaneous concepts, while distinct, interact with and influence each other: One might say that the development of the child’s spontaneous concepts proceeds upwards, and the development of his scientific concepts downwards, to a more elementary and concrete level ... The inception of a spontaneous concept can usually be traced to a face-to-face meeting with a concrete situation, while a scientific concept

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involves from the first, a “mediated’ attitude towards its object (Vygotsky, 1986, pp. 172-173, 193-194).

In order to deal with the interaction of scientlfc and spontaneous concepts, and to further elaborate his cultural law of development, Vygotsky posited his now wellknown notion of the Zone of Proximal Development (ZPD), that “distance” between: the actual development level as determined by independent problem-solving and the level of potential development as determined through problem-solving under adult guidance or in collaboration with more capable peers (1978, p. 86).

Tied to the ZPD is Vygotsky’s understandmg that there is no learning that is not in advance of development. Formal instruction “brings forth the zone of proximal development” (p. 89). Vygotsky did little to elaborate the concept of the ZPD himself. This has, however, been done by Wertsch. In Wertsch’s (1984; 1991) discussion of activity, he distinguishes three components to functioning in the ZPD: situation definition; intersubjectivity and semiotic mediation. As learners begin a task constructed by their teacher, they adopt an orientation to the task that requires, in the first instance, what can be called the situation definition of the task — how the task is situated and defined by the learners and teacher. What motives, goals, needs and values are read into the task by the learners and the teacher? How is the task understood in relation to its specific classroom context? When the situation definition of a task is shared, then intersubjectivity (between the teacher and learners) in relation to the task is easily established. It is when situation definition is not shared (either within a group of learners or between a learner and the teacher) lhat mediation is required for intersubjectivity to be established. The issue in adult-child interactions is the changing of the child’s situation definition and the kind of mediation that is required to establish intersubjectivity in relation to the task at hand. Issues of power and control and whose knowledge enter here, but pertinent to this chapter is that if situation definition is shared then intersubjectivity is easier to establish and semiotic mediation easier too. If situation definition is not shared, then establishing intersubjectivity through semiotic mediation is a more complex process. Brodie (1995a), in a detailed study of a group of Grade 9 learners in a South African school working on a task related to the concept of area, showed how the situation definitions differed both within the group and between the group and the teacher. Importantly, and in addition to specific motives and goals, their spatial orientation to a geoboard which had them focus on the pegs rather than the spaces or distances between pegs, resulted in interesting but problematic attempts by the group to generate effective scientific concepts in relation to area. This was compounded by the interactions in the group itself and their limited interaction with the teacher. In the time the teacher had with this group, given her need to manage the whole class, she listened to and questioned the learners in order to understand their orientation to the task. Her interactions with them, however, did not progress to establishing intersubjective meanings of the task. As mentioned above, such mediation is more complex when task orientations are different. The effect was that the teacher did not

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manage to mediate the learners’ spontaneous approach in a way that could have facilitated their “scientific” understanding of area. The zone of proximal development thus highlights the teaching-learning dialectic and the issues of challenging yet supporting diverse and spontaneous student conceptions and orientations. In more general terms, the ZPD brings to the fore the issues of effective mediation of scientific concepts, and so too, learning to talk in the manner of the practice of school mathematics. In Vygotskian theory, the development of “scientific” thinking resides only in the context of schooling. It is beyond the scope of this chapter to argue fully that not only has schooling fared rather poorly in this regard, but that it has distributed schooled knowledge in socially unequal ways (see, for example, Apple, 1982; Bowles and Gintis, 1976). There are extensive arguments that Vygotsky did not problematise schooling sufficiently (Bernstein, 1993; Levine, 1993; Daniels, 1993; Ivic. 1989). Sociocultural developmental theory. nevertheless, provides significant insight into issues that confront the complexity of teaching and learning mathematics in the context of school. School mathematics requires mediation, and specifically mediation between everyday and scientific concepts — between previously acquired mathematics and new mathematics. Thus, it is not only learning to talk that needs to be problematised but also learning from talk. The use of tools in classrooms and particularly the language resources made available for learning need to come under scrutiny. In multilingual classrooms, this becomes a particularly interesting question: how language is and is not made use of and why. In the language of the previous chapter, the journey from informal talk in the learners’ main language to formal written mathematics in English requires careful navigation by the teacher. Sue, and other teachers who tried to make their classes more participatory and enquiry-based faced complex dilemmas of mediation. and specifically the languagerelated dilemma of listening to and validating diverse perspectives (spontaneous concepts) vs. developing mathematical communicative competence (scientific concepts); and of moving effectively between learners’ more informal expression of their mathematical thinking and more formalised school mathematical discourse Sue talked about these at length in her initial interview. Of course, what teachers articulate is only a partial window on what they know. What happens in practice? In the remainder of this chapter I move into Sue’s classroom and explore with her whether and how she actually faced the dilemmas she talked about, and if so, how were these managed in instances of practice? Were there trade-offs, and if so, what were they? Was there a specificity to these dilemmas of mediation in a multilingual setting? I begin with a discussion that contextualises Sue’s work, including a brief description of her school, and her approach to teaching and learning mathematics as illustrated in the lesson(s) in focus through the chapter.

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3. THE CONTEXT

3.1 Sue’s school Sue’s school was a well-resourced private school in Johannesburg, South Africa. Classes were small (around 20 students) and multilingual. The vast majority of students were black and they brought a variety of main languages to class. English, the LoLT, was an additional language for most learners. Learners were, nevertheless, in a supportive additional language learning environment — and exposed to English in use in and around the school. Many students were on bursaries, and thus not necessarily from economically affluent families. Most teachers (including Sue) were white and English-speaking. All were academically and professionally qualified and a culture of professionalism and enquiry permeated the school and staffroom. Sue’s participatory-enquiry approach was thus supported in her school. This is important because the difficulties she might faced occurred despite this support. It is important to note here that Sue’s teaching context was very different from the over-crowded, under-resourced reality of many other schools in South Africa. That she worked in optimum conditions is one reason why her experience and struggles are pertinent and illuminating. Of course, Sue’s school was located withing a broader schooling system where traditional approaches to mathematics teaching were dominant, and she had to work with the canonical school mathematics curriculum. 3.2 A lesson For most of the 40 minute lesson, the 16 students worked in pairs on part of a worksheet (Figure 3 below) where the questions were designed to elaborate the concept of the angles of a triangle.

FORM I: GEOMETRY If any of these is impossible, explain why, otherwise draw it. 1. Draw a triangle with 3 acute angles. 2. Draw a triangle with 1 obtuse angle. 3. Draw a triangle with 2 obtuse angles. 4. Draw a triangle with 1 reflex angle. 5. Draw a triangle with 1 right angle.

Figure 3. Extract from worksheet in Sue’s Grade 8 class

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The lesson began with brief instructions related to the worksheet. Students were to discuss their ideas with their partners, and illustrate (draw a diagram) and write (in words) an explanation of their answers in their notebooks. They were also told that later in the lesson, one from each pair would be called on to “explain” to the class what they had done. 3.3 Sue’s approach 3.3.1 Participation and enquiry While students were working in pairs on the worksheet, Sue circulated. Her interaction with learners was largely through questions that asked for clarification and justification of their ideas, and that encouraged student-student interaction. So questions like “Why?”, “How do you know?”, “Do you understand what (your partner) says? Ask him a question?” predominated. In addition, she also reminded learners to write their explanations in words and/or to draw a picture — thus reinforcing the multiple representations built into the worksheet, as well as Sue’s explicit attention to the journey from informal spoken mathematics to formal written mathematics evident in the task instructions. An enquiry approach was further evidenced in the actions of her learners. I observed four lessons on angles, during which students asked her a wide range of questions, for example: Can we have a curved angle? If we have a right angle is there also a left angle? In a triangle, why don’t we include straight angles (referring to any point and a 180 degree angle on one of the sides). Sue also spontaneously asked questions, for example: How many triangles are there in the world? In addition, Sue fostered student-student interaction. Students interacted with each other while on task, and then during report back. Students had learnt that they were expected to ask themselves ‘why?’, to explain and ask ‘why?’ of others and to interact verbally with each other. These interactions, while controlled by Sue, also reflected her skill in listening to, valuing and pushing students in her interactions with them and the task in hand. What Sue valued (mathematics as “something you talk about”, “have your own ideas about”, “ask questions about”; learning as social “interactive”; and knowledge as personal and problematic), she accomplished. This participatory-enquiry approach stands in sharp contrast to many mathematics classrooms where teacher-initiated recall-type questions and I-R-F interactions (initiation-response-feedback) predominate and where students “go for an answer” (Campbell, 1986). Sue’s lessons are better described by students’ “going for a question”. 3.3.2 Talking within and about mathematics In Lave and Wenger’s (1991) terms, there was ample opportunity for talking within (i.e. for exploratory talk in) the mathematical practices in Sue’s class. While working on tasks, Sue’s students were talking within their mathematical practice.

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Then, and less so in this part of the lesson, they also talked about their mathematical ideas, either to the teacher, or their partner. In report-back time, individual students, while talking within an overall classroom practice, explicitly talked to the class about their mathematical ideas. They were thus required to formalise their mathematical ideas in speech and in writing. 3.3.3 Scientific concepts as implicit goal In addition to descriptions, questions, explanations and justifications, Sue also encouraged some students to consider and extend the generalisability of their answers. As discussed, in Vygotskian terms, generalisation is a scientific concept, linked to other concepts and acquired through mediated systematic instruction. Sue’s interaction with Joe and Rose (the two students focussed on in the report-back incident below), while they are working on their tasks, reflected how at an individual level she worked to scaffold this scientific concept and at the same time listened to and validated their diverse perspectives. Rose drew a shape like this: and answered “impossible” to 3 (the possibility of two obtuse angles in a triangle). Sue validated the response and then asked: “Will this always be the case? that they don’t join?”. Joe explained to her how he started with a triangle with an angle of 89 degrees. The other two angles were then both acute. If he made the 89 degree angle obtuse, “you like stretch this a bit, and then while you are stretching this, these other two angles will get smaller”. Sue validated this, and said (extracted from videotape): Sue: That is an okay explanation — but I am not sure if it covers all the possibilities, because what if you start off with an obtuse angle? You started off with 89 and it became 91 — and I think you should write that explanation. But when you have finished writing that think about what happens if you start off with an obtuse angle, like 125 degrees. Could you then have the triangle with another obtuse angle?

[Sue then turned to Joe’s partner to see if she shared Joe‘s view and then encouraged them to “together think about the 125 degree starting angle”.]

Sue offered Joe and his partner another particular case for them to try. She thus encouraged both Rose and Joe, though in different ways, to consider the generalisability of their answers. These individual interactions suggested that the development of the concept of generalisation was included in Sue’s lesson goals. The incident below occurred in whole class report-back time and concerns the worksheet question already discussed: the possibility of drawing a triangle with two obtuse angles. The incident is focussed on Joe’s reporting back and the ensuing interactions between him, Sue and Rose.

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4. A VIGNETTE: AN INCIDENT AND OBSERVATIONS

4.1 From stretching to labelling angles: an incident KEY: (brackets within a data extract — research commentary) (()) - inaudible utterance [] - unnecessary utterances edited out () - short pause ... - longer pause Rose has just drawn and explained to the whole class why it is impossible to draw a triangle with two obtuse angles — that you get a quadrilateral Joe’s reporting of his explanation followed. While talking, he drew the following two triangles on the board: Joe: I said all the A’s must be like more than () they must, uh, be the biggest in the triangle, um, so that if, uh, if this A here, say, is like 89, () and then these are say 37 and [mumbling to himself, ja, ja] 44, ja. And then in this one, number two, () it w ll be an obtuse angle. I said 91 and this is 41, () and this here is 46, no [crosses it out and puts 45 - all “labels” are outside the triangle]. And I said like if A, if A is going io stretch, () if A is going to stretch [pointing to 91] then these two angles here ... if it has to stretch then these two, like these two they are going to contract. i

Figure 4. Joe’s first drawing

He drew another 90 degree triangle below, and re-explained:

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Figure 5. Joe’s second drawing If this here, if this is A, if A is here now miss and if it has to stretch, like these two we gonna have to (()) them both ... if this is 90, andyou if you, if you, if it is gonna (()), turn to be lets say 110 or something, () (drawing the obtuse angle) then this one here (pointing to top angle) will be smaller than it was before, it was before, so, so if it was, say, 40 here then it is going to be 30 here, uh () then A is going to be taking that 90 degrees, uh, that 10 degrees, let’s say B had ... uh uh if if if one angle stretches then, uh, the the two angles, the two other angles have to contract Sue:

Okay what do other people think? Any questions? Rose?

Rose:

Isn’t that triangle the same as the other one if you measure (())

Joe: I was just doing an example, I forgot what angles I was using in my book [ ] but [ ] they are supposed io add up somewhere near to 180 degrees [ ]

After some teacher-mediated interaction between Rose and Joe during which Rose was able to clarify that her question was whether the one triangle is the “same as the other turned upside down’?, Sue said: Sue: I think Joe maybe the first problem is that you haven’t shown these angles on the picture and lots of people do this - they write the angles outside the picture. OK Now you know what you mean and I know what you mean and maybe some people know what you mean. But to be clear (she writes the angle sizes inside the triangle), do that. Put it inside. [] Now, are these two triangles the same just turned upside down?

She continued interacting with Rose and the rest of the class to ensure they understood that while the triangles in figure 4 “look the same”, they are not. SO, Joe was not “wrong”. The bell rang but she continued: Sue: [] it does not really matter what they really measure - we still get what he is trying to tell us because he has shown us an example of what he has done [] we will come back to this tomorrow.

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In the recap at the beginning of the following lesson the next day, Joe’s partner volunteered and summarised his reasoning quite clearly to the class. ... He said miss, um, you stretch A miss, then B and C will get smaller.

Sue’s question about starting with 125 degrees, that is, trying another case, did not resurface. 4.2 My initial observations At the outset, while observing the lesson in process, this incident caught my interest. Firstly, I was impressed by Joe’s dynamic, relational conception of the angles of a triangle — how angles change in relation to each other.2 Yet he struggled to explain himself clearly, to find the words and illustrations to express his ideas publicly. Rose’s question and Joe’s response suggested that they did not understand each other and Sue’s mediation focussed on clarifying Rose’s question, and then on how to label angle size clearly, on estimated angle values in the diagram, and away from the actual mathematical content of the task — away from “stretching” in order to form an obtuse angle to labelling angles. I noted this as an instance of problematic communicative competence so as to ensure I discussed it in the reflective interview with Sue. At its most obvious, the communication difficulties could have been indicative of students communicating in English when this was not their main language. Neither Joe nor Rose had English as their main language. Moreover, they did not share the same main language. In addition, however, there was clearly a difficulty for both Joe and Rose with mathematical English. I also noted that while Joe battled to explain himself to the class, earlier he had managed to convey his reasoning to both Sue and his partner, albeit with lots of particularist language (such as ‘‘this one here”) and pointing. Sue was not entirely happy with his explanation. Yet, at the public level in the class, her concern whether ‘‘it covers all possibilities” and that Joe’s response was a particular case, did not resurface. From my perspective as researcher, this incident promised to provide insight into diverse communicative competence within and across learners in a multilingual mathematics classroom. It also promised to provide insight into how teacher actions are shaped by problematic communication, into the kinds of effects of such actions on learners’ development of mathematical knowledge. Close viewing of the videotape supported the sense that I had gained when I was physically present in the lesson, that in the report-back session constraints were operating on whole-class student-student interaction in relation to the content of the mathematical tasks. Most discussion and mediation related to questions of clarification of elements of the report given. Such questioning is, of course, part of learning mathematics. The point here is that if questions are all of this type, deeper mathematical thinking could be constrained.

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4.3 Sue’s Reflections 4.3.1 On student-student interaction As mentioned earlier, there are common sense assumptions about the benefits of interactive learning: that it will democratise access; that it will make communication less problematic. Indeed, the Cockcroft Report (Cockcroft, 1982) stimulated what quickly became a new commonsense notion (though not necessarily common practice): that mathematical learning is enhanced by conversation. It took some 15 years for this assumption to be interrogated by what I referred to in Chapter 1 as the dominant mathematics education research community. Sfard et al. present an interesting debate on “Mathematical conversation: is it as good as they say?”, concluding that the question is not “whether to teach [mathematics] through conversation but rather how’’, and that ... communication skills cannot he taken for granted... if conversation is to be effective and conducive to learning, the art of communication has to be taught. How this is to be done, and what exactly should he learned by the students remains a question to which the mathematics education community has yet to give much thought. (Sfard et al., 1998, p. 50)

As a practising mathematics teacher in her multilingual classroom in South Africa, Sue had indeed given this a great deal of thought. It was obvious to her that communication skills could not be taken for granted. After far fewer than 15 years, and through intense experience in promoting interaction and discussion (conversation) in her multilingual mathematics classes; Sue had pedagogical concerns (‘how’ questions) that reflect dilemmas of mediation specific to conversation and communication, and they were writ large: Sue’s opening point in her reflective interview was: Sue: ... the thing that worries me the most is that I am not sure whether, I am not sure to what extent it helps them learn. I think that talking to each other is not unproblematic. I think a lot of the kids don‘t listen. Maybe they are too young. I think. You can see it with the questions [] they’ll ask a question and say “I don’t understand” and then the one who is up will try to explain and it doesn‘t really help but they are being polite and they are not quite sure and they say “OK fine”. I am not sure they understand.

She particularised her concerns later in the interview when we viewed Joe’s explanation of “stretching” angles: He doesn't really answer her question. They are not communicating - and that happens a lot! He can't hear her question and she can't hear his explanation.

As the tape reached the point where she taught angle labelling, Sue commented critically on her actions: Now I am deflecting more.

Sue's opening general comment above pertains to the incident and mirrors many of my observations. We shared concerns that communication was problematic and that her actions were indeed “deflecting” off the mathematical substance of task. Sue

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was acutely aware of the difficulties her students had. On Joe’s difficulties in articulating his thinking, she said that Joe “does okay” but, more significantly she didn’t know “how to move them on. I don’t know how to develop the language”. It was not only Joe and Rose who had difficulty understanding each other. In this research, there were three layers of interaction and communication about Joe’s “stretching”, each with difficulties in mathematical communication that extended beyond the level of the multilingual nature of the classroom. There was Joe and Rose (two students), Joe and Sue (student and teacher), and Sue and myself (teacher and researcher). While Sue plainly saw the difficulties between Joe and Rose, it was in discussion with me that the dynamism of, and limitations in, Joe’s response became apparent to her. And, not surprisingly. it was only in discussion with a colleague on this text, that I could acknowledge my differences with Sue as to what was and was not a general or good explanation and justification of the task.3 4.3.2 A good explanation is a generalised one On the content of Joe’s thinking, Sue said: Sue: I think what I was saying to him is you starred with one triangle and you explained it - so now start with a different triangle [] I am definitely pushing Joe more ‘cause I felt the others are more generalised explanations. He was starting off with a specific triangle I did want him to generalise more.

It is difficult for teachers always to see student perspectives: JA:

What is so fascinating about this is how do you see everything

Sue: You can’t ... especially when he is not very good at explaining himself [ ]

This affected what was affirmed: Sue: They must, they have to. I mean in terms of affirmation: how do I know that something is good, how do I say that is a good question. Its because it is a question I would have asked. So it is bringing up what I think is mathematical thinking and that is my own view, so it definitely does. And often you don‘t hear what a child is saying because it doesn’t match []

Sue certainly did have a notion of a good explanation — one that was general. I thus read two intentions for her into thls task: the development of the mathematical properties of a triangle, that it cannot be made of two obtuse angles; and the development of the scientific concept of generalisation through justifications that are general. The interesting observation here is that the first of Sue’s intentions was clearly evident in the construction of the task and the instructions students were given. The second intention was, however, more implicit, evident only through her separate interactions with Joe and Rose and her reflections. 4.3.3 Deflection or teaching in context In the reflective interview, Sue and I did not specifically discuss her “deflection” to labelling angles. This came up in a later conversation, however, after she had read my description and interpretation of her classroom practice and her reflections.

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Bearing in mind the time-lag, she said that she had been “thinking about deflection’’ and noticed more and more that she used it frequently to “teach in context”. This was especially the case now as a materials-writer. She often deflected to teach in context. The issue then is one of refocusing — getting back to the task. In the middle of whole class discussion of Joe’s “stretching” explanation the bell rang. Sue returned to the task the next day, and Joe’s partner expressed his idea clearly. Hence the “mathematical properties of a triangle” were explicitly dealt with. But the mathematical limitations of Joe’s response from Sue’s perspective were not publicly explored. Sue’s notion of what would qualify as a more generalised, and therefore a better explanation remained implicit and at the level of individual mediation. It did not enter public discussion — whole-class mediation — inreport-back time. Does this matter? To whom? Is this a case of unintentionally enabling only some (Joe and Rose)? Or, is it a pragmatic and appropriate way of mediating in diversity, and of creating a culture where personal and diverse knowledges are valued encouraged? 5. DILEMMAS OF MEDIATION In the complexity of teaching mathematics in a multilingual classroom, a participatory-enquiry approach makes particular demands on learners’ communicative competence. Although Sue had developed a culture of meaningful mathematical enquiry in her class, at times students struggled to explain their mathematical thinking. Students also had difficulty understanding each other Student-student interaction (verbal communication) was thus not a taken-for-granted given. These communicative difficulties with public articulation and whole-class student-student interaction shaped Sue’s actions and interacted with what explicit and what she left implicit and, in turn, with the mathematical knowledge made available to students. What were the mediation dilemmas at play here and how did Sue understand and explain her management of them and the trade-offs she might have made? Of interest in this chapter is where and how the multilingual context, in foregrounding linguistic diversity in a mathematics classroom, brings to light dilemmas of mediation pertinent to, but perhaps not as visible in, all classrooms i.e. including classrooms where linguistic diversity is not so obvious. 5.1 From talking within to talking about mathematical practice Joe had difficulty articulating his thinking in his report back to the class. His language was littered with “ums”, repetitive phrases and hesitancies. A simplistic explanation is that Joe, and others who display similar behaviour, are not main language English-speakers. Their task is thus one of double attention — to their new mathematical ideas and to a language they are still learning. However, difficulties in “speaking mathematically” (Pimm, 1987) are not unique to learners who are studying in an additional language. As suggested at the start of this chapter, the

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multilingual context highlights communicative issues in the classroom. Sue was acutely aware of Joe’s difficulties in articulating his thinking but “doesn’t know how to move him on — to develop the language”. Pimm’s (1992, 1994) work explains why the issue is not simply about proficiency in or access to English. Reporting mathematical thinking, even for main language English-speakers, is not a simple process because of the linguistic and communicative demands entailed. “Skills of reflection and selection” and a “sense of audience” are important to successful report back. This could explain why Joe could convey his meaning to the teacher and his partner but struggled with his whole class report-back Interacting individually with Sue and his partner, Joe was able to point to his work in his book and thus did not have to select in the same way as when he worked on the board in front of the whole class. While he displayed a sense of audience by trying to recount the process of his thinking, his loose selection of angles was confusing. His more formalised written and spoken mathematics was not clear. To some extent, Sue was aware of the issues of selection and audience. In her reflective interview, she described a situation where students in another class had worked on an investigative task for a few days, where exciting mathematics had been evident You see what happened in another lesson: I gave them an investigation to do - and it was a two to three lesson investigation: given a fixed perimeter, which shape has the greatest area? And they all did wonderful things and in different ways, and after three lessons the time was to present it and they presented appallingly - no-one could understand what the other group had done on the board. I knew and was able to draw it out but they just weren’t able io present and partly ‘cause I have never told them how to present, never told them you can use diagrams and structure it in this way. And also I have never structured explaining to them.

And they don‘t know what it is others need to know about their thinking ...

What is also visible here is that talking within (discussion with other learners) and talking about (reporting ideas publicly to others) mathematics within the classroom, while deeply related, do not place the same communicative demands on the speaker. Here is empirical support for my earlier argument that Lave and Wenger’s seamless web of becoming a full participant through learning in a community of practice is problematic. In this community of practice, Sue’s mathematics classroom, the move between talking within and talking about is not spontaneously or tacitly learnt. The journey and/or movement between informal exploratory talk and more formal spoken and written subject-specific discourse requires mediation. Sue knew that there was problem, that she needed to mediate learners’ reporting. But knowing that does not mean knowing how to act. Sue’s tacit knowledge was to “deflect” to labelling, which, as she herself noted, had a real purpose. Sue’s labelling focussed, to some extent, on this issue as she highlighted for Joe and others, how you need to label effectively to be able to point with words if others are to understand you. Being explicit about what was required was necessary. But this required language teaching and Sue did not know how to “develop the language”.

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Moreover, as others have pointed out, a didactic tension arises as teachers attend to being more explicit about what is required of learners. The more explicit, the more learners will take form for substance. The less explicit, the less learners are likely to notice what is going on, what is intended (Brousseau, 1989; Pimm, 1992; 1994). This vignette of Sue’s practice suggests that teachers could be usefully informed about aspects and issues of reporting-back as discussed by Pimm, so as to become aware that the shift from talking within to talking about is not necessarily spontaneous. Mathematical communicative competence needs to be mediated by the teacher. Related teaching dilemmas are inevitable. Reporting skills as part of mathematical communicative competence is important. However, they are not the only issue here. The multilingual setting foregrounds a deeper question, that is, whether the underlying problem for Joe was a communicative or an epistemic one and how these problems might be inter-related. The situation here needs a reminder. In Sue’s multilingual classroom, most classroom talk in the public domain as well as in group or paried discussion took place in English, the LoLT. However, her classroom was multilingual. Many learners had greater fluency in a language other than English, and some had limited proficiency in English, particularly in Grade 8, the first year in secondary school. The multilingual character of Sue’s class was highly visible to her, but not necessarily so to readers of a reconstructed account of her practice. In classes like Sue’s, the specificity of the multilingual context slips in and out of view as issues in communicating mathematics are explored. Yet for Sue it was a significant component of her knowledge of her learners, and thus of the complexity of her practice. The question as to whether Joe’s difficulty was epistemic or communicative reflects back on questions posed by all the teachers in their initial interviews. They said that it was often difficult in their multilingual classroom to discern whether the problem a learner appeared to be having was “the language” or “the concept”. Mindlanguage interrelatedness is often at issue in student-student interactions with mathematics. Sue’s multilingual setting brings this issue sharply into view. 5.2 Student-student interaction and the development of scientific concepts Why, when Joe reported, could Rose not hear his explanation and he her question? Sue’s insights from her reflections on her practice were very instructive. Students in this class struggled to hear and to engage each other. Sue reflected on this, providing practice-based understanding of metacognitive issues. Sue understood how hard it was for learners to step out of their own ideas and frames of reference to engage others’ mathematical thinking. Indeed, it was hard for her as teacher, and myself as researcher. The question begging is: How might students have the vantage point that one expects of the teacher — a vantage point from which to interpret and engage a range of ideas different from your own. Wertsch’s elaboration of activity and Vygotsky‘s Zone of Proximal Development (ZPD) provide ways of describing Sue’s insights. As discussed earlier, activity in the ZPD involves: situation definition; intersubjectivity and semiotic mediation

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(Wertsch, 1984; 1991). When situation definition (motivation and/or orientation to the task) is not shared, mediation for establishing intersubjectivity becomes complex. Joe and Rose did not share the same situation definition. Rose started by drawing obtuse angles and then could not form a triangle, only a quadrilateral. Joe started with a triangle and saw that if he stretched one angle into an obtuse angle, the others would contract. Rose and Joe‘s orientations to the task, their objects of attention, were different and they struggled to see past their own orientations to establish intersubjectivity. There was so much potential for quality learning in this situation, for Joe and Rose and the class to reflect on their diverse conceptions. But they did not do it on their own. The teacher could assist in the creation of a “construction zone” (Newman, Griffin and Cole, 1989, p. 153), by mediating these differences publicly, bringing to attention different orientations and starting points, their connections and relative mathematical strengths. For here, discussion amongst the students is not about who is right or wrong — both Rose and Joe’s approaches make sense and answer the question — it is about how they are similar and different, how they are related, and then too, which is better mathematically. Sue’s mediation of these student-student interactions was to ensure that questions were clarified and that the different explanations of why a triangle cannot have two obtuse angles were each individually understood by the class. In other words, her focus of attention was that students understood what other students were attempting to communicate. In Joe’s case, she deflected to teach labelling and did not refocus back onto the mathematical substance ofthe task. Sue saw her deflection as an opportunity for teaching in context, for example, teaching labelling in a context where such mathematical skill had meaning. A major issue for Sue, however, was that as a ‘participatory-enquiry’ teacher, she would have liked her students to engage each other. She wanted them to ask each other more effective questions, perhaps like those she asked Joe and Rose as she interacted with them individually while they were on task. But there appeared to be no effective construction zone between Joe and Rose. The irony here is that Sue’s desire for students to engage each other was perhaps simultaneously part of, and undermined by. her participatory pedagogical approach — and this appeared to be obscured from her. What were Sue’s purposes behind not refocusing? It is arguable that Sue’s concern to encourage participation and enquiry interacted with the difficulties students had in explaining their thinking and engaging each other, and her repeated point that she “does not know how to move them on”. Together these mitigated against her mediating across differences and evaluating the differing substance and content of what students offered. Sue’s actions and reflections here are similar to those described and interpreted by Jaworksi (1999) as evidence of a dilemma for ‘investigative’ teachers between eliciting and inculcating mathematics. The effect of Sue’s actions was that rich mathematical opportunities were thus simultaneously created and partially lost, the trade-off for sustaining what she had worked hard to build — a culture of meaningful enquiry where students perspectives were valued and knowledge was treated as personal and problematic.

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A second explanation as to why Sue did not refocus is bound up in the tension between her goals for participation (negotiation and development of meaning) on the one hand and her implicit intention — the development of the scientific concept of generalisation — on the other. As Bartolini-Bussi (1995) has argued from a Vygotskian perspective, a scientific concept is neither a natural development of an everyday concept nor a matter of negotiation, but is acquired through instruction Bartolini-Bussi analysed classroom activity where a teaching intention was the scientific concept of “patterning”. However, teacher mediation of this concept was nowhere evident in teacher-student interactions on the task set. This then accounts for difficulties students had in generating and working with the patterns in the task. I have described how Sue’s idea of what constitutes a “good explanation”, one that is more general, was not made explicit. This implicit intention was evident in individual interactions and absent in Sue’s interaction with the whole class in public report-back time. However, it was her responsibility to provide appropriate mediation of a more generalised response. But doing this produces a profound teaching dilemma. Sue would need to focus attention on, and provide a scaffolding process for, how Joe and Rose differ. In addition, she would need to attend to why, in her view, Joe’s response was not a generalised one and therefore limited, and why Rose’s response was more general. This could have been done in a way that continued to encourage student participation and interaction but such activity on Sue’s part might well have undermined her goal for her students to build confidence in the value of their own ideas. Joe might have felt that his thinking was not good enough because it was not like Rose or Sue’s. This could inhibit his willingness to participate in future or negatively shape his goals. Yet, if Sue does not mediate publicly how and why Rose’s response was a general answer, but Joe’s a specific case, then her intention to develop “good explanations” through student-student interactions will be thwarted. Pragmatically, given the time constraints, the question of whether, at this level, the concept of generalisation could only be emergent and individually mediated must also be asked. 5.3 Communicative competence (talking about mathematics) and scientific concepts This vignette of Sue’s practice provides insight into how learners’ difficulties with engaging each other mathematically in a multilingual class were metacognitive on the one hand and on the other, related to their ability to express their mathematical thinking. Their difficulties were bound up with the teaching and learning approach in the class and with access to English, mathematical discourse and classroom discourse, that is, the three-dimensional dynamic of learning and teaching mathematics in multilingual classrooms. What is illustrated by Sue is that she knew that expression and engagement were problems, but not how to deal with these. Sue has long been confronting the challenge Sfard (1998) made to the mathematics education community, that we do not know how to teach the art of communication so that conversation in and for mathematics in school could be effective and conducive to learning. This gives

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substance to the criticism I made of dominant research in language and mathematics, that it has ignored research with a specific focus on bi-/mutlilingual mathematics classrooms. If Sfard had attended to reports of such research (e.g. Adler, 1997; Moschkovich, 1996; Khisty, 1995), she would have seen that the need for the art of conversation in mathematics classrooms to be explicitly mediated has long been recognised by those working in conversation-rich multilingual mathematics classrooms. However, there are teaching dilemmas in developing learners’ mathematical communicative competence, and a dilemma as it emerged from Sue’s practice was between developing communicative competence vs. negotiating and developing meaning. Sue was aware of the need to “develop their language”. In contrast, she did not appear to be aware of how her approach, embedded as it was in her actions, was implicated in her mediation of whole-class student-student interactions and the scientific concept of generalisation. Her reluctance to work on how her learners expressed their mathematical thinking on the one hand, and to distinguish better mathematical responses by particular learners on the other, was bound up with her goal to build learners’ confidence in their own mathematical ideas. 6. IN CONCLUSION A culture of meaningful enquiry, student-student interaction and multiple perspectives on mathematics was encouraged and achieved in Sue’s classes, both while learners were working on a mathematical task and when they publicly reported on their work. The practice included talking within and about mathematics. But talking about mathematics was hard, and evidently so in a multilingual classroom where many students were learning in a language that was not their primary language. Sue knew that mathematical communicative competence was a problem but she did not know what to do about it. From a sociocultural perspective on teaching and learning, and on the basis of the empirical evidence in Sue’s classroom, I have argued that the shift from talking within to talking about mathematics required explicit mediation. While there are tensions in managing this, teachers would be better placed if their decision-making was informed by an understanding of the boundary between such practices and of their role in assisting learners to move back and forth across it. Furthermore, this vignette has illustrated that it was precisely when perspectives were not shared (and this will occur in any classroom, not only multilingual classrooms) that talking about mathematics in public or whole-class student-student communication was problematic (questions are restrictive and confusing). An effective construction zone was not spontaneously and necessarily present. The effect it had on Sue’s actions was to deflect her from the substance of the task to dealing with confusions. Refocusing was constrained by a number of factors, including her participatory-enquiry approach. The unintended consequence of this was to constrain possibilities for deepening engagement with the mathematical task at hand and hence the development of mathematical knowledge.

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The dilemma of validating diverse learner meanings vs. developing mathematical communicative competence was a profound one for Sue who worked hard to develop a classroom culture that valued and supported learners’ personal meanings and that treated knowledge in mathematics as problematic. This is a dilemma of mediation. As mentioned earlier, it has similarities with the dilemma between “eliciting vs. inculcating” identified and discussed by Jaworski (1999) in her study of teachers who had adopted an investigative approach in their mathematics classrooms. For Jaworski, the eliciting/inculcating dilemma can be explained by teachers having to work with competing epistemological paradigms (constructivist and objectivist) or culturally different pedagogies (investigative or didactic). In contrast, the spotlight on and from a multilingual setting foregrounds specific language dimensions of dilemmas of mediation. Here, mediation entails recognising and working with the boundary between talking within and about mathematics. It also entails recognising teaching intentions in relation to the development of scientific concepts and the mediating role this implies. A mediating or scaffolding role is further entailed when there is no zone for effective studentstudent interaction. Mediating roles are in tension with a desire to elicit, encourage and validate students’ conceptions. For Sue, trade-offs like deflecting to teach in context were a function of both her personal identity as a mathematics teacher (and her goals for enquiry, participation and student voice) and the contextual forces at play (in particular, her multilingual context). In the description and discussion of Sue’s lesson and her knowledge of her practice. I have identified and described issues and teaching dilemmas related to communicative competence and the development of mathematical knowledge. The implications for teaching are not new. In the past decade in particular, the ideology of constructivism in mathematics learning, its pervasive focus on learning and its wide-scale take-up by the mathematics education community have worked to position the mathematics teacher as facilitator. In its more extreme form, the constructivist teacher is present in the class setting new norms of interaction, yet non-interventionist in relation to learners’ mathematical constructions. What is suggested by the language-related dilemma of developing learners mathematical communicative competence vs. negotiating and developing meaning, is that while the withdrawal of the teacher as continual intermediary and reference point for learners enables a participatory classroom culture, teacher mediation is essential to improving the substance of communication about mathematics and the development of scientific concepts. That is, both are required, and managing the tension is the challenge! As Bartolini Bussi argues: ... no meaning can be taught directly ... On the other hand, no meaning can be a matter of negotiation, since scientific concepts are not to be created anew in school, but are to be assimilated as products of centuries of development by human kind. This situation generates a well-known paradox. Developing a better and better definition of the teacher’s role in specific situations seems to be necessary for creating new models of teaching ...(Bartolini Bussi, 1998, p. 83)

The tensions behind dilemmas of mediation are in all classrooms where learners are working on rich mathematical tasks, and where teachers are working between learners’ communication of the mathematical ideas they bring to these tasks and the

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more formalised mathematical concepts, skills and values in the curriculum. All mathematics classrooms have learners with diverse linguistic competence, even if the classroom is homogeneous in relation to learners’ main language. The spotlight on Sue’s classroom, precisely because it was multilingual, produced a gaze on linguistic diversity. As such, it cast a broad enough beam to illuminate dilemmas of mediation pertinent to all mathematics classrooms. Indeed, the spotlight on Sue’s classroom practice illuminates significant challenges for mathematics classroom practice, and so too for mathematics teacher education in South Africa, where new curriculum and language-in-education policies advocate learner-centred, communicatively rich and multilingual classroom practices. Sue worked in conditions highly conducive to educational innovation, and to a participatory-enquiry approach to teaching and learning. She had access to a wide range of material resources, small classes, time and collegial support. Indeed, much of what she was able to accomplish was tied to her extensive resource-base. She nevertheless faced teaching dilemmas, dilemmas that are inherent in the practice, and not simply a function of resources as is often argued (Adler, 1998c; 2000a). The mathematics reform movement in general, and curriculum policy in South Africa in particular, need to take far more cognisance of the complex practices proposed for better quality mathematics learning. Aside from assumptions about resources, a view from a multilingual classroom context indicates that there need to be opportunities in mathematics teacher education for teachers to focus on and engage explicitly with forms of classroom communication and their interrelationship with conceptual knowledge in the mathematics class. As argued in Chapter 6, and developed further in the concluding chapter of this book, a language of dilemmas is a source of praxis. Teachers can engage with the tensions they will and do face as they negotiate the journey from informal mathematical talk to formal written mathematics, and so too between learners’ mathematical conceptions and those recognised and valued in the curriculum. Because dilemmas are just that — means by which to capture and work on the complexity of teaching — working on mathematical communicative competence, or what I have described as explicit mathematics language teaching in multilingual classrooms, creates further dilemmas, and these are brought into focus in Chapter 8.

CHAPTER 8

THE DILEMMA OF TRANSPARENCY: LANGUAGE VISIBILITY IN THE MULTILINGUAL CLASSROOM 1. INTRODUCTION This chapter explores the benefits and contraints of explicit mathematics language teaching, or what I have described as a dilemma of transparency for teachers in multilingual secondary mathematics classrooms. In their initial interviews, Helen, Sara and Clive, all English-speaking teachers in recently deracialised suburban schools talked about the value and benefit to all learners of what I called explicit mathematics language teaching. Once their classes included students whose main language was not English, it became obvious to these teachers that they needed to be explicit about instructions for tasks, as well as mathematical terms and ideas. To their surprise, they found that explicit mathematics language teaching benefited all learners in their mathematics classes, irrespective of their language background. Explicit language teaching implies that language itself becomes the object of attention in the mathematics class and a resource in the teaching-learning process. As the study progressed Helen specifically problematised the issue of explicit language teaching. For Helen, successful mathematics learning was related to learners saying what they think in concise and precise mathematical language. She had tried to develop mathematical language teaching as part of her practice in her multilingual classroom. As she reflected on her teaching during the study, however, she became aware of instances in which her explicit language teaching, in her terms, went on “too long”. There was too much focus on what and how something was said, and the mathematics under consideration got lost. She began to question what explicit mathematics language teaching meant in practice and whether and how it actually helped. Helen’s experiences and reflections provoked questions like HOW does one pay attention to appropriate ways of speaking mathematically without conflating medium and message? and How does a mathematics teacher focus attention on the form of speech in class without losing mathematical meaning and conceptual focus? In this chapter I argue that Lave and Wenger’s (1991) idea that access to a practice requires its resources to be “transparent”, though this idea is not usually applied to language as a resource or developed in school settings, it can be useful and illuminating when applied to the use of language in schools. I will illustrate a teaching dilemma of transparency. The horns of this dilemma are that, on the one side, explicit mathematics language teaching, in which teachers attend to learners’ verbal expressions as a public resource for class teaching, appears to be a primary condition for access to mathematics, particularly for learners whose main language is not the language of instruction. On the other side, however, there is always the possibility in explicit language teaching of focusing too much on what is said and how it is said.

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Teachers’ decision-making at critical moments, while always a reflection of both their personal identity and their teaching context, requires the ability to shift focus between language per se and the mathematical problem under consideration The challenge, of course, is to judge when and how such shifts are best for whom and for what purpose. The chapter proceeds with an elaboration of Lave and Wenger’s (1991) concept of “transparency” as this provides a framework for interpreting Helen’s practical understanding of explicit language teaching, and for illuminating the dilemmas she faced in her teaching. 2. TALK AS A TRANSPARENT RESOURCE As discussed in the previous chapter, for Lave and Wenger becoming knowledgeable about a practice means becoming a full participant in the practice, which includes learning to talk in the manner of the practice. They argue further that increasing participation entails having access to a wide range of ongoing activity in the practice — to other members in the community, to information, resources and opportunities for participation. Such access hinges on the concept of transparency. The significance of artifacts in the full complexity of their relations with the practice can he more or less transparent to learners. Transparency in its simplest form may imply that the inner workings of an artifact are available for the learner’s inspection ... [T]ransparency refers to the way in which using artifacts and understanding their significance interact to become one learning process (Lave and Wenger, 1991. pp. 102103).

If an apprentice carpenter, for example, is to become a full participant in the practice of carpentry, it is not sufficient that he or she learns to use a particular cutting tool — a carpentry resource. He or she also needs to understand how and where this tool developed in the practice of carpentry, why it is used now and for what purpose. Thus, access to artifacts in the community through both their use and understanding of their significance is crucial. Artifacts (which include material tools and technologies) are often treated as given, as if their histories and significance are selfevident. Yet artifacts embody inner workings that are tied up with the history and development of the practice and that are hidden. These inner workings need to be made available. Lave and Wenger elaborate the concept of transparency as involving the dual characteristics of invisibility and visibility: ... invisibility in the form of unproblematic interpretation and integration (of the artifact) into activity, and visibility in the form of extended access to information. This is not a simple dichotomous distinction, since these two crucial characteristics are in a complex interplay (Lave and Wenger, 1991, p. 102).

Access to a practice is through its resources, and these need to be both visible and invisible. Lave and Wenger used the metaphor of a window to clarify these ideas. A window’s invisibility is what makes it a window. It is an object through which the outside world becomes visible. However, in a wall, it is simultaneously highlyvisible. In other words, that we can see through it is precisely what also makes it

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highly visible. For Lave and Wenger, the “mediating technologies” (p. 103) in a practice, like the carpentry tool, need to be visible so that they can be noticed and used, and they need to be simultaneously invisible so that attention is focused on the subject matter, the object of attention in the practice (e.g. the cupboard being made by the carpenter). Managing this duality of visibility and invisibility of resources for mathematics learning in school can create dilemmas for teachers. Student discussion of a mathematical task illuminates this duality if one understands talk as a resource in the practice of school mathematics. Discussion of a task should enable the mathematical learning and so be invisible. It is the window through which the mathematics can be seen. At the same time, the specificity of mathematical discourse inevitably enters such discussion and can require explicit attention; that is, it needs to be visible. Learners need to understand the significance of mathematical talk. These are the dual characteristics of a transparent resource. It is possible, however, that in the mathematics class the discussion itself becomes the focus and object of attention rather than a means to the mathematics. Then it obscures access to mathematics by becoming too visible itself. This possibility might well be exaggerated in multilingual situations to which learners bring a number of different main languages. Thus, practices that are more or less transparent can enable, obstruct, or even deny participation and, hence, access to the practice. While the notion of resource transparency is not usually applied to talk as a resource, nor to learning in school, it is illuminating of classroom language practices. As the vignette following will reveal, the notion of language as a transparent resource is particularly illuminating of Helen and her exploration of explicit language teaching in her multilingual secondary mathematics classroom. 3. HELEN AND HER FOCUS ON EXPLICIT LANGUAGE TEACHING Helen is white and multilingual. She speaks English, French and some Afrikaans and isiZulu. English is her main language. At the time of the study, she had 6 years experience as a secondary mathematics teacher. During the research workshops she invited the other participating teachers to struggle with her over whether or not explicit language teaching actually helps. over whether and how working on students’ competence to talk mathematics is a good thing. In the language of this chapter, she thus raised the issue of talk as a transparent resource in the mathematics classroom. That the dilemma of transparency was particularly strong for Helen was not surprising. She held a particularly strong view of “mathematics as language” and of language as a crucial resource in the practices in her classroom. Helen appeared to share Lave and Wenger’s notion that becoming knowledgeable means learning to talk mathematically. In her initial interview she said that her “greatest thrill” was when students could express themselves, their thinking, “in clear mathematical language”. She repeated this view in her reflective interview: ‘Cause if they start to describe something to me in accurate mathematical language, it does seem to reflect some kind of mastery.

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Through her reflections, and in her discussions with the other teachers during the workshops, explicit mathematics language teaching came to mean more than the teacher making mathematical and classroom discourse explicit. It included teachers encouraging and working on students’ verbalisations in the mathematics classroom. For Helen, explicit language teaching in her mathematics classroom included: (a) attention to pronunciation and clarity of instructions. When she discussed one of her videoed lessons Helen said. “One of the issues was linguistic, . . . the sound issue between ‘sides’ with an ‘s’ and ‘sizzzze’. A lot were hearing ‘size’ when I was saying ‘sides’ and we picked up on that issue.” She pointed out that the pronunciation of particular words by either or both students and teacher could be a problem in a multilingual mathematics classroom. Teachers’ instructions could be misunderstood. For Helen, clear speech and clear instructions were important and could improve clarity for all learners, not just learners whose main language was not English. (b)

student verbalisation (putting things into words) as a tool for thinking.1 Helen saw mathematical conversations as “good for your thinking”. She raised for discussion with the other teachers, her view that if students said what they were thinking, this would help them know the mathematics they were working with. Referring to Debbie, one of her students she said: Debbie, who did that very nice summary at the end of the last lesson, has got absolutely no idea at this stage. For me it seemed that if she had done this great summary the day before, that she should have been able to do that.

This assumption in Helen’s commitment to learning mathematics through talk was challenged by her reflection on her practices in her multilingual context. She took this up in her own action research, and, as will be shown later in the chapter, this came to illuminate the dilemma of transparency. (c) verbalisation of mathematical thinking as a display of mathematical knowledge, Helen articulated on numerous occasions that if students could clearly say what they were thinking, then they knew the mathematics they were working with. Her description of a student below reflects her view that ability to speak is indicative of knowing. Now listen io how clearly Rosie verbalises that, . . . and she is a successful student. There must be a relationship.

As Sfard (in Sfard et al., 1998) points out there is a problematic underlying assumption in the cognitivist argument that the ability to talk is indicative of knowing. Helen came to see the problems with this assumption in her thinking. (d) student verbalisation as a tool for teaching. Together with Helen, all the teachers agreed that at least if students said what they were thinking, this helped the teacher to know what learners were construing and so to respond appropriately. Sara summed this up in the workshop discussion, when she said “Hearing what it is students think and articulate can help you [the teacher] see what they understand”.

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In sum, Helen’s explicit language teaching was multi-faceted. She clearly regarded students’ verbalisation in the mathematics classroom as a resource. In (b) and (c) above we see talk as important for exploring and displaying mathematical knowledge (Barnes, 1976). The latter acknowledges the fact that all a teacher has access to is forms of language (Pimm, 1996). We also have language functioning as psychological tool in (b) and as a cultural tool2 for the sharing and joint construction of knowledge in (d) (Vygotsky, 1978; Mercer, 1995). Thus, although for Helen the practice of explicit language teaching entailed being explicit about mathematical discourse, explicit language teaching was bound up with her view of a strong and complex relationship between language and learning. 4. THE CONTEXT

4.1 The school and class Helen taught in a girls-only suburban state school. This school deracialised faster than many other suburban schools in Johannesburg, a function perhaps of its proximity to Alexandra, an African township on the North-East of Johannesburg. At the time of the research, fewer than 50% of the students were white. The school was well-resourced. The Grade 10 (15 – 16 years) class in which observation and videotaping were carried out was a mixed-ability class of 30 students. English, Sesotho, and isiZulu all now official languages, were some of the main languages spoken by students in this class. There were also immigrant students, one of whom had arrived in the country recently from Taiwan and spoke no English. The LoLT in the school was English, and all interaction in the public domain in Helen’s classes was in English. A range of languages was used by learners during group work and in social talk with other learners. 4.2 Helen’s approach Helen’s classes, although largely teacher directed, were also interactive and task based. Group-based tasks were followed by whole-class, teacher-directed reaction to reports students gave. Helen’s approach included situations in which learners talked with each other during their interaction on tasks, reported verbally on these tasks and interactions, and engaged with Helen in public verbal interactions. It was during these public interactions that Helen paid explicit attention to mathematical language. Helen’s approach and resulting classroom culture that included learner-learner discussion and verbalisation was not surprising in the light of her views of mathematics as language as well as her concern that mathematics should be contextualised and learning should be meaningful and lasting. Moreover, her approach, like Sue’s, reflected a significant shift away from the ‘drill and practice’ model dominant in South African mathematics classrooms. Helen also held strong views on access to mathematics for both girls and the racially disadvantaged in

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South Africa It is thus important to note here that Helen engaged with the issues of code-switching, particularly by her learners, and of effective mediation. Her overarching concern, however, and thus the focus of this chapter, was whether or not explicit mathematical language teaching actually does help students — whether it makes mathematics more accessible. Helen introduced trigonometry to one of her Grade 10 classes with an outdoor activity investigating shadow length caused by the sun at different times of the day. This activity was followed by a classroom-based task in which groups of learners measured and compared the ratios of the lengths of sides of a right-angled triangle having one angle of 40 degrees. Working in class later when groups presented their reports of what they had found out, Helen attempted to develop their understanding of constant ratios and related these ratios to the programming of trigonometric ratios into a calculator. While Helen argued for the value of explicit language teaching (Chapter 5), like Sue she was concerned with the actual benefits of learner-learner discussion. For Helen this included worries and dilemmas around exploratory learning and both the performance and mathematical knowledge-growth of her learners. In her reflective interview she said: Helen: I am not hopeful. I had some of this group — many — last year, and we spent ages measuring circle circumference etc and then this year I asked “what’s pi — give me some idea of where it comes from? ... meaning?” and not a clue. How about area of a triangle? Nothing ... and this was a group that I did extensive work with, and for nothing! So what I am saying is that why didn’t I just get them to learn the formula for the area of a circle in Std 7 [Gr 9] and then you expect them to have forgotten it instead of having spent three double periods on measuring the circle etc. That is the dilemma. They don’t remember the string either. You are exhausted at the end of a day of this. And “What for?” you sometimes ask? JA:

Could you teach in another way?

Helen: Nnnooo ... I have made a lot of the changes academically for myself, and then it has resulted in this kind of thing. Then I become more critical of what is going on and it helped me make decisions I wanted to make but didn’t have the skills to do so, or awareness or whatever to do it. JA:

But why did you want to do it?

Helen: I think it is a general ideological framework that I come from Um ... I am very authoritarian and strict in the general school life But I like, I am comfortable with the idea of respect for the individual and that kind of thing. And the other kind of teaching doesn’t do that. I have taught with teachers who teach in other ways and I am horrifed at how they talk of pupils — so it is very much an ideological thing. JA:

Which then is difficult to bring into successful maths teaching?

Helen: Like this year in my matrics — I’ve never punished. But they got something wrong in the prelims and five hundred times they had to write it out and I felt nothing. It is an external exam. It doesn’t fit with my ideas of what maths is. They are going to have to learn if they want to pass. I don’t feel comfortable with that because I don’t see it as respected On the other hand you treat a bunch like this with full respect and you just

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get trampled on. It works at a lot of different levels. But my teaching is always changing and hopefully always improving.

This extract from her reflective interview shows that Helen struggled to take on what she believed were more politically and educationally progressive pedagogical approaches, approaches that were respectful of learners and that brought more language and exploratory activity into her classroom. She did this within her view of mathematics as language and within the constraints of a content-driven examination system forcibly at play in the senior secondary school. Her struggle was captured in her closing comments in the interview: Helen: It was strange to watch. Some things I liked what I was seeing because I knew that I was explicitly trying to do those things but didn’t know that I was actually managing to get them done ... and also seeing the extent to which they worked in groups () like seeing the whole class doing that with me as part of it was also quite interesting. It helped in that it made me more critical of some of things I do, and one of the times when I was watching it more carefully I, picked up inaccuracies in my questions, or not finishing a question. ... ... It was interesting, very interesting to watch it, to see it all happening — like a sociological interest — this little hour in the middle in the world of this difficult class and this person who thought too much about her teaching — it’s just interesting.

In the first workshop with the other teachers in the study, Helen expressed her firm commitment to explicit language teaching. After she had observed and reflected on her video, she asked the other teachers to help her grapple with whether “saying it” actually is indicative of understanding, of knowing. Helen then followed up her question with her own action research. She planned a double lesson (one hour) on trigonometry for the same class who were then in Grade 11. She organised the lesson around group discussion of a set of tasks, audio-taped the discussions of two of the student groups, and invited me to observe and videotape the lesson. She wanted to listen carefully to how learners engaged in discussion on mathematical tasks, and to reflect more systematically on her assumptions about a strong relationship between language and learning and about the values of explicit mathematics language teaching. After Helen had observed the videotape and listened to the tape-recordings of the student groups, she brought her reflections from this action research to the second workshop with the other teachers in the research study. Episodes in the videotape of Helen’s action research lesson, her reflections on the lesson, together with some of my own commentary are woven into an analytic narrative vignette (Erickson, 1986). The vignette provides insight first into how Helen coped in practice with learners’ meanings and with their mathematical expression, and second, into reflections on her practice. In so doing, it illuminates the dilemma of transparency. The episode and reflections in the vignette are neither typical nor rare. Rather they are instances that illustrate and create a space for opening dialogue on an important element of teachers’ knowledge of their practices in multilingual classrooms — an element quite apparent in newly deracialised schools in South Africa.

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5. AVIGNETTE —CLASS ROOM EPISODES

5.1 Focus on mathematical discourse The episode below took place in the first trigonometry lesson of Grade 11 and was part of Helen’s action research in the year following the initial interviews and videotaping of her teaching trigonometry to her Grade 10 class. In this lesson Helen asked learners in groups of four to discuss what trigonometry meant to them, and then to report back their meanings to the rest of the class in a ‘‘maximum of 2 minutes per group . . . using key words and putting across your main ideas.” Seven minutes into the lesson, the first group reported, explaining that “trigonometry is used to determine the size and sides of the angles”, and that there are six ratios. The transcription key follows that used in Chapters 6 and 7, but I have emphasised, through bold type, utterances that will be referred in the discussion following. Emphasised utterances by Helen are in bold caps. EPISODE 1 GRP 1: OK, um, ... our group said that algebra is the mixture of geometry ... um No, it said trigonometry is the mixture of algebra and geometry. And () trigonometry is used to determine the size and sides of the angles. Therefore it also has six ratios in trigonometry. For instance, you have your triangle [she draws triangle below], the ratios, like you have your hypotenuse, if you have an angle, this is the adjacent, this is the opposite.

Figure 6. Group 1’s triangle

So six ratios are sin, which has a reciprocal of cosec, and your cos, which has a reciprocal of sec, and your tan, which has a reciprocal cot. So therefore () uh we said the ratio of two angles is independent to the size of the angle in the other two triangles That means if you have two triangles, there you have a small triangle and a bigger triangle [and she draws]

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Figure 7: Group 1’s second drawing. it doesn’t matter of the size of the of the two triangles, whether its small or big. But then you,find that the two angles are the same. So they work hand in hand.

Groups 2, 3 and 5’s explanations of trigonometry also focused on the six ratios, and how these, “with Pythagoras’ theorem”, can be used to determine the sides and angles of triangles, especially right-angled triangles. Specifically, trigonometric ratios are used to solve problems related to heights and distances. Group 4’s articulation of the meaning of “trigonometry” was in similar language to Group 1’s: Grp 4: Okay uh () what you were talking about is actually is what is actually true. You said that trigonometry is the measurement of triangles. It can also be the measurement of right angled triangles. Say, for example, you got uh() two triangles right angle triangles [and she draws them] of different sizes.

Figure 8: Group 4’s first drawing. As long as your theta, which is the angle, is equal () the size, the side uh, uh, the ratio of each angle will be equal. Therefore, we came to the same thing that the ratio of two sides is independent to the size of the tri. of the angle in two triangles. So this angle will equal to that angle because of this theta there, and ... uh, if you got a triangle like this [draws]

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Figure 9. Group 4's second drawing and you're asked to find or say what sin, cot and tan is, you can use by using the hypotenuse which is the angle opposite, which is, um, the side opposite your right angle, and your adjacent and your opposite. And there will be your theta which is the angle you must find So trigonometry is actually the measurement of angles, it can also be the measurement of right angle triangles.

At the end of all the presentations, some 14 minutes into the lesson, Helen, whose only role (and talk) during this time was to select the order of group presentations, moved to the front of the class and said: EPISODE 2 Helen: I want to pick up on () some words that you’ve been using () um, that concerned me. ... Right [writes 'size', 'side' on board]. What is the difference between those two words? () Just look at the words. Ss:

Size // side [mumblings in unison of both words].

Helen:

One can be measured? ... Penny say those two words for me.

Penny:

[reads] Size, Side.

Helen: [slowly] Size, side. When you speak, are you clear of the difference between the two? Ss:

Yes

Helen:

Are you sure?

Ss:

Yes //No [both said in unison]

Helen: Then how come you say the side, the size of the, the sizes of the triangles, you said to me the side of the angles and you said all sorts of funny things? Ss:

Sides.

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Helen: Who can distinguish for me very carefully, the difference between sizzzze [she emphasises ‘zze’] and siddde [emphasises ‘dde’]? ‘Cause, when I say sides, which word do you hear, the first or the second? Ss:

Size// sides// first// second

Helen:

Are you hearing the first?

S1:

I am hearing the second.

Ss:

Second, (laugh]

Helen: Have I made my point? OK, when I say this [points to ‘side ] to you, some of you hear it as [she draws a circle around ‘size ] Ss:

Size.

Helen then repeated the importance of being clear in what is being said and referred to, and towards the end of the lesson she suggested to students that it is more accurate to refer to the “length” of sides, rather than the “size” of sides, as sides could have thickness. Helen then raised the next expression that she was concerned about, those expressions from Groups 1 and 4 in bold above. She stated that it was still not clear to her whether they “understand what trigonometry is saying”. She repeated that most reports talked of two triangles with the same angle in it. But, she asks, where do we go from there? Offerings were made repeating the expressions that she was concerned about: EPISODE 3 H:

Say that to me slowly, the

S6: [H writes as student talks] The ratios of the two sides () is independent to the size of the angles () in the two triangles ... H:

Is independent to ...?

S6:

The two tri.., is independent, no, the two sides is independent ...

H:

The ratio of the two sides is independent to?

S6:

The size ofthe angles in the two triangles [and H finishes writing].

H: Let‘s look at that statement carefully I need some distance. [And she moves back from the board, and then says slowly] The ratios of the two sides is independent () to the size of the angle () in the two triangles. What does that statement mean to, uh, to anyone?

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Now listen to what you said: how long or short the triangles are?

S6:

The length, the length of the triangle.

H:

Triangle is a shape.

Ss:

[mumblings] The length of the sides.

H: The length of the sides of the triangle. OK. You know. Let’s just look at this word “independent”. OK. Now I know when I teach this, I use the word independent and then you think, well that’s a nice fancy word to use. If I just repeat it nicely in the right sentence then she’ll be very impressed. But, when you use the word independent you‘ve got to know what it means. What does it mean? Phindiwe? Phin: [some mumbling] It stands on its own.

After distinguishing “length of sides” from “sides of triangle” Helen pulled the word independent out on its own, and attended to its meaning. She then returned to focus on the sentence in which it was placed: The ratios of the two sides is independent () to the size of the angle () in the two triangles. Helen:

OK. All right. Is that statement true?

Ss:

No//Yes. ()

Helen:

Must I put a true or a false at the end of it?

Ss:

True//false

Helen:

OK. Who says it’s true?

S6:

[puts her hand up]

Helen:

S6 says its true ‘cause she said it.

Ss:

[laugh]

Helen:

OK, who says its false?

Ss:

[laugh]

Helen:

What do you think?

Phin:

I don’t know, I don’t understand the sentence.

Helen: OK, let’s try and sort out the sentence. The ratios of two sides, that’s a true part of the line, uh, of the sentence. Does that make sense?

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Yes

Helen: OK. Ratios of two sides, we know we always talk about opposite to hypotenuse, or adjacent to opposite or something () we are talking about a ratio and we are talking about two sides. () Is independent. OK. Wait. The most important word in the sentence is independent? Right. So one thing is independent of another. So maybe if I just change this to of () we can start. So the ratio is independent from what? Size of the angle in the two triangles? () It's true, who says it's true? Why? S7: Because, mam, um, I think it means that, no, uh, if if you, if you have, uh, one big triangle and you have one small triangle and you have the same angle in both of them, uh, the the size of the angles is equal, then the ratio of the, of the sides won't change. Helen: Now listen to what you're saying. You're saying you've got (), you said to me [and H links the bold words below to related words on the board as she speaks] you've got the size of two triangles and then you said that the angle inside them is the same, OK. So if we want to, is what she said different to what is on the board at the moment. Ss:

No//yes ()

Helen: She said to me the ratio of the two sides is independent of the SIZE of the triangle, WHEN you’ve got the same angle in all of them. So it’s NOT true to say that the ratios are independent of the size of the ANGLE. The size of the angle is EXACTLY what makes the FUNDAMENTAL DIFFERENCE. Because if I’ve got two triangles, these two beautiful triangles over here, 40, 40 [and she fills in 40 degrees into two similar triangles on the board], and these two over here, 20, 20 [and again fills in these angle sizes onto another set of similar triangles on the board]. () Would I get if I say spoke about () sin here and sin here? OK? Will I get the same answer?

Figure10. Helen draws two right-angled triangles with an angle of 40°, and two with an angle of 20° Ss:

No

Helen: No! I'll get two different answers. So it is not true to say to me it is independent of the size of the angle — because the angle if it is 40, makes the difference to 20, right. It's the size of the TRIANGLE that makes the difference. () Does that make sense to you? Ss:

No

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What doesn’t make sense?

S2:

Mam?

Helen:

Ja

S2:

It makes a difference to what?

Helen:

It makes a difference ... to ... [clearly unsettled]

Ss:

[laugh]

Helen:

Where was I starting off? ... um, let me start again...

Helen then recapped by drawing attention to diagrams on the board, to how two different right-angled triangles each with 40 degree angles will have the same ratios between their sides, as will two different right-angled triangles each with 20 degree angles. But the two sets of ratios will be different precisely because the angles across the triangle pairs are different. And then she asked the student who first articulated the sentence to tell her what she understood in her own words. 5.2 My observations My observational notes taken during this lesson reflected my strong sense that despite problematic verbalisation by some students, there was a sense of shared meaning that ratios were constant in similar right-angled triangles. I was also confronted, for the first time, with the issue of pronunciation. I was struck by Helen’s insight into possible confusion between “size” and “side” in Episode 2. In the literature on language and mathematics there is little focus on difficulties that might arise through different pronunciations. I suppose this reflects the assumption that pronunciation is attended to and further, that in use, sense is made of differing pronunciations and hence that it is not in and of itself a problem. Helen’s practice challenged my taken-for-granted assumptions. I became aware that different mathematical terms, or different words used in mathematical discourse that nevertheless sound similar, could cause confusion particularly in a classroom where learner talk was encouraged. Moreover, the particular pronunciation of some English words by learners whose main language is an African language could cause confusion. For example, observing a student teacher a few months later, I noticed some students had written “the sights of the triangle”. This is perhaps an effect of the stress in some African languages being on the first or middle syllable of a word, and a dropping of emphasis at the end of the word. While sight, size and side are all monosyllabic, with emphasis at the beginning of a word, they all come to sound roughly the same. Similarly, white and wide, forty and fourteen, multiple and multiply all create possibilities for confusion. In this context, that Helen explicitly distinguished the spoken words “dependent” and “independent” was appropriate. I noted further that Helen worked explicitly in this lesson with both pronunciation and correct mathematical expression of meanings of trigonometry. I

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was aware too that as part of her own research, her attention to language issues was probably heightened, bringing more emphasis on explicit language teaching in her practice here than might have been otherwise. Both in this lesson, and those videoed and observed the year previously, Helen was notably more directive than Sue during report-back time. While Sue invited discussion from the class after each report, Helen had all groups report and then she directed whole class teacher-student interaction on what had been presented. She focused attention on problems and reformulated and recapped where necessary. It is interesting to note that recent research from bilingual settings in the USA have pointed to the importance of strategies like revoicing learner expressions (Msochkovich, 1999), and whole class teaching so that learners can be inducted via the teacher into scientific discourse (Fradd and Lee, 1999). It was in this whole class reporting back part of the lesson that Helen’s explicit language teaching was evident. On reflection on my own observations and with the hindsight brought by the study as a whole. I am not sure in this instance that “sides” and “size” actually caused any confusion of meaning in students. Strictly speaking in mathematics, we do talk of the side opposite or adjacent to an angle, but we do not refer to “sides of angles”. What became more and more interesting for me in the context of the study and Helen’s views and concerns was her intentional shifting of focus in the classroom onto the mathematical language at play. She was working hard to reduce her authoritarianism and control, to allow and encourage learners’ voices and meanings, and to provide opportunity for language as a social mode of thinking. And she did this while making explicit what counted as mathematical language, for example, being precise. In Episode 3 in particular Helen asked what the statement that included “independent to the size of the angles” meant, inviting rethinking, and further elaboration. She tried to engage students in making sense of the statement. When S7 expressed a clear explanation she focused on this, reformulated, and asked the class to compare the two versions. She assisted by recapping and stressing that the “angle makes the fundamental difference” only to find that the focus of the mathematical discussion was lost on the students. And so she reformulated and recapped again, by which time, in her view, she had “gone on too long”. Helen’s practice came to include periodic focusing of her and her students’ attention onto how to speak mathematics and on possible language/pronunciation confusions. This created what I have called the dilemma of transparency, of language as a resource in the classroom carrying the dual characteristics of visibility and invisibility. As a result, Helen faced a new set of challenges. 5.3 Helen’s Refections Opening the second workshop, and before showing some of the extract above to the other teachers, she said: One of the issues was linguistic ... the sound issue between sides with an s and size. A lot were hearing size when I was saying sides and we picked up on that issue and then

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CHAPTER 8 (on what trig is) on the ratio being the same for a given angle in different sized triangles. When Jill and I discussed this we talked about the one part where a child put forward what she thinks is going on in relation to that issue and it is a question of even though her language is not clear is there understanding amongst the rest of the students and it seems like the rest of the students do understand even though she is using incorrect language. So we can watch and think around that.

She then played the video from the point where the student said: the ratio of the two sides is independent to the size of the angles in the two triangles and Helen was writing what was being said word for word on the board for the class to think about. She continued her reflection: Just after the sentence is written on the board and I ask: “What do you understand by this statement?” The one child puts forward a perfect explanation. She talks about the angle being the same in both triangles and then she talks about the depth of the triangles or whatever and Ipick up on that ... and then this child [getting to the place on the video where a second pupil is responding] now does it absolutely perfectly. So, that is two very good expressions of what is going on. And yet when you ask the class: “Is this sentence correct?” [Pointing to the sentence she has written verbatim from the first student on the board], there is this complete silence. So the question for me is: even in the minds of those two children who put forward such consistent explanations, what’s going on with them? () that they cannot ... um ... pick up incorrectness in the sentence?

And she revisited her question in the first workshop with the other teachers: if they can say it do they know it? and moreover the recognition that mathematical expression is often very difficult ... I think that that sentence came out of something that the group was working with ... if you actually take a sentence like that which is supposed to be concise, and it carries a whole lot of meaning there is difficulty ... They can talk to you about it and they can give you a long explanation of what to do ... so its seems to me to be also a problem of expressing a lot of maths in one clear sentence. For me that is also linked to the issue of how we transmit maths to each other. If you make a mathematical statement you are involved in getting it down to a simple, short-hand language that we can all share .. ,

Finally, she posed a central question on verbalisation and the dilemma of transparency: ... in retrospect, when I look at that lesson, I went on but much too long [laughter] on and on and on andI keep saying the same thing and I repeat myself; on and on .. and I watch the video and I think I wonder why they are still sitting in their seats and I am falling on the floor falling asleep. But the thing is then if you have a sense that there is a shared meaning amongst the group can you go with it? um ... when the sentence is completely wrong? ... Can you let it go? Can a teacher use a sense of shared meaning to move on? I think this is a central question in terms of the verbalisation and discussion.

Helen also remarked on, and remembered that, in her attempt to teach mathematical language explicitly, the mathematical focus of the lesson was lost. She remembered being thrown by a student’s interjection: “It makes a difference to what?”. Helen’s working assumptions of a strong relationshp between language and thought were seriously challenged as she experienced and observed: students expressing their thinking on one day then not the next; students expressing clear and correct mathematical thinking but not being able to discern problematic expression in/of others; and students saying things ‘wrong’ but creating a sense that they have

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some grasp of the mathematics in play. She also saw how in her focus on language teaching, in her attention to their use of “dependent” and “independent”, the students lost their focus on the mathematical and trigonometric problem from which this use arose.

6. DILEMMAS IN EXPLICIT MATHEMATICS LANGUAGE TEACHING Helen’s action research revealed the tensions in whole class interaction as attention was focused both on pronunciation and on students’ mathematical verbalisations. It highlights the dilemmas that explicit mathematics language teaching creates for teachers. Episodes in Helen’s class and her reflections highlight pronunciation in a multilingual mathematics class and the role it might play in meaning-making. She also demonstrates what is well known, that some mathematics is difficult for students to say precisely and with meaning, and that Helen on a number of occasions reformulated. The specific dilemma for Helen was in moving back and forth between language used for thinking and language used as a display of knowledge, between talking within and about mathematics, or what Mercer distinguished as educational and educated discourses. Educational discourse is the discourse of teaching and learning in the classroom (e.g. ways of asking and answering questions in class). Educated discourse is new ways of using language (e.g. in algebra “let x be any number”), “ways with words” that would enable students to become active members of wider communities that use this educated discourse (Mercer: 1995, p. 82). In Mercer’s terms, educated discourse in school mathematics will include the mathematics register (Halliday, 1978). Mercer argues that there can be a mismatch between the educational discourse in use (the ways in which words are being used in the classroom) and the educated discourse learners are meant to be acquiring. In relation to mathematical discourse, the teacher’s role then is to translate what is being said into mathematical discourse, to help frame discussion, pose questions, suggest real life connections, probe arguments and ask for evidence. The language practices of the classroom (educational discourse) must scaffold students’ entry into mathematical (educated) discourse: [Teachers] have to use educational discourse to organise, energise and maintain a local mini-community of educated discourse. We can think of each teacher as a discourse guide and each classroom as a discourse village, a small language outpost from which roads lead to larger communities of educated discourse. ... Teachers are expected to help their students develop ways of talking, writing and thinking which will enable them to travel on wider intellectual journeys ... : but they have to start from where learners are, ... and help them go back and forth across the bridge from “everyday” discourse into “educated discourse”. (Mercer, 1995, p. 83-84)

In the previous chapter I argued that teaching and learning mathematics entails this moving back and forth, and that Sue, for example, needed to be more explicit in this regard. In contrast to Sue, Helen did work explicitly on students talking about mathematics. She provided opportunity for students, amongst themselves, to elaborate and then share their meanings of “trigonometry”. This elicitation of students’ thinking suggested to her that there was confusion and she moved to

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clarify this through a particular scaffolding process where she questioned, brought into focus the incorrect use of the concept and term “independent”. and finally reformulated and recapped emphasising what she saw as most significant in the descriptions of trigonometry that emerged from the students. But this explicit language teaching was a struggle for her and her learners. Helen’s knowledge helps us identify a fundamental pedagogic tension between implicit and explicit practices with respect to language issues in her multilingual mathematics class. She harnessed language as a resource in her classroom. As a resource in the practice, its transparency, i.e. its enabling use by learners, was related to both its visibility and invisibility. Specifically, Helen attended to students’ expression as a shared public resource for class teaching. This is a characteristic of classrooms that is not shared by many other speech settings (Pimm, 1996). The language itself becomes visible and the explicit focus of attention. It is no longer the medium of expression, but the message itself — that to which the students now attend. Episode 3 shows how Helen struggled to mediate the scientific concepts of constant ratios and “dependence” and “independence” as they arise in school trigonometry. She did this in her multilingual classroom where the complex three dimensional dynamic intersected with her educational and political beliefs as well as her view of mathematics as language. Helen focused on pronunciation and correct ways of speaking mathematically, thus attempting to provide access to English and to mathematical discourse. However, these attempts occurred within her classroom culture where language was used simultaneously to explore and display mathematical knowledge. And problems emerged. On reflection, Helen felt that her attempt to enable access to mathematical (educated) discourse brought the problem of “going on too long”. In explicitly making mathematical language visible, it became opaque, obscuring the mathematical problem. The dilemma of transparency arose: of whether (and when) to make mathematical language explicit or leave it more implicit. Again there were both political and educational dimensions to this dilemma for Helen. If she “goes on too long”, she diminishes students’ opportunities to use educational discourse and inadvertently obscures the mathematics at play. If she leaves too much implicit then she runs the risk of losing or alienating those who most need opportunity for access to educated discourse. Helen wondered about the possible effects of leaving in play a shared sense of trigonometric ratios but a public display of incorrect mathematical language: “If they don’t say it right, can I let it go?” Of course, there is a world of difference between “what they are saying is wrong” and “I can‘t get at what they are trying to say to me” (Pimm, 1996). This is the difference Sue tried to get at in the third workshop when the teachers were discussing how a student can talk about sin 40 on one day but not sin theta the next. Sue said she would ask herself what it was she (the teacher) was not understanding and she would try to ask the student another question, something that would help Sue understand what the student was trying to say. Helen was working more explicitly on the form of what the students were saying, and that that form was wrong. Helen provides insight into a teacher with a strong and particular set of educational and political beliefs about language and learning, who was struggling to

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understand the potential benefits for students of more interactive practices and hence more language in the mathematics classroom. Underpinned by her view of mathematics as a language, and her deep concern with the politics of access in South Africa, Helen moved between taking “size” and “side”, and “dependent” and “independent” out for scrutiny — making them visible. She then re-inserted them in their particular mathematical use in the lesson. But this was a struggle, and deeply bound with using language (attending to students’ language expression) as a shared public resource. If the mathematical problem was to be addressed, the language itself needed to be invisible. It could not remain the focus of conscious attention. This struggle over visibility and invisibility was intertwined with using language both to explore mathematics and as a display of knowledge, with developing both educational and educated discourse. 7. IN CONCLUSION Helen’s experience and her reflections on it shows that explicit mathematics language teaching, although beneficial, is not necessarily always appropriate. This kind of explicit teaching can result in a language-related dilemma of transparency with its dual characteristics of visibility and invisibility. Helen’s particular questions and reflections, and the discussion they provoked in the workshops, highlight tensions teachers can experience as they try to initiate new and different forms of instruction. Lave and Wenger’s notion of transparency illuminates classroom processes. Transparency involves both visibility and invisibility. Both visibility and invisibility are part of transparency in the practice of teaching mathematics. Resources need to be seen to be used. They also need to be invisible to illuminate aspects of practice. For talk to be a resource for mathematics learning it needs to be transparent; learners must be able to see it and use it. They must be able to focus on language per se when necessary, but they must also be able to render it invisible while using it as a means for building mathematical knowledge. Helen’s particular questions, reflections and the discussion they provoked in the workshops reflect the real worries of teachers as they try to mount something new and different in their circumstances which particularly at the more senior levels, include passing external examinations in a wider context where traditional teaching practice remains dominant. Helen reveals the value of explicit mathematics language teaching some of the time. This is in relation to both pronunciation and to the dilemma of transparency in practice where attempts to make mathematical language visible can actually obscure. For school mathematics teachers, it is not simply a matter of going on too long but of managing and mediating the shift of focus between mathematical language and the mathematical problem (which of course are intertwined). Together with Chapter 7, this chapter reveals the fundamental tension between implicit and explicit practices with respect to language issues in multilingual mathematics classrooms. As was argued in Chapter 7, these issues are present in all classrooms, but are present in particularly heightened form in multilingual

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classrooms. Helen attended to students’ mathematical language expression as a shared public resource for class teaching. She also saw it as providing a display of mathematical knowledge. As Pimm (1996) points out, the latter attention acknowledges that all I, as teacher, have access to, is the forms. And Helen posed these questions: “If they can say it ‘right’, do they know it?”, and: “If they don’t say it ‘right’, can I let it go?’ As in Chapters 6 and 7, there is no simple answer here, rather, an instance of a teacher grappling with the issue as she tries to embrace new practices and make mathematical knowledge available in her particular multilingual classroom. An important question for mathematics teacher education if it is to address the three dimensional dynamic of a multilingual classroom is: what significance do teachers attach to students use of a “wrong” word or expression? Opportunities to grapple with implicit and explicit practices related to language issues like those highlighted by Helen will enable mathematics teachers’ informed decision-making in moments of their practice in their multilingual classrooms.

CHAPTER 9

CENTRAL DILEMMAS AS CURRICULUM AND RESEARCH AGENDA The notion of a teaching dilemma constitutes the key to capturing and opening up teachers’ knowledge of the elusive, complex and dialectical nature of teaching and learning mathematics in multilingual classrooms. The stories told in the chapters of this book show how teaching dilemmas are at once explanatory tools and analytic devices for teaching. They make explicit the tensions inherent in teaching. At the same time a language of dilemmas can function as a source of praxis. Teachers can use a language of dilemmas to reflect on and transform their practices so as to meet the mathematical needs of their linguistically diverse learners. The notion of a teaching dilemma did not originate with me. Berlak and Berlak found in the notion of dilemmas, the means for capturing complex teaching practice. As they struggled to organise their observations, it became clear that there were no simple dichotomous ways of capturing the teaching they observed nor how teachers themselves understood their work. Teachers behaved in complex and often contradictory ways, and their understandings of their actions and teaching exhibited similar complexities and contradictions. Possible descriptions and explanations did not lie within teachers, nor their classrooms, nor the wider context, but rather in their inter-relations. A language of dilemmas became a means for capturing the complex relationship between the context of schooling and the dilemmas of teaching. Lampert came to the notion of teaching dilemmas through her concern with the schism between educational theory and practice, and her sense that in complex moments of practice, no amount of theory can guide decision-making. Such direction came from the personal goals of the teacher and the practical setting of teaching. The notion of a teaching dilemma developed in this book moves beyond the polarisation between Berlaks’ contextual emphasis in their dialectical account of dilemmas in schooling, and Lampert’s emphasis on the personal and the practical in managing dilemmas. Teaching dilemmas are at once personal, practical and contextual. Teaching dilemmas, and the language of practice that is produced when describing dilemmas, are also sources of praxis, what Jaworski described as possibilities for knowledge growth about teaching. These empirically derived observations can be explained by a social theory of mind which effectively combines the teacher’s context and his or her biography, at the same time as recognising a specificity to school subject knowledge. As noted in Chapter 3, sociocultural and social practice theory have indeed been “deeply infused” throughout the book. From this perspective, learning and so knowledgeability, be it about mathematics, or about teaching, is constituted by and constitutive of persons acting in the world. Knowing is deeply intertwined with being. Knowledgeability is not some decontextualised static set of cognitive tools but rather how these fashion and are fashioned by identities, which in turn shape and are shaped by social practices and so the context in which learning takes place. It is

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this knowing as tied to being which also explains displays of knowledge as shaped by identity and practice. Accessing teachers’ tacit and articulated knowledge of their practice was a first step to opening it up. Interpreting this knowledge through a language of dilemmas has provided an account which demonstrates this kind of knowledgeability. But social practice theory as developed by Lave and Wenger required moulding and shaping to accommodate the specificity of knowing about the Beaming and teaching mathematics in school. Vygotsky’s sociocultural theory, with his attention to school as a leading activity in learning and development, together with Mercer’s elaboration of talk as a social thinking tool and of the teacher’s role in working across the boundary between educational and educated discourses in school classrooms, enabled interpretation and description of the dilemmas of mediation and transparency. The theoretical contribution of this book thus lies in broadening the conception of teaching dilemmas and in extending the language of dilemmas to capture the specificity of the multilingual mathematics classroom. Three key dilemmas emerged in this study. The dilemma of code-switching, and its extension to include the dilemma of whether or not to model mathematical English, simultaneously captures, categorises and allows for explanation of the teacher’s actions and reflections in a situation where she and most, but not necessarily all, her learners share a main language that is not the LoLT. Here decisions in the classroom often revolve around the tension between developing learners’ English vs. harnessing learners’ main language(s) as a resource to ensure learners‘ understanding of the mathematics; and around whether the tacit practice of modelling mathematical English is in effect, the teacher “talking too much”. In a context of curriculum reform where teacher talk is devalued, this is a significant challenge for mathematics education. Furthermore, the extension of the dilemma of code-switching beyond the urban secondary classroom indicated that while the dilemma persists across primary and non-urban mathematics classrooms, it takes on new and more acute forms, thus supporting the notion of teachers’ knowledge, and so too teaching dilemmas, as situated. Over time, the dilemma of code-switching was a powerful source of praxis for Thandi. Thandi interviewed her learners to find out their views of group work, and specifically their use of their main languages in group discussions. Through her action research and her use of the notion of a teaching dilemma, Thandi was able to identify and then transcend her dilemma. She also embraced the need not only for code-switching but for change in her mathematical activities. Together with her reflections on her video, Thandi’s action research is a convincing illustration of dilemmas as both analytic devices and sources of praxis. The dilemma of mediation enables analysis of action and reflection in learnercentred classrooms where a participatory-enquiry approach has effectively been developed. Here, emphases and decisions in a multilingual classroom revolve around validating students meanings vs. developing mathematical communicative competence, and around the shifts between talking within vs. talking about mathematics. These are specific language-related dilemmas of mediation. The dilemma of mediation also entails recognising teaching intentions in relation to the

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development of scientific concepts and the mediating role this implies. A mediating or scaffolding role is further required when there is no effective construction zone for student-student interaction. These mediating roles for the teacher are in tension with a desire to elicit, encourage and validate students’ conceptions. This range of dilemmas of mediation emerged for Sue precisely because hers was a context of changing pedagogy and a multilingual context. Sue was always aware that in her classroom not all learners were proficient in English. Sue’s multilingual, participatory-enquiry classroom became a wide angle lens that could foreground language-specific dilemmas while keeping more general dilemmas of mediation continually in focus. The dilemma of transparency (of the visibility vs. invisibility of language in the process of mathematical classroom activity) was highlighted in classrooms where the racial composition of the student body changed rapidly and so too its multilingual character. Teachers in these classrooms advocated the benefits of explicit mathematical language teaching, not only for learners whose main language was not the LoLT. Yet episodes in Helen’s classroom revealed that attempts to harness language as a public resource by explicitly focusing on its form, can inadvertently obscure rather than provide smooth entry into mathematics. It is not simply a matter of “going on too long” or talking too much but of managing to shift focus between mathematical language and the mathematical problem at play, of managing both implicit and explicit practices (and of course these are intertwined). Together, the vignettes in Sue’s and Helen’s classrooms reveal the fundamental tension between implicit and explicit practices with respect to language issues in multilingual mathematics classrooms. As has been argued these issues are present in all classrooms, but are present in particularly heightened form in multilingual classrooms. Research on conversation and communication in the mathematics classroom that ignores research in multilingual settings is in danger of silencing linguistic diversity in their accounts of practice when linguistic diversity is clearly a significant dimension of classroom practice. Each of the three key dilemmas and their extended forms were constantly analysed within a conception of a multilingual mathematics classroom as dynamic and three-dimensional. For example, dilemmas around access to English emerged in practice in an ongoing and dynamic relationship with access to mathematical discourse as well as access to classroom cultural processes. In the logic of practice, it is sometimes difficult for teachers to see how classroom cultural processes, such as a participatory-enquiry approach, factor into classroom interactions and the quality of mathematical activity and mathematical meaning-making. Moreover, through three distinct vignettes on how teachers in multilingual mathematics classrooms faced and worked with dilemmas in their teaching, this study has shown that the dilemmas emphasised by particular teachers are a function of both their contexts and their biography, of the social and the political, the personal and the practical. For Thandi, the dilemma of code-switching was tied up with her own ability to switch, her political aspirations for her students and her view of mathematics. Sue’s strong conviction was to participatory and meaningful learning, and for Helen mathematical expression was a sign of mathematical competence and skill. Hence Helen’s emphasis and then doubts about explicit

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mathematics language teaching. and Sue’s concerns about meaning negotiation through student-student discussion, and about how to “develop their language”. Managing dilemmas also intersected with broader social and political factors such as the hegemony of English as the language of power in South Africa and the rhetoric of democratic educational practice that wove through progressive educational circles from the late 1980s. The implications of the findings in this study for teacher education are profound yet simple. A programme in teacher education would be enriched by documented cases that highlight dilemmas teachers face. the kinds of snapshots provided in the vignettes in this study. A language of diIemmas can assist teachers to recognise, talk about and act on the tensions in their practice Such engagement could empower teachers to make informed and contextually appropriate decisions to maximise learning possibilities for their diverse learners. Teaching dilemmas, despite their expression as binary opposites, are useful in teacher education precisely because they highlight fundamental pedagogic tensions that cannot be resolved once and for all. They are potential sources of praxis. They can be managed, and awareness of them can lead to transformatory practice. The teachers in this study reveal that there is no simple route to effective high quality mathematics teaching and learning in complex multilingual classrooms. Learning about teaching involves enabling teachers to be appropriately discerning at critical moments, for example, to help Sue move on from a poignant comment in her initial interview: Sue. ... There are some kids who are really not good at explaining themselves, and I don’t do anything to address that ... maybe what I am saying is that there is a purpose to being in the classroom and that is to learn and I am there to guide learning. So it is not good enough for all of us to say how we feel. We‘ve got to go somewhere with that and I am not sure how to do that

As is the case in any moment of educational reform, teachers in transitional South Africa are coping with a difficult, dynamic and shifting policy environment. Not only are there new national norms and standards for teacher education (DoE, 2000), but policy for curriculum reform and language-in-education are in constant formation. A commitment to learner-centredness and to multilingual teaching and learning remains clear. Moreover, there is recognition, at least at the level of policy, of teachers as key to any improvements and change in the quality of learning in schools and other educational institutions (DOE, 1996; 1999). But, as discussed in Chapter 2, these commitments and shifting policies are being mapped onto the apartheid legacy of diversity coupled with extensive inequality and poverty. The road from the apartheid past to quality education for all South Africa’s children is long and complex. Current policy is for the development of two languages in the curriculum for all learners, and of qualification requirements that teachers study two languages (English and one other) in their preparation to teach. There is also explicit encouragement for teachers to adopt multilingual practices. But as yet there is little guidance and understanding of how to develop subject knowledge using all languages in the class as a resource for learning. Hence the suggestion above that reflecting on teachers’ actions in a range of multilingual contexts as part of an

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interacting three-dimensional dynamic should indeed become part of mathematics teacher preparation and development programmes. 1. DILEMMAS AS LEARNING CURRICULUM FOR LANGUAGE AND MATHEMATICS TEACHER EDUCATION In her discussion of “math practice in school classrooms”, Lave argued that problem-solving in general (and thus in school) is “dilemma-motivated”. The problem with mathematics in school, in her analysis, is that the dilemmas that motivate learners’ practices are not substantively mathematical. They are more about passing tests and examinations. Mathematics “is the official activity but not the central dilemma” and so the issues that are lively for learners are more about performance than about mathematics (Lave, 1990, p. 324). This in itself is a central dilemma for all mathematics teachers, particularly in contexts like South Africa, where there is a high stakes final exit examination from school in Grade 12. This examination is as much, if not more, about selection into further education and employment rather than assessment of knowledge. Performance then inevitably becomes a driving force. In Lave’s terms, social practices are driven by problem-solving around central dilemmas in the practice. She argues that learning curricula rather than teaching curricula are what are needed to guide educational practice. I have argued elsewhere (Adler, 1998b) that in the context of schooling teaching and learning cannot and should not be dichotomised. I would nevertheless argue with Lave that a learning and teaching curriculum in mathematics teacher education that foregrounds language issues for teaching and learning mathematics should be structured around the three key dilemmas identified in this book. In South Africa, whether prepared for their task or not, mathematics teachers are also language teachers. Teacher education, in addition to attending to the languages teachers need to be able to speak, needs to attend at the same time to the language demands of specific subjects. A learning curriculum in mathematics teacher education would benefit from being organised around the dilemmas identified in this book as constitutive of, and constituted by, the multilingual setting. This suggestion resonates deeply with a wider movement towards cases in teacher education practice. Current debate in teacher education is crucially concerned with teachers’ knowledge-bases, and in particular, what does and should constitute a mathematics teacher’s subject knowledge-base (Wilson and Berne, 1999; Lampert and Ball, 1998; Adler et al., 1999). There are increasing arguments for professional learning (and so knowledge growth about teaching) to be situated in practice, or in Lampert and Ball’s terms, to be practice-based. The contribution of this book is that it identifies and describes three key teaching dilemmas in multilingual mathematics classrooms around which a learning and teaching curriculum in language and mathematics teacher education could be built. In other words, these cases are critical resources, exemplars, for professional learning and hence for mathematics teacher education.

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Of course, further cases need to be constructed across primary and non-urban school classrooms in particular, cases that start from the key dilemma of codeswitching but are illustrate of diverse possible journeys from informal talk in the learners’ main language to formal written mathematics, and so of different ways in which such dilemmas are experienced. Cases could include issues that arise when the journey is abbreviated, as well as cases where the journey appears more complete. From these, teachers through professional development activity, can engage with where and how code-switching and explicit induction into mathematical discourse, for example, facilitate or hinder the journey. Such engagement would not only serve as critical advocacy for practices like code-switching and modelling mathematical discourse, but it would provide opportunity for teachers to work on the benefits and constraints of learning English as subject, and English or learners’ main language as LoLT; and for critical reflection on the meaning of learner-centred practice in a multilingual mathematics classroom. 2. WHAT THEN OF A RESEARCH AGENDA? In this book I have been centrally concerned with teachers’ knowledgeability, as a window into the complexities of teaching and learning secondary level mathematics in multilingual classrooms, and so teachers’ understanding of their practice. Of course, teachers’ concerns inevitably focus on their learners, and in this way, teaching dilemmas are about teaching and learning. The teaching dilemmas developed in this book are, however, explicitly and intentionally, those that teachers expressed, acted and reflected on. Such a focus, like any study, is partial, leaving much still to be done, not least of which is studies of learners’ experiences in multilingual classrooms, and of how learners are positioned by teachers’ actions in relation to the dilemmas that infuse their work. Moreover, any research inevitably leaves some questions unanswered and at the same time inspires new and different questions. A crucial advantage of the methodological choice of working across three multilingual contexts, in contrast to an ethnographic study of one such context, was the identification of presences and silences across teachers in different contexts and hence empirical support for a conceptualisation of teachers’ knowledge as personal, practical and situated, and an understanding of teaching dilemmas as similarly situated. At the same time, questions have been left and others raised in the various chapters of the book, pointing out specific focuses for a research agenda in each of the three different multilingual contexts analysed, and in the wider field of language and mathematics education. Emerging from the exploration of code-switching is the need for research into the tacit practice of modelling mathematical English in multilingual classrooms, especially in classrooms where neither learners nor the teacher have English as their main language. An ethnographic study of such classrooms could illuminate the purposes and effects of such practices for both teacher and learners. Furthermore, the management of code-switching in an enquiry-based, task-based multilingual

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secondary mathematics classrooms requires further study. Both areas of study, in turn, could extend the language of dilemmas of code-switching. In addition, the analysis and discussion of code-switching in Chapter 6 focused on teachers and students whose main language is not English, but who are teaching and learning in English. Much of the research on bi-/multilingual classrooms has a bi-/multilingual teacher in focus. As Helen provocatively suggested: ... being a bilingual student of mathematics is not in itself a determining disadvantage. In the context of south Africa where the norm is to be bilingual, we could be researching the negative effects on students of monolingual teachers. Instead, we research the disadvantages for mathematics peformance lying in the student. In a time of reconstruction in South Africa, we must develop strategies for the mathematics classroom that maximise the possibilities for the bilingual student to develop to her fullest potential.

What would be the teaching and learning benefits in South Africa (and so too elsewhere) if English-speaking teachers could speak an African language? Of course, knowledge of one African language does not fully confront the reality of multilingual South Africa and its eleven official languages, nine of which are African languages. Nevertheless research on English-speaking teachers who do have an additional African language could illuminate whether such teachers, with their plurality of languages, significantly impact on communication and hence mathematical activity and learning in their multilingual classrooms. Similarly, Helen’s comment raises questions about bi-/multilingual classrooms selected elsewhere for studying teaching and learning mathematics. In the more obvious multilingual classrooms in suburban schools in South Africa, the actual effects of explicit language teaching, of language being used as a public resource for teaching needs further study. Specifically, the following question needs to be answered: to what extent and how does explicit attention to pronunciation on the one hand and to formal mathematical expression on the other provide smoother access to mathematical practices? Moreover, a further question that remains unanswered is whether the practice of, and a concern with, explicit mathematics language teaching is and should be exaggerated in multilingual classrooms. In Chapter 7 I argued that a participatory-enquiry approach creates a qualitatively different learning environment from dominant traditional teachercentred environments. Sue’s approach created a mathematically rich environment where tensions in the teaching-learning dialectic are clearly revealed. That an instance in Sue’s practice constrained mathematical knowledge development motivates ethnographic study of participatory-enquiry approaches. I located Sue’s dilemmas of mediation within the broader democracy-development tension. It is in Sue’s classroom, where participation and diversity are encouraged and enabled, that possibilities for deepening mathematical knowledge are created, and then partially lost. Further research is needed to develop an understanding of teaching and learning mathematics in school by building on and learning from the possibilities and constraints in participatory-enquiry multilingual classrooms such as Sue’s. In addition, as with the dilemma of code-switching, it is likely that dilemmas of mediation will take on specific forms in participatory-enquiry-based classrooms

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across school contexts. A great deal more could be learned from further research across a range of classrooms: for example, well-resourced, under-resourced, multilingual, bilingual and monolingual. In their epilogue to Language and Communication in the Mathematics Classroom: Steinbring et al point out: The instances of mathematically rich and meaningful communication reported in this book all show at how high a cost they have been obtained, in terms of the teacher’s involvement, students’ intellectual and emotional investment, time and organisational skills. (1998, p. 342).

In the light of the dilemmas of mediation and transparency in this book, I ask: What comes to count as “mathematically rich and meaningful communication”? For whom are these then available? Sue’s classroom was one where there were clear were instances of mathematically rich and meaningful communication. Yet, these instances also reveal partial losses of meaning. I also pointed out that Sue worked in a supportive and well-resourced school, one that enabled teacher involvements, learners’ investments, and time. What counts mathematically and pedagogically for Steinbring et al. assumes a well-resourced classroom. Aside from learning materials, time and organisational skills necessary to support such pedagogy, the resources required include a pedagogical mathematical knowledge-base of teachers that is not easily produced outside of labour-intensive, and hence costly, professional development activity (Wilson and Berne, 1999). Moreover, sociological analyses of schooling have long ago revealed social and political constraints on students willingness to “invest”, either emotionally or intellectually, in school learning (e.g. Apple, 1982). Herein lies a profound challenge for mathematics education research and practice. If the costs of obtaining meaningful mathematical communication are so high, can they possibly be made widely available? Or does meaningful mathematics conversation as a route to mathematical learning, become, however unintentionally, the preserve of the privileged few? Expressed in more political terms: in whose interests is the dominant construction of mathematically rich and meaningful communication? The empirical base for the kinds of arguments put forward for promoting rich mathematical communication in school classrooms needs to be extended to situations where the kinds of resources described above are not readily available, and where learners bring a range of linguistic competencies in the LoLT to their mathematics class. Moreover, in Valero and Vithal’s terms, the kinds of insights gained from contexts in “the South” should be used to reflect critically on research ‘findings’ in the dominant world, to interrogate assumptions made in such research, rather than simply to relegate the issues raised in such research as “developmental”. Other questions arose through the research and were not addressed, and so point to further research in the field of mathematics and language. For example, in the initial interviews, three teachers pointed to instances of communication breakdown that they felt were intuitively part of their multilingual context. I called these “dilemmas of explanation’‘ (Chapter 5) and explained that the teachers could not easily describe or ascribe these incidents. Further research, through situated teacher

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stories (Clandinin and Connelly, 1996) could provide insight (possibly language of dilemmas) into instances of communication breakdown. These could, at least potentially, contribute to reducing communication breakdown and/or enhancing teachers’ capabilities in recognising and using such instances productively and constructively in their practice. In particular, the silence on communication breakdown in urban township schools raised the question of whether the assumptions about homogeneity by teachers in these classrooms was appropriate. Like any classroom, classrooms of African learners are also heterogeneous. Further study on communcation within seemingly homogeneous allAfrican secondary mathematics classrooms is needed. Time, in particular, emerged as a significant additional dimension requiring further research. Four of the teachers talked about time constraints in their initial interviews. Teachers’ decisions and reflections are indeed constrained by the timespace relations in school classrooms. Helen worried about “going on too long”; Thandi worried that if she has to work continually in more than one language, this will take time; and Sue’s reflections include a concern that one of her whole-class discussions was worthwhile, but perhaps not in relation to the amount of time she gave it. These issues were not taken further in this study. In the language of this book, there was no explicit attention to developing a language of dilemmas related to time and mathematics teaching in multilingual calssrooms. Clearly, time is a significant fourth dimension of the dynamics of teaching and learning mathematics in multilingual classrooms and thus should be factored in to any further study in mathematics and language in school. Indeed, time emerged yet again as a significant dimension of teaching and learning in the FDE study discussed in Chapter 6. Time is significant when working in two languages, and when developing learner-centred practices (See Adler et al., 1999). Finally, this study worked with language-in-use and analysed classroom data as they related to teachers’ reflections and as they illuminated teachers’ dilemmas within the three-dimensional dynamic (access to the LoLT, to mathematical discourse and to classroom discourse) in multilingual classrooms. A linguisticallyfocused analysis of the classroom transcripts will undoubtedly throw additional and different light onto the complexities of teaching and learning in multilingual mathematics classrooms. As will discursive analysis of the teacher interviews and workshops. Further study of multilingual mathematics classrooms, of the relationshipbetweenmathematics, education and langiage could well be enriched by an interdisciplinary, multilingual team where mathematics educators and sociolinguists combine their respective academic and linguistic skills and insights. My hope for this book is that through its theoretical and empirical contributions its implications for teacher education and suggestions and questions for further research, it offers generative research and development possibilities in the field of mathematics education.

ENDNOTES

CHAPTER 1 These are pseudonyms. The names of all teachers whose voices are heard in this book have been changed. 2. Throughout the book, school students are referred to either as students or learners. In new education policy documentation in South Africa, learner has come to replace both pupil and student, the ideology behind such a discursive shift being that learning is a life-long process and terms like student and pupil have an age and level connotation. However, in the story told here, I have found it inappropriate and sometimes cumbersome to restrict my reference to learners, and hence I have taken some liberty in using students and learners interchangeably to refer to school learners. 3. JA refers to myself, as interviewer. 1.

CHAPTER 2 For a more comprehensive consideration of the changes that have occurred in South Africa, see Lodge (1999) and Adam et al. (1997). 2. Such a description has also been elaborated in Reality Check, a representative national household survey undertaken by the Independent Newspaper Group on the 5th anniversary of the first democratic election, and published on 28 April, 2000. 3. Presentation by Z Desai at Language in the Classroom National Colloquium, Department of Education, Pretoria, 9-10 July 2000. 4. http://education.pwv.gov.za/Media-Statements/Dec99_Folder/99Report.html, Report on the 1999 Senior Certificate Examination. 5. Interested readers will find further discussion in Hartshorne (1992), Taylor and Vinjevold (1999), and Heugh (1995). 6. Newspaper reports in May 2000 highlight the issues, and refer to the work of Praesa (Project for the Study of Alternative Education in South Africa) and the Pan South African Language Board (Pansalb), a statutory body and successor to LANTAG. The points made here are reflected in interviews with the Chair and director of Pansalb and Praesa respectively (Simmonds, 2000). 7. Heugh made this point forcefully in her presentation at the Language in the Classroom National Colloquium, Department of Education, Pretoria, 9-10 July 2000. She pin-pointed 1976 as a downward turning point in school performance for Africans in South Africa as evidence for the detrimental effects of language policy away from ‘mother tongue’ instruction post 1976. 8.. I am drawing here on a recent visit to Sweden (March, 1999) where I worked with a group of mathematics teachers all of whom are teaching in schools whose students are drawn largely from immigrant communities. Swedish schooling conditions are vastly different from those that are described in this book. The research I discuss here resonated with the Swedish teachers, and so provided a 1.

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strong basis for the workshop. Moreover, the challenges Swedish teachers are facing in schools with large immigrant student bodies appeared similar to those in South African urban schools. Second generation immigrant students appear more isolated from the dominant culture and language and so disadvantaged in the dominant Swedish school context. I learnt that not only are there numerous main languages present in these schools, and thus constraints on supporting main language maintenance, but expansion of immigrant communities together with satellite TV made it possible, in fact likely, that students in these schools had limited contact with Swedish outside of the formal school context.

CHAPTER 3 1.

What of course is obscured in the discussion of these three multilingual contexts is that support for English as LoLT — the English language infrastructure — does not operate in isolation from other social dimensions of schooling. While the historically white schools might provide the strongest support for English as LoLT, as black students entered these schools, they found themselves in a culturally alienating environment. Cultural alienation must thus intersect with possibilities for learners’ to benefit from their immersion in an English language environment. As students are not at the centre of the study, I am not able to explore this interesting dynamic. The issue is raised in the next chapter, though only briefly. Clearly, this dynamic needs to be followed up in further research.

CHAPTER 5 1.

Clive (and Thandi later in this chapter) use “vernac” here as short for “vernacular language”. What they are referring to is learners’ or teachers’ use of learners’ main language in class. 2. This common sense assumption is problematised in debate in language teaching on the distinction between ‘acquisition’ and ‘learning’. The issue is how much, and what language skills and competencies are acquired in use, and which are learned as a function of instruction (Baker, 1993). I do not take it up any further in this book. 3. It is important to add here that English-speaking white South African teachers all had to study Afrikaans (the other official language in the apartheid era) and in this sense they are bilingual. However, as discussed in Chapter 2, only a few learnt an African language. 4. Personal communication with Professor Hilary Janks, Head of the Applied English Language Studies Department at the University of the Witwatersrand.

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CHAPTER 6 In order to preserve anonymity for the teachers I am not able to provide a published reference here. 2. This example from Walkerdine’s analysis does not do her work sufficient justice. Social practices are infused with relations of power, all of which enter the classroom with the meanings children bring with them. ‘More’ might well bring up very different responses and positionings for learners from different class backgrounds, for example. Poor families are more likely to admonish (as her own mother did to her) a child for continually wanting ‘more’, when there was ‘no more’. It is nevertheless beyond the scope of the analysis here to draw in any further, these important elements embedded in her use of the notions of “chains of signification”, and “regimes of meaning”. 3. As clear from Chapter 2, socio-economic conditions are the most significant factor in the urban/non-urban divide. Non-urban schools are largely impoverished contexts, with many having been denied basic resources like electricity and water. The focus on the linguistic context in this research is not to deny these additional contextual issues but to highlight the particular language and learning challenges produced across contexts. 4. The project involved 9 researchers in addition to myself, K. Brodie, P. Dikgomo, T. Nyabanyaba and M. Setati in Mathematics Education; A. Lelliott, A. Bapoo in Science Education and Y. Reed, H. Davis and L. Slonimsky in English Language Education 1.

CHAPTER 7 There are many labels for more open, learner-centred mathematics classrooms. “Constructivist”, “investigative” are two that have current currency in mathematics education. I am concerned here with accurate description of the classroom that is in focus in this chapter and hence have specifically avoided any label that might attach other meanings or set up particular expectations. 2. Some time after closely observing the video and the reflective interview with the teacher, I learnt in discussion with a mathematics colleague that Joe’s answer closely resembles an attempt to prove the sum of the angles of a triangle is 180 degrees in the early part of this century. (Finlow-Bates et al., 1993) 3. I am grateful to Lyn Slonimsky for sharpening my awareness here. I was taken by the dynamism of Joe’s response and initially less concerned with its generalisability 1.

CHAPTER 8 In sociocultural terms, this is the dialectic between language and thought, where paraphrasing is associated with personal appropriation of cultural concepts and ideas (i.e. within a community of practice) (Leontiev and Luria, 1968). 1.

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It is important to note here (see Bernstein, 1993) that language as a cultural tool, is a tool for learning. But language itself is a producer of relations of power. This point is also made by Ivic (1989). While language is a resource in the classroom, it is does not function in any simple, unproblematic way. 2.

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SUBJECT INDEX

access, 114, 122, 124, 127, 140 democratise, 111 politics of access, 141 to English, 145 to mathematical discourse, 145 to a practice, 124 action research, 79 ,87, 88, 89, 129, 139 Additional Language Learning Environment (ALLE), 31, 91, 93, 105 additional language, 60, 77, 90, 91, 114, 149 analytic devices, 143 analytic narrative vignette, 43, 129 apprenticeships, 102

92, 93, 94, 98, 99, 127, 143, 147, 148 communication breakdown, 150 communicative competence, 4, 10, 72, 96, 101, 111, 114, 116, 119, 120,124, 144 community of practice, 102, 115 conceptual mathematical discourse, 9,78 construction zone, 117, 120, 144 constructivism, 121 control dilemmas, 54 conversation, 112, 119 cultural tool, 126

articulated (knowledge), 14, 59, 143 artifacts, 123

culture and communication, 69 Curriculum 2005 (C2005), 33 curriculum dilemmas, 54

at least, 80, 81, 84, 87

curriculum policy, 33

at most, 80, 83, 85, 86, 87

curriculum reform, 97, 98, 146

bi-/multilingual mathematics classroom, 75, 76 bi-/multilingual(ism), 6, 7, 77, 78, 97 additive bilingualism, 8, 78 additive multilingualism, 8, 26 bilingual education, 8

data disruption, 43, 45

calculational discourse, 9,78 central dilemma(s), 142, 146 chains of signification, 84 classroom cultural processes, 87 classroom discourse(s), 6, 119 code-borrowing, 77 code-mixing, 77 code-switch(ing), 1, 26, 43, 55, 63, 64, 65, 76, 77, 78, 88, 89, 90, 91

deflection/deflecting, 113, 115, 117, 120 democracy-development tension, 149 didactic tension, 1 15 dilemma(s), 1, 56, 79, 86, 90, 95, 127 control dilemmas, 54 curriculum dilemmas, 54 central dilemma(s), 142, 146 dilemma(s) of code-switching, 70, 76, 78, 79, 85, 87, 88, 89, 90, 94, 95, 96, 98, 143 dilemma of explanation, 69,73, 150 dilemmas of mediation, 3 5,7 1, 98, 100, 101, 105, 112, 114, 120, 121, 143, 144, 149 dilemmas of teaching, 54

156

SUBJECT INDEX

dilemma of transparency, 72, 122, 125, 137, 138, 140, 142, 144 language of dilemmas, 1, 51, 54, 55, 57, 89, 122, 142, 145 language-related dilemma(s), 101, 121, 141, 144 managing dilemmas, 52, 101, 142 mathematical dilemmas, 56 practical dilemmas, 51 societal dilemmas, 54 teaching dilemma(s), 1, 43, 51, 54, 101, 116, 118, 120, 121, 122, 142,145 discourse(s), 78, 84 calculational, 9,78 classroom, 6, 119 conceptual, 9, 78 educated, 139, 141, 143 educational, 139, 141, 143 mathematical, 4, 42, 68, 74, 97, 101,119, 124, 139 pedagogic, 84, 85 school, 89 scientific, 97, 136 subject-specific, 87, 96 discourse-specific (talk), 42, 75, 77 English language infrastructure, 29, 32, 91, 95, 99 English Language Learning Environments, 29 equity, 80, 97 espoused theory, 41 ethics, 89 everyday, 84, 85, 87 everyday concepts, 104 everyday knowledge, 102, 103 explicit (mediation), 97, 114, 118,119, 126, 137,

explicit (mathematical) language practices, 4, 43, 67, 72, 122, 124, 127, 129, 139, 140, 141, 144, 149 exploratory activities, 77,97,128 exploratory learning, 127 exploratory talk, 67, 75, 77, 78, 90, 92, 96, 97, 98 Foreign Language Learning Environment (FLLE), 30, 41, 91, 93 foreign language, 77,91 formal mathematical language, 2,75, 88, 90, 98, 101, 106, 122, 149 formalised instruction, 103 fundamental pedagogic tension(s), 140,145 generalisation, 107, 108, 113, 118 greater than, 85 groundedness, 42 group discussion(s), 87, 88 group work, 88, 92, 95, 96, 97, 104, 127,143 hegemony (of English), 145 identities, 49, 142 implicit (practices), 43, 67, 114, 118, 140, 142, 144 informal and formal mathematical talk, 102 informal expressions, 88 informal mathematical language, 2, 75, 88, 90, 101, 102, 106, 122 integration, 33, 34 intersubjectivity, 117

SUBJECT INDEX

157

invisible/invisibility, 123, 124, 137, 141, 144

mathematical discourse(s), 4, 42, 68, 75, 97, 101, 119, 124, 139

I-R-F interactions (initiationresponse-feedback), 83, 107

mathematical English, 1 11

knowledge(ability), 42, 53, 55, 58, 102, 123, 125, 142, 143, 148 language and learning, 129 language as a resource, 140 language journey, 96 language of description, 15, 48, 49 language of dilemmas, 1, 51, 54, 55, 57, 89, 122, 142, 145 language-in-education, 24, 98, 146 language-in-education policy(ies), 24, 64, 91, 121 language-related dilemma(s), 101, 121, 141, 144 learner-centered (practice), 3, 34, 68, 72, 90, 91, 92, 96, 97, 99, 100, 104, 146, 147, learning as situated, 102 learning curriculum, 147 learning from talk, 104 learning talk, 75,98 learning to talk, 104 linguistic capital, 14 linguistic competence, 121 linguistic diversity, 114, 142, 144 LoLT, 24, 26, 42, 61, 75, 77, 143 main language, 75, 76, 111, 114, 143 managing dilemmas, 52, 101, 114, 142 mathematical communicative competence, 3, 71, 100, 116, 119, 122 mathematical dilemmas, 56

mathematics as language, 125 mathematics register, 10, 28, 59, 62, 68, 70, 85, 89, 139 mathematics teacher education, 16 mathematics, science and technology, 28 mediate/mediation, 1, 43, 55, 96, 101, 113, 114, 116, 117, 120, 127, 144 model(ling) mathematical language, 79, 83, 86, 89, 93, 94, 97, 143, 147 mother tongue instruction, 78 multilingual(ism), 11, 76, 77,114, 116,121,139,144,146 multingual (context), 43,44,114, 120, 129, 142 new curriculum, 121 non-urban (schools), 76, 90, 91 not less than, 80, 81 not more than, 80, 81, 84 outcomes-based approach (OBE), 33 participatory-enquiry (approach), 43, 100, 101, 114, 117, 144, 145, 149 pedagogic discourse, 84, 85 pedagogy, 59, 67, 85, 95, 144 politics/political, 89, 95, 140, 141, 145 post-colonial, 77, 96 practical dilemmas, 5 1 practice-based knowledge, 8 praxis, 1,58,76,88 source of, 89, 122, 142, 145 pronunciation(s), 125, 136, 137, 139, 142, 149

158

SUBJECT INDEX

psychological tool, 126 public domain, 78 public speech channel, 78

subject-specific discourse, 87, 96 substractive (bi-/mutilingualism), 8, 27

recapp(ing), 136, 137 reciprocity, 47

tacit (practices). 14, 83, 101, 143, 148 talk(ing), 101,102,107,115,116: 119, 120, 124, 125, 139, 143,

reformulate/reformulation, 83, 85, 86, 89, 91, 136, 137, 139, 140 register(s), 28, 59, 85 repeat/repetition, 83, 85, 86, 89

teacher education, 16, 76, 90, 145 teacher talk, 143 teaching curriculum, 147

report(ing)-back, 108,111,113.114, 115,116, 127, 129, 136,137 research relationships, 43,47

teaching dilemma(s), 1, 43, 51, 54, 101, 116, 118, 120, 121, 122, 142, 145

resource(s), 9, 17, 35, 65, 76, 78, 88, 89, 121, 122, 123, 124, 140, 141, 142,146,149

theory-practice, 54, 142 three dimensional dynamic, 6. 42, 86, 95, 119, 140, 142, 145, 151,

revoice/revoicing, 9, 70, 88, 97, 136

threshold hypothesis, 8 time, 151 trade-offs, 114, 120 transparency, 1, 55, 98, 122, 123, 124, 140, 141, 143

safetalk, 86 scaffold(ing), 77, 83, 107, 118, 139 school mathematics, 89, 102 school science, 97 school(ing), 102, 103 scientific concept(s), 12, 85, 87, 103, 104, 107,113, 118, 120, 140, 144 scientific discourse, 97, 136 semiotic mediation, 117 situated theory, 39, 143 situation definition, 1 17 social practice theory, 38, 142 social talk, 127, 143 societal dilemmas, 54 sociocultural theory, 38, 142 spontaneous concepts, 102, 103

triangulation, 42 unit of study, 8 urban schools, 76 verbalisation(s), 125, 126 vignette(s), 43, 101, 116, 118, 120, 129,145 visible/visibility, 123, 124, 137, 141, 144 voice, 3, 43, 48, 137 Zone of Proximal Development (ZPD), 103, 117

INDEX OF NAMES

Adler, J. 8, 13, 29, 30, 35, 38, 79, 91, 96, 97, 102, 119, 121, 149, 153 Alexander, N. 27, 28 Apple, M. 104, 152 Arthur, J. 8, 78, Austin, J. 10

Heugh, K. 28 Hutchinson, S. 42

Baker. C. 28, 78 Barnes, D. 76, 127 Bartolini Bussi, M. 12, 72, 76, 118, 121 Barton, B. 28 Berlak, A. and Berlak, H. 51, 53, 54, 56, 57, 58, 71, 72, 73, 74, 75, 144 Bernstein, B. 104 Boaler, J. 38 Bourdieu, P. 27, 70 Brodie, K. 104 Brousseau. G. 115

Keitel, C. 23, Khisty, L. 8, 9, 79, 119

Campbell, D. 107 Clarkson, P. 7 Clements, M. 64 Cobb, P. 9, 10 Cocking, R. 7 Cooper, B. and Dunne, M. 23 Cummins, J. 8 Daniels, H. 104 Dawe, L, 7 Dowling, P. 10 Durkin, K. and Shire, B. 7 Edwards, D. and Mercer, N. 51, 57, 58 Erickson, F. 130

Jansen, J. 33 Jaworski, B. 47, 48, 51, 56, 57, 58, 72,75,120, 144

Lampert. M. 8, 51, 52, 53, 54, 55, 56, 57, 58, 74, 144, 149 Lave, J. 37, 38, 44, 102, 107, 115, 123, 124, 125, 126, 142 Lerman, S. 38, 46 Levine, J. 38, 67, 104 Lin, A. 75 Macdonald, C. 27 Maxwell, J. 43 Mercer, N. 51, 57, 58, 72, 101, 145 Morgan, C. 10, 13 Moschkovich, J. 9, 11, 13, 71, 119 Mouseley, J. 69, 72 Ndayipfukamiye, L. 79 Newman, D., Griffin, P. and Cole, M. 117 Pimm, D. 10, 64, 68, 72, 76, 86, 114, 115, 127, 141, 143 Pirie, S. 4, 10 Polanyi, M. 41

Fradd, S. and Lee, 0. 98, 137

Ringbom. H. 29 Rose, G. 39,48 Rhubaguyma, C. 65,66 Ruthven, K. 46

Gee, J. 14 Gerofsky, S. 10

Secada, W. 8 Setati, M. 8, 9, 30, 47, 48, 79, 80, 92,

Halliday, M. 10, 140

Sfard, A. 11, 12, 72, 111, 119, 127

94

160

Sierpinska, A 12 Silverman, D. 41,48 Stein, M. and Brown, C. 38 Steinbring, H. 12, 35, 151, 152 Taylor, N. and Vinjevold, P. 26 Valero, P. and Vithal, R. 45, 46, 152 Vygotsky, L. 38, 103, 104, 116, 127, 145 Walkerdine, V. 85,86 Wells, G. 76 Wenger, E. 37, 38, 44, 102, 107, 115, 123, 124, 142, 145 Wertsch, J. 103, 116, 117 Wilson, S. and Berne, J. 149, 152 Wittgenstein, L. 14 Zepp, R. 7

INDEX OF NAMES

APPENDIX 1

GLOSSARY OF TERMS This glossary provides definitions of various terms and acronyms used in discussion of the socio-economic, curriculum, language-in-education and language policy and practice in South Africa. Many of these terms are problematic and contested in South Africa. Where appropriate, I have signalled competing meanings as well as my own preferences, thus enabling readers to interpret the meanings I am attaching to some specific terms I have chosen to use through the book. The glossary is organised into two main sections — Geographic and School Context and Language and Language-in-Education. Terms in each of these sections are arranged alphabetically for ease of reference. In relation to language and language-in-education terminology, I am indebted to Granville et al. (1998), as a significant portion of what appears here is drawn from the Glossary provided in their analysis of language-in-education policy in a changing South Africa. 1. GEOGRAPHIC AND SCHOOL CONTEXT TERMINOLOGY Black, white, Indian, coloured, African are all racial terminology developed and used in the apartheid era. For clarity of description through the book it becomes necessary at times to refer to apartheid-defined racial groups. Here ‘white’ implies South Africans who are historically European. ‘Black’ is used generically to refer to all South Africans who were disenfranchised during the apartheid era and thus includes ‘Africans’ (South Africans who are historically African), so-called ‘coloured’ South Africans (‘mixed’ race), as well as Indians (historically Asian). Through the book, I use ‘black’ in its generic sense, and I use ‘African’, ‘coloured’, ‘Indian’ and ‘white’ when I wish to identify a person or persons’ origin. The continued use of apartheid-defined racial descriptors is controversial. Naming is productive. At the same time, refusing to identify difference in South Africa, particularly as it relates to the legacy of apartheid, can be equally problematic. For example, school enrolments and pass rates are no longer officially dis-aggregated along apartheid-defined racial lines. It becomes impossible to determine in a multi-racial school, for example, whether the situation of black learners in those schools has improved. Through the book I use the racial categories advisedly, and particularly when I need to point to persons’ primary language(s).

162

APPENDIX 1

Historically white schools are those schools in cities designated ‘white’ during the apartheid era. These schools are typically in the suburbs of major cities, and now have multiracial student bodies. Informal settlement refers to large settlements which occur in both urban and non-urban settings but which generally have arisen spontaneously without formal planning processes, and as a result have few services such as water and electricity. Learner and student are used interchangeably to refer to scholars or pupils. current education policy documentation in South Africa, the term ‘learner’ used in an attempt to capture life-long learning and so not restrict learning young scholars. I have used both student and learner through the book preference to ‘pupil’, unless the latter is used within a quotation by a teacher.

In is to in

NCCRD - National Centre for Curriculum Research and Development — is a centre located within the National Department of Education, and responsible for undertaking mational research and development projects, particularly in relation to new curriculum policy. NGEO stands for Non governmental educational organisations Non-urban is used in preference to ‘rural’ in South Africa. ‘Rural’, as it typically refers to agrarian, is too narrow a concept to capture the range of living and working conditions outside the urban areas of South Africa. ‘Progressive’ schools refer to those few schools that developed post 1976 with a dual progressive agenda: firstly, they were explicitly non-racial, and secondly they adopted, though in diverse ways, alternative approaches to curriculum to those prescribed by the state system. Typically interpretations of progressive education involved some or all of integrated curricula activity, enquiry-based and learner-centred pedagogy. Townships refer to those areas around major cities designated for separate development during the apartheid era. There were African townships (e.g. Soweto south-west of Johannesburg), so-called ‘coloured’ townships (e.g. Riverlea, also south-west of Johannesburg) and Indian townships (e.g. Lenasia, south-west of Johannesburg). Township schools are schools in those areas designated for separate development just outside major cities during apartheid (See township above). Post-apartheid organisation of cities and school districts now includes townships into greater city areas. For example, Soweto is now part of Greater Johannesburg.

APPENDIX 1

163

2. LANGUAGE AND LANGUAGE-IN-EDUCATION TERMINOLOGY Additional language refers to any language which an individual adds to his or her first language or main/primary language(s) (see below). With Granville et al., I use the term ‘additional language’ rather than ‘second language’ (see below) to avoid the pejorification and because this term complements an additive approach to multilingualism, which I support. Additional Language Learning Environment (ALLE) is a term developed in the book (Chapters 2 and 6) to refer to those schools where English as language of learning and teaching (LoLT, see below) in the school can be understood as an additional language for most learners. This is in contrast to a Foreign Language Learning Environment (FLLE) where English is barely used or seen outside the school classroom, yet English remains the LoLT. Additive bilingualism is used in language-in-education policy to refer to a situation in which individuals add a language to their main language but at the same time continue to develop their main language. African languages is used in the book to refer to primary languages spoken by Africans in South Africa. The primary languages of Africans in South Africa are formally called ‘Bantu’ languages. The association with the Bantu Education Act in 1953 forces a redescription. I thus use ‘African languages’ to mean the Bantu languages spoken in South Africa. There is hotly contested debate in the country as to what ‘African’ means. President Mbeki opened his inaugural address to the nation in 1999 with what is called his “I am an African” speech, where he did not restrict his meaning to the apartheid-use of ‘African’ (See ‘African’ in the above section of the glossary). The notion ‘native South African’ in this view is not a racial determinant, but refers to those born and bred in South Africa and who are commited to the development and growth of Africa. In this context, some would argue that Afrikaans is an African language, produced and used in South Africa. The opposing view is that African means the racial description of African as used under apartheid. It does not include Indians and so-called ‘coloureds’ and certainly not whites, whenever and wherever they might have been born. Afrikaans would not qualify as an African language, but one derived from Dutch and thus European. Bilingual/bilingualism refers to proficiency in two languages, but not necessarily equal proficiency. Prior to eleven official languages policy, enshrined in the 1996 South African Constitution, South Africa had only two official languages: English and Afrikaans. ‘Bilingulaism’ was often used to refer to only Afrikaans/English bilingualism. Like Granville et al., I do not use the term in this way. One of the consequences of the two official languages policy is that monolingualism is rare in South Africa.

164

APPENDIX 1

Classroom discourse. See discourse below. Discourse. I use discourse to mean, loosely, ways with words or ways of using words, including the purpose to which language is put, as well as the values that are embedded in the use of language and the power relations and attitudes to knowledge. Hence classroom discourse and mathematical discourse are ways of using words in the classroom and in mathematics respectively, together with values that accompany schooling on the one hand, and mathematics on the other. Dominant language (or languages) refers to a language most widely used in a particular context or community. In South Africa, English and Afrikaans have been the dominant languages of schooling beyond the junior primary level. For example, English and Afrikaans have been the languages in which textbooks and examination papers have been written. In local community contexts the dominant language is the one most widely used in that community. English as second language (ESL) see 'second language' below. First language, natice language, mother tongue and home language are all terms used to describe the language that a child acquires from birth and in which he or she is most proficient (i.e. advanced in ability to use). However, the terms are problematic in a multilingual society for several reasons. Firstly, many children begin to acquire more than one language from their earliest childhood so that they do not have one first or native (in the sense of original) or home language. Secondly, for some people, the language(s) that they acquired first do not continue to be the languages in which they are most competent as adults or which they feel most comfortable using. Thirdly, the main language(s) of a family may be the language(s) of the father or the dominant language of the community in which the family lives rather than mother's language(s). Foreign language refers to any language which learners are unlikely to hear or read outside the classroom in which they are learning it, because it is not in use in the wider community. European languages such as Russian and Spanish would be examples of foreign languages in South Africa. It has been argued that English is a foreign language for students in some parts of South Africa, because they neither hear nor read the language outside the classroom. This notion is developed in the book, and used to refer to different English language learning environments. See FLLE andALLE above. Home Language is discussed under First language above LANGTAG refers to the Language Task Action Group delegated by the Minister of Education, in 1995, to establish South Africa’s language related needs and policies.

APPENDIX 1

165

Language as subject refers to any language that is offered as one of the subjects that students may study at school. The old curriculum offered English and Afrikaans as either first or second languages and African languages as either first or third languages. In the new curriculum African languages have the same status as English and Afrikaans for the first time. Foreign languages are also offered as part of the language-as-subject curriculum. Language ofchoice refers to any language (or variation of a language) which an individual selects for use in a particular context. Language(s) of Learning and Teaching (LoLT) is the current term used in South Africa in preference to ‘language of instruction’ or ‘medium of instruction’, both of which refer only to teaching, and not learning. LoLT refers to the language or languages used for both learning and teaching across the curriculum and gives equal importance to both learning and teaching. It is also used to refer to the language/s used in textbooks, other classroom materials and the language/s used for examination papers and answers across the curriculum. LoLT recognises that teachers and learners should use whatever languages are necessary to ensure that students understand what they are learning. LoLT also recognises the importance of allowing students to use the full range of their linguistic codes and resources. See also Medium of instruction, below. Main language and primary language both refer to the language most often used by an individual, in which he or she becomes proficient. Some people who are fully bilingual or multilingual (see below) may use two or more languages on an approximately equal basis and thus have more than one main/primary language. These people may choose to use one of their main/primary languages in some contexts and another main/primary language in other contexts. Again; like Granville et al., I prefer to use these terms and to avoid ‘first language’ or ‘mother tongue’. Mathematical discourse. See discourse above. Mathematics register. See register below. Medium of instruction is the term previously used in South Africa to refer to the language of instruction for subjects across the curriculum. Previous medium of instruction policy proscribed code-switching and required teachers to use the prescribed language inflexibility. ‘Medium of instruction’ tends to focus on the language of teaching and to background the role that language plays in learning. Medium of instruction also refers to the language used in textbooks, other classroom materials, and the language used for examination papers and examination answers across the curriculum. See also LoLT above.

166

APPENDIX 1

Monolingual/monolingualism refers to advanced proficiency in only one language. Mother Tongue is discussed under First language above multilingual/multilingualism of a speaker refers to his or her competence or proficiency in more than two languages. Many South Africans, whose main language is neither English nor Afrikaans (the former official languages), are multilingual. They speak three or more languages. In this book I also use ‘multilingual’ to describe a classroom and South African society as a whole. The ‘multilingual classroom’ refers to the situation in many South African classrooms where learners bring into one class, a range (i.e. more than two) of primary languages. The ‘multilingual classroom’ does not imply that all learners and/or teachers in the class are themselves necessarily multilingual. Similarly, in referring to South Africa as ‘a multilingual society’, the meaning here is that there are many languages spoken and now legitimised in the country. At the same time, many, but not all South Africans, are able to speak more than two languages. Multilingual classroom is explained under multilingual/multilingualism above. Native language is discussed under First language above Official languages refers to state recognised languages in South Africa. The post-apartheid constitution recognises eleven official languages: English, Afrikaans, isiNdebele, isiXhosa, isiZulu, Sepedi, Sesotho, Setswana siSwati, Tshivenda, Xitsonga. The names of the nine African languages here are their formal names. However, they are more often referred to by shortened anglicised versions e.g. Ndebele, Xhosa, Zulu, Pedi, Sotho, Tswana, Venda, Tsonga. In the book, including interview extracts, I use both Tswana and Setswana as I talk about language-in-use in some teachers’ classrooms. Primary language is discussed under Main language above Register refers to varieties of language and the different functions to which they are put. So the mathematics register is, within say English, the use of English words with particular mathematical functions. Register signals topic, purpose and social relations of the communicative act. For example, hence within the mathematics register wouid typically be found in written texts, and signals deductive reasoning as often found in Euclidean geometry. This use is different from hence in ordinary English. There is thus some overlap between discourse and register. I view the latter as subsumed in the former.

APPENDIX 1

167

Secund language is a term used to describe a language which an individual adds to a first language, often in a formal learning context. It differs from foreign language (see definition above) in that it describes a language which is widely used in the society in which some individuals are learning the second language as an additional language. ‘Second language’ is also problematic in a multilingual society in which children begin to acquire two or more languages from earliest childhood. The label ‘English Second Language’ for a school subject is misleading when it may be the third, fourth or fifth language of some learners and where it may be a foreign language for learners in contexts where they have little or no contact with it outside the classroom. In South Africa, ‘second language’ is sometimes associated with ‘second rate’ and has thus acquired pejorative overtones. Subtractive bilingualism refers to a situation in which individuals become unable to proficiently use their main language because this language has been replaced by an additional language which is dominant in school or workplace or local community or all of these. In other words, much of the ability to use the main language is subtracted or lost.

Target language refers to any language which an individual aims to learn in addition to the language(s) acquired in early childhood. Vernacular languages is a term that has been commonly used to refer to African languages in South Africa. There are occasional references in the book by teachers themselves to ‘vernac’. Outside of such reference, I do not use this term.

APPENDIX 2

APPENDIX 2

169

APPENDIX 3 Methods of data collection in the order in which they occurred in the study. (a) an initial semi-structured, in-depth, interactive individual interview The teachers were invited to talk about the context in which they worked, the particular challenges they believed they faced as a mathematics teachers, and what they experienced as language issues in their mathematics teaching. (b) a feedback session back and discussion of a first level analysis of the initial interviews with the group All six teachers participated in a group discussion after the initial interviews were completed. The intention here was for teachers to validate my initial analysis and interpretation of their interviews. As it turned out, this discussion became another context for the teachers to talk about language issues in their practice. In addition to ‘checking’ their accounts, the group discussion became an additional site for discussion, and contestation, over what was an appropriate “extraction out” of key language issues from their collective interviews. (c) videotaped classroom observation Each teacher was observed and videotaped in one or two of their mathematics classes. At least two consecutive lessons in the same class were observed. Each teacher was observed for up to three hours. (d) individual rejective interviews with each teacher An individual lengthy and in-depth reflective interview on the videotape was conducted with each teacher. It was structured so that the teacher or I could stop the tape at any point to discuss something of interest. (e) three follow-up workshops, and related action research projects These took place during 1993 and the year following all the data collection above. At these workshops teachers selected aspects of the data that they themselves wanted to discuss with each other, and to pursue as part of their practice. In preparation for these workshops, some of the teachers, including those in focus in Chapters 6, 7 and 8, undertook small action-research projects so as to further explore issues that had arisen for them during the research process.

Mathematics Education Library Managing Editor: A.J. . Bishop, Melbourne, Australia 1.

H. Freudenthal: Didactical Phenomenology of Mathematical Structures. 1983 ISBN 90-277-1535-1; Pb 90-277-2261-7

2.

B. Christiansen, A. G. Howson and M. Otte (eds.): Perspectives on Mathematics Education. Papers submitted by Members of the Bacomet Group. 1986. ISBN 90-277-1929-2; Pb 90-277-21 18-1

3.

A. Treffers: Three Dimensions. A Model of Goal and Theory Description in MathISBN 90-277-2165-3 ematics Instruction The Wiskobas Project. 1987

4.

S. Mellin-Olsen: The Politics of Mathematics Education. 1987 ISBN 90-277-2350-8

5.

E. Fischbein: Intuition in Science and Mathematics. An Educational Approach. 1987 ISBN 90-277-2506-3

6.

A.J. Bishop: Mathematical Enculturation. A Cultural Perspective on Mathematics ISBN 90-277-2646-9; Pb (1991) 0-7923-1270-8 Education. 1988

7.

E. von Glasersfeld (ed.): Radical Constructivism in Mathematics Education. 1991 ISBN 0-7923-1257-0

8.

L. Streefland: Fractions in Realistic Mathematics Education. A Paradigm of Developmental Research. 1991 ISBN 0-7923-1282-1

9.

H. Freudenthal: Revisiting Mathematics Education. China Lectures. 199 1 ISBN 0-7923-1299-6

10.

A.J. Bishop, S. Mellin-Olsen and J. van Dormolen (eds.): Mathematical Knowledge: ISBN 0-7923-1344-5 Its Growth Through Teaching. 1991

11.

D. Tall (ed.): Advanced Mathematical Thinking. 1991

12.

R. Kapadia and M. Borovcnik (eds.): Chance Encounters: Probability in Education. 1991 ISBN 0-7923-1474-3

13.

R. Biehler, R.W. Scholz, R. Sträßer and B. Winkelmann (eds.): Didactics ofMathISBN 0-7923-2613-X ematics as a Scientific Discipline. 1994

14.

S. Lerman (ed.): Cultural Perspectives on the Mathematics Classroom. 1994 ISBN 0-7923-293 1-7

15.

O. Skovsmose: Towards a Philosophy of Critical Mathematics Education. 1994 ISBN 0-7923-2932-5

16.

H. Mansfield, N.A. Pateman and N. Bednarz (eds.): Mathematics for Tomorrow’s Young Children. International Perspectives on Curriculum. 1996 ISBN 0-7923-3998-3

17.

R. Noss and C. Hoyles: Windows on Mathematical Meanings. Learning Cultures and Computers. 1996 ISBN 0-7923-4073-6; Pb 0-7923-4074-4

ISBN 0-7923-1456-5

18.

N. Bednarz, C. Kieran and L. Lee (eds.): Approaches to Algebra. Perspectives for ISBN 0-7923-4145-7; Pb ISBN 0-7923-4168-6 Research and Teaching. 1996

19.

G. Brousseau: Theory of Didactical Situations in Mathematics. Didactique des Mathématiques 19701990. Edited and translated by N. Balacheff, M. Cooper, R. Sutherland and V. Warfield. 1997 ISBN 0-7923-4526-6

20.

T. Brown: Mathematics Education and Language. Interpreting Hermeneutics and Post-Structuralism. 1997 ISBN 0-7923-4554-1 Second Revised Edition. 2001 Pb ISBN 0-7923-6969-6

21.

D. Coben, J. O’Donoghue and G.E. FitzSimons (eds.): Perspectives on Adults LearnISBN 0-7923-6415-5 ing Mathematics. Research and Practice. 2000 R. Sutherland, T. Rojano, A. Bell and R. Lins (eds.): Perspectives on School Algebra. 2000 ISBN 0-7923-6462-7

22. 23.

J.-L. Dorier (ed.): On the Teaching of Linear Algebra. 2000 ISBN 0-7923-6539-9

24.

A. Bessot and J. Ridgway (eds.): Education for Mathematics in the Workplace. 2000 ISBN 0-7923-6663-8

25.

D. Clarke (ed.): Perspectives on Practice and Meaning in Mathematics and Science Classrooms.2001 ISBN 0-7923-6938-6; Pb ISBN 0-7923-6939-4

26.

J. Adler: Teaching Mathematics in Multilingual Classrooms. 2001 ISBN 0-7923-7079-1; Pb ISBN 0-7923-7080-5

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